ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
m : ℕ
hm : Nat.card G = p ^ m

⊢ p ^ n ∣ Nat.card G
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
this✝ : Fact (Nat.Prime p)
this : Finite G
m : ℕ
hn : p ^ n ≤ p ^ m
hm : Nat.card G = p ^ m

⊢ p ^ n ∣ p ^ m
-- Mathlib/MeasureTheory/Integral/Asymptotics.lean
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : LinearOrder α
inst✝¹ : CompactIccSpace α
g' : α → F
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrableOn f (Ici a) μ
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ

⊢ IntegrableOn f (Ici a) μ
-- Mathlib/MeasureTheory/Integral/Asymptotics.lean
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁷ : MeasurableSpace α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f : α → E
g : α → F
a b : α
μ : Measure α
l : Filter α
inst✝⁴ : TopologicalSpace α
inst✝³ : SecondCountableTopology α
inst✝² : LinearOrder α
inst✝¹ : CompactIccSpace α
g' : α → F
inst✝ : atTop.IsMeasurablyGenerated
hf : LocallyIntegrableOn f (Ici a) μ
ho : f =O[atTop] g
hg : IntegrableAtFilter g atTop μ

⊢ StronglyMeasurableAtFilter f atTop μ
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B

⊢ mulRothNumber B ≤ mulRothNumber A
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s

⊢ mulRothNumber B ≤ mulRothNumber A
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s
hsA : invFunOn f ↑A '' ↑s ⊆ ↑A

⊢ mulRothNumber B ≤ mulRothNumber A
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s
hsA : invFunOn f ↑A '' ↑s ⊆ ↑A
hfsA : Set.SurjOn f ↑A ↑s

⊢ mulRothNumber B ≤ mulRothNumber A
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s
hsA : invFunOn f ↑A '' ↑s ⊆ ↑A
hfsA : Set.SurjOn f ↑A ↑s

⊢ (image (invFunOn f ↑A) s).card ≤ mulRothNumber A
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s
hsA : invFunOn f ↑A '' ↑s ⊆ ↑A
hfsA : Set.SurjOn f ↑A ↑s

⊢ ThreeGPFree ↑(image (invFunOn f ↑A) s)
-- Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : DecidableEq α
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : DecidableEq β
A : Finset α
B : Finset β
f : α → β
hf : IsMulFreimanHom 2 (↑A) (↑B) f
hf' : Set.BijOn f ↑A ↑B
s : Finset β
hsB : s ⊆ B
hcard : s.card = mulRothNumber B
hs : ThreeGPFree ↑s
hsA : invFunOn f ↑A '' ↑s ⊆ ↑A
hfsA : Set.SurjOn f ↑A ↑s

⊢ ThreeGPFree (invFunOn f ↑A '' ↑s)
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
m n : ℕ

⊢ ↑(m * (n + 1)) = ↑m * ↑(n + 1)
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c

⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ

⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)

⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)
H₂ : map (↑e.symm) (𝓝 (↑e c)) = 𝓝 c

⊢ ∀ᶠ (y : M) in 𝓝 c, ‖f y‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hd : ∀ᶠ (z : M) in 𝓝 c, MDifferentiableAt I 𝓘(ℂ, F) f z
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)
H₂ : map (↑e.symm) (𝓝 (↑e c)) = 𝓝 c

⊢ ∀ᶠ (a : E) in 𝓝 (↑e c), ‖f (↑e.symm a)‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)
H₂ : map (↑e.symm) (𝓝 (↑e c)) = 𝓝 c
hd : ∀ᶠ (y : E) in 𝓝 (↑e c), DifferentiableAt ℂ (f ∘ ↑e.symm) y

⊢ ∀ᶠ (a : E) in 𝓝 (↑e c), ‖f (↑e.symm a)‖ = ‖f c‖
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)
H₂ : map (↑e.symm) (𝓝 (↑e c)) = 𝓝 c
hd : ∀ᶠ (y : E) in 𝓝 (↑e c), DifferentiableAt ℂ (f ∘ ↑e.symm) y
x✝ : E

⊢ f c = (f ∘ ↑e.symm) (↑e c)
-- Mathlib/Geometry/Manifold/Complex.lean
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
F : Type u_2
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℂ F
H : Type u_3
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℂ E H
inst✝⁴ : I.Boundaryless
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : CompactSpace M
inst✝¹ : ChartedSpace H M
inst✝ : SmoothManifoldWithCorners I M
f : M → F
c : M
hc : IsLocalMax (norm ∘ f) c
e : PartialEquiv M E := extChartAt I c
hI : range ↑I = univ
H₁ : 𝓝[range ↑I] ↑e c = 𝓝 (↑e c)
H₂ : map (↑e.symm) (𝓝 (↑e c)) = 𝓝 c
hd : ∀ᶠ (y : E) in 𝓝 (↑e c), DifferentiableAt ℂ (f ∘ ↑e.symm) y

⊢ IsLocalMax (norm ∘ f ∘ ↑e.symm) (↑e c)
-- 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 Q : Fin 3 → F
hQz : Q z = 0

⊢ addU P Q = 0
-- Mathlib/Data/Set/NAry.lean
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Set α
t t' : Set β
u u' : Set γ
v : Set δ
a a' : α
b b' : β
c c' : γ
d d' : δ

⊢ image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u
-- Mathlib/RingTheory/Localization/NormTrace.lean
R : Type u_1
S : Type u_2
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
Rₘ : Type u_3
Sₘ : Type u_4
inst✝¹¹ : CommRing Rₘ
inst✝¹⁰ : Algebra R Rₘ
inst✝⁹ : CommRing Sₘ
inst✝⁸ : Algebra S Sₘ
M : Submonoid R
inst✝⁷ : IsLocalization M Rₘ
inst✝⁶ : IsLocalization (algebraMapSubmonoid S M) Sₘ
inst✝⁵ : Algebra Rₘ Sₘ
inst✝⁴ : Algebra R Sₘ
inst✝³ : IsScalarTower R Rₘ Sₘ
inst✝² : IsScalarTower R S Sₘ
ι : Type u_5
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι R S
a : S
i j : ι

⊢ (algebraMap R Rₘ).mapMatrix ((leftMulMatrix b) a) i j = (leftMulMatrix (Basis.localizationLocalization Rₘ M Sₘ b)) ((algebraMap S Sₘ) a) i j
-- Mathlib/Topology/Algebra/Monoid.lean
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : l.NeBot
h₁ : Tendsto (fun x => ↑(f x)) l (𝓝 r₁)
h₂ : Tendsto (fun x => ↑(f x)⁻¹) l (𝓝 r₂)

⊢ 1 = r₁ * r₂
-- Mathlib/Topology/Algebra/Monoid.lean
ι : Type u_1
α : Type u_2
M : Type u_3
N : Type u_4
X : Type u_5
inst✝⁸ : TopologicalSpace X
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Mul M
inst✝⁵ : ContinuousMul M
inst✝⁴ : TopologicalSpace N
inst✝³ : Monoid N
inst✝² : ContinuousMul N
inst✝¹ : T2Space N
f : ι → Nˣ
r₁ r₂ : N
l : Filter ι
inst✝ : l.NeBot
h₁ : Tendsto (fun x => ↑(f x)) l (𝓝 r₁)
h₂ : Tendsto (fun x => ↑(f x)⁻¹) l (𝓝 r₂)

⊢ 1 = r₂ * r₁
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α

⊢ (⋃ x ∈ s₁, t x) \ ⋃ x ∈ s₂, t x ⊆ ⋃ x ∈ s₁ \ s₂, t x
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α

⊢ ⋃ x ∈ s₁, t x ⊆ ⋃ x ∈ s₂ ∪ s₁ \ s₂, t x
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α

⊢ s₁ ⊆ s₂ ∪ s₁ \ s₂
-- Mathlib/Data/Set/Lattice.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
t : α → Set β
s₁ s₂ : Set α

⊢ s₁ ⊆ s₂ ∪ s₁
-- Mathlib/Order/CompleteLattice.lean
α : Type u_1
β : Type u_2
β₂ : Type u_3
γ : Type u_4
ι : Sort u_5
ι' : Sort u_6
κ : ι → Sort u_7
κ' : ι' → Sort u_8
inst✝ : CompleteLattice α
f✝ g s✝ t : ι → α
a b : α
s : Set β
f : β → α

⊢ sSup (f '' s) = ⨆ a ∈ s, f a
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ

⊢ (W.Φ ↑n).natDegree ≤ n ^ 2 ∧ (W.Φ ↑n).coeff (n ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le

⊢ (W.Φ ↑n).natDegree ≤ n ^ 2 ∧ (W.Φ ↑n).coeff (n ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n

⊢ (W.Φ ↑n).natDegree ≤ n ^ 2 ∧ (W.Φ ↑n).coeff (n ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le

⊢ (W.Φ ↑n).natDegree ≤ n ^ 2 ∧ (W.Φ ↑n).coeff (n ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n

⊢ (W.Φ ↑n).natDegree ≤ n ^ 2 ∧ (W.Φ ↑n).coeff (n ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ

⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3

⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0

⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc : 1 = (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) - WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1

⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc : 1 = (1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) - WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1

⊢ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) - W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).natDegree ≤ (n + 1 + 1) ^ 2 ∧ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) - W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).coeff ((n + 1 + 1) ^ 2) = 1
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : PartialOrder α
inst✝ : OrderTop α
h : IsAtomic αᵒᵈ
b : α

⊢ b = ⊤ ∨ ∃ a, IsCoatom a ∧ b ≤ a
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝¹ : PartialOrder α
inst✝ : OrderTop α
h : IsCoatomic α
b : αᵒᵈ

⊢ b = ⊥ ∨ ∃ a, IsAtom a ∧ a ≤ b
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b

⊢ fourierCoeffOn hab f n = 1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b

⊢ fourierCoeffOn hab f n = 1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)

⊢ fourierCoeffOn hab f n = 1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n)
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)

⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x = 1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f' x))
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)

⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x = 1 / (-2 * ↑π * I * ↑n) * ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))
-- Mathlib/Analysis/Fourier/AddCircle.lean
T a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
hT : Fact (0 < b - a)

⊢ 1 / ↑(b - a) * (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) - f a * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑a) - ∫ (x : ℝ) in a..b, f' x * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑x)) = 1 / (-2 * ↑π * I * ↑n) * ((f b - f a) * (fourier (-n)) ↑a - ↑(b - a) * (1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f' x * (fourier (-n)) ↑x))