ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
\u22a2 smul α φ (r + r') m = smul α φ r m + smul α φ r' m
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
\u22a2 smul α φ (r + r') m = smul α φ r m + smul α φ r' m
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y : Cᵒᵖ
π : X ⟶ Y
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
\u22a2 ∀ (Y : C) (f : Y ⟶ Opposite.unop X), S.arrows f → (A.val.map f.op) (smul α φ (r + r') m) = (A.val.map f.op) (smul α φ r m + smul α φ r' m)
-- Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
inst✝ : Presheaf.IsLocallySurjective J φ
X Y✝ : Cᵒᵖ
π : X ⟶ Y✝
r r' : ↑(R.val.obj X)
m m' : ↑(A.val.obj X)
S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m
hS : S ∈ J.sieves (Opposite.unop X)
Y : C
f : Y ⟶ Opposite.unop X
r₀ : ↑(R₀.obj (Opposite.op Y))
hr₀ : (α.app (Opposite.op Y)) r₀ = (R.val.map f.op) r
r₀' : ↑(R₀.obj (Opposite.op Y))
hr₀' : (α.app (Opposite.op Y)) r₀' = (R.val.map f.op) r'
m₀ : (forget AddCommGrp).obj (M₀.presheaf.obj (Opposite.op Y))
hm₀ : (φ.app (Opposite.op Y)) m₀ = (A.val.map f.op) m
\u22a2 (A.val.map f.op) (smul α φ (r + r') m) = (A.val.map f.op) (smul α φ r m + smul α φ r' m)
-- Mathlib/RingTheory/Valuation/Basic.lean
K : Type u_1
F : Type u_2
R : Type u_3
inst✝³ : DivisionRing K
Γ₀ : Type u_4
Γ'₀ : Type u_5
Γ''₀ : Type u_6
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v✝ : Valuation R Γ₀
x✝ y z : R
v : Valuation K Γ₀
x : K
h : x ≠ 0
\u22a2 v x < 1 ↔ 1 < v x⁻¹
-- Mathlib/LinearAlgebra/Span.lean
R : Type u_1
R₂ : Type u_2
K : Type u_3
M : Type u_4
M₂ : Type u_5
V : Type u_6
S : Type u_7
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
x : M
p p' : Submodule R M
inst✝⁴ : Semiring R₂
σ₁₂ : R →+* R₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module R₂ M₂
F : Type u_8
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F σ₁₂ M M₂
s t : Set M
x✝ : M
\u22a2 x✝ ∈ span R {0} ↔ x✝ ∈ ⊥
-- Mathlib/Data/Real/Sqrt.lean
x y : ℝ
h : x ≤ 0
\u22a2 √x = 0
-- Mathlib/SetTheory/Ordinal/Exponential.lean
b x : Ordinal.{u_1}
hx : x ≠ 0
\u22a2 b ^ log b x ≤ x
-- Mathlib/SetTheory/Ordinal/Exponential.lean
x : Ordinal.{u_1}
hx : x ≠ 0
\u22a2 0 ^ log 0 x ≤ x
-- Mathlib/SetTheory/Ordinal/Exponential.lean
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
\u22a2 b ^ log b x ≤ x
-- Mathlib/SetTheory/Ordinal/Exponential.lean
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
\u22a2 b ^ log b x ≤ x
-- Mathlib/SetTheory/Ordinal/Exponential.lean
x : Ordinal.{u_1}
hx : x ≠ 0
b0 : 1 ≠ 0
\u22a2 1 ^ log 1 x ≤ x
-- Mathlib/Order/LiminfLimsup.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
ι' : Type u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
l : Filter γ
u : γ → α
a : α
\u22a2 (∀ᶠ (x : β) in map (fun x => e (u x)) l, (fun x x_1 => x ≤ x_1) x (e a)) ↔ ∀ᶠ (x : α) in map u l, (fun x x_1 => x ≤ x_1) x a
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
inst✝ : TopologicalSpace β
a b c : α
f : α → β
\u22a2 IsOpen (Ioi a)
-- Mathlib/Topology/Order/OrderClosed.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
inst✝ : TopologicalSpace β
a b c : α
f : α → β
\u22a2 IsOpen (Iic a)ᶜ
-- Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁸ : Nonempty ι
inst✝⁷ : Nonempty ι'
inst✝⁶ : NormedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : NormedField 𝕜₂
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
p : SeminormFamily 𝕜 E ι
t : TopologicalSpace E
hp : WithSeminorms p
e : ι' ≃ ι
\u22a2 WithSeminorms (p ∘ ⇑e)
-- Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
Ω : Type u_1
Ω' : Type u_2
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : MeasurableSpace Ω'
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω'
inst✝ : BorelSpace Ω'
f : Ω → Ω'
f_cont : Continuous f
\u22a2 Continuous fun ν => ν.map ⋯
-- Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
Ω : Type u_1
Ω' : Type u_2
inst✝⁵ : MeasurableSpace Ω
inst✝⁴ : MeasurableSpace Ω'
inst✝³ : TopologicalSpace Ω
inst✝² : OpensMeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω'
inst✝ : BorelSpace Ω'
f : Ω → Ω'
f_cont : Continuous f
\u22a2 ∀ (x : ProbabilityMeasure Ω), ContinuousAt (fun ν => ν.map ⋯) x
-- Mathlib/GroupTheory/Submonoid/Inverses.lean
M : Type u_1
inst✝ : CommMonoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : ↥S
\u22a2 ↑x * ↑((S.leftInvEquiv hS).symm x) = 1
-- Mathlib/CategoryTheory/Comma/StructuredArrow.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
S S' S'' : D
Y Y' Y'' : C
T T' : C ⥤ D
A B : StructuredArrow S T
f : A ⟶ B
\u22a2 A.hom ≫ T.map f.right = B.hom
-- Mathlib/CategoryTheory/Comma/StructuredArrow.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
S S' S'' : D
Y Y' Y'' : C
T T' : C ⥤ D
A B : StructuredArrow S T
f : A ⟶ B
this : (Functor.fromPUnit S).map f.left ≫ B.hom = A.hom ≫ T.map f.right
\u22a2 A.hom ≫ T.map f.right = B.hom
-- Mathlib/Data/Sym/Sym2.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a : α
z : Sym2 α
h : a ∈ z
\u22a2 Mem.other h ∈ z
-- Mathlib/Data/Sym/Sym2.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a : α
z : Sym2 α
h : a ∈ z
\u22a2 z = s(a, Mem.other h)
-- Mathlib/Data/Finsupp/NeLocus.lean
α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁴ : DecidableEq α
inst✝³ : DecidableEq N
inst✝² : DecidableEq M
inst✝¹ : Zero M
inst✝ : Zero N
f g : α →₀ N
F : N → M
F0 : F 0 = 0
hF : Function.Injective F
a✝ : α
\u22a2 a✝ ∈ (mapRange F F0 f).neLocus (mapRange F F0 g) ↔ a✝ ∈ f.neLocus g
-- Mathlib/Algebra/BigOperators/Finprod.lean
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M : Type u_5
N : Type u_6
inst✝¹ : CommMonoid M
inst✝ : CommMonoid N
f✝ g : α → M
a b : α
s✝ t : Set α
s : Finset (α × β)
f : α × β → M
\u22a2 ∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab = ∏ᶠ (a : α) (b : β) (_ : (a, b) ∈ s), f (a, b)
-- Mathlib/Data/NNReal/Basic.lean
ι : Sort u_1
f : ι → ℝ≥0
a : ℝ≥0
g : ι → ℝ≥0
h : ℝ≥0
H : ∀ (i : ι), g i * h ≤ a
\u22a2 iSup g * h ≤ a
-- Mathlib/Data/NNReal/Basic.lean
ι : Sort u_1
f : ι → ℝ≥0
a : ℝ≥0
g : ι → ℝ≥0
h : ℝ≥0
H : ∀ (i : ι), g i * h ≤ a
\u22a2 ⨆ i, g i * h ≤ a
-- Mathlib/Data/Int/Bitwise.lean
n : ℕ
\u22a2 Nat.shiftLeft' false 1 n = 2 ^ n
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
\u22a2 ((starRingEnd ℂ) x).arg = if x.arg = π then π else -x.arg
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re < 0
hi : x.im < 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re < 0
hi : x.im = 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re < 0
hi : 0 < x.im
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re = 0
hi : x.im < 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re = 0
hi : x.im = 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : x.re = 0
hi : 0 < x.im
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : 0 < x.re
hi : x.im < 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : 0 < x.re
hi : x.im = 0
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ
hr : 0 < x.re
hi : 0 < x.im
\u22a2 (if 0 ≤ x.re then -Real.arcsin (x.im / abs x) else if 0 ≤ -x.im then Real.arcsin (x.im / abs x) + π else Real.arcsin (x.im / abs x) - π) = if x.re < 0 ∧ x.im = 0 then π else -if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then -Real.arcsin (x.im / abs x) + π else -Real.arcsin (x.im / abs x) - π
-- Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P : Fin 3 → R
hPz : P z = 0
\u22a2 W'.neg P = ![P x, -P y, 0]
-- Mathlib/FieldTheory/Perfect.lean
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
y : R
\u22a2 (X ^ p ^ n - C y).roots = p ^ n • {(iterateFrobeniusEquiv R p n).symm y}
-- Mathlib/FieldTheory/Perfect.lean
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
p n : ℕ
inst✝¹ : ExpChar R p
f : R[X]
inst✝ : PerfectRing R p
y : R
H : ((expand R (p ^ n)) (X - C y)).roots = p ^ n • Multiset.map (⇑(iterateFrobeniusEquiv R p n).symm) (X - C y).roots
\u22a2 (X ^ p ^ n - C y).roots = p ^ n • {(iterateFrobeniusEquiv R p n).symm y}
-- Mathlib/Algebra/Lie/Submodule.lean
R : Type u
L : Type v
L' : Type w₂
M : Type w
M' : Type w₁
inst✝¹⁵ : CommRing R
inst✝¹⁴ : LieRing L
inst✝¹³ : LieAlgebra R L
inst✝¹² : LieRing L'
inst✝¹¹ : LieAlgebra R L'
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : LieRingModule L M
inst✝⁷ : LieModule R L M
inst✝⁶ : AddCommGroup M'
inst✝⁵ : Module R M'
inst✝⁴ : LieRingModule L M'
inst✝³ : LieModule R L M'
f : M →ₗ⁅R,L⁆ M'
N N₂ : LieSubmodule R L M
N' : LieSubmodule R L M'
M'' : Type u_1
inst✝² : AddCommGroup M''
inst✝¹ : Module R M''
inst✝ : LieRingModule L M''
g : M' →ₗ⁅R,L⁆ M''
\u22a2 ↑(map (g.comp f) N) = ↑(map g (map f N))
-- Mathlib/RingTheory/Ideal/AssociatedPrime.lean
R : Type u_1
inst✝⁵ : CommRing R
I J : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommGroup M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : Subsingleton M
hI : I.IsPrime
x : M
hx : I = (Submodule.span R {x}).annihilator
\u22a2 False
-- Mathlib/RingTheory/Ideal/AssociatedPrime.lean
R : Type u_1
inst✝⁵ : CommRing R
I J : Ideal R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommGroup M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : Subsingleton M
hI : I.IsPrime
x : M
hx : I = (Submodule.span R {x}).annihilator
\u22a2 I = ⊤
-- Mathlib/Algebra/Module/Defs.lean
α : Type u_1
R : Type u_2
k : Type u_3
S : Type u_4
M : Type u_5
M₂ : Type u_6
M₃ : Type u_7
ι : Type u_8
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
r s : R
x y : M
\u22a2 -r • -x = r • x
-- Mathlib/Algebra/CubicDiscriminant.lean
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
ha : P.a = 0
hb : P.b = 0
hc : P.c = 0
\u22a2 P.toPoly.natDegree = 0
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
\u22a2 (if h : { byteIdx := utf8Len l } < { byteIdx := utf8Len l + utf8Len (c :: m) } then if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l } else let_fun this := ⋯; { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } ({ data := l ++ c :: m ++ r }.next { byteIdx := utf8Len l }) else { byteIdx := utf8Len l }) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
\u22a2 (if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l } else let_fun this := ⋯; { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } ({ data := l ++ c :: m ++ r }.next { byteIdx := utf8Len l })) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
h1 : { data := l ++ (c :: m ++ r) }.get { byteIdx := utf8Len l } = (c :: m ++ r).headD default
\u22a2 (if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l } else let_fun this := ⋯; { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } ({ data := l ++ c :: m ++ r }.next { byteIdx := utf8Len l })) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
h1 : { data := l ++ (c :: m ++ r) }.get { byteIdx := utf8Len l } = (c :: m ++ r).headD default
h2 : { data := l ++ c :: (m ++ r) }.next { byteIdx := utf8Len l } = { byteIdx := utf8Len l + c.utf8Size }
\u22a2 (if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l } else let_fun this := ⋯; { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } ({ data := l ++ c :: m ++ r }.next { byteIdx := utf8Len l })) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
h1 : { data := l ++ c :: (m ++ r) }.get { byteIdx := utf8Len l } = c
h2 : { data := l ++ c :: (m ++ r) }.next { byteIdx := utf8Len l } = { byteIdx := utf8Len l + c.utf8Size }
\u22a2 (if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l } else let_fun this := ⋯; { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } ({ data := l ++ c :: m ++ r }.next { byteIdx := utf8Len l })) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
p : Char → Bool
l : List Char
c : Char
m r : List Char
h1 : { data := l ++ c :: (m ++ r) }.get { byteIdx := utf8Len l } = c
h2 : { data := l ++ c :: (m ++ r) }.next { byteIdx := utf8Len l } = { byteIdx := utf8Len l + c.utf8Size }
\u22a2 (if p c = true then { byteIdx := utf8Len l } else { data := l ++ c :: (m ++ r) }.findAux p { byteIdx := utf8Len l + (utf8Len m + c.utf8Size) } { byteIdx := utf8Len l + c.utf8Size }) = { byteIdx := utf8Len l + utf8Len (match !p c with | true => c :: List.takeWhile (fun x => !p x) m | false => []) }
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : α
h : a < b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : α
h : a < b
c : α
ha : inl a < inl c
hb : inl c < inl b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : α
h : a < b
c : β
ha : inl a < inr c
hb : inr c < inl b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : β
h : a < b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : β
h : a < b
c : α
ha : inr a < inl c
hb : inl c < inr b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/Data/Sum/Order.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹ : LT α
inst✝ : LT β
x✝ : DenselyOrdered (α ⊕ β)
a b : β
h : a < b
c : β
ha : inr a < inr c
hb : inr c < inr b
\u22a2 ∃ a_1, a < a_1 ∧ a_1 < b
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝⁴ : Semiring S
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Field K
p : R[X]
r : R
inst✝ : NoZeroDivisors R
a : R[X]
\u22a2 p.scaleRoots r ∣ (p * a).scaleRoots r
-- Mathlib/RingTheory/Polynomial/ScaleRoots.lean
R : Type u_1
S : Type u_2
A : Type u_3
K : Type u_4
inst✝⁴ : Semiring S
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Field K
p : R[X]
r : R
inst✝ : NoZeroDivisors R
a : R[X]
\u22a2 p.scaleRoots r ∣ p.scaleRoots r * a.scaleRoots r
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : Splits (RingHom.id K) f
\u22a2 ⊤ = ⊥
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : Splits (RingHom.id K) f
\u22a2 ↑(Multiset.map (⇑(algebraMap K L)) f.roots).toFinset ⊆ ↑⊥
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : Splits (RingHom.id K) f
y : L
hy : y ∈ ↑(Multiset.map (⇑(algebraMap K L)) f.roots).toFinset
\u22a2 y ∈ ↑⊥
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : Splits (RingHom.id K) f
y : L
hy : ∃ a ∈ f.roots.toFinset, (algebraMap K L) a = y
\u22a2 y ∈ ↑⊥
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : Splits (RingHom.id K) f
y : L
x : K
hxy : (algebraMap K L) x = y
\u22a2 (algebraMap K L) x ∈ ↑⊥
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : ⊤ = ⊥
\u22a2 Splits ((RingEquiv.ofBijective (algebraMap K L) ⋯).trans (RingEquiv.ofBijective (algebraMap K L) ⋯).symm).toRingHom f
-- Mathlib/FieldTheory/SplittingField/IsSplittingField.lean
F : Type u
K : Type v
L : Type w
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Field F
inst✝⁴ : Algebra K L
inst✝³ : Algebra F K
inst✝² : Algebra F L
inst✝¹ : IsScalarTower F K L
f : K[X]
inst✝ : IsSplittingField K L f
h : ⊤ = ⊥
\u22a2 Splits ((RingEquiv.ofBijective (algebraMap K L) ⋯).symm.toRingHom.comp (RingEquiv.ofBijective (algebraMap K L) ⋯).toRingHom) f
-- Mathlib/Probability/Martingale/OptionalStopping.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
τ π : Ω → ℕ
inst✝ : IsFiniteMeasure μ
h : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
\u22a2 Submartingale (stoppedProcess f τ) 𝒢 μ
-- Mathlib/Probability/Martingale/OptionalStopping.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
τ π : Ω → ℕ
inst✝ : IsFiniteMeasure μ
h : Submartingale f 𝒢 μ
hτ : IsStoppingTime 𝒢 τ
\u22a2 ∀ (τ_1 π : Ω → ℕ), IsStoppingTime 𝒢 τ_1 → IsStoppingTime 𝒢 π → τ_1 ≤ π → (∃ N, ∀ (x : Ω), π x ≤ N) → ∫ (x : Ω), stoppedValue (stoppedProcess f τ) τ_1 x ∂μ ≤ ∫ (x : Ω), stoppedValue (stoppedProcess f τ) π x ∂μ