ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
X : (c d : C) → Set (c ⟶ d)
\u22a2 generated X ≤ generatedNormal X
-- Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
X : (c d : C) → Set (c ⟶ d)
\u22a2 {S | (∀ (c d : C), X c d ⊆ S.arrows c d) ∧ S.IsNormal} ⊆ {S | ∀ (c d : C), X c d ⊆ S.arrows c d}
-- Mathlib/Algebra/Order/Field/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : LinearOrderedSemifield α
a b c d e : α
m n : ℤ
ha : 0 < a
hb : 0 < b
\u22a2 1 / a < b ↔ 1 / b < a
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u
inst✝⁴ : CommSemiring R
M₁ : Type v
M₂ : Type v'
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
f : Dual R R
r : R
\u22a2 f 1 * r = f r
-- Mathlib/LinearAlgebra/Dual.lean
R : Type u
inst✝⁴ : CommSemiring R
M₁ : Type v
M₂ : Type v'
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R M₂
f : Dual R R
r : R
\u22a2 f 1 * r = f (r • 1)
-- Mathlib/Logic/Equiv/Basic.lean
β : Sort ?u.114339
α : Sort ?u.114344
P : β → Sort w
e✝ : α ≃ β
ι'' : Sort u_2
ι : Type u_3
ι' : Type u_4
π : ι'' → Type u_1
e : ι ⊕ ι' ≃ ι''
f : (i : ι) → π (e (inl i))
g : (i : ι') → π (e (inr i))
j : ι'
\u22a2 (piCongrLeft π e) ((sumPiEquivProdPi fun x => π (e x)).symm (f, g)) (e (inr j)) = g j
-- Mathlib/Order/Category/FinBoolAlg.lean
α β : FinBoolAlg
e : ↑α.toBoolAlg ≃o ↑β.toBoolAlg
x✝ : (forget FinBoolAlg).obj α
\u22a2 ((let __src := { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ }) ≫ let __src := { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ }) x✝ = (𝟙 α) x✝
-- Mathlib/Order/Category/FinBoolAlg.lean
α β : FinBoolAlg
e : ↑α.toBoolAlg ≃o ↑β.toBoolAlg
x✝ : (forget FinBoolAlg).obj β
\u22a2 ((let __src := { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ }) ≫ let __src := { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }; { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ }) x✝ = (𝟙 β) x✝
-- Mathlib/Algebra/Lie/Quotient.lean
R : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
x✝ : L ⧸ f.ker
x : L
y✝ : L ⧸ f.ker
y : L
\u22a2 { toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun ⁅Quot.mk Setoid.r x, Quot.mk Setoid.r y⁆ = ⁅{ toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun (Quot.mk Setoid.r x), { toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun (Quot.mk Setoid.r y)⁆
-- Mathlib/Algebra/Lie/Quotient.lean
R : Type u_1
L : Type u_2
L' : Type u_3
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
inst✝¹ : LieRing L'
inst✝ : LieAlgebra R L'
f : L →ₗ⁅R⁆ L'
x✝ : L ⧸ f.ker
x : L
y✝ : L ⧸ f.ker
y : L
\u22a2 ↑({ toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun ⁅Quot.mk Setoid.r x, Quot.mk Setoid.r y⁆) = ⁅↑({ toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun (Quot.mk Setoid.r x)), ↑({ toFun := ⇑(↑f).quotKerEquivRange, map_add' := ⋯, map_smul' := ⋯ }.toFun (Quot.mk Setoid.r y))⁆
-- Mathlib/LinearAlgebra/Dimension/Constructions.lean
R S : Type u
M : Type v
M' : Type v'
M₁ : Type v
ι : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝¹⁰ : Ring R
inst✝⁹ : CommRing S
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup M'
inst✝⁶ : AddCommGroup M₁
inst✝⁵ : Module R M
inst✝⁴ : Module R M'
inst✝³ : Module R M₁
F : Type u_2
E : Type u_3
inst✝² : CommRing F
inst✝¹ : Ring E
inst✝ : Algebra F E
\u22a2 Module.rank F ↥⊤ = Module.rank F E
-- Mathlib/Order/OmegaCompletePartialOrder.lean
α : Type u
α' : Type u_1
β✝ : Type v
β' : Type u_2
γ✝ : Type u_3
φ : Type u_4
inst✝⁵ : OmegaCompletePartialOrder α
inst✝⁴ : OmegaCompletePartialOrder β✝
inst✝³ : OmegaCompletePartialOrder γ✝
inst✝² : OmegaCompletePartialOrder φ
inst✝¹ : OmegaCompletePartialOrder α'
inst✝ : OmegaCompletePartialOrder β'
β γ : Type v
c : Chain α
f : α →o Part β
g : α →o β → Part γ
\u22a2 ωSup (c.map (f.partBind g)) = ωSup (c.map f) >>= ωSup (c.map g)
-- Mathlib/Order/OmegaCompletePartialOrder.lean
α : Type u
α' : Type u_1
β✝ : Type v
β' : Type u_2
γ✝ : Type u_3
φ : Type u_4
inst✝⁵ : OmegaCompletePartialOrder α
inst✝⁴ : OmegaCompletePartialOrder β✝
inst✝³ : OmegaCompletePartialOrder γ✝
inst✝² : OmegaCompletePartialOrder φ
inst✝¹ : OmegaCompletePartialOrder α'
inst✝ : OmegaCompletePartialOrder β'
β γ : Type v
c : Chain α
f : α →o Part β
g : α →o β → Part γ
\u22a2 ∀ (c_1 : Part γ), ωSup (c.map (f.partBind g)) ≤ c_1 ↔ ωSup (c.map f) >>= ωSup (c.map g) ≤ c_1
-- Mathlib/Order/OmegaCompletePartialOrder.lean
α : Type u
α' : Type u_1
β✝ : Type v
β' : Type u_2
γ✝ : Type u_3
φ : Type u_4
inst✝⁵ : OmegaCompletePartialOrder α
inst✝⁴ : OmegaCompletePartialOrder β✝
inst✝³ : OmegaCompletePartialOrder γ✝
inst✝² : OmegaCompletePartialOrder φ
inst✝¹ : OmegaCompletePartialOrder α'
inst✝ : OmegaCompletePartialOrder β'
β γ : Type v
c : Chain α
f : α →o Part β
g : α →o β → Part γ
x : Part γ
\u22a2 ωSup (c.map (f.partBind g)) ≤ x ↔ ωSup (c.map f) >>= ωSup (c.map g) ≤ x
-- Mathlib/Order/OmegaCompletePartialOrder.lean
α : Type u
α' : Type u_1
β✝ : Type v
β' : Type u_2
γ✝ : Type u_3
φ : Type u_4
inst✝⁵ : OmegaCompletePartialOrder α
inst✝⁴ : OmegaCompletePartialOrder β✝
inst✝³ : OmegaCompletePartialOrder γ✝
inst✝² : OmegaCompletePartialOrder φ
inst✝¹ : OmegaCompletePartialOrder α'
inst✝ : OmegaCompletePartialOrder β'
β γ : Type v
c : Chain α
f : α →o Part β
g : α →o β → Part γ
x : Part γ
\u22a2 (∀ (i : ℕ), (c.map (f.partBind g)) i ≤ x) ↔ ∀ a ∈ ωSup (c.map f), ωSup (c.map g) a ≤ x
-- Mathlib/Analysis/BoxIntegral/Partition/Split.lean
ι : Type u_1
M : Type u_2
n : ℕ
I✝ J✝ : Box ι
i : ι
x : ℝ
I J Js : Box ι
s : Finset (ι × ℝ)
H : ∀ (i : ι), {(i, J.lower i), (i, J.upper i)} ⊆ s
HJs : Js ∈ splitMany I s
Hn : ¬Disjoint ↑J ↑Js
\u22a2 Js ≤ J
-- Mathlib/Analysis/BoxIntegral/Partition/Split.lean
ι : Type u_1
M : Type u_2
n : ℕ
I✝ J✝ : Box ι
i : ι
x : ℝ
I J Js : Box ι
s : Finset (ι × ℝ)
HJs : Js ∈ splitMany I s
Hn : ¬Disjoint ↑J ↑Js
H : ∀ (i : ι), (i, J.lower i) ∈ s ∧ (i, J.upper i) ∈ s
\u22a2 Js ≤ J
-- Mathlib/Analysis/BoxIntegral/Partition/Split.lean
ι : Type u_1
M : Type u_2
n : ℕ
I✝ J✝ : Box ι
i : ι
x✝ : ℝ
I J Js : Box ι
s : Finset (ι × ℝ)
HJs : Js ∈ splitMany I s
Hn : ¬Disjoint ↑J ↑Js
H : ∀ (i : ι), (i, J.lower i) ∈ s ∧ (i, J.upper i) ∈ s
x : ι → ℝ
hx : x ∈ ↑J
hxs : x ∈ ↑Js
\u22a2 Js ≤ J
-- Mathlib/Algebra/Polynomial/Coeff.lean
R : Type u
S : Type v
a b : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ q r✝ p : R[X]
n d : ℕ
r : R
\u22a2 ((monomial n) r * p).coeff (d + n) = r * p.coeff d
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s₁ s₂ : Set α
hd : Disjoint s₁ s₂
h₁ : P s₁
h₂ : P s₂
\u22a2 extend m (s₁ ∪ s₂) = extend m s₁ + extend m s₂
-- Mathlib/MeasureTheory/OuterMeasure/Induced.lean
α : Type u_1
P : Set α → Prop
m : (s : Set α) → P s → ℝ≥0∞
P0 : P ∅
m0 : m ∅ P0 = 0
PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)
mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯
msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯
m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂
s₁ s₂ : Set α
hd : Disjoint s₁ s₂
h₁ : P s₁
h₂ : P s₂
\u22a2 ∑ b : Bool, extend m (bif b then s₁ else s₂) = extend m s₁ + extend m s₂
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b x y : ℝ
hb : 1 < b
hx : 0 < x
hy : 0 < y
\u22a2 logb b x < logb b y ↔ x < y
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b x y : ℝ
hb : 1 < b
hx : 0 < x
hy : 0 < y
\u22a2 log x < log y ↔ x < y
-- Mathlib/Data/Complex/Basic.lean
z : ℂ
h : z ≠ 0
\u22a2 z * z⁻¹ = 1
-- Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
a b c : ℕ
hb0 : 0 < b
hc : a / b ≤ c
x : ℕ
\u22a2 x ∈ Ico 1 (a / b).succ ↔ x ∈ filter (fun x => x * b ≤ a) (Ico 1 c.succ)
-- Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
a b c : ℕ
hb0 : 0 < b
hc : a / b ≤ c
x : ℕ
this : x * b ≤ a → x ≤ c
\u22a2 x ∈ Ico 1 (a / b).succ ↔ x ∈ filter (fun x => x * b ≤ a) (Ico 1 c.succ)
-- Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
a b c : ℕ
hb0 : 0 < b
hc : a / b ≤ c
x : ℕ
this : x * b ≤ a → x ≤ c
\u22a2 x * b ≤ a → 1 ≤ x → x ≤ c
-- Mathlib/Analysis/Normed/MulAction.lean
α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖
a : α
b₁ b₂ : β
\u22a2 dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂
-- Mathlib/Analysis/Normed/MulAction.lean
α : Type u_1
β : Type u_2
inst✝² : SeminormedRing α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : Module α β
h : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖
a₁ a₂ : α
b : β
\u22a2 dist (a₁ • b) (a₂ • b) ≤ dist a₁ a₂ * dist b 0
-- Mathlib/RingTheory/Derivation/Basic.lean
R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring A
inst✝⁶ : CommSemiring B
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
inst✝² : Module A M
inst✝¹ : Module B M
inst✝ : Module R M
D D1 D2 : Derivation R A M
r : R
a b : A
\u22a2 D ((algebraMap R A) r) = 0
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
\u22a2 ∀ x ≥ x₀, P x
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
\u22a2 ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
P : ℝ → Prop
x₀ r : ℝ
hr : 1 < r
hx₀ : 0 < x₀
base : ∀ x ∈ Ico x₀ (r * x₀), P x
step : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z
n : ℕ
\u22a2 ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x
-- Mathlib/Algebra/Group/Units.lean
α : Type u
M : Type u_1
N : Type u_2
inst✝ : DivisionMonoid α
b c : α
u : αˣ
\u22a2 ↑u * (↑u)⁻¹ = 1
-- Mathlib/Topology/Maps/Basic.lean
X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hs : ∀ (s : Set X), f '' interior s ⊆ interior (f '' s)
u : Set X
hu : IsOpen u
\u22a2 f '' u = f '' interior u
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ^ Group.rank ↥(closureCommutatorRepresentatives G)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 : Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ ((center ↥(closureCommutatorRepresentatives G)).index * Nat.card ↑(commutatorSet ↥(closureCommutatorRepresentatives G)) + 1)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 : Nat.card ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ ((center ↥(closureCommutatorRepresentatives G)).index * Nat.card ↑(commutatorSet G) + 1)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ Group.rank ↥(closureCommutatorRepresentatives G)
h2 : Nat.card ↥(_root_.commutator G) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ ((center ↥(closureCommutatorRepresentatives G)).index * Nat.card ↑(commutatorSet G) + 1)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h2 : Nat.card ↥(_root_.commutator G) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ ((center ↥(closureCommutatorRepresentatives G)).index * Nat.card ↑(commutatorSet G) + 1)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 : Nat.card ↥(_root_.commutator G) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G)) * Nat.card ↑(commutatorSet G) + 1)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 : Nat.card ↥(_root_.commutator G) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)
\u22a2 Nat.card ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
h1 : (center ↥(closureCommutatorRepresentatives G)).index ≤ Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G))
h2 : Nat.card ↥(_root_.commutator G) ∣ (center ↥(closureCommutatorRepresentatives G)).index ^ (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)
\u22a2 0 < (center ↥(closureCommutatorRepresentatives G)).index ^ (Nat.card ↑(commutatorSet G) ^ (2 * Nat.card ↑(commutatorSet G) + 1) + 1)
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P : Karoubi C
\u22a2 (G.map P.decompId_p).f = ((((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).obj P).p ≫ (G.map P.decompId_p).f ≫ (((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).obj P).p
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P Q : Karoubi C
f : P ⟶ Q
\u22a2 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).map f ≫ (fun P => { f := (G.map P.decompId_p).f, comm := ⋯ }) Q = (fun P => { f := (G.map P.decompId_p).f, comm := ⋯ }) P ≫ ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).map f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P : Karoubi C
\u22a2 (G.map P.decompId_i).f = (((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).obj P).p ≫ (G.map P.decompId_i).f ≫ ((((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).obj P).p
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P Q : Karoubi C
f : P ⟶ Q
\u22a2 ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).map f ≫ (fun P => { f := (G.map P.decompId_i).f, comm := ⋯ }) Q = (fun P => { f := (G.map P.decompId_i).f, comm := ⋯ }) P ≫ (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).map f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P : Karoubi C
\u22a2 (({ app := fun P => { f := (G.map P.decompId_p).f, comm := ⋯ }, naturality := ⋯ } ≫ { app := fun P => { f := (G.map P.decompId_i).f, comm := ⋯ }, naturality := ⋯ }).app P).f = ((𝟙 (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G)).app P).f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
G : Karoubi C ⥤ Karoubi D
P : Karoubi C
\u22a2 (({ app := fun P => { f := (G.map P.decompId_i).f, comm := ⋯ }, naturality := ⋯ } ≫ { app := fun P => { f := (G.map P.decompId_p).f, comm := ⋯ }, naturality := ⋯ }).app P).f = ((𝟙 ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G)).app P).f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
X Y : Karoubi C ⥤ Karoubi D
φ : X ⟶ Y
P : Karoubi C
\u22a2 ((X.map ((toKaroubi C).map P.p)).f ≫ (φ.app ((toKaroubi C).obj P.X)).f) ≫ (Y.map P.decompId_p).f = (X.map P.decompId_p).f ≫ (φ.app P).f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
X Y : Karoubi C ⥤ Karoubi D
φ : X ⟶ Y
P : Karoubi C
\u22a2 ((X.map (P.decompId_p ≫ P.decompId_i)).f ≫ (φ.app ((toKaroubi C).obj P.X)).f) ≫ (Y.map P.decompId_p).f = (X.map P.decompId_p).f ≫ (X.map P.decompId_i ≫ φ.app { X := P.X, p := 𝟙 P.X, idem := ⋯ } ≫ Y.map P.decompId_p).f
-- Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{?u.48051, u_1} C
inst✝¹ : Category.{?u.48055, u_2} D
inst✝ : Category.{?u.48059, u_3} E
X Y : Karoubi C ⥤ Karoubi D
φ : X ⟶ Y
P : Karoubi C
\u22a2 (X.map P.decompId_p).f ≫ (X.map P.decompId_i).f ≫ (φ.app ((toKaroubi C).obj P.X)).f ≫ (Y.map P.decompId_p).f = (X.map P.decompId_p).f ≫ (X.map P.decompId_i).f ≫ (φ.app { X := P.X, p := 𝟙 P.X, idem := ⋯ }).f ≫ (Y.map P.decompId_p).f
-- Mathlib/Data/Complex/Exponential.lean
x y : ℝ
\u22a2 cos x ^ 2 ≤ 1
-- Mathlib/Data/Complex/Exponential.lean
x y : ℝ
\u22a2 cos x ^ 2 ≤ sin x ^ 2 + cos x ^ 2
-- Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
C : Type u_1
inst✝¹ : Category.{?u.114018, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
\u22a2 rightHomologyMap' e.hom h₁ h₂ ≫ rightHomologyMap' e.inv h₂ h₁ = 𝟙 h₁.H
-- Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
C : Type u_1
inst✝¹ : Category.{?u.114018, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
\u22a2 rightHomologyMap' e.inv h₂ h₁ ≫ rightHomologyMap' e.hom h₁ h₂ = 𝟙 h₂.H
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
d : DynkinSystem α
s : Set (Set α)
hs : IsPiSystem s
\u22a2 ofMeasurableSpace ((generate s).toMeasurableSpace ⋯) ≤ ofMeasurableSpace (generateFrom s)
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
d : DynkinSystem α
s : Set (Set α)
hs : IsPiSystem s
\u22a2 generate s ≤ ofMeasurableSpace (generateFrom s)
-- Mathlib/MeasureTheory/Integral/Average.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 ν : Measure α
s t N : Set α
f : α → ℝ≥0∞
inst✝ : IsProbabilityMeasure μ
hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤
hN : μ N = 0
\u22a2 ∃ x ∉ N, ∫⁻ (a : α), f a ∂μ ≤ f x
-- Mathlib/Geometry/Manifold/Algebra/LieGroup.lean
𝕜 : Type u_1
inst✝¹⁹ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
E : Type u_3
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
F : Type u_4
inst✝¹⁵ : NormedAddCommGroup F
inst✝¹⁴ : NormedSpace 𝕜 F
J : ModelWithCorners 𝕜 F F
G : Type u_5
inst✝¹³ : TopologicalSpace G
inst✝¹² : ChartedSpace H G
inst✝¹¹ : Group G
inst✝¹⁰ : LieGroup I G
E' : Type u_6
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H' M
E'' : Type u_9
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_10
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M' : Type u_11
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H'' M'
n : ℕ∞
f g : M → G
x₀ : M
hf : ContMDiffAt I' I n f x₀
hg : ContMDiffAt I' I n g x₀
\u22a2 ContMDiffAt I' I n (fun x => f x / g x) x₀
-- Mathlib/Geometry/Manifold/Algebra/LieGroup.lean
𝕜 : Type u_1
inst✝¹⁹ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁸ : TopologicalSpace H
E : Type u_3
inst✝¹⁷ : NormedAddCommGroup E
inst✝¹⁶ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
F : Type u_4
inst✝¹⁵ : NormedAddCommGroup F
inst✝¹⁴ : NormedSpace 𝕜 F
J : ModelWithCorners 𝕜 F F
G : Type u_5
inst✝¹³ : TopologicalSpace G
inst✝¹² : ChartedSpace H G
inst✝¹¹ : Group G
inst✝¹⁰ : LieGroup I G
E' : Type u_6
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_7
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H' M
E'' : Type u_9
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_10
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M' : Type u_11
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H'' M'
n : ℕ∞
f g : M → G
x₀ : M
hf : ContMDiffAt I' I n f x₀
hg : ContMDiffAt I' I n g x₀
\u22a2 ContMDiffAt I' I n (fun x => f x * (g x)⁻¹) x₀