ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
\u22a2 ∀ (x : C(X, 𝕜)), x ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
h_f_re : I f_re ∈ A.topologicalClosure
h_f_im : I f_im ∈ A.topologicalClosure
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
h_f_re : I f_re ∈ A.topologicalClosure
h_f_im : I f_im ∈ A.topologicalClosure
this : I f_re + RCLike.I • I f_im ∈ A.topologicalClosure.toSubalgebra
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
h_f_re : I f_re ∈ A.topologicalClosure
h_f_im : I f_im ∈ A.topologicalClosure
this : I f_re + RCLike.I • I f_im ∈ A.topologicalClosure
\u22a2 f ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
h_f_re : I f_re ∈ A.topologicalClosure
h_f_im : I f_im ∈ A.topologicalClosure
this : I f_re + RCLike.I • I f_im ∈ A.topologicalClosure
a✝ : X
\u22a2 f a✝ = (I f_re + RCLike.I • I f_im) a✝
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
𝕜 : Type u_1
X : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : StarSubalgebra 𝕜 C(X, 𝕜)
hA : A.SeparatesPoints
I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM
key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure
f : C(X, 𝕜)
f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f
f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f
h_f_re : I f_re ∈ A.topologicalClosure
h_f_im : I f_im ∈ A.topologicalClosure
this : I f_re + RCLike.I • I f_im ∈ A.topologicalClosure
a✝ : X
\u22a2 (I f_re + RCLike.I • I f_im) a✝ = f a✝
-- Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
\u22a2 Nonempty (G →g F)
-- Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
\u22a2 Nonempty (G →g F)
-- Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
this : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')
\u22a2 Nonempty (G →g F)
-- Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
this✝ : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')
this : (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G')
\u22a2 Nonempty (G →g F)
-- Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
this✝ : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')
this : (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F).obj G')
u : (j : G.Finsubgraphᵒᵖ) → (G.finsubgraphHomFunctor F).obj j
hu : u ∈ (G.finsubgraphHomFunctor F).sections
\u22a2 Nonempty (G →g F)
-- Mathlib/Data/Multiset/Powerset.lean
α : Type u_1
n : ℕ
a : α
l : List α
\u22a2 powersetCardAux (n + 1) (a :: l) = powersetCardAux (n + 1) l ++ List.map (cons a) (powersetCardAux n l)
-- Mathlib/Tactic/Group.lean
G : Type u_1
inst✝ : Group G
a b : G
n : ℤ
\u22a2 a * b ^ n * b = a * b ^ (n + 1)
-- Mathlib/Data/Matroid/Closure.lean
ι : Type u_1
α : Type u_2
M✝ : Matroid α
F X Y : Set α
e : α
M : Matroid α
hXY : X ⊆ Y
hX : autoParam (X ⊆ M.E) _auto✝
\u22a2 X ⊆ M.closure Y
-- Mathlib/Data/Matroid/Closure.lean
ι : Type u_1
α : Type u_2
M✝ : Matroid α
F X Y : Set α
e : α
M : Matroid α
hXY : X ⊆ Y
hX : autoParam (X ⊆ M.E) _auto✝
\u22a2 X ⊆ M.closure (Y ∩ M.E)
-- .lake/packages/batteries/Batteries/Data/RBMap/WF.lean
α : Type u_1
β : Type u_2
cmpα : α → α → Ordering
cmpβ : β → β → Ordering
f : α → β
inst✝ : IsMonotone cmpα cmpβ f
c✝ : RBColor
a : RBNode α
x : α
b : RBNode α
ax : All (fun x_1 => cmpLT cmpα x_1 x) a
xb : All (fun x_1 => cmpLT cmpα x x_1) b
ha : Ordered cmpα a
hb : Ordered cmpα b
\u22a2 Ordered cmpβ (map f (node c✝ a x b))
-- Mathlib/Computability/Reduce.lean
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
q : β → Prop
h₂ : ComputablePred q
f : α → β
c : Computable f
hf : ∀ (a : α), p a ↔ q (f a)
\u22a2 ComputablePred p
-- Mathlib/Computability/Reduce.lean
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
q : β → Prop
h₂ : ComputablePred q
f : α → β
c : Computable f
hf : ∀ (a : α), p a ↔ q (f a)
\u22a2 ComputablePred fun a => q (f a)
-- Mathlib/Computability/Reduce.lean
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
p : α → Prop
f : α → β
c : Computable f
g : β → Bool
hg : Computable g
h₂ : ComputablePred fun a => g a = true
hf : ∀ (a : α), p a ↔ (fun a => g a = true) (f a)
\u22a2 ComputablePred fun a => (fun a => g a = true) (f a)
-- Mathlib/Analysis/MeanInequalities.lean
ι✝ : Type u
s✝ : Finset ι✝
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 < z i
\u22a2 (∑ i ∈ s, w i) / ∑ i ∈ s, w i / z i ≤ (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹
-- Mathlib/Analysis/MeanInequalities.lean
ι✝ : Type u
s✝ : Finset ι✝
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 < z i
this : (∑ i ∈ s, (fun i => w i / ∑ i ∈ s, w i) i / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (fun i => w i / ∑ i ∈ s, w i) i
\u22a2 (∑ i ∈ s, w i) / ∑ i ∈ s, w i / z i ≤ (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹
-- Mathlib/Analysis/MeanInequalities.lean
ι✝ : Type u
s✝ : Finset ι✝
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 < z i
\u22a2 ∀ i ∈ s, 0 < (fun i => w i / ∑ i ∈ s, w i) i
-- Mathlib/Analysis/MeanInequalities.lean
ι✝ : Type u
s✝ : Finset ι✝
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : 0 < ∑ i ∈ s, w i
hz : ∀ i ∈ s, 0 < z i
\u22a2 ∑ i ∈ s, (fun i => w i / ∑ i ∈ s, w i) i = 1
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
\u22a2 (PowerSeries.coeff A (n + 1)) ((PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (exp A - 1)) = (PowerSeries.coeff A (n + 1)) (PowerSeries.X * (rescale t) (exp A))
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
\u22a2 ∑ x ∈ range (n + 1), (PowerSeries.coeff A (x, n + 1 - x).1) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (x, n + 1 - x).2) (exp A - 1) + (PowerSeries.coeff A (n + 1, n + 1 - (n + 1)).1) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1, n + 1 - (n + 1)).2) (exp A - 1) = t ^ n * (algebraMap ℚ A) (1 / ↑n !)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
\u22a2 ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (1 / ↑n !)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (1 / ↑n !)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 (algebraMap ℚ A) ↑(n + 1)! * ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = (algebraMap ℚ A) ↑(n + 1)! * (t ^ n * (algebraMap ℚ A) (1 / ↑n !))
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 (algebraMap ℚ A) ↑(n + 1)! * ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (↑(n + 1)! * (1 / ↑n !))
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 (algebraMap ℚ A) ↑(n + 1)! * ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (↑((n + 1) * n !) * (1 / ↑n !))
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 (algebraMap ℚ A) ↑(n + 1)! * ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = (aeval t) ((monomial n) (↑n + 1))
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 ∑ i ∈ range (n + 1), (algebraMap ℚ A) ↑(n + 1)! * ((PowerSeries.coeff A i) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - i)) (exp A - 1)) = ∑ x ∈ range (n + 1), (aeval t) (↑((n + 1).choose x) • bernoulli x)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
\u22a2 ∀ x ∈ range (n + 1), (algebraMap ℚ A) ↑(n + 1)! * ((PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1)) = (aeval t) (↑((n + 1).choose x) • bernoulli x)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
i : ℕ
hi : i ∈ range (n + 1)
\u22a2 (algebraMap ℚ A) ↑(n + 1)! * ((PowerSeries.coeff A i) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - i)) (exp A - 1)) = (aeval t) (↑((n + 1).choose i) • bernoulli i)
-- Mathlib/NumberTheory/BernoulliPolynomials.lean
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
t : A
n : ℕ
hnp1 : IsUnit ↑(n + 1)!
i : ℕ
hi : i ∈ range (n + 1)
\u22a2 (algebraMap ℚ A) (↑(n + 1)! * (↑i !)⁻¹ * (↑(n + 1 - i)!)⁻¹) * (aeval t) (bernoulli i) = (algebraMap ℚ A) (↑(n + 1)! / (↑i ! * ↑(n + 1 - i)!)) * (aeval t) (bernoulli i)
-- Mathlib/Analysis/Calculus/FDeriv/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
h : x ∉ tsupport f
\u22a2 HasStrictFDerivAt f 0 x
-- Mathlib/Analysis/Calculus/FDeriv/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
h : f =ᶠ[𝓝 x] 0
\u22a2 HasStrictFDerivAt f 0 x
-- Mathlib/CategoryTheory/Sites/LocallySurjective.lean
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
H : ∀ (U : Cᵒᵖ), Function.Surjective ⇑(f.app U)
U : C
t : (forget A).obj (F.obj (op U))
\u22a2 imageSieve f ((f.app (op U)) t) ∈ J.sieves U
-- Mathlib/CategoryTheory/Sites/LocallySurjective.lean
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : ConcreteCategory A
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
H : ∀ (U : Cᵒᵖ), Function.Surjective ⇑(f.app U)
U : C
t : (forget A).obj (F.obj (op U))
\u22a2 ⊤ ∈ J.sieves U
-- Mathlib/Analysis/Convex/Integral.lean
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : CompleteSpace E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
μ : Measure α
s : Set E
t : Set α
f : α → E
g : E → ℝ
C : ℝ
inst✝ : IsFiniteMeasure μ
hg : StrictConcaveOn ℝ s g
hgc : ContinuousOn g s
hsc : IsClosed s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
hgi : Integrable (g ∘ f) μ
\u22a2 f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ⨍ (x : α), g (f x) ∂μ < g (⨍ (x : α), f x ∂μ)
-- Mathlib/Order/Filter/Cofinite.lean
ι : Type u_1
α : Type u_2
β : Type u_3
l : Filter α
\u22a2 cofinite = atTop
-- Mathlib/Order/Filter/Cofinite.lean
ι : Type u_1
α : Type u_2
β : Type u_3
l : Filter α
\u22a2 cofinite ≤ atTop
-- Mathlib/Order/Filter/Cofinite.lean
ι : Type u_1
α : Type u_2
β : Type u_3
l : Filter α
N : ℕ
x✝ : True
\u22a2 Ici N ∈ cofinite
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ
\u22a2 (aeval x) (hermite n) = (-1) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2))
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ
\u22a2 (aeval x) (hermite n) = (-1) ^ n * ((-1) ^ n * (aeval x) (hermite n) * Real.exp (-(x ^ 2 / 2))) / Real.exp (-(x ^ 2 / 2))
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ
\u22a2 (aeval x) (hermite n) * Real.exp (-x ^ 2 / 2) = (-1) ^ n * ((-1) ^ n * (aeval x) (hermite n) * Real.exp (-x ^ 2 / 2))
-- Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean
n : ℕ
x : ℝ
\u22a2 (-1) ^ n ≠ 0
-- Mathlib/Algebra/Polynomial/Mirror.lean
R : Type u_1
inst✝ : Semiring R
p q : R[X]
n : ℕ
\u22a2 p.mirror.mirror.coeff n = p.coeff n
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a : α
\u22a2 ⋃ n, Ico (a + n • p) (a + (n + 1) • p) = univ
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
\u22a2 ∃ i, b ∈ Ico (a + i • p) (a + (i + 1) • p)
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
\u22a2 ∃ i, b ∈ Ico (a + i • p) (a + (i + 1) • p)
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
\u22a2 b < a + (toIcoDiv hp a b + 1) • p
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
\u22a2 b < a + toIcoDiv hp a b • p + p
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
p : α
hp : 0 < p
a b : α
hl : a ≤ b - toIcoDiv hp a b • p
hr : b - toIcoDiv hp a b • p < a + p
\u22a2 a + toIcoDiv hp a b • p + p = a + p + toIcoDiv hp a b • p
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
c : Cardinal.{u}
\u22a2 lift.{max (u + 1) v, u} c < univ.{u, v}
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
c : Cardinal.{u}
this : lift.{max (u + 1) v, u + 1} (lift.{u + 1, u} c) < lift.{max (u + 1) v, u + 1} univ.{u, u + 1}
\u22a2 lift.{max (u + 1) v, u} c < univ.{u, v}
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
c : Cardinal.{u}
this : lift.{max (u + 1) v, u} c < univ.{u, v}
\u22a2 lift.{max (u + 1) v, u} c < univ.{u, v}
-- Mathlib/Algebra/BigOperators/Fin.lean
α : Type u_1
β : Type u_2
n : ℕ
R : Type u_3
inst✝ : CommSemiring R
a b : R
\u22a2 ∑ s : Finset (Fin n), a ^ s.card * b ^ (n - s.card) = (a + b) ^ n
-- Mathlib/Algebra/Group/Submonoid/Membership.lean
M : Type u_1
A : Type u_2
B : Type u_3
inst✝ : AddMonoid A
\u22a2 closure {0} = ⊥
-- Mathlib/GroupTheory/Perm/Cycle/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f g : Perm α
s✝ t : Set α
a b x y : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
hb : b ∈ s
\u22a2 ∃ n < s.card, (f ^ n) a = b
-- Mathlib/Topology/UniformSpace/Equicontinuity.lean
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
inst✝ : Finite ι
F : ι → β → α
S : Set β
\u22a2 UniformEquicontinuousOn F S ↔ ∀ (i : ι), UniformContinuousOn (F i) S
-- Mathlib/Topology/UniformSpace/Equicontinuity.lean
ι : Type u_1
κ : Type u_2
X : Type u_3
X' : Type u_4
Y : Type u_5
Z : Type u_6
α : Type u_7
α' : Type u_8
β : Type u_9
β' : Type u_10
γ : Type u_11
𝓕 : Type u_12
tX : TopologicalSpace X
tY : TopologicalSpace Y
tZ : TopologicalSpace Z
uα : UniformSpace α
uβ : UniformSpace β
uγ : UniformSpace γ
inst✝ : Finite ι
F : ι → β → α
S : Set β
\u22a2 (∀ (y : ι), ∀ x ∈ 𝓤 α, ∀ᶠ (x_2 : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), (F y x_2.1, F y x_2.2) ∈ x) ↔ ∀ (i : ι), UniformContinuousOn (F i) S
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
f g : Ordinal.{u} → Ordinal.{u}
hf : IsNormal f
hg : IsNormal g
x✝ : f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
h₁ : f 0 = g 0
h₂ : ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
\u22a2 f 0 = g 0
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
f g : Ordinal.{u} → Ordinal.{u}
hf : IsNormal f
hg : IsNormal g
x✝ : f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
h₁ : f 0 = g 0
h₂ : ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
o✝ : Ordinal.{u}
a✝ : f o✝ = g o✝
\u22a2 f (succ o✝) = g (succ o✝)
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
f g : Ordinal.{u} → Ordinal.{u}
hf : IsNormal f
hg : IsNormal g
x✝ : f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
h₁ : f 0 = g 0
h₂ : ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
o✝ : Ordinal.{u}
ho : o✝.IsLimit
H : ∀ o' < o✝, f o' = g o'
\u22a2 f o✝ = g o✝
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
f g : Ordinal.{u} → Ordinal.{u}
hf : IsNormal f
hg : IsNormal g
x✝ : f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
h₁ : f 0 = g 0
h₂ : ∀ (a : Ordinal.{u}), f a = g a → f (succ a) = g (succ a)
o✝ : Ordinal.{u}
ho : o✝.IsLimit
H : ∀ o' < o✝, f o' = g o'
\u22a2 (o✝.bsup fun x x_1 => f x) = o✝.bsup fun x x_1 => g x
-- Mathlib/Algebra/Group/Action/Defs.lean
M : Type u_1
N : Type u_2
G : Type u_3
H : Type u_4
A : Type u_5
B : Type u_6
α : Type u_7
β : Type u_8
γ : Type u_9
δ : Type u_10
inst✝⁴ : SMul M α
inst✝³ : SMul N α
inst✝² : SMul M β
inst✝¹ : SMul N β
inst✝ : SMulCommClass M N α
f : α → β
hf : Surjective f
h₁ : ∀ (c : M) (x : α), f (c • x) = c • f x
h₂ : ∀ (c : N) (x : α), f (c • x) = c • f x
c₁ : M
c₂ : N
x : α
\u22a2 c₁ • c₂ • f x = c₂ • c₁ • f x
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R : Type u
R' : Type u'
M M₁ : Type v
M' : Type v'
inst✝⁹ : Ring R
inst✝⁸ : Ring R'
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : Module R M
inst✝³ : Module R M'
inst✝² : Module R M₁
inst✝¹ : Module R' M'
inst✝ : Module R' M₁
f : M →ₗ[R] M'
h : Surjective ⇑f
\u22a2 lift.{v, v'} (Module.rank R M') ≤ lift.{v', v} (Module.rank R M)
-- Mathlib/LinearAlgebra/Dimension/Basic.lean
R : Type u
R' : Type u'
M M₁ : Type v
M' : Type v'
inst✝⁹ : Ring R
inst✝⁸ : Ring R'
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : Module R M
inst✝³ : Module R M'
inst✝² : Module R M₁
inst✝¹ : Module R' M'
inst✝ : Module R' M₁
f : M →ₗ[R] M'
h : Surjective ⇑f
\u22a2 lift.{v, v'} (Module.rank R ↥(range f)) ≤ lift.{v', v} (Module.rank R M)