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exercise_1_13a
e14de2442fcda3e1
validation
import Mathlib theorem exercise_1_13a {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω) (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).re = c) : f a = f b := by
f : ℂ → ℂ Ω : Set ℂ a b : ↑Ω h : IsOpen Ω hf : DifferentiableOn ℂ f Ω hc : ∃ c, ∀ z ∈ Ω, (f z).re = c ⊢ f ↑a = f ↑b
exercise_1_13b
1f057164b821dc73
test
import Mathlib theorem exercise_1_13b {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω) (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).im = c) : f a = f b := by
f : ℂ → ℂ Ω : Set ℂ a b : ↑Ω h : IsOpen Ω hf : DifferentiableOn ℂ f Ω hc : ∃ c, ∀ z ∈ Ω, (f z).im = c ⊢ f ↑a = f ↑b
exercise_1_13c
e533b1d034f2ab2b
validation
import Mathlib theorem exercise_1_13c {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω) (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, abs (f z) = c) : f a = f b := by
f : ℂ → ℂ Ω : Set ℂ a b : ↑Ω h : IsOpen Ω hf : DifferentiableOn ℂ f Ω hc : ∃ c, ∀ z ∈ Ω, Complex.abs (f z) = c ⊢ f ↑a = f ↑b
exercise_1_19a
686c0e351483e877
test
import Mathlib theorem exercise_1_19a (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ) (h : s = (λ n => ∑ i in (range n), i * z ^ i)) : ¬ ∃ y, Tendsto s atTop (𝓝 y) := by
z : ℂ hz : Complex.abs z = 1 s : ℕ → ℂ h : s = fun n => ∑ i ∈ range n, ↑i * z ^ i ⊢ ¬∃ y, Tendsto s atTop (𝓝 y)
exercise_1_19b
f84bea6a06de7140
validation
import Mathlib theorem exercise_1_19b (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ) (h : s = (λ n => ∑ i in (range n), i * z / i ^ 2)) : ∃ y, Tendsto s atTop (𝓝 y) := by
z : ℂ hz : Complex.abs z = 1 s : ℕ → ℂ h : s = fun n => ∑ i ∈ range n, ↑i * z / ↑i ^ 2 ⊢ ∃ y, Tendsto s atTop (𝓝 y)
exercise_1_19c
2c250a906c7d5747
test
import Mathlib theorem exercise_1_19c (z : ℂ) (hz : abs z = 1) (hz2 : z ≠ 1) (s : ℕ → ℂ) (h : s = (λ n => ∑ i in (range n), i * z / i)) : ∃ z, Tendsto s atTop (𝓝 z) := by
z : ℂ hz : Complex.abs z = 1 hz2 : z ≠ 1 s : ℕ → ℂ h : s = fun n => ∑ i ∈ range n, ↑i * z / ↑i ⊢ ∃ z, Tendsto s atTop (𝓝 z)
exercise_1_26
5147969a440b69e7
validation
import Mathlib theorem exercise_1_26 (f F₁ F₂ : ℂ → ℂ) (Ω : Set ℂ) (h1 : IsOpen Ω) (h2 : IsConnected Ω) (hF₁ : DifferentiableOn ℂ F₁ Ω) (hF₂ : DifferentiableOn ℂ F₂ Ω) (hdF₁ : ∀ x ∈ Ω, deriv F₁ x = f x) (hdF₂ : ∀ x ∈ Ω, deriv F₂ x = f x) : ∃ c : ℂ, ∀ x, F₁ x = F₂ x + c := by
f F₁ F₂ : ℂ → ℂ Ω : Set ℂ h1 : IsOpen Ω h2 : IsConnected Ω hF₁ : DifferentiableOn ℂ F₁ Ω hF₂ : DifferentiableOn ℂ F₂ Ω hdF₁ : ∀ x ∈ Ω, deriv F₁ x = f x hdF₂ : ∀ x ∈ Ω, deriv F₂ x = f x ⊢ ∃ c, ∀ (x : ℂ), F₁ x = F₂ x + c
exercise_2_2
0b08976e0310633e
test
import Mathlib theorem exercise_2_2 : Tendsto (λ y => ∫ x in (0 : ℝ)..y, Real.sin x / x) atTop (𝓝 (Real.pi / 2)) := by
⊢ Tendsto (fun y => ∫ (x : ℝ) in 0 ..y, x.sin / x) atTop (𝓝 (Real.pi / 2))
exercise_2_9
8cd220f0354864ba
validation
import Mathlib theorem exercise_2_9 {f : ℂ → ℂ} (Ω : Set ℂ) (b : Bornology.IsBounded Ω) (h : IsOpen Ω) (hf : DifferentiableOn ℂ f Ω) (z : Ω) (hz : f z = z) (h'z : deriv f z = 1) : ∃ (f_lin : ℂ →L[ℂ] ℂ), ∀ x ∈ Ω, f x = f_lin x := by
f : ℂ → ℂ Ω : Set ℂ b : Bornology.IsBounded Ω h : IsOpen Ω hf : DifferentiableOn ℂ f Ω z : ↑Ω hz : f ↑z = ↑z h'z : deriv f ↑z = 1 ⊢ ∃ f_lin, ∀ x ∈ Ω, f x = f_lin x
exercise_2_13
d8f5b47b63585740
test
import Mathlib theorem exercise_2_13 {f : ℂ → ℂ} (hf : ∀ z₀ : ℂ, ∃ (s : Set ℂ) (c : ℕ → ℂ), IsOpen s ∧ z₀ ∈ s ∧ ∀ z ∈ s, Tendsto (λ n => ∑ i in range n, (c i) * (z - z₀)^i) atTop (𝓝 (f z₀)) ∧ ∃ i, c i = 0) : ∃ (c : ℕ → ℂ) (n : ℕ), f = λ z => ∑ i in range n, (c i) * z ^ n := by
f : ℂ → ℂ hf : ∀ (z₀ : ℂ), ∃ s c, IsOpen s ∧ z₀ ∈ s ∧ ∀ z ∈ s, Tendsto (fun n => ∑ i ∈ range n, c i * (z - z₀) ^ i) atTop (𝓝 (f z₀)) ∧ ∃ i, c i = 0 ⊢ ∃ c n, f = fun z => ∑ i ∈ range n, c i * z ^ n
exercise_3_3
e37c100992b82429
validation
import Mathlib theorem exercise_3_3 (a : ℝ) (ha : 0 < a) : Tendsto (λ y => ∫ x in -y..y, Real.cos x / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * (Real.exp (-a) / a))) := by
a : ℝ ha : 0 < a ⊢ Tendsto (fun y => ∫ (x : ℝ) in -y..y, x.cos / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * ((-a).exp / a)))
exercise_3_4
8c803ad2f3327b07
test
import Mathlib theorem exercise_3_4 (a : ℝ) (ha : 0 < a) : Tendsto (λ y => ∫ x in -y..y, x * Real.sin x / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * (Real.exp (-a)))) := by
a : ℝ ha : 0 < a ⊢ Tendsto (fun y => ∫ (x : ℝ) in -y..y, x * x.sin / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * (-a).exp))
exercise_3_9
67e9880f6ffc7ce7
validation
import Mathlib theorem exercise_3_9 : ∫ x in (0 : ℝ)..(1 : ℝ), Real.log (Real.sin (Real.pi * x)) = - Real.log 2 := by
⊢ ∫ (x : ℝ) in 0 ..1, (Real.pi * x).sin.log = -Real.log 2
exercise_3_14
13198380d5f67154
test
import Mathlib theorem exercise_3_14 {f : ℂ → ℂ} (hf : Differentiable ℂ f) (hf_inj : Function.Injective f) : ∃ (a b : ℂ), f = (λ z => a * z + b) ∧ a ≠ 0 := by
f : ℂ → ℂ hf : Differentiable ℂ f hf_inj : Injective f ⊢ ∃ a b, (f = fun z => a * z + b) ∧ a ≠ 0
exercise_3_22
811c9315e920273a
validation
import Mathlib theorem exercise_3_22 (D : Set ℂ) (hD : D = ball 0 1) (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f D) (hfc : ContinuousOn f (closure D)) : ¬ ∀ z ∈ (sphere (0 : ℂ) 1), f z = 1 / z := by
D : Set ℂ hD : D = ball 0 1 f : ℂ → ℂ hf : DifferentiableOn ℂ f D hfc : ContinuousOn f (closure D) ⊢ ¬∀ z ∈ sphere 0 1, f z = 1 / z
exercise_5_1
75820696b6fd32d0
test
import Mathlib theorem exercise_5_1 (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f (ball 0 1)) (hb : Bornology.IsBounded (Set.range f)) (h0 : f ≠ 0) (zeros : ℕ → ℂ) (hz : ∀ n, f (zeros n) = 0) (hzz : Set.range zeros = {z | f z = 0 ∧ z ∈ (ball (0 : ℂ) 1)}) : ∃ (z : ℂ), Tendsto (λ n => (∑ i in range n, (1 - zeros i))) atT...
f : ℂ → ℂ hf : DifferentiableOn ℂ f (ball 0 1) hb : Bornology.IsBounded (Set.range f) h0 : f ≠ 0 zeros : ℕ → ℂ hz : ∀ (n : ℕ), f (zeros n) = 0 hzz : Set.range zeros = {z | f z = 0 ∧ z ∈ ball 0 1} ⊢ ∃ z, Tendsto (fun n => ∑ i ∈ range n, (1 - zeros i)) atTop (𝓝 z)
exercise_1_1a
724e9263d2bbdc39
validation
import Mathlib theorem exercise_1_1a (x : ℝ) (y : ℚ) : ( Irrational x ) -> Irrational ( x + y ) := by
x : ℝ y : ℚ ⊢ Irrational x → Irrational (x + ↑y)
exercise_1_1b
8c1daeb8964e576b
test
import Mathlib theorem exercise_1_1b (x : ℝ) (y : ℚ) (h : y ≠ 0) : ( Irrational x ) -> Irrational ( x * y ) := by
x : ℝ y : ℚ h : y ≠ 0 ⊢ Irrational x → Irrational (x * ↑y)
exercise_1_2
a044a3e2d39116da
validation
import Mathlib theorem exercise_1_2 : ¬ ∃ (x : ℚ), ( x ^ 2 = 12 ) := by
⊢ ¬∃ x, x ^ 2 = 12
exercise_1_4
2c252ef331f8aae1
test
import Mathlib theorem exercise_1_4 (α : Type*) [PartialOrder α] (s : Set α) (x y : α) (h₀ : Set.Nonempty s) (h₁ : x ∈ lowerBounds s) (h₂ : y ∈ upperBounds s) : x ≤ y := by
α : Type u_1 inst✝ : PartialOrder α s : Set α x y : α h₀ : s.Nonempty h₁ : x ∈ lowerBounds s h₂ : y ∈ upperBounds s ⊢ x ≤ y
exercise_1_5
4e8c2122011b8451
validation
import Mathlib theorem exercise_1_5 (A minus_A : Set ℝ) (hA : A.Nonempty) (hA_bdd_below : BddBelow A) (hminus_A : minus_A = {x | -x ∈ A}) : Inf A = Sup minus_A := by
A minus_A : Set ℝ hA : A.Nonempty hA_bdd_below : BddBelow A hminus_A : minus_A = {x | -x ∈ A} ⊢ Inf ↑A = Sup ↑minus_A
exercise_1_8
80de722946058f4f
test
import Mathlib theorem exercise_1_8 : ¬ ∃ (r : ℂ → ℂ → Prop), IsLinearOrder ℂ r := by
⊢ ¬∃ r, IsLinearOrder ℂ r
exercise_1_11a
4e7e95c2fc999016
validation
import Mathlib theorem exercise_1_11a (z : ℂ) : ∃ (r : ℝ) (w : ℂ), abs w = 1 ∧ z = r * w := by
z : ℂ ⊢ ∃ r w, Complex.abs w = 1 ∧ z = ↑r * w
exercise_1_12
17ba7ca749e67928
test
import Mathlib theorem exercise_1_12 (n : ℕ) (f : ℕ → ℂ) : abs (∑ i in range n, f i) ≤ ∑ i in range n, abs (f i) := by
n : ℕ f : ℕ → ℂ ⊢ Complex.abs (∑ i ∈ range n, f i) ≤ ∑ i ∈ range n, Complex.abs (f i)
exercise_1_13
43ecd62b2af93a45
validation
import Mathlib theorem exercise_1_13 (x y : ℂ) : |(abs x) - (abs y)| ≤ abs (x - y) := by
x y : ℂ ⊢ |Complex.abs x - Complex.abs y| ≤ Complex.abs (x - y)
exercise_1_14
30b8e342c237c8ca
test
import Mathlib theorem exercise_1_14 (z : ℂ) (h : abs z = 1) : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 := by
z : ℂ h : Complex.abs z = 1 ⊢ Complex.abs (1 + z) ^ 2 + Complex.abs (1 - z) ^ 2 = 4
exercise_1_16a
db59df1dfd19624c
validation
import Mathlib theorem exercise_1_16a (n : ℕ) (d r : ℝ) (x y z : EuclideanSpace ℝ (Fin n)) -- R^n (h₁ : n ≥ 3) (h₂ : ‖x - y‖ = d) (h₃ : d > 0) (h₄ : r > 0) (h₅ : 2 * r > d) : Set.Infinite {z : EuclideanSpace ℝ (Fin n) | ‖z - x‖ = r ∧ ‖z - y‖ = r} := by
n : ℕ d r : ℝ x y z : EuclideanSpace ℝ (Fin n) h₁ : n ≥ 3 h₂ : ‖x - y‖ = d h₃ : d > 0 h₄ : r > 0 h₅ : 2 * r > d ⊢ {z | ‖z - x‖ = r ∧ ‖z - y‖ = r}.Infinite
exercise_1_17
29c896bb6047fadc
test
import Mathlib theorem exercise_1_17 (n : ℕ) (x y : EuclideanSpace ℝ (Fin n)) -- R^n : ‖x + y‖^2 + ‖x - y‖^2 = 2*‖x‖^2 + 2*‖y‖^2 := by
n : ℕ x y : EuclideanSpace ℝ (Fin n) ⊢ ‖x + y‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * ‖x‖ ^ 2 + 2 * ‖y‖ ^ 2
exercise_1_18a
bf73de070e24500f
validation
import Mathlib theorem exercise_1_18a (n : ℕ) (h : n > 1) (x : EuclideanSpace ℝ (Fin n)) -- R^n : ∃ (y : EuclideanSpace ℝ (Fin n)), y ≠ 0 ∧ (inner x y) = (0 : ℝ) := by
n : ℕ h : n > 1 x : EuclideanSpace ℝ (Fin n) ⊢ ∃ y, y ≠ 0 ∧ ⟪x, y⟫_ℝ = 0
exercise_1_18b
74a09bc9a90a8712
test
import Mathlib theorem exercise_1_18b : ¬ ∀ (x : ℝ), ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0 := by
⊢ ¬∀ (x : ℝ), ∃ y, y ≠ 0 ∧ x * y = 0
exercise_1_19
a01c1222b82d12f8
validation
import Mathlib theorem exercise_1_19 (n : ℕ) (a b c x : EuclideanSpace ℝ (Fin n)) (r : ℝ) (h₁ : r > 0) (h₂ : 3 • c = 4 • b - a) (h₃ : 3 * r = 2 * ‖x - b‖) : ‖x - a‖ = 2 * ‖x - b‖ ↔ ‖x - c‖ = r := by
n : ℕ a b c x : EuclideanSpace ℝ (Fin n) r : ℝ h₁ : r > 0 h₂ : 3 • c = 4 • b - a h₃ : 3 * r = 2 * ‖x - b‖ ⊢ ‖x - a‖ = 2 * ‖x - b‖ ↔ ‖x - c‖ = r
exercise_2_19a
1f0284985242f68b
test
import Mathlib theorem exercise_2_19a {X : Type*} [MetricSpace X] (A B : Set X) (hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) : SeparatedNhds A B := by
X : Type u_1 inst✝ : MetricSpace X A B : Set X hA : IsClosed A hB : IsClosed B hAB : Disjoint A B ⊢ SeparatedNhds A B
exercise_2_24
ee959c2cb628d664
validation
import Mathlib theorem exercise_2_24 {X : Type*} [MetricSpace X] (hX : ∀ (A : Set X), Infinite A → ∃ (x : X), x ∈ closure A) : SeparableSpace X := by
X : Type u_1 inst✝ : MetricSpace X hX : ∀ (A : Set X), Infinite ↑A → ∃ x, x ∈ closure A ⊢ SeparableSpace X
exercise_2_25
41227786f06dc834
test
import Mathlib theorem exercise_2_25 {K : Type*} [MetricSpace K] [CompactSpace K] : ∃ (B : Set (Set K)), Set.Countable B ∧ IsTopologicalBasis B := by
K : Type u_1 inst✝¹ : MetricSpace K inst✝ : CompactSpace K ⊢ ∃ B, B.Countable ∧ IsTopologicalBasis B
exercise_2_27a
55d92d55a331d7a0
validation
import Mathlib theorem exercise_2_27a (k : ℕ) (E P : Set (EuclideanSpace ℝ (Fin k))) (hE : E.Nonempty ∧ ¬ Set.Countable E) (hP : P = {x | ∀ U ∈ 𝓝 x, ¬ Set.Countable (P ∩ E)}) : IsClosed P ∧ P = {x | ClusterPt x (𝓟 P)} := by
k : ℕ E P : Set (EuclideanSpace ℝ (Fin k)) hE : E.Nonempty ∧ ¬E.Countable hP : P = {x | ∀ U ∈ 𝓝 x, ¬(P ∩ E).Countable} ⊢ IsClosed P ∧ P = {x | ClusterPt x (𝓟 P)}
exercise_2_27b
404d42d51b35bac5
test
import Mathlib theorem exercise_2_27b (k : ℕ) (E P : Set (EuclideanSpace ℝ (Fin k))) (hE : E.Nonempty ∧ ¬ Set.Countable E) (hP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬ Set.Countable (P ∩ E)}) : Set.Countable (E \ P) := by
k : ℕ E P : Set (EuclideanSpace ℝ (Fin k)) hE : E.Nonempty ∧ ¬E.Countable hP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬(P ∩ E).Countable} ⊢ (E \ P).Countable
exercise_2_28
8acd23fe2492c047
validation
import Mathlib theorem exercise_2_28 (X : Type*) [MetricSpace X] [SeparableSpace X] (A : Set X) (hA : IsClosed A) : ∃ P₁ P₂ : Set X, A = P₁ ∪ P₂ ∧ IsClosed P₁ ∧ P₁ = {x | ClusterPt x (𝓟 P₁)} ∧ Set.Countable P₂ := by
X : Type u_1 inst✝¹ : MetricSpace X inst✝ : SeparableSpace X A : Set X hA : IsClosed A ⊢ ∃ P₁ P₂, A = P₁ ∪ P₂ ∧ IsClosed P₁ ∧ P₁ = {x | ClusterPt x (𝓟 P₁)} ∧ P₂.Countable
exercise_2_29
99d9b35150132018
test
import Mathlib theorem exercise_2_29 (U : Set ℝ) (hU : IsOpen U) : ∃ (f : ℕ → Set ℝ), (∀ n, ∃ a b : ℝ, f n = {x | a < x ∧ x < b}) ∧ (∀ n, f n ⊆ U) ∧ (∀ n m, n ≠ m → f n ∩ f m = ∅) ∧ U = ⋃ n, f n := by
U : Set ℝ hU : IsOpen U ⊢ ∃ f, (∀ (n : ℕ), ∃ a b, f n = {x | a < x ∧ x < b}) ∧ (∀ (n : ℕ), f n ⊆ U) ∧ (∀ (n m : ℕ), n ≠ m → f n ∩ f m = ∅) ∧ U = ⋃ n, f n
exercise_3_1a
fd2d515123bec387
validation
import Mathlib theorem exercise_3_1a (f : ℕ → ℝ) (h : ∃ (a : ℝ), Tendsto (λ (n : ℕ) => f n) atTop (𝓝 a)) : ∃ (a : ℝ), Tendsto (λ (n : ℕ) => |f n|) atTop (𝓝 a) := by
f : ℕ → ℝ h : ∃ a, Tendsto (fun n => f n) atTop (𝓝 a) ⊢ ∃ a, Tendsto (fun n => |f n|) atTop (𝓝 a)
exercise_3_2a
4dd25a1090ec708f
test
import Mathlib theorem exercise_3_2a : Tendsto (λ (n : ℝ) => (sqrt (n^2 + n) - n)) atTop (𝓝 (1/2)) := by
⊢ Tendsto (fun n => √(n ^ 2 + n) - n) atTop (𝓝 (1 / 2))
exercise_3_3
089658608145e6f4
validation
import Mathlib theorem exercise_3_3 : ∃ (x : ℝ), Tendsto f atTop (𝓝 x) ∧ ∀ n, f n < 2 := by
⊢ ∃ x, Tendsto f atTop (𝓝 x) ∧ ∀ (n : ℕ), f n < 2
exercise_3_5
7b9df33537f61ce6
test
import Mathlib theorem exercise_3_5 (a b : ℕ → ℝ) (h : limsup a + limsup b ≠ 0) : limsup (λ n => a n + b n) ≤ limsup a + limsup b := by
a b : ℕ → ℝ h : limsup a + limsup b ≠ 0 ⊢ (limsup fun n => a n + b n) ≤ limsup a + limsup b
exercise_3_6a
707a167ceaab8166
validation
import Mathlib theorem exercise_3_6a : Tendsto (λ (n : ℕ) => (∑ i in range n, g i)) atTop atTop := by
⊢ Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop
exercise_3_7
ca1ecc72f707c17d
test
import Mathlib theorem exercise_3_7 (a : ℕ → ℝ) (h : ∃ y, (Tendsto (λ n => (∑ i in (range n), a i)) atTop (𝓝 y))) : ∃ y, Tendsto (λ n => (∑ i in (range n), sqrt (a i) / n)) atTop (𝓝 y) := by
a : ℕ → ℝ h : ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i) atTop (𝓝 y) ⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, √(a i) / ↑n) atTop (𝓝 y)
exercise_3_8
58e8abbd2f1b5eb2
validation
import Mathlib theorem exercise_3_8 (a b : ℕ → ℝ) (h1 : ∃ y, (Tendsto (λ n => (∑ i in (range n), a i)) atTop (𝓝 y))) (h2 : Monotone b) (h3 : Bornology.IsBounded (Set.range b)) : ∃ y, Tendsto (λ n => (∑ i in (range n), (a i) * (b i))) atTop (𝓝 y) := by
a b : ℕ → ℝ h1 : ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i) atTop (𝓝 y) h2 : Monotone b h3 : Bornology.IsBounded (Set.range b) ⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i * b i) atTop (𝓝 y)
exercise_3_13
2b543003cd76c468
test
import Mathlib theorem exercise_3_13 (a b : ℕ → ℝ) (ha : ∃ y, (Tendsto (λ n => (∑ i in (range n), |a i|)) atTop (𝓝 y))) (hb : ∃ y, (Tendsto (λ n => (∑ i in (range n), |b i|)) atTop (𝓝 y))) : ∃ y, (Tendsto (λ n => (∑ i in (range n), λ i => (∑ j in range (i + 1), a j * b (i - j)))) atTop (𝓝 y)) := by
a b : ℕ → ℝ ha : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |a i|) atTop (𝓝 y) hb : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |b i|) atTop (𝓝 y) ⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, fun i => ∑ j ∈ range (i + 1), a j * b (i - j)) atTop (𝓝 y)
exercise_3_20
4098f6a3c7e1610a
validation
import Mathlib theorem exercise_3_20 {X : Type*} [MetricSpace X] (p : ℕ → X) (l : ℕ) (r : X) (hp : CauchySeq p) (hpl : Tendsto (λ n => p (l * n)) atTop (𝓝 r)) : Tendsto p atTop (𝓝 r) := by
X : Type u_1 inst✝ : MetricSpace X p : ℕ → X l : ℕ r : X hp : CauchySeq p hpl : Tendsto (fun n => p (l * n)) atTop (𝓝 r) ⊢ Tendsto p atTop (𝓝 r)
exercise_3_21
9f191ca99f76a6d9
test
import Mathlib theorem exercise_3_21 {X : Type*} [MetricSpace X] [CompleteSpace X] (E : ℕ → Set X) (hE : ∀ n, E n ⊃ E (n + 1)) (hE' : Tendsto (λ n => Metric.diam (E n)) atTop (𝓝 0)) : ∃ a, Set.iInter E = {a} := by
X : Type u_1 inst✝¹ : MetricSpace X inst✝ : CompleteSpace X E : ℕ → Set X hE : ∀ (n : ℕ), E n ⊃ E (n + 1) hE' : Tendsto (fun n => Metric.diam (E n)) atTop (𝓝 0) ⊢ ∃ a, Set.iInter E = {a}
exercise_3_22
38d8e032c920914e
validation
import Mathlib theorem exercise_3_22 (X : Type*) [MetricSpace X] [CompleteSpace X] (G : ℕ → Set X) (hG : ∀ n, IsOpen (G n) ∧ Dense (G n)) : ∃ x, ∀ n, x ∈ G n := by
X : Type u_1 inst✝¹ : MetricSpace X inst✝ : CompleteSpace X G : ℕ → Set X hG : ∀ (n : ℕ), IsOpen (G n) ∧ Dense (G n) ⊢ ∃ x, ∀ (n : ℕ), x ∈ G n
exercise_4_1a
b25ce1a7983540ed
test
import Mathlib theorem exercise_4_1a : ∃ (f : ℝ → ℝ), (∀ (x : ℝ), Tendsto (λ y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬ Continuous f := by
⊢ ∃ f, (∀ (x : ℝ), Tendsto (fun y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬Continuous f
exercise_4_2a
a15e62dea80e6d4c
validation
import Mathlib theorem exercise_4_2a {α : Type} [MetricSpace α] {β : Type} [MetricSpace β] (f : α → β) (h₁ : Continuous f) : ∀ (x : Set α), f '' (closure x) ⊆ closure (f '' x) := by
α : Type inst✝¹ : MetricSpace α β : Type inst✝ : MetricSpace β f : α → β h₁ : Continuous f ⊢ ∀ (x : Set α), f '' closure x ⊆ closure (f '' x)
exercise_4_3
11d3f97c1fcf8660
test
import Mathlib theorem exercise_4_3 {α : Type} [MetricSpace α] (f : α → ℝ) (h : Continuous f) (z : Set α) (g : z = f⁻¹' {0}) : IsClosed z := by
α : Type inst✝ : MetricSpace α f : α → ℝ h : Continuous f z : Set α g : z = f ⁻¹' {0} ⊢ IsClosed z
exercise_4_4a
4d8eba9716856240
validation
import Mathlib theorem exercise_4_4a {α : Type} [MetricSpace α] {β : Type} [MetricSpace β] (f : α → β) (s : Set α) (h₁ : Continuous f) (h₂ : Dense s) : f '' Set.univ ⊆ closure (f '' s) := by
α : Type inst✝¹ : MetricSpace α β : Type inst✝ : MetricSpace β f : α → β s : Set α h₁ : Continuous f h₂ : Dense s ⊢ f '' Set.univ ⊆ closure (f '' s)
exercise_4_4b
27abd6835e4f22e7
test
import Mathlib theorem exercise_4_4b {α : Type} [MetricSpace α] {β : Type} [MetricSpace β] (f g : α → β) (s : Set α) (h₁ : Continuous f) (h₂ : Continuous g) (h₃ : Dense s) (h₄ : ∀ x ∈ s, f x = g x) : f = g := by
α : Type inst✝¹ : MetricSpace α β : Type inst✝ : MetricSpace β f g : α → β s : Set α h₁ : Continuous f h₂ : Continuous g h₃ : Dense s h₄ : ∀ x ∈ s, f x = g x ⊢ f = g
exercise_4_5a
ddaa08e101c56ae9
validation
import Mathlib theorem exercise_4_5a (f : ℝ → ℝ) (E : Set ℝ) (h₁ : IsClosed E) (h₂ : ContinuousOn f E) : ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x := by
f : ℝ → ℝ E : Set ℝ h₁ : IsClosed E h₂ : ContinuousOn f E ⊢ ∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x
exercise_4_5b
58abeb81b7193e4f
test
import Mathlib theorem exercise_4_5b : ∃ (E : Set ℝ) (f : ℝ → ℝ), (ContinuousOn f E) ∧ (¬ ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x) := by
⊢ ∃ E f, ContinuousOn f E ∧ ¬∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x
exercise_4_6
e69e19fb8f410a59
validation
import Mathlib theorem exercise_4_6 (f : ℝ → ℝ) (E : Set ℝ) (G : Set (ℝ × ℝ)) (h₁ : IsCompact E) (h₂ : G = {(x, f x) | x ∈ E}) : ContinuousOn f E ↔ IsCompact G := by
f : ℝ → ℝ E : Set ℝ G : Set (ℝ × ℝ) h₁ : IsCompact E h₂ : G = {x | ∃ x_1 ∈ E, (x_1, f x_1) = x} ⊢ ContinuousOn f E ↔ IsCompact G
exercise_4_8a
3ba2c78ff3b09984
test
import Mathlib theorem exercise_4_8a (E : Set ℝ) (f : ℝ → ℝ) (hf : UniformContinuousOn f E) (hE : Bornology.IsBounded E) : Bornology.IsBounded (Set.image f E) := by
E : Set ℝ f : ℝ → ℝ hf : UniformContinuousOn f E hE : Bornology.IsBounded E ⊢ Bornology.IsBounded (f '' E)
exercise_4_8b
0bb7a8e9d3f49182
validation
import Mathlib theorem exercise_4_8b (E : Set ℝ) : ∃ f : ℝ → ℝ, UniformContinuousOn f E ∧ ¬ Bornology.IsBounded (Set.image f E) := by
E : Set ℝ ⊢ ∃ f, UniformContinuousOn f E ∧ ¬Bornology.IsBounded (f '' E)
exercise_4_11a
944da115bef03e50
test
import Mathlib theorem exercise_4_11a {X : Type*} [MetricSpace X] {Y : Type*} [MetricSpace Y] (f : X → Y) (hf : UniformContinuous f) (x : ℕ → X) (hx : CauchySeq x) : CauchySeq (λ n => f (x n)) := by
X : Type u_1 inst✝¹ : MetricSpace X Y : Type u_2 inst✝ : MetricSpace Y f : X → Y hf : UniformContinuous f x : ℕ → X hx : CauchySeq x ⊢ CauchySeq fun n => f (x n)
exercise_4_12
a284dcd4c9e28200
validation
import Mathlib theorem exercise_4_12 {α β γ : Type*} [UniformSpace α] [UniformSpace β] [UniformSpace γ] {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous (g ∘ f) := by
α : Type u_1 β : Type u_2 γ : Type u_3 inst✝² : UniformSpace α inst✝¹ : UniformSpace β inst✝ : UniformSpace γ f : α → β g : β → γ hf : UniformContinuous f hg : UniformContinuous g ⊢ UniformContinuous (g ∘ f)
exercise_4_15
06d9cb86cd921423
test
import Mathlib theorem exercise_4_15 {f : ℝ → ℝ} (hf : Continuous f) (hof : IsOpenMap f) : Monotone f := by
f : ℝ → ℝ hf : Continuous f hof : IsOpenMap f ⊢ Monotone f
exercise_4_19
a72ceaa68465619b
validation
import Mathlib theorem exercise_4_19 {f : ℝ → ℝ} (hf : ∀ a b c, a < b → f a < c → c < f b → ∃ x, a < x ∧ x < b ∧ f x = c) (hg : ∀ r : ℚ, IsClosed {x | f x = r}) : Continuous f := by
f : ℝ → ℝ hf : ∀ (a b c : ℝ), a < b → f a < c → c < f b → ∃ x, a < x ∧ x < b ∧ f x = c hg : ∀ (r : ℚ), IsClosed {x | f x = ↑r} ⊢ Continuous f
exercise_4_21a
114c1b3514aa2bfe
test
import Mathlib theorem exercise_4_21a {X : Type*} [MetricSpace X] (K F : Set X) (hK : IsCompact K) (hF : IsClosed F) (hKF : Disjoint K F) : ∃ (δ : ℝ), δ > 0 ∧ ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ := by
X : Type u_1 inst✝ : MetricSpace X K F : Set X hK : IsCompact K hF : IsClosed F hKF : Disjoint K F ⊢ ∃ δ > 0, ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ
exercise_4_24
02d5e134d02d76aa
validation
import Mathlib theorem exercise_4_24 {f : ℝ → ℝ} (hf : Continuous f) (a b : ℝ) (hab : a < b) (h : ∀ x y : ℝ, a < x → x < b → a < y → y < b → f ((x + y) / 2) ≤ (f x + f y) / 2) : ConvexOn ℝ (Set.Ioo a b) f := by
f : ℝ → ℝ hf : Continuous f a b : ℝ hab : a < b h : ∀ (x y : ℝ), a < x → x < b → a < y → y < b → f ((x + y) / 2) ≤ (f x + f y) / 2 ⊢ ConvexOn ℝ (Set.Ioo a b) f
exercise_5_1
89d4e44810710ba8
test
import Mathlib theorem exercise_5_1 {f : ℝ → ℝ} (hf : ∀ x y : ℝ, |(f x - f y)| ≤ (x - y) ^ 2) : ∃ c, f = λ x => c := by
f : ℝ → ℝ hf : ∀ (x y : ℝ), |f x - f y| ≤ (x - y) ^ 2 ⊢ ∃ c, f = fun x => c
exercise_5_2
b5f9fe8b41e61e21
validation
import Mathlib theorem exercise_5_2 {a b : ℝ} {f g : ℝ → ℝ} (hf : ∀ x ∈ Set.Ioo a b, deriv f x > 0) (hg : g = f⁻¹) (hg_diff : DifferentiableOn ℝ g (Set.Ioo a b)) : DifferentiableOn ℝ g (Set.Ioo a b) ∧ ∀ x ∈ Set.Ioo a b, deriv g x = 1 / deriv f x := by
a b : ℝ f g : ℝ → ℝ hf : ∀ x ∈ Set.Ioo a b, deriv f x > 0 hg : g = f⁻¹ hg_diff : DifferentiableOn ℝ g (Set.Ioo a b) ⊢ DifferentiableOn ℝ g (Set.Ioo a b) ∧ ∀ x ∈ Set.Ioo a b, deriv g x = 1 / deriv f x
exercise_5_3
ad8d996f2353a7d5
test
import Mathlib theorem exercise_5_3 {g : ℝ → ℝ} (hg : Continuous g) (hg' : ∃ M : ℝ, ∀ x : ℝ, |deriv g x| ≤ M) : ∃ N, ∀ ε > 0, ε < N → Function.Injective (λ x : ℝ => x + ε * g x) := by
g : ℝ → ℝ hg : Continuous g hg' : ∃ M, ∀ (x : ℝ), |deriv g x| ≤ M ⊢ ∃ N, ∀ ε > 0, ε < N → Function.Injective fun x => x + ε * g x
exercise_5_4
21ffb0cc2a360491
validation
import Mathlib theorem exercise_5_4 {n : ℕ} (C : ℕ → ℝ) (hC : ∑ i in (range (n + 1)), (C i) / (i + 1) = 0) : ∃ x, x ∈ (Set.Icc (0 : ℝ) 1) ∧ ∑ i in range (n + 1), (C i) * (x^i) = 0 := by
n : ℕ C : ℕ → ℝ hC : ∑ i ∈ range (n + 1), C i / (↑i + 1) = 0 ⊢ ∃ x ∈ Set.Icc 0 1, ∑ i ∈ range (n + 1), C i * x ^ i = 0
exercise_5_5
c62ad09b34e5a1ae
test
import Mathlib theorem exercise_5_5 {f : ℝ → ℝ} (hfd : Differentiable ℝ f) (hf : Tendsto (deriv f) atTop (𝓝 0)) : Tendsto (λ x => f (x + 1) - f x) atTop atTop := by
f : ℝ → ℝ hfd : Differentiable ℝ f hf : Tendsto (deriv f) atTop (𝓝 0) ⊢ Tendsto (fun x => f (x + 1) - f x) atTop atTop
exercise_5_6
71b5926558088014
validation
import Mathlib theorem exercise_5_6 {f : ℝ → ℝ} (hf1 : Continuous f) (hf2 : ∀ x, DifferentiableAt ℝ f x) (hf3 : f 0 = 0) (hf4 : Monotone (deriv f)) : MonotoneOn (λ x => f x / x) (Set.Ioi 0) := by
f : ℝ → ℝ hf1 : Continuous f hf2 : ∀ (x : ℝ), DifferentiableAt ℝ f x hf3 : f 0 = 0 hf4 : Monotone (deriv f) ⊢ MonotoneOn (fun x => f x / x) (Set.Ioi 0)
exercise_5_7
757359474abf89e3
test
import Mathlib theorem exercise_5_7 {f g : ℝ → ℝ} {x : ℝ} (hf' : DifferentiableAt ℝ f 0) (hg' : DifferentiableAt ℝ g 0) (hg'_ne_0 : deriv g 0 ≠ 0) (f0 : f 0 = 0) (g0 : g 0 = 0) : Tendsto (λ x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x)) := by
f g : ℝ → ℝ x : ℝ hf' : DifferentiableAt ℝ f 0 hg' : DifferentiableAt ℝ g 0 hg'_ne_0 : deriv g 0 ≠ 0 f0 : f 0 = 0 g0 : g 0 = 0 ⊢ Tendsto (fun x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x))
exercise_5_15
168540424b23ca24
validation
import Mathlib theorem exercise_5_15 {f : ℝ → ℝ} (a M0 M1 M2 : ℝ) (hf' : DifferentiableOn ℝ f (Set.Ici a)) (hf'' : DifferentiableOn ℝ (deriv f) (Set.Ici a)) (hM0 : M0 = sSup {(|f x|) | x ∈ (Set.Ici a)}) (hM1 : M1 = sSup {(|deriv f x|) | x ∈ (Set.Ici a)}) (hM2 : M2 = sSup {(|deriv (deriv f) x|) | x ∈ (Set.Ici...
f : ℝ → ℝ a M0 M1 M2 : ℝ hf' : DifferentiableOn ℝ f (Set.Ici a) hf'' : DifferentiableOn ℝ (deriv f) (Set.Ici a) hM0 : M0 = sSup {x | ∃ x_1 ∈ Set.Ici a, |f x_1| = x} hM1 : M1 = sSup {x | ∃ x_1 ∈ Set.Ici a, |deriv f x_1| = x} hM2 : M2 = sSup {x | ∃ x_1 ∈ Set.Ici a, |deriv (deriv f) x_1| = x} ⊢ M1 ^ 2 ≤ 4 * M0 * M2
exercise_5_17
34e73e60098ca214
test
import Mathlib theorem exercise_5_17 {f : ℝ → ℝ} (hf' : DifferentiableOn ℝ f (Set.Icc (-1) 1)) (hf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1)) (hf''' : DifferentiableOn ℝ (deriv (deriv f)) (Set.Icc 1 1)) (hf0 : f (-1) = 0) (hf1 : f 0 = 0) (hf2 : f 1 = 1) (hf3 : deriv f 0 = 0) : ∃ x, x ∈ Set.Ioo (...
f : ℝ → ℝ hf' : DifferentiableOn ℝ f (Set.Icc (-1) 1) hf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1) hf''' : DifferentiableOn ℝ (deriv (deriv f)) (Set.Icc 1 1) hf0 : f (-1) = 0 hf1 : f 0 = 0 hf2 : f 1 = 1 hf3 : deriv f 0 = 0 ⊢ ∃ x ∈ Set.Ioo (-1) 1, deriv (deriv (deriv f)) x ≥ 3
exercise_2_1_18
1b5c346d3bd577de
test
import Mathlib theorem exercise_2_1_18 {G : Type*} [Group G] [Fintype G] (hG2 : Even (card G)) : ∃ (a : G), a ≠ 1 ∧ a = a⁻¹ := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG2 : Even (card G) ⊢ ∃ a, a ≠ 1 ∧ a = a⁻¹
exercise_2_1_21
0a7908cf5cd2c04a
validation
import Mathlib def exercise_2_1_21 (G : Type*) [Group G] [Fintype G] (hG : card G = 5) : CommGroup G := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG : card G = 5 ⊢ CommGroup G
exercise_2_1_26
fe0a2c1fa5e9b3de
test
import Mathlib theorem exercise_2_1_26 {G : Type*} [Group G] [Fintype G] (a : G) : ∃ (n : ℕ), a ^ n = 1 := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G a : G ⊢ ∃ n, a ^ n = 1
exercise_2_1_27
5c17ccff6806a104
validation
import Mathlib theorem exercise_2_1_27 {G : Type*} [Group G] [Fintype G] : ∃ (m : ℕ), ∀ (a : G), a ^ m = 1 := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G ⊢ ∃ m, ∀ (a : G), a ^ m = 1
exercise_2_2_3
8da61620ac643d35
test
import Mathlib def exercise_2_2_3 {G : Type*} [Group G] {P : ℕ → Prop} {hP : P = λ i => ∀ a b : G, (a*b)^i = a^i * b^i} (hP1 : ∃ n : ℕ, P n ∧ P (n+1) ∧ P (n+2)) : CommGroup G := by
G : Type u_1 inst✝ : Group G P : ℕ → Prop hP : P = fun i => ∀ (a b : G), (a * b) ^ i = a ^ i * b ^ i hP1 : ∃ n, P n ∧ P (n + 1) ∧ P (n + 2) ⊢ CommGroup G
exercise_2_2_5
0815be3b09c52ae9
validation
import Mathlib def exercise_2_2_5 {G : Type*} [Group G] (h : ∀ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5) : CommGroup G := by
G : Type u_1 inst✝ : Group G h : ∀ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5 ⊢ CommGroup G
exercise_2_2_6c
ae478dd7d2ed9761
test
import Mathlib theorem exercise_2_2_6c {G : Type*} [Group G] {n : ℕ} (hn : n > 1) (h : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n) : ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1 := by
G : Type u_1 inst✝ : Group G n : ℕ hn : n > 1 h : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n ⊢ ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1
exercise_2_3_17
57b7eaea8f24e998
validation
import Mathlib theorem exercise_2_3_17 {G : Type*} [Mul G] [Group G] (a x : G) : centralizer {x⁻¹*a*x} = (λ g : G => x⁻¹*g*x) '' (centralizer {a}) := by
G : Type u_1 inst✝¹ : Mul G inst✝ : Group G a x : G ⊢ {x⁻¹ * a * x}.centralizer = (fun g => x⁻¹ * g * x) '' {a}.centralizer
exercise_2_3_16
d11e2c17d5866dda
test
import Mathlib theorem exercise_2_3_16 {G : Type*} [Group G] (hG : ∀ H : Subgroup G, H = ⊤ ∨ H = ⊥) : IsCyclic G ∧ ∃ (p : ℕ) (Fin : Fintype G), Nat.Prime p ∧ @card G Fin = p := by
G : Type u_1 inst✝ : Group G hG : ∀ (H : Subgroup G), H = ⊤ ∨ H = ⊥ ⊢ IsCyclic G ∧ ∃ p Fin, p.Prime ∧ card G = p
exercise_2_4_36
4f851eb79a8d7b11
validation
import Mathlib theorem exercise_2_4_36 {a n : ℕ} (h : a > 1) : n ∣ (a ^ n - 1).totient := by
a n : ℕ h : a > 1 ⊢ n ∣ (a ^ n - 1).totient
exercise_2_5_23
82a6e4d0b8f63132
test
import Mathlib theorem exercise_2_5_23 {G : Type*} [Group G] (hG : ∀ (H : Subgroup G), H.Normal) (a b : G) : ∃ (j : ℤ) , b*a = a^j * b := by
G : Type u_1 inst✝ : Group G hG : ∀ (H : Subgroup G), H.Normal a b : G ⊢ ∃ j, b * a = a ^ j * b
exercise_2_5_30
1a562ef0b11a45da
validation
import Mathlib theorem exercise_2_5_30 {G : Type*} [Group G] [Fintype G] {p m : ℕ} (hp : Nat.Prime p) (hp1 : ¬ p ∣ m) (hG : card G = p*m) {H : Subgroup G} [Fintype H] [H.Normal] (hH : card H = p): Subgroup.Characteristic H := by
G : Type u_1 inst✝³ : Group G inst✝² : Fintype G p m : ℕ hp : p.Prime hp1 : ¬p ∣ m hG : card G = p * m H : Subgroup G inst✝¹ : Fintype ↥H inst✝ : H.Normal hH : card ↥H = p ⊢ H.Characteristic
exercise_2_5_31
657857cd270c2db7
test
import Mathlib theorem exercise_2_5_31 {G : Type*} [CommGroup G] [Fintype G] {p m n : ℕ} (hp : Nat.Prime p) (hp1 : ¬ p ∣ m) (hG : card G = p^n*m) {H : Subgroup G} [Fintype H] (hH : card H = p^n) : Subgroup.Characteristic H := by
G : Type u_1 inst✝² : CommGroup G inst✝¹ : Fintype G p m n : ℕ hp : p.Prime hp1 : ¬p ∣ m hG : card G = p ^ n * m H : Subgroup G inst✝ : Fintype ↥H hH : card ↥H = p ^ n ⊢ H.Characteristic
exercise_2_5_37
379528093e7f57fb
validation
import Mathlib def exercise_2_5_37 (G : Type*) [Group G] [Fintype G] (hG : card G = 6) (hG' : IsEmpty (CommGroup G)) : G ≃* Equiv.Perm (Fin 3) := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG : card G = 6 hG' : IsEmpty (CommGroup G) ⊢ G ≃* Equiv.Perm (Fin 3)
exercise_2_5_43
00057c2f023e4dff
test
import Mathlib def exercise_2_5_43 (G : Type*) [Group G] [Fintype G] (hG : card G = 9) : CommGroup G := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG : card G = 9 ⊢ CommGroup G
exercise_2_5_44
a7137a89dc8438c1
validation
import Mathlib theorem exercise_2_5_44 {G : Type*} [Group G] [Fintype G] {p : ℕ} (hp : Nat.Prime p) (hG : card G = p^2) : ∃ (N : Subgroup G) (Fin : Fintype N), @card N Fin = p ∧ N.Normal := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G p : ℕ hp : p.Prime hG : card G = p ^ 2 ⊢ ∃ N Fin, card ↥N = p ∧ N.Normal
exercise_2_5_52
dcd34a57fcbddbc1
test
import Mathlib theorem exercise_2_5_52 {G : Type*} [Group G] [Fintype G] (φ : G ≃* G) {I : Finset G} (hI : ∀ x ∈ I, φ x = x⁻¹) (hI1 : (0.75 : ℚ) * card G ≤ card I) : ∀ x : G, φ x = x⁻¹ ∧ ∀ x y : G, x*y = y*x := by
G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G φ : G ≃* G I : Finset G hI : ∀ x ∈ I, φ x = x⁻¹ hI1 : 0.75 * ↑(card G) ≤ ↑(card { x // x ∈ I }) ⊢ ∀ (x : G), φ x = x⁻¹ ∧ ∀ (x y : G), x * y = y * x
exercise_2_6_15
1a8aadcce86502b6
validation
import Mathlib theorem exercise_2_6_15 {G : Type*} [CommGroup G] {m n : ℕ} (hm : ∃ (g : G), orderOf g = m) (hn : ∃ (g : G), orderOf g = n) (hmn : m.Coprime n) : ∃ (g : G), orderOf g = m * n := by
G : Type u_1 inst✝ : CommGroup G m n : ℕ hm : ∃ g, orderOf g = m hn : ∃ g, orderOf g = n hmn : m.Coprime n ⊢ ∃ g, orderOf g = m * n
exercise_2_7_7
5163df03893bda94
test
import Mathlib theorem exercise_2_7_7 {G : Type*} [Group G] {G' : Type*} [Group G'] (φ : G →* G') (N : Subgroup G) [N.Normal] : (Subgroup.map φ N).Normal := by
G : Type u_1 inst✝² : Group G G' : Type u_2 inst✝¹ : Group G' φ : G →* G' N : Subgroup G inst✝ : N.Normal ⊢ (Subgroup.map φ N).Normal
exercise_2_8_12
96fc2706704eb445
validation
import Mathlib def exercise_2_8_12 {G H : Type*} [Fintype G] [Fintype H] [Group G] [Group H] (hG : card G = 21) (hH : card H = 21) (hG1 : IsEmpty (CommGroup G)) (hH1 : IsEmpty (CommGroup H)) : G ≃* H := by
G : Type u_1 H : Type u_2 inst✝³ : Fintype G inst✝² : Fintype H inst✝¹ : Group G inst✝ : Group H hG : card G = 21 hH : card H = 21 hG1 : IsEmpty (CommGroup G) hH1 : IsEmpty (CommGroup H) ⊢ G ≃* H
exercise_2_8_15
0fb2b3c507acdaa5
test
import Mathlib def exercise_2_8_15 {G H: Type*} [Fintype G] [Group G] [Fintype H] [Group H] {p q : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (h : p > q) (h1 : q ∣ p - 1) (hG : card G = p*q) (hH : card G = p*q) : G ≃* H := by
G : Type u_1 H : Type u_2 inst✝³ : Fintype G inst✝² : Group G inst✝¹ : Fintype H inst✝ : Group H p q : ℕ hp : p.Prime hq : q.Prime h : p > q h1 : q ∣ p - 1 hG hH : card G = p * q ⊢ G ≃* H
exercise_2_9_2
dc04e2dc301bf277
validation
import Mathlib theorem exercise_2_9_2 {G H : Type*} [Fintype G] [Fintype H] [Group G] [Group H] (hG : IsCyclic G) (hH : IsCyclic H) : IsCyclic (G × H) ↔ (card G).Coprime (card H) := by
G : Type u_1 H : Type u_2 inst✝³ : Fintype G inst✝² : Fintype H inst✝¹ : Group G inst✝ : Group H hG : IsCyclic G hH : IsCyclic H ⊢ IsCyclic (G × H) ↔ (card G).Coprime (card H)
exercise_2_10_1
b89f958d167e5868
test
import Mathlib theorem exercise_2_10_1 {G : Type*} [Group G] (A : Subgroup G) [A.Normal] {b : G} (hp : Nat.Prime (orderOf b)) : A ⊓ (Subgroup.closure {b}) = ⊥ := by
G : Type u_1 inst✝¹ : Group G A : Subgroup G inst✝ : A.Normal b : G hp : (orderOf b).Prime ⊢ A ⊓ Subgroup.closure {b} = ⊥
exercise_2_11_6
035563ebf1d2c35d
validation
import Mathlib theorem exercise_2_11_6 {G : Type*} [Group G] {p : ℕ} (hp : Nat.Prime p) {P : Sylow p G} (hP : P.Normal) : ∀ (Q : Sylow p G), P = Q := by
G : Type u_1 inst✝ : Group G p : ℕ hp : p.Prime P : Sylow p G hP : (↑P).Normal ⊢ ∀ (Q : Sylow p G), P = Q
exercise_2_11_7
6a62075dc330b261
test
import Mathlib theorem exercise_2_11_7 {G : Type*} [Group G] {p : ℕ} (hp : Nat.Prime p) {P : Sylow p G} (hP : P.Normal) : Subgroup.Characteristic (P : Subgroup G) := by
G : Type u_1 inst✝ : Group G p : ℕ hp : p.Prime P : Sylow p G hP : (↑P).Normal ⊢ (↑P).Characteristic
exercise_2_11_22
c98ffd7ad5540dc0
validation
import Mathlib theorem exercise_2_11_22 {p : ℕ} {n : ℕ} {G : Type*} [Fintype G] [Group G] (hp : Nat.Prime p) (hG : card G = p ^ n) {K : Subgroup G} [Fintype K] (hK : card K = p ^ (n-1)) : K.Normal := by
p n : ℕ G : Type u_1 inst✝² : Fintype G inst✝¹ : Group G hp : p.Prime hG : card G = p ^ n K : Subgroup G inst✝ : Fintype ↥K hK : card ↥K = p ^ (n - 1) ⊢ K.Normal
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ProofNet-SATP — Lean 4 (371 problems, SATP-normalized)

ProofNet (undergraduate-level mathematics theorem-proving benchmark, 371 problems from Rudin / Munkres / Dummit-Foote / Axler / Herstein / Ireland-Rosen / Artin) normalized for the SATP-DSP-Eval pipeline.

Upstream: deepseek-ai/DeepSeek-Prover-V1.5 datasets/proofnet.jsonl — the Lean 4 port of hoskinson-center/proofnet shipped with DeepSeek-Prover V1.5.

Differences from upstream

Field Upstream This repo
formal_statement bare theorem import Mathlib\n\n<theorem> (header baked in)
uuid absent sha256(canonical(formal_statement))[:16] after baking
split valid / test validation / test (matches minif2f-satp naming)
goal_state shipped as goal renamed; value byte-identical

Dropped: informal_prefix (the /-- ... -/ English statement). Mirrors ChristianZ97/minif2f-satp which drops informal_* fields. The SATP eval pipeline reads only formal_statement.

Dropped from baked header: the per-row open / open scoped / set_option directives. Only import Mathlib is baked in. Same pattern as minif2f-satp. Consumer scripts strip the import preamble then layer their own canonical opens.

Schema

Field Type Description
name str Stable identifier, upstream verbatim (e.g. exercise_1_13a)
uuid str sha256(canonical(formal_statement))[:16]
split str "validation" (185 rows) or "test" (186 rows)
formal_statement str import Mathlib\n\n<theorem>, ends in := by
goal_state str Initial proof goal, byte-identical to DSP V1.5

Citation 

@inproceedings{azerbayev2023proofnet,
  title={ProofNet: Autoformalizing and Formally Proving
         Undergraduate-Level Mathematics},
  author={Zhangir Azerbayev and Bartosz Piotrowski and Hailey Schoelkopf
          and Edward W. Ayers and Dragomir Radev and Jeremy Avigad},
  booktitle={Proceedings of the 12th International Conference on Learning
             Representations (ICLR)},
  year={2024}
}
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