ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Order/Fin/Tuple.lean
m n : ℕ
α : Fin (n + 1) → Type u_1
x✝ : α 0
q : (i : Fin (n + 1)) → α i
p✝ : (i : Fin n) → α i.succ
i✝ : Fin n
y : α i✝.succ
z : α 0
inst✝ : (i : Fin (n + 1)) → Preorder (α i)
i : Fin (n + 1)
x : α i
q₁ q₂ : (j : Fin (n + 1)) → α j
hx : x ∈ Icc (q₁ i) (q₂ i)
p : (j : Fin n) → α (i.succAbove j)
\u22a2 p ∈ i.insertNth x ⁻¹' Icc q₁ q₂ ↔ p ∈ Icc (fun j => q₁ (i.succAbove j)) fun j => q₂ (i.succAbove j)
-- Mathlib/RingTheory/PowerSeries/Derivative.lean
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
n✝ : ℕ
\u22a2 (coeff R n✝) (↑f).derivativeFun = (coeff R n✝) ↑(derivative f)
-- Mathlib/Algebra/Group/Pi/Basic.lean
I : Type u
α✝ : Type u_1
β : Type u_2
γ : Type u_3
f : I → Type v₁
g : I → Type v₂
h : I → Type v₃
x✝ y : (i : I) → f i
i✝ : I
α : Type u_4
inst✝² : DecidableEq I
inst✝¹ : Subsingleton I
inst✝ : One α
i : I
x : α
j : I
\u22a2 Pi.mulSingle i x j = x
-- Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : CompleteSpace E
inst✝⁴ : NormedSpace ℝ E
a✝ b✝ : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : ℝ
φ : ℝ → F →L[𝕜] E
hφ : IntervalIntegrable φ μ a b
v : F
\u22a2 (∫ (x : ℝ) in a..b, φ x ∂μ) v = ∫ (x : ℝ) in a..b, (φ x) v ∂μ
-- Mathlib/GroupTheory/GroupAction/Blocks.lean
G : Type u_1
X : Type u_2
inst✝ : SMul G X
B : Set X
\u22a2 IsBlock G B ↔ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
-- Mathlib/Algebra/GroupWithZero/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : GroupWithZero G₀
a✝ b c a : G₀
\u22a2 a / a * a = a
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
\u22a2 @replaceF = @replaceFTR
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
p : α → Option α
l : List α
\u22a2 replaceF p l = replaceFTR p l
-- .lake/packages/batteries/Batteries/Data/List/Basic.lean
α : Type u_1
p : α → Option α
l : List α
\u22a2 replaceF p l = replaceFTR.go p l #[]
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 padicValNat 2 (x ^ n - y ^ n) + 1 = padicValNat 2 (x + y) + padicValNat 2 (x - y) + padicValNat 2 n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 ↑(padicValNat 2 (x ^ n - y ^ n)) + ↑1 = ↑(padicValNat 2 (x + y)) + ↑(padicValNat 2 (x - y)) + ↑(padicValNat 2 n)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 multiplicity 2 (x ^ n - y ^ n) + ↑1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 0 < n
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 0 < x - y
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 0 < x + y
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n✝ x y : ℕ
hyx : y < x
hxy : 2 ∣ x - y
hx : ¬2 ∣ x
n : ℕ
hn : n ≠ 0
hneven : Even n
\u22a2 0 < x ^ n - y ^ n
-- 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
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
\u22a2 o.bmex f < (succ o.card).ord
-- 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
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{u}
\u22a2 o.bmex f < (succ #(Quotient.out o).α).ord
-- Mathlib/Analysis/SpecialFunctions/Exp.lean
α : Type u_1
x y z : ℝ
l : Filter α
\u22a2 Tendsto rexp atTop atTop
-- Mathlib/Analysis/SpecialFunctions/Exp.lean
α : Type u_1
x y z : ℝ
l : Filter α
A : Tendsto (fun x => x + 1) atTop atTop
\u22a2 Tendsto rexp atTop atTop
-- Mathlib/Analysis/SpecialFunctions/Exp.lean
α : Type u_1
x y z : ℝ
l : Filter α
A : Tendsto (fun x => x + 1) atTop atTop
B : ∀ᶠ (x : ℝ) in atTop, x + 1 ≤ rexp x
\u22a2 Tendsto rexp atTop atTop
-- Mathlib/Analysis/Calculus/FDeriv/Prod.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : 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
ι : Type u_6
inst✝² : Fintype ι
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
φ' : (i : ι) → E →L[𝕜] F' i
Φ : E → (i : ι) → F' i
Φ' : E →L[𝕜] (i : ι) → F' i
i : ι
f : (i : ι) → F' i
\u22a2 HasFDerivAt (fun f => f i) (proj i) f
-- Mathlib/Analysis/Calculus/FDeriv/Prod.lean
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : 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
ι : Type u_6
inst✝² : Fintype ι
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
φ' : (i : ι) → E →L[𝕜] F' i
Φ : E → (i : ι) → F' i
Φ' : E →L[𝕜] (i : ι) → F' i
i : ι
f : (i : ι) → F' i
\u22a2 HasStrictFDerivAt (fun f => f i) (proj i) f
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
\u22a2 (stalkFunctor C x).map (𝟙 X).c ≫ stalkPushforward C (𝟙 ↑X) X.presheaf x = 𝟙 (X.presheaf.stalk x)
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
\u22a2 (stalkFunctor C x).map (𝟙 X).c ≫ (stalkFunctor C x).map (Pushforward.id X.presheaf).hom = 𝟙 (X.presheaf.stalk x)
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
\u22a2 (stalkFunctor C x).map ((𝟙 X).c ≫ (Pushforward.id X.presheaf).hom) = 𝟙 (X.presheaf.stalk x)
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (X.presheaf.stalk x ⟶ X.presheaf.stalk x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
\u22a2 (𝟙 X).c ≫ (Pushforward.id X.presheaf).hom = 𝟙 X.presheaf
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (X.presheaf.stalk x ⟶ X.presheaf.stalk x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
U✝ : Opens ↑↑X
\u22a2 ((𝟙 X).c ≫ (Pushforward.id X.presheaf).hom).app (op U✝) = (𝟙 X.presheaf).app (op U✝)
-- Mathlib/Geometry/RingedSpace/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : PresheafedSpace C
x : ↑↑X
e_1✝ : (X.presheaf.stalk x ⟶ X.presheaf.stalk x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)
U✝ : Opens ↑↑X
\u22a2 𝟙 (((pushforward C (𝟙 ↑X)).obj X.presheaf).obj (op U✝)) = (𝟙 X.presheaf).app (op U✝)
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
a : Ordinal.{u_1}
ha : 0 < a
\u22a2 nfp (fun x => a * x) 1 = a ^ ω
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
a : Ordinal.{u_1}
ha : 0 < a
\u22a2 (fun a_1 => sup fun n => (fun x => a * x)^[n] a_1) 1 = sup fun n => a ^ ↑n
-- Mathlib/SetTheory/Ordinal/FixedPoint.lean
a : Ordinal.{u_1}
ha : 0 < a
\u22a2 0 < a
-- Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
f : ℝ → E
c ca cb : E
l l' la la' lb lb' : Filter ℝ
lt : Filter ι
a b z : ℝ
u v ua ub va vb : ι → ℝ
inst✝¹ : FTCFilter a la la'
inst✝ : FTCFilter b lb lb'
hab : IntervalIntegrable f volume a b
hmeas : StronglyMeasurableAtFilter f lb' volume
hf : Tendsto f (lb' ⊓ ae volume) (𝓝 c)
hu : Tendsto u lt lb
hv : Tendsto v lt lb
\u22a2 (fun t => ((∫ (x : ℝ) in a..v t, f x) - ∫ (x : ℝ) in a..u t, f x) - (v t - u t) • c) =o[lt] (v - u)
-- Mathlib/Tactic/Abel.lean
α : Type u_1
inst✝ : AddCommMonoid α
n : ℕ
x a k a' : α
h : a + k = a'
\u22a2 term n x a + k = term n x a'
-- Mathlib/Algebra/Polynomial/Monic.lean
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p q r : R[X]
hp : p.Monic
h : p.natDegree = 0
\u22a2 p = 1
-- Mathlib/Algebra/Polynomial/Monic.lean
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p q r : R[X]
hp : p.Monic
h : p.natDegree = 0
this : p = C (p.coeff 0)
\u22a2 p = 1
-- Mathlib/Algebra/Polynomial/Monic.lean
R : Type u
S : Type v
a b : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p q r : R[X]
hp : p.Monic
h : p.natDegree = 0
this : p = C (p.coeff 0)
\u22a2 C (p.coeff 0) = 1
-- Mathlib/Analysis/Analytic/Within.lean
α : Type u_1
𝕜 : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
e : E
f : E → F
g : E → G
s : Set E
w✝¹ : FormalMultilinearSeries 𝕜 E F
hf : HasFPowerSeriesWithinAt f w✝¹ s e
w✝ : FormalMultilinearSeries 𝕜 E G
hg : HasFPowerSeriesWithinAt g w✝ s e
\u22a2 AnalyticWithinAt 𝕜 (fun x => (f x, g x)) s e
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
\u22a2 ((e₁.cmp e₂).orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe : (e₁.cmp e₂).Compares e₁ e₂
\u22a2 ((e₁.cmp e₂).orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
IHe : match e₁.cmp e₂, e₁, e₂ with | Ordering.lt, a, b => a < b | Ordering.eq, a, b => a = b | Ordering.gt, a, b => b < a
\u22a2 ((e₁.cmp e₂).orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
\u22a2 (match Ordering.lt, e₁, e₂ with | Ordering.lt, a, b => a < b | Ordering.eq, a, b => a = b | Ordering.gt, a, b => b < a) → (Ordering.lt.orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
\u22a2 (match Ordering.eq, e₁, e₂ with | Ordering.lt, a, b => a < b | Ordering.eq, a, b => a = b | Ordering.gt, a, b => b < a) → (Ordering.eq.orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/SetTheory/Ordinal/Notation.lean
o₁ e₁ : ONote
n₁ : ℕ+
a₁ : ONote
h✝¹ : o₁ = e₁.oadd n₁ a₁
o₂ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h✝ : o₂ = e₂.oadd n₂ a₂
h₁ : (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹).NF
h₂ : (namedPattern o₂ (e₂.oadd n₂ a₂) h✝).NF
\u22a2 (match Ordering.gt, e₁, e₂ with | Ordering.lt, a, b => a < b | Ordering.eq, a, b => a = b | Ordering.gt, a, b => b < a) → (Ordering.gt.orElse ((_root_.cmp ↑n₁ ↑n₂).orElse (a₁.cmp a₂))).Compares (namedPattern o₁ (e₁.oadd n₁ a₁) h✝¹) (namedPattern o₂ (e₂.oadd n₂ a₂) h✝)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean
R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W : WeierstrassCurve R
\u22a2 W.Ψ 1 = 1
-- Mathlib/Algebra/Group/Subsemigroup/Operations.lean
M : Type u_1
N : Type u_2
P : Type u_3
σ : Type u_4
inst✝² : Mul M
inst✝¹ : Mul N
inst✝ : Mul P
S : Subsemigroup M
\u22a2 ↑(⊥.prod ⊥) = ↑⊥
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
\u22a2 G.IsUniform ε {a} {b} ↔ 0 < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
s' : Finset α
hs' : s' ⊆ {a}
t' : Finset α
ht' : t' ⊆ {b}
hs : ↑{a}.card * ε ≤ ↑s'.card
ht : ↑{b}.card * ε ≤ ↑t'.card
\u22a2 |↑(G.edgeDensity s' t') - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
s' : Finset α
hs' : s' ⊆ {a}
t' : Finset α
ht' : t' ⊆ {b}
hs : ε ≤ ↑s'.card
ht : ε ≤ ↑t'.card
\u22a2 |↑(G.edgeDensity s' t') - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
t' : Finset α
ht' : t' ⊆ {b}
ht : ε ≤ ↑t'.card
hs' : ∅ ⊆ {a}
hs : ε ≤ ↑∅.card
\u22a2 |↑(G.edgeDensity ∅ t') - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
t' : Finset α
ht' : t' ⊆ {b}
ht : ε ≤ ↑t'.card
hs' : {a} ⊆ {a}
hs : ε ≤ ↑{a}.card
\u22a2 |↑(G.edgeDensity {a} t') - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
hs' : {a} ⊆ {a}
hs : ε ≤ ↑{a}.card
ht' : ∅ ⊆ {b}
ht : ε ≤ ↑∅.card
\u22a2 |↑(G.edgeDensity {a} ∅) - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
hε : 0 < ε
hs' : {a} ⊆ {a}
hs : ε ≤ ↑{a}.card
ht' : {b} ⊆ {b}
ht : ε ≤ ↑{b}.card
\u22a2 |↑(G.edgeDensity {a} {b}) - ↑(G.edgeDensity {a} {b})| < ε
-- Mathlib/Data/Set/Pointwise/Basic.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : MulZeroClass α
s✝ t s : Set α
\u22a2 0 * s ⊆ 0
-- .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean
α : Type u_1
β : Type u_2
γ : Type u_3
as : Array α
bs : Array β
f : α → β → γ
\u22a2 (as.zipWith bs f).size = min as.size bs.size
-- Mathlib/Data/Part.lean
α : Type u_1
β : Type u_2
γ : Type u_3
o : Part α
h : o.Dom
f : α → Part β
b : β
\u22a2 b ∈ o.bind f ↔ b ∈ f (o.get h)
-- Mathlib/Data/Part.lean
α : Type u_1
β : Type u_2
γ : Type u_3
o : Part α
h : o.Dom
f : α → Part β
b : β
\u22a2 (∃ a ∈ o, b ∈ f a) ↔ b ∈ f (o.get h)
-- Mathlib/Data/Part.lean
α : Type u_1
β : Type u_2
γ : Type u_3
o : Part α
h : o.Dom
f : α → Part β
b : β
\u22a2 (∃ a ∈ o, b ∈ f a) → b ∈ f (o.get h)
-- Mathlib/Data/Part.lean
α : Type u_1
β : Type u_2
γ : Type u_3
o : Part α
h : o.Dom
f : α → Part β
b : β
a : α
ha : a ∈ o
hb : b ∈ f a
\u22a2 b ∈ f (o.get h)
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
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_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
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
hg : DifferentiableOn 𝕜 f s
\u22a2 DifferentiableOn 𝕜 (fun y => f y - g y) s ↔ DifferentiableOn 𝕜 g s
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommRing R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : TopologicalRing R
inst✝⁷ : ContinuousStar R
inst✝⁶ : TopologicalSpace A
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : Module R A
inst✝² : IsScalarTower R A A
inst✝¹ : SMulCommClass R A A
inst✝ : NonUnitalContinuousFunctionalCalculus R p
f g : R → R
a : A
hf : autoParam (ContinuousOn f (σₙ R a)) _auto✝
hf0 : autoParam (f 0 = 0) _auto✝
hg : autoParam (ContinuousOn g (σₙ R a)) _auto✝
hg0 : autoParam (g 0 = 0) _auto✝
ha : autoParam (p a) _auto✝
\u22a2 cfcₙ (fun x => -x) a = -a
-- Mathlib/GroupTheory/NoncommPiCoprod.lean
G : Type u_1
inst✝ : Group G
ι : Type u_2
hdec : DecidableEq ι
hfin : Fintype ι
H : ι → Subgroup G
f g : (i : ι) → ↥(H i)
hcomm : Pairwise fun i j => ∀ (x y : G), x ∈ H i → y ∈ H j → Commute x y
\u22a2 (noncommPiCoprod hcomm).range = ⨆ i, H i
-- Mathlib/Data/Nat/ModEq.lean
m✝ n a✝ b✝ c✝ d a b c m : ℕ
hc : c ≠ 0
\u22a2 a * c ≡ b * c [MOD m * c] → a ≡ b [MOD m]
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : Path F
a : F.A
f : F.B a → F.M
a' : F.A
f' : F.B a' → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a', f'⟩)
hh : IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a', f'⟩)
\u22a2 iselect ps (M.mk ⟨a, f⟩) = iselect ps (M.mk ⟨a', f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a', f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : Path F
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
hh : IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩)
\u22a2 iselect ps (M.mk ⟨a, f⟩) = iselect ps (M.mk ⟨a, f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : List F.Idx
ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → iselect ps (M.mk ⟨a, f⟩) = iselect ps (M.mk ⟨a, f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
a' : F.A
i : F.B a'
hh : IsPath (⟨a', i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a', i⟩ :: ps) (M.mk ⟨a, f'⟩)
\u22a2 iselect (⟨a', i⟩ :: ps) (M.mk ⟨a, f⟩) = iselect (⟨a', i⟩ :: ps) (M.mk ⟨a, f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree (⟨a', i⟩ :: ps) (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree (⟨a', i⟩ :: ps) (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : List F.Idx
ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → iselect ps (M.mk ⟨a, f⟩) = iselect ps (M.mk ⟨a, f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
i : F.B a
hh : IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)
\u22a2 iselect (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = iselect (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩) ∧ ∃ a_1 f_1 f'_1, isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : List F.Idx
ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → (isubtree ps (M.mk ⟨a, f⟩)).head = (isubtree ps (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
i : F.B a
hh : IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)
\u22a2 (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩)).head = (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : List F.Idx
ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → (isubtree ps (M.mk ⟨a, f⟩)).head = (isubtree ps (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
i : F.B a
hh : IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)
h₁ : R (f i) (f' i)
\u22a2 (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩)).head = (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
-- Mathlib/Data/PFunctor/Univariate/M.lean
F : PFunctor.{u}
X : Type u_1
f✝ : X → ↑F X
R : F.M → F.M → Prop
inst✝ : Inhabited F.M
bisim : IsBisimulation R
ps : List F.Idx
ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → (isubtree ps (M.mk ⟨a, f⟩)).head = (isubtree ps (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)
a : F.A
f f' : F.B a → F.M
h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)
i : F.B a
hh : IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)
h₁ : R (f i) (f' i)
a₀ : F.A
f₀ : F.B a₀ → F.M
h : f i = M.mk ⟨a₀, f₀⟩
a₁ : F.A
f₁ : F.B a₁ → F.M
h' : f' i = M.mk ⟨a₁, f₁⟩
\u22a2 (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩)).head = (isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)