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exercise_1_13a | valid | /-- Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Re}(f)$ is constant, then $f$ is constant.-/
| 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:= sorry | 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 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
open scoped BigOperators Topology
| Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Re}(f)$ is constant, then $f$ is constant. |
exercise_1_13c | valid | /-- Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $|f|$ is constant, then $f$ is constant.-/
| 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:= sorry | 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 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $|f|$ is constant, then $f$ is constant. |
exercise_1_19b | valid | /-- Prove that the power series $\sum zn/n^2$ converges at every point of the unit circle.-/
| 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):= sorry | 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) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Prove that the power series $\sum zn/n^2$ converges at every point of the unit circle. |
exercise_1_26 | valid | /-- Suppose $f$ is continuous in a region $\Omega$. Prove that any two primitives of $f$ (if they exist) differ by a constant.-/
| 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:= sorry | 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 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Suppose $f$ is continuous in a region $\Omega$. Prove that any two primitives of $f$ (if they exist) differ by a constant. |
exercise_2_9 | valid | /-- Let $\Omega$ be a bounded open subset of $\mathbb{C}$, and $\varphi: \Omega \rightarrow \Omega$ a holomorphic function. Prove that if there exists a point $z_{0} \in \Omega$ such that $\varphi\left(z_{0}\right)=z_{0} \quad \text { and } \quad \varphi^{\prime}\left(z_{0}\right)=1$ then $\varphi$ is linear.-/
| 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:= sorry | 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 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Let $\Omega$ be a bounded open subset of $\mathbb{C}$, and $\varphi: \Omega \rightarrow \Omega$ a holomorphic function. Prove that if there exists a point $z_{0} \in \Omega$ such that $\varphi\left(z_{0}\right)=z_{0} \quad \text { and } \quad \varphi^{\prime}\left(z_{0}\right)=1$ then $\varphi$ is linear. |
exercise_3_3 | valid | /-- Show that $ \int_{-\infty}^{\infty} \frac{\cos x}{x^2 + a^2} dx = \pi \frac{e^{-a}}{a}$ for $a > 0$.-/
| 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))):= sorry | 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))) | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Show that $ \int_{-\infty}^{\infty} \frac{\cos x}{x^2 + a^2} dx = \pi \frac{e^{-a}}{a}$ for $a > 0$. |
exercise_3_9 | valid | /-- Show that $\int_0^1 \log(\sin \pi x) dx = - \log 2$.-/
| theorem exercise_3_9 : ∫ x in (0 : ℝ)..(1 : ℝ),
Real.log (Real.sin (Real.pi * x)) = - Real.log 2:= sorry | ⊢ ∫ (x : ℝ) in 0 ..1, (Real.pi * x).sin.log = -Real.log 2 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Show that $\int_0^1 \log(\sin \pi x) dx = - \log 2$. |
exercise_3_22 | valid | /-- Show that there is no holomorphic function $f$ in the unit disc $D$ that extends continuously to $\partial D$ such that $f(z) = 1/z$ for $z \in \partial D$.-/
| 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:= sorry | 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 | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
| Show that there is no holomorphic function $f$ in the unit disc $D$ that extends continuously to $\partial D$ such that $f(z) = 1/z$ for $z \in \partial D$. |
exercise_1_1a | valid | /-- If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational.-/
| theorem exercise_1_1a
(x : ℝ) (y : ℚ) :
( Irrational x ) -> Irrational ( x + y ):= sorry | x : ℝ
y : ℚ
⊢ Irrational x → Irrational (x + ↑y) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational. |
exercise_1_2 | valid | /-- Prove that there is no rational number whose square is $12$.-/
| theorem exercise_1_2 : ¬ ∃ (x : ℚ), ( x ^ 2 = 12 ):= sorry | ⊢ ¬∃ x, x ^ 2 = 12 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Prove that there is no rational number whose square is $12$. |
exercise_1_5 | valid | /-- Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A=-\sup (-A)$.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A=-\sup (-A)$. |
exercise_1_11a | valid | /-- If $z$ is a complex number, prove that there exists an $r\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$.-/
| theorem exercise_1_11a (z : ℂ) :
∃ (r : ℝ) (w : ℂ), abs w = 1 ∧ z = r * w:= sorry | z : ℂ
⊢ ∃ r w, Complex.abs w = 1 ∧ z = ↑r * w | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $z$ is a complex number, prove that there exists an $r\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$. |
exercise_1_13 | valid | /-- If $x, y$ are complex, prove that $||x|-|y|| \leq |x-y|$.-/
| theorem exercise_1_13 (x y : ℂ) :
|(abs x) - (abs y)| ≤ abs (x - y):= sorry | x y : ℂ
⊢ |Complex.abs x - Complex.abs y| ≤ Complex.abs (x - y) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $x, y$ are complex, prove that $||x|-|y|| \leq |x-y|$. |
exercise_1_16a | valid | /-- Suppose $k \geq 3, x, y \in \mathbb{R}^k, |x - y| = d > 0$, and $r > 0$. Prove that if $2r > d$, there are infinitely many $z \in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$.-/
| 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}:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $k \geq 3, x, y \in \mathbb{R}^k, |x - y| = d > 0$, and $r > 0$. Prove that if $2r > d$, there are infinitely many $z \in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$. |
exercise_1_18a | valid | /-- If $k \geq 2$ and $\mathbf{x} \in R^{k}$, prove that there exists $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$-/
| 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):= sorry | n : ℕ
h : n > 1
x : EuclideanSpace ℝ (Fin n)
⊢ ∃ y, y ≠ 0 ∧ ⟪x, y⟫_ℝ = 0 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $k \geq 2$ and $\mathbf{x} \in R^{k}$, prove that there exists $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$ |
exercise_1_19 | valid | /-- Suppose $a, b \in R^k$. Find $c \in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $a, b \in R^k$. Find $c \in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$. |
exercise_2_24 | valid | /-- Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.-/
| theorem exercise_2_24 {X : Type*} [MetricSpace X]
(hX : ∀ (A : Set X), Infinite A → ∃ (x : X), x ∈ closure A) :
SeparableSpace X:= sorry | X : Type u_1
inst✝ : MetricSpace X
hX : ∀ (A : Set X), Infinite ↑A → ∃ x, x ∈ closure A
⊢ SeparableSpace X | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable. |
exercise_2_27a | valid | /-- Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect.-/
| 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)}:= sorry | 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)} | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect. |
exercise_2_28 | valid | /-- Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. |
exercise_3_1a | valid | /-- Prove that convergence of $\left\{s_{n}\right\}$ implies convergence of $\left\{\left|s_{n}\right|\right\}$.-/
| 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):= sorry | f : ℕ → ℝ
h : ∃ a, Tendsto (fun n => f n) atTop (𝓝 a)
⊢ ∃ a, Tendsto (fun n => |f n|) atTop (𝓝 a) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Prove that convergence of $\left\{s_{n}\right\}$ implies convergence of $\left\{\left|s_{n}\right|\right\}$. |
exercise_3_3 | valid | /-- If $s_{1}=\sqrt{2}$, and $s_{n+1}=\sqrt{2+\sqrt{s_{n}}} \quad(n=1,2,3, \ldots),$ prove that $\left\{s_{n}\right\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \ldots$.-/
| theorem exercise_3_3
: ∃ (x : ℝ), Tendsto f atTop (𝓝 x) ∧ ∀ n, f n < 2:= sorry | ⊢ ∃ x, Tendsto f atTop (𝓝 x) ∧ ∀ (n : ℕ), f n < 2 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
noncomputable def f : ℕ → ℝ
| 0 => sqrt 2
| (n + 1) => sqrt (2 + sqrt (f n))
| If $s_{1}=\sqrt{2}$, and $s_{n+1}=\sqrt{2+\sqrt{s_{n}}} \quad(n=1,2,3, \ldots),$ prove that $\left\{s_{n}\right\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \ldots$. |
exercise_3_6a | valid | /-- Prove that $\lim_{n \rightarrow \infty} \sum_{i<n} a_i = \infty$, where $a_i = \sqrt{i + 1} -\sqrt{i}$.-/
| theorem exercise_3_6a
: Tendsto (λ (n : ℕ) => (∑ i in range n, g i)) atTop atTop:= sorry | ⊢ Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
noncomputable section
def g (n : ℕ) : ℝ := sqrt (n + 1) - sqrt n
| Prove that $\lim_{n \rightarrow \infty} \sum_{i<n} a_i = \infty$, where $a_i = \sqrt{i + 1} -\sqrt{i}$. |
exercise_3_8 | valid | /-- If $\Sigma a_{n}$ converges, and if $\left\{b_{n}\right\}$ is monotonic and bounded, prove that $\Sigma a_{n} b_{n}$ converges.-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $\Sigma a_{n}$ converges, and if $\left\{b_{n}\right\}$ is monotonic and bounded, prove that $\Sigma a_{n} b_{n}$ converges. |
exercise_3_20 | valid | /-- Suppose $\left\{p_{n}\right\}$ is a Cauchy sequence in a metric space $X$, and some sequence $\left\{p_{n l}\right\}$ converges to a point $p \in X$. Prove that the full sequence $\left\{p_{n}\right\}$ converges to $p$.-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $\left\{p_{n}\right\}$ is a Cauchy sequence in a metric space $X$, and some sequence $\left\{p_{n l}\right\}$ converges to a point $p \in X$. Prove that the full sequence $\left\{p_{n}\right\}$ converges to $p$. |
exercise_3_22 | valid | /-- Suppose $X$ is a nonempty complete metric space, and $\left\{G_{n}\right\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\bigcap_{1}^{\infty} G_{n}$ is not empty.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $X$ is a nonempty complete metric space, and $\left\{G_{n}\right\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\bigcap_{1}^{\infty} G_{n}$ is not empty. |
exercise_4_2a | valid | /-- If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $f(\overline{E}) \subset \overline{f(E)}$ for every set $E \subset X$. ($\overline{E}$ denotes the closure of $E$).-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $f(\overline{E}) \subset \overline{f(E)}$ for every set $E \subset X$. ($\overline{E}$ denotes the closure of $E$). |
exercise_4_4a | valid | /-- Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$.-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$. |
exercise_4_5a | valid | /-- If $f$ is a real continuous function defined on a closed set $E \subset \mathbb{R}$, prove that there exist continuous real functions $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$.-/
| 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:= sorry | f : ℝ → ℝ
E : Set ℝ
h₁ : IsClosed E
h₂ : ContinuousOn f E
⊢ ∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $f$ is a real continuous function defined on a closed set $E \subset \mathbb{R}$, prove that there exist continuous real functions $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$. |
exercise_4_6 | valid | /-- If $f$ is defined on $E$, the graph of $f$ is the set of points $(x, f(x))$, for $x \in E$. In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $f$ is defined on $E$, the graph of $f$ is the set of points $(x, f(x))$, for $x \in E$. In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact. |
exercise_4_8b | valid | /-- Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$.-/
| theorem exercise_4_8b
(E : Set ℝ) :
∃ f : ℝ → ℝ, UniformContinuousOn f E ∧ ¬ Bornology.IsBounded (Set.image f E):= sorry | E : Set ℝ
⊢ ∃ f, UniformContinuousOn f E ∧ ¬Bornology.IsBounded (f '' E) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$. |
exercise_4_12 | valid | /-- A uniformly continuous function of a uniformly continuous function is uniformly continuous.-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| A uniformly continuous function of a uniformly continuous function is uniformly continuous. |
exercise_4_19 | valid | /-- Suppose $f$ is a real function with domain $R^{1}$ which has the intermediate value property: if $f(a)<c<f(b)$, then $f(x)=c$ for some $x$ between $a$ and $b$. Suppose also, for every rational $r$, that the set of all $x$ with $f(x)=r$ is closed. Prove that $f$ is continuous.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $f$ is a real function with domain $R^{1}$ which has the intermediate value property: if $f(a)<c<f(b)$, then $f(x)=c$ for some $x$ between $a$ and $b$. Suppose also, for every rational $r$, that the set of all $x$ with $f(x)=r$ is closed. Prove that $f$ is continuous. |
exercise_4_24 | valid | /-- Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$ for all $x, y \in(a, b)$. Prove that $f$ is convex.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$ for all $x, y \in(a, b)$. Prove that $f$ is convex. |
exercise_5_2 | valid | /-- Suppose $f^{\prime}(x)>0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)} \quad(a<x<b)$.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $f^{\prime}(x)>0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)} \quad(a<x<b)$. |
exercise_5_4 | valid | /-- If $C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0,$ where $C_{0}, \ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1.-/
| 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:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| If $C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0,$ where $C_{0}, \ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1. |
exercise_5_6 | valid | /-- Suppose (a) $f$ is continuous for $x \geq 0$, (b) $f^{\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\prime}$ is monotonically increasing. Put $g(x)=\frac{f(x)}{x} \quad(x>0)$ and prove that $g$ is monotonically increasing.-/
| 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):= sorry | 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) | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose (a) $f$ is continuous for $x \geq 0$, (b) $f^{\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\prime}$ is monotonically increasing. Put $g(x)=\frac{f(x)}{x} \quad(x>0)$ and prove that $g$ is monotonically increasing. |
exercise_5_15 | valid | /-- Suppose $a \in R^{1}, f$ is a twice-differentiable real function on $(a, \infty)$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $|f(x)|,\left|f^{\prime}(x)\right|,\left|f^{\prime \prime}(x)\right|$, respectively, on $(a, \infty)$. Prove that $M_{1}^{2} \leq 4 M_{0} M_{2} .$-/
| 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 a)}) :
(M1 ^ 2) ≤ 4 * M0 * M2:= sorry | 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 | import Mathlib
open Topology Filter Real Complex TopologicalSpace Finset
open scoped BigOperators
| Suppose $a \in R^{1}, f$ is a twice-differentiable real function on $(a, \infty)$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $|f(x)|,\left|f^{\prime}(x)\right|,\left|f^{\prime \prime}(x)\right|$, respectively, on $(a, \infty)$. Prove that $M_{1}^{2} \leq 4 M_{0} M_{2} .$ |
exercise_2_1_21 | valid | /-- Show that a group of order 5 must be abelian.-/
| def exercise_2_1_21 (G : Type*) [Group G] [Fintype G]
(hG : card G = 5) :
CommGroup G:= sorry | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
hG : card G = 5
⊢ CommGroup G | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Show that a group of order 5 must be abelian. |
exercise_2_1_27 | valid | /-- If $G$ is a finite group, prove that there is an integer $m > 0$ such that $a^m = e$ for all $a \in G$.-/
| theorem exercise_2_1_27 {G : Type*} [Group G]
[Fintype G] : ∃ (m : ℕ), ∀ (a : G), a ^ m = 1:= sorry | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
⊢ ∃ m, ∀ (a : G), a ^ m = 1 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $G$ is a finite group, prove that there is an integer $m > 0$ such that $a^m = e$ for all $a \in G$. |
exercise_2_2_5 | valid | /-- Let $G$ be a group in which $(a b)^{3}=a^{3} b^{3}$ and $(a b)^{5}=a^{5} b^{5}$ for all $a, b \in G$. Show that $G$ is abelian.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $G$ be a group in which $(a b)^{3}=a^{3} b^{3}$ and $(a b)^{5}=a^{5} b^{5}$ for all $a, b \in G$. Show that $G$ is abelian. |
exercise_2_3_17 | valid | /-- If $G$ is a group and $a, x \in G$, prove that $C\left(x^{-1} a x\right)=x^{-1} C(a) x$-/
| 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}):= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $G$ is a group and $a, x \in G$, prove that $C\left(x^{-1} a x\right)=x^{-1} C(a) x$ |
exercise_2_4_36 | valid | /-- If $a > 1$ is an integer, show that $n \mid \varphi(a^n - 1)$, where $\phi$ is the Euler $\varphi$-function.-/
| theorem exercise_2_4_36 {a n : ℕ} (h : a > 1) :
n ∣ (a ^ n - 1).totient:= sorry | a n : ℕ
h : a > 1
⊢ n ∣ (a ^ n - 1).totient | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $a > 1$ is an integer, show that $n \mid \varphi(a^n - 1)$, where $\phi$ is the Euler $\varphi$-function. |
exercise_2_5_30 | valid | /-- Suppose that $|G| = pm$, where $p \nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Suppose that $|G| = pm$, where $p \nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic. |
exercise_2_5_37 | valid | /-- If $G$ is a nonabelian group of order 6, prove that $G \simeq S_3$.-/
| def exercise_2_5_37 (G : Type*) [Group G] [Fintype G]
(hG : card G = 6) (hG' : IsEmpty (CommGroup G)) :
G ≃* Equiv.Perm (Fin 3):= sorry | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
hG : card G = 6
hG' : IsEmpty (CommGroup G)
⊢ G ≃* Equiv.Perm (Fin 3) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $G$ is a nonabelian group of order 6, prove that $G \simeq S_3$. |
exercise_2_5_44 | valid | /-- Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$. |
exercise_2_6_15 | valid | /-- If $G$ is an abelian group and if $G$ has an element of order $m$ and one of order $n$, where $m$ and $n$ are relatively prime, prove that $G$ has an element of order $mn$.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $G$ is an abelian group and if $G$ has an element of order $m$ and one of order $n$, where $m$ and $n$ are relatively prime, prove that $G$ has an element of order $mn$. |
exercise_2_8_12 | valid | /-- Prove that any two nonabelian groups of order 21 are isomorphic.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Prove that any two nonabelian groups of order 21 are isomorphic. |
exercise_2_9_2 | valid | /-- If $G_1$ and $G_2$ are cyclic groups of orders $m$ and $n$, respectively, prove that $G_1 \times G_2$ is cyclic if and only if $m$ and $n$ are relatively prime.-/
| 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):= sorry | 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) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $G_1$ and $G_2$ are cyclic groups of orders $m$ and $n$, respectively, prove that $G_1 \times G_2$ is cyclic if and only if $m$ and $n$ are relatively prime. |
exercise_2_11_6 | valid | /-- If $P$ is a $p$-Sylow subgroup of $G$ and $P \triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $P$ is a $p$-Sylow subgroup of $G$ and $P \triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$. |
exercise_2_11_22 | valid | /-- Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$.-/
| 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:= sorry | 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 | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$. |
exercise_4_1_19 | valid | /-- Show that there is an infinite number of solutions to $x^2 = -1$ in the quaternions.-/
| theorem exercise_4_1_19 : Infinite {x : Quaternion ℝ | x^2 = -1}:= sorry | ⊢ Infinite ↑{x | x ^ 2 = -1} | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Show that there is an infinite number of solutions to $x^2 = -1$ in the quaternions. |
exercise_4_2_5 | valid | /-- Let $R$ be a ring in which $x^3 = x$ for every $x \in R$. Prove that $R$ is commutative.-/
| def exercise_4_2_5 {R : Type*} [Ring R]
(h : ∀ x : R, x ^ 3 = x) : CommRing R:= sorry | R : Type u_1
inst✝ : Ring R
h : ∀ (x : R), x ^ 3 = x
⊢ CommRing R | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $R$ be a ring in which $x^3 = x$ for every $x \in R$. Prove that $R$ is commutative. |
exercise_4_2_9 | valid | /-- Let $p$ be an odd prime and let $1 + \frac{1}{2} + ... + \frac{1}{p - 1} = \frac{a}{b}$, where $a, b$ are integers. Show that $p \mid a$.-/
| theorem exercise_4_2_9 {p : ℕ} (hp : Nat.Prime p) (hp1 : Odd p) :
∃ (a b : ℤ), (a / b : ℚ) = ∑ i in Finset.range p, 1 / (i + 1) → ↑p ∣ a:= sorry | p : ℕ
hp : p.Prime
hp1 : Odd p
⊢ ∃ a b, ↑a / ↑b = ↑(∑ i ∈ Finset.range p, 1 / (i + 1)) → ↑p ∣ a | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $p$ be an odd prime and let $1 + \frac{1}{2} + ... + \frac{1}{p - 1} = \frac{a}{b}$, where $a, b$ are integers. Show that $p \mid a$. |
exercise_4_3_25 | valid | /-- Let $R$ be the ring of $2 \times 2$ matrices over the real numbers; suppose that $I$ is an ideal of $R$. Show that $I = (0)$ or $I = R$.-/
| theorem exercise_4_3_25 (I : Ideal (Matrix (Fin 2) (Fin 2) ℝ)) :
I = ⊥ ∨ I = ⊤:= sorry | I : Ideal (Matrix (Fin 2) (Fin 2) ℝ)
⊢ I = ⊥ ∨ I = ⊤ | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $R$ be the ring of $2 \times 2$ matrices over the real numbers; suppose that $I$ is an ideal of $R$. Show that $I = (0)$ or $I = R$. |
exercise_4_5_16 | valid | /-- Let $F = \mathbb{Z}_p$ be the field of integers $\mod p$, where $p$ is a prime, and let $q(x) \in F[x]$ be irreducible of degree $n$. Show that $F[x]/(q(x))$ is a field having at exactly $p^n$ elements.-/
| theorem exercise_4_5_16 {p n: ℕ} (hp : Nat.Prime p)
{q : Polynomial (ZMod p)} (hq : Irreducible q) (hn : q.degree = n) :
∃ is_fin : Fintype $ Polynomial (ZMod p) ⧸ span ({q} : Set (Polynomial $ ZMod p)),
@card (Polynomial (ZMod p) ⧸ span {q}) is_fin = p ^ n ∧
IsField (Polynomial $ ZMod p):= sorry | p n : ℕ
hp : p.Prime
q : (ZMod p)[X]
hq : Irreducible q
hn : q.degree = ↑n
⊢ ∃ is_fin, card ((ZMod p)[X] ⧸ span {q}) = p ^ n ∧ IsField (ZMod p)[X] | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $F = \mathbb{Z}_p$ be the field of integers $\mod p$, where $p$ is a prime, and let $q(x) \in F[x]$ be irreducible of degree $n$. Show that $F[x]/(q(x))$ is a field having at exactly $p^n$ elements. |
exercise_4_5_25 | valid | /-- If $p$ is a prime, show that $q(x) = 1 + x + x^2 + \cdots x^{p - 1}$ is irreducible in $Q[x]$.-/
| theorem exercise_4_5_25 {p : ℕ} (hp : Nat.Prime p) :
Irreducible (∑ i : Finset.range p, X ^ p : Polynomial ℚ):= sorry | p : ℕ
hp : p.Prime
⊢ Irreducible (∑ i : { x // x ∈ Finset.range p }, X ^ p) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| If $p$ is a prime, show that $q(x) = 1 + x + x^2 + \cdots x^{p - 1}$ is irreducible in $Q[x]$. |
exercise_4_6_3 | valid | /-- Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$.-/
| theorem exercise_4_6_3 :
Infinite {a : ℤ | Irreducible (X^7 + 15*X^2 - 30*X + (a : Polynomial ℚ) : Polynomial ℚ)}:= sorry | ⊢ Infinite ↑{a | Irreducible (X ^ 7 + 15 * X ^ 2 - 30 * X + ↑a)} | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$. |
exercise_5_2_20 | valid | /-- Let $V$ be a vector space over an infinite field $F$. Show that $V$ cannot be the set-theoretic union of a finite number of proper subspaces of $V$.-/
| theorem exercise_5_2_20 {F V ι: Type*} [Infinite F] [Field F]
[AddCommGroup V] [Module F V] {u : ι → Submodule F V}
(hu : ∀ i : ι, u i ≠ ⊤) :
(⋃ i : ι, (u i : Set V)) ≠ ⊤:= sorry | F : Type u_1
V : Type u_2
ι : Type u_3
inst✝³ : Infinite F
inst✝² : Field F
inst✝¹ : AddCommGroup V
inst✝ : Module F V
u : ι → Submodule F V
hu : ∀ (i : ι), u i ≠ ⊤
⊢ ⋃ i, ↑(u i) ≠ ⊤ | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Let $V$ be a vector space over an infinite field $F$. Show that $V$ cannot be the set-theoretic union of a finite number of proper subspaces of $V$. |
exercise_5_3_10 | valid | /-- Prove that $\cos 1^{\circ}$ is algebraic over $\mathbb{Q}$.-/
| theorem exercise_5_3_10 : IsAlgebraic ℚ (cos (Real.pi / 180)):= sorry | ⊢ IsAlgebraic ℚ (π / 180).cos | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Prove that $\cos 1^{\circ}$ is algebraic over $\mathbb{Q}$. |
exercise_5_5_2 | valid | /-- Prove that $x^3 - 3x - 1$ is irreducible over $\mathbb{Q}$.-/
| theorem exercise_5_5_2 : Irreducible (X^3 - 3*X - 1 : Polynomial ℚ):= sorry | ⊢ Irreducible (X ^ 3 - 3 * X - 1) | import Mathlib
open Fintype Set Real Ideal Polynomial
open scoped BigOperators
| Prove that $x^3 - 3x - 1$ is irreducible over $\mathbb{Q}$. |
exercise_2_12a | valid | /-- Let $(p_n)$ be a sequence and $f:\mathbb{N}\to\mathbb{N}$. The sequence $(q_k)_{k\in\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement.-/
| theorem exercise_2_12a (f : ℕ → ℕ) (p : ℕ → ℝ) (a : ℝ)
(hf : Injective f) (hp : Tendsto p atTop (𝓝 a)) :
Tendsto (λ n => p (f n)) atTop (𝓝 a):= sorry | f : ℕ → ℕ
p : ℕ → ℝ
a : ℝ
hf : Injective f
hp : Tendsto p atTop (𝓝 a)
⊢ Tendsto (fun n => p (f n)) atTop (𝓝 a) | import Mathlib
open Filter Real Function
open scoped Topology
| Let $(p_n)$ be a sequence and $f:\mathbb{N}\to\mathbb{N}$. The sequence $(q_k)_{k\in\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement. |
exercise_2_29 | valid | /-- Let $\mathcal{T}$ be the collection of open subsets of a metric space $\mathrm{M}$, and $\mathcal{K}$ the collection of closed subsets. Show that there is a bijection from $\mathcal{T}$ onto $\mathcal{K}$.-/
| theorem exercise_2_29 (M : Type*) [MetricSpace M]
(O C : Set (Set M))
(hO : O = {s | IsOpen s})
(hC : C = {s | IsClosed s}) :
∃ f : O → C, Bijective f:= sorry | M : Type u_1
inst✝ : MetricSpace M
O C : Set (Set M)
hO : O = {s | IsOpen s}
hC : C = {s | IsClosed s}
⊢ ∃ f, Bijective f | import Mathlib
open Filter Real Function
open scoped Topology
| Let $\mathcal{T}$ be the collection of open subsets of a metric space $\mathrm{M}$, and $\mathcal{K}$ the collection of closed subsets. Show that there is a bijection from $\mathcal{T}$ onto $\mathcal{K}$. |
exercise_2_41 | valid | /-- Let $\|\cdot\|$ be any norm on $\mathbb{R}^{m}$ and let $B=\left\{x \in \mathbb{R}^{m}:\|x\| \leq 1\right\}$. Prove that $B$ is compact.-/
| theorem exercise_2_41 (m : ℕ) {X : Type*} [NormedSpace ℝ ((Fin m) → ℝ)] :
IsCompact (Metric.closedBall 0 1):= sorry | m : ℕ
X : Type u_1
inst✝ : NormedSpace ℝ (Fin m → ℝ)
⊢ IsCompact (Metric.closedBall 0 1) | import Mathlib
open Filter Real Function
open scoped Topology
| Let $\|\cdot\|$ be any norm on $\mathbb{R}^{m}$ and let $B=\left\{x \in \mathbb{R}^{m}:\|x\| \leq 1\right\}$. Prove that $B$ is compact. |
exercise_2_57 | valid | /-- Show that if $S$ is connected, it is not true in general that its interior is connected.-/
| theorem exercise_2_57 {X : Type*} [TopologicalSpace X]
: ∃ (S : Set X), IsConnected S ∧ ¬ IsConnected (interior S):= sorry | X : Type u_1
inst✝ : TopologicalSpace X
⊢ ∃ S, IsConnected S ∧ ¬IsConnected (interior S) | import Mathlib
open Filter Real Function
open scoped Topology
| Show that if $S$ is connected, it is not true in general that its interior is connected. |
exercise_2_126 | valid | /-- Suppose that $E$ is an uncountable subset of $\mathbb{R}$. Prove that there exists a point $p \in \mathbb{R}$ at which $E$ condenses.-/
| theorem exercise_2_126 {E : Set ℝ}
(hE : ¬ Set.Countable E) : ∃ (p : ℝ), ClusterPt p (𝓟 E):= sorry | E : Set ℝ
hE : ¬E.Countable
⊢ ∃ p, ClusterPt p (𝓟 E) | import Mathlib
open Filter Real Function
open scoped Topology
| Suppose that $E$ is an uncountable subset of $\mathbb{R}$. Prove that there exists a point $p \in \mathbb{R}$ at which $E$ condenses. |
exercise_3_4 | valid | /-- Prove that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$.-/
| theorem exercise_3_4 (n : ℕ) :
Tendsto (λ n => (sqrt (n + 1) - sqrt n)) atTop (𝓝 0):= sorry | n : ℕ
⊢ Tendsto (fun n => √(n + 1) - √n) atTop (𝓝 0) | import Mathlib
open Filter Real Function
open scoped Topology
| Prove that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$. |
exercise_3_63b | valid | /-- Prove that $\sum 1/k(\log(k))^p$ diverges when $p \leq 1$.-/
| theorem exercise_3_63b (p : ℝ) (f : ℕ → ℝ) (hp : p ≤ 1)
(h : f = λ (k : ℕ) => (1 : ℝ) / (k * (log k) ^ p)) :
¬ ∃ l, Tendsto f atTop (𝓝 l):= sorry | p : ℝ
f : ℕ → ℝ
hp : p ≤ 1
h : f = fun k => 1 / (↑k * (↑k).log ^ p)
⊢ ¬∃ l, Tendsto f atTop (𝓝 l) | import Mathlib
open Filter Real Function
open scoped Topology
| Prove that $\sum 1/k(\log(k))^p$ diverges when $p \leq 1$. |
exercise_2_2_9 | valid | /-- Let $H$ be the subgroup generated by two elements $a, b$ of a group $G$. Prove that if $a b=b a$, then $H$ is an abelian group.-/
| theorem exercise_2_2_9 {G : Type} [Group G] {a b : G} (h : a * b = b * a) :
∀ x y : closure {x| x = a ∨ x = b}, x * y = y * x:= sorry | G : Type
inst✝ : Group G
a b : G
h : a * b = b * a
⊢ ∀ (x y : ↥(Subgroup.closure {x | x = a ∨ x = b})), x * y = y * x | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $H$ be the subgroup generated by two elements $a, b$ of a group $G$. Prove that if $a b=b a$, then $H$ is an abelian group. |
exercise_2_4_19 | valid | /-- Prove that if a group contains exactly one element of order 2 , then that element is in the center of the group.-/
| theorem exercise_2_4_19 {G : Type*} [Group G] {x : G}
(hx : orderOf x = 2) (hx1 : ∀ y, orderOf y = 2 → y = x) :
x ∈ center G:= sorry | G : Type u_1
inst✝ : Group G
x : G
hx : orderOf x = 2
hx1 : ∀ (y : G), orderOf y = 2 → y = x
⊢ x ∈ center G | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that if a group contains exactly one element of order 2 , then that element is in the center of the group. |
exercise_2_11_3 | valid | /-- Prove that a group of even order contains an element of order $2 .$-/
| theorem exercise_2_11_3 {G : Type*} [Group G] [Fintype G](hG : Even (card G)) :
∃ x : G, orderOf x = 2:= sorry | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
hG : Even (card G)
⊢ ∃ x, orderOf x = 2 | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that a group of even order contains an element of order $2 .$ |
exercise_3_5_6 | valid | /-- Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite.-/
| theorem exercise_3_5_6 {K V : Type*} [Field K] [AddCommGroup V]
[Module K V] {S : Set V} (hS : Set.Countable S)
(hS1 : span K S = ⊤) {ι : Type*} (R : ι → V)
(hR : LinearIndependent K R) : Countable ι:= sorry | K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
S : Set V
hS : S.Countable
hS1 : Submodule.span K S = ⊤
ι : Type u_3
R : ι → V
hR : LinearIndependent K R
⊢ Countable ι | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite. |
exercise_6_1_14 | valid | /-- Let $Z$ be the center of a group $G$. Prove that if $G / Z$ is a cyclic group, then $G$ is abelian and hence $G=Z$.-/
| theorem exercise_6_1_14 (G : Type*) [Group G]
(hG : IsCyclic $ G ⧸ (center G)) :
center G = ⊤:= sorry | G : Type u_1
inst✝ : Group G
hG : IsCyclic (G ⧸ center G)
⊢ center G = ⊤ | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $Z$ be the center of a group $G$. Prove that if $G / Z$ is a cyclic group, then $G$ is abelian and hence $G=Z$. |
exercise_6_4_3 | valid | /-- Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.-/
| theorem exercise_6_4_3 {G : Type*} [Group G] [Fintype G] {p q : ℕ}
(hp : Prime p) (hq : Prime q) (hG : card G = p^2 *q) :
IsSimpleGroup G → false:= sorry | G : Type u_1
inst✝¹ : Group G
inst✝ : Fintype G
p q : ℕ
hp : Prime p
hq : Prime q
hG : card G = p ^ 2 * q
⊢ IsSimpleGroup G → false = true | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple. |
exercise_6_8_1 | valid | /-- Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$.-/
| theorem exercise_6_8_1 {G : Type*} [Group G]
(a b : G) : closure ({a, b} : Set G) = Subgroup.closure {b*a*b^2, b*a*b^3}:= sorry | G : Type u_1
inst✝ : Group G
a b : G
⊢ Subgroup.closure {a, b} = Subgroup.closure {b * a * b ^ 2, b * a * b ^ 3} | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$. |
exercise_10_2_4 | valid | /-- Prove that in the ring $\mathbb{Z}[x],(2) \cap(x)=(2 x)$.-/
| theorem exercise_10_2_4 :
span ({2} : Set $ Polynomial ℤ) ⊓ (span {X}) =
span ({2 * X} : Set $ Polynomial ℤ):= sorry | ⊢ Ideal.span {2} ⊓ Ideal.span {X} = Ideal.span {2 * X} | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that in the ring $\mathbb{Z}[x],(2) \cap(x)=(2 x)$. |
exercise_10_4_6 | valid | /-- Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \cap J$ in $R / I J$ is nilpotent.-/
| theorem exercise_10_4_6 {R : Type*} [CommRing R]
[NoZeroDivisors R] (I J : Ideal R) (x : ↑(I ⊓ J)) :
IsNilpotent ((Ideal.Quotient.mk (I*J)) x):= sorry | R : Type u_1
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
I J : Ideal R
x : ↥(I ⊓ J)
⊢ IsNilpotent ((Ideal.Quotient.mk (I * J)) ↑x) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \cap J$ in $R / I J$ is nilpotent. |
exercise_10_7_10 | valid | /-- Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$.-/
| theorem exercise_10_7_10 {R : Type*} [Ring R]
(M : Ideal R) (hM : ∀ (x : R), x ∉ M → IsUnit x)
(hProper : ∃ x : R, x ∉ M) :
IsMaximal M ∧ ∀ (N : Ideal R), IsMaximal N → N = M:= sorry | R : Type u_1
inst✝ : Ring R
M : Ideal R
hM : ∀ x ∉ M, IsUnit x
hProper : ∃ x, x ∉ M
⊢ M.IsMaximal ∧ ∀ (N : Ideal R), N.IsMaximal → N = M | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$. |
exercise_11_4_1b | valid | /-- Prove that $x^3 + 6x + 12$ is irreducible in $\mathbb{Q}$.-/
| theorem exercise_11_4_1b {F : Type*} [Field F] [Fintype F] (hF : card F = 2) :
Irreducible (12 + 6 * X + X ^ 3 : Polynomial F):= sorry | F : Type u_1
inst✝¹ : Field F
inst✝ : Fintype F
hF : card F = 2
⊢ Irreducible (12 + 6 * X + X ^ 3) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that $x^3 + 6x + 12$ is irreducible in $\mathbb{Q}$. |
exercise_11_4_6b | valid | /-- Prove that $x^2+1$ is irreducible in $\mathbb{F}_7$-/
| theorem exercise_11_4_6b {F : Type*} [Field F] [Fintype F] (hF : card F = 31) :
Irreducible (X ^ 3 - 9 : Polynomial F):= sorry | F : Type u_1
inst✝¹ : Field F
inst✝ : Fintype F
hF : card F = 31
⊢ Irreducible (X ^ 3 - 9) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that $x^2+1$ is irreducible in $\mathbb{F}_7$ |
exercise_11_4_8 | valid | /-- Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\mathbb{Q}[x]$.-/
| theorem exercise_11_4_8 (p : ℕ) (hp : Prime p) (n : ℕ) :
-- p ∈ ℕ can be written as p ∈ ℚ[X]
Irreducible (X ^ n - (p : Polynomial ℚ) : Polynomial ℚ):= sorry | p : ℕ
hp : Prime p
n : ℕ
⊢ Irreducible (X ^ n - ↑p) | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\mathbb{Q}[x]$. |
exercise_13_4_10 | valid | /-- Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.-/
| theorem exercise_13_4_10
{p : ℕ} {hp : Nat.Prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) :
∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1:= sorry | p : ℕ
hp : p.Prime
h : ∃ r, p = 2 ^ r + 1
⊢ ∃ k, p = 2 ^ 2 ^ k + 1 | import Mathlib
open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd
open scoped BigOperators
| Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$. |
exercise_1_3 | valid | /-- Prove that $-(-v) = v$ for every $v \in V$.-/
| theorem exercise_1_3 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] {v : V} : -(-v) = v:= sorry | F : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field F
inst✝ : Module F V
v : V
⊢ - -v = v | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Prove that $-(-v) = v$ for every $v \in V$. |
exercise_1_6 | valid | /-- Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), but $U$ is not a subspace of $\mathbf{R}^2$.-/
| theorem exercise_1_6 : ∃ U : Set (ℝ × ℝ),
(U ≠ ∅) ∧
(∀ (u v : ℝ × ℝ), u ∈ U ∧ v ∈ U → u + v ∈ U) ∧
(∀ (u : ℝ × ℝ), u ∈ U → -u ∈ U) ∧
(∀ U' : Submodule ℝ (ℝ × ℝ), U ≠ ↑U'):= sorry | ⊢ ∃ U, U ≠ ∅ ∧ (∀ (u v : ℝ × ℝ), u ∈ U ∧ v ∈ U → u + v ∈ U) ∧ (∀ u ∈ U, -u ∈ U) ∧ ∀ (U' : Submodule ℝ (ℝ × ℝ)), U ≠ ↑U' | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), but $U$ is not a subspace of $\mathbf{R}^2$. |
exercise_1_8 | valid | /-- Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$.-/
| theorem exercise_1_8 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] {ι : Type*} (u : ι → Submodule F V) :
∃ U : Submodule F V, (⋂ (i : ι), (u i).carrier) = ↑U:= sorry | F : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field F
inst✝ : Module F V
ι : Type u_3
u : ι → Submodule F V
⊢ ∃ U, ⋂ i, (u i).carrier = ↑U | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$. |
exercise_3_1 | valid | /-- Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname{dim} V=1$ and $T \in \mathcal{L}(V, V)$, then there exists $a \in \mathbf{F}$ such that $T v=a v$ for all $v \in V$.-/
| theorem exercise_3_1 {F V : Type*}
[AddCommGroup V] [Field F] [Module F V] [FiniteDimensional F V]
(T : V →ₗ[F] V) (hT : finrank F V = 1) :
∃ c : F, ∀ v : V, T v = c • v:= sorry | F : Type u_1
V : Type u_2
inst✝³ : AddCommGroup V
inst✝² : Field F
inst✝¹ : Module F V
inst✝ : FiniteDimensional F V
T : V →ₗ[F] V
hT : finrank F V = 1
⊢ ∃ c, ∀ (v : V), T v = c • v | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname{dim} V=1$ and $T \in \mathcal{L}(V, V)$, then there exists $a \in \mathbf{F}$ such that $T v=a v$ for all $v \in V$. |
exercise_4_4 | valid | /-- Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p^{\prime}$ have no roots in common.-/
| theorem exercise_4_4 (p : Polynomial ℂ) :
p.degree = @card (rootSet p ℂ) (rootSetFintype p ℂ) ↔
Disjoint
(@card (rootSet (derivative p) ℂ) (rootSetFintype (derivative p) ℂ))
(@card (rootSet p ℂ) (rootSetFintype p ℂ)):= sorry | p : ℂ[X]
⊢ p.degree = ↑(card ↑(p.rootSet ℂ)) ↔ Disjoint (card ↑((derivative p).rootSet ℂ)) (card ↑(p.rootSet ℂ)) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $p \in \mathcal{P}(\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p^{\prime}$ have no roots in common. |
exercise_5_4 | valid | /-- Suppose that $S, T \in \mathcal{L}(V)$ are such that $S T=T S$. Prove that $\operatorname{null} (T-\lambda I)$ is invariant under $S$ for every $\lambda \in \mathbf{F}$.-/
| theorem exercise_5_4 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] (S T : V →ₗ[F] V) (hST : S ∘ T = T ∘ S) (c : F):
Submodule.map S (ker (T - c • LinearMap.id)) = ker (T - c • LinearMap.id):= sorry | F : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field F
inst✝ : Module F V
S T : V →ₗ[F] V
hST : ⇑S ∘ ⇑T = ⇑T ∘ ⇑S
c : F
⊢ Submodule.map S (ker (T - c • LinearMap.id)) = ker (T - c • LinearMap.id) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose that $S, T \in \mathcal{L}(V)$ are such that $S T=T S$. Prove that $\operatorname{null} (T-\lambda I)$ is invariant under $S$ for every $\lambda \in \mathbf{F}$. |
exercise_5_12 | valid | /-- Suppose $T \in \mathcal{L}(V)$ is such that every vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity operator.-/
| theorem exercise_5_12 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] {S : End F V}
(hS : ∀ v : V, ∃ c : F, v ∈ eigenspace S c) :
∃ c : F, S = c • LinearMap.id:= sorry | F : Type u_1
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Field F
inst✝ : Module F V
S : End F V
hS : ∀ (v : V), ∃ c, v ∈ S.eigenspace c
⊢ ∃ c, S = c • LinearMap.id | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $T \in \mathcal{L}(V)$ is such that every vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity operator. |
exercise_5_20 | valid | /-- Suppose that $T \in \mathcal{L}(V)$ has $\operatorname{dim} V$ distinct eigenvalues and that $S \in \mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $S T=T S$.-/
| theorem exercise_5_20 {F V : Type*} [AddCommGroup V] [Field F]
[Module F V] [FiniteDimensional F V] {S T : End F V}
(h1 : card (T.Eigenvalues) = finrank F V)
(h2 : ∀ v : V, ∃ c : F, v ∈ eigenspace S c ↔ ∃ c : F, v ∈ eigenspace T c) :
S * T = T * S:= sorry | F : Type u_1
V : Type u_2
inst✝³ : AddCommGroup V
inst✝² : Field F
inst✝¹ : Module F V
inst✝ : FiniteDimensional F V
S T : End F V
h1 : card T.Eigenvalues = finrank F V
h2 : ∀ (v : V), ∃ c, v ∈ S.eigenspace c ↔ ∃ c, v ∈ T.eigenspace c
⊢ S * T = T * S | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose that $T \in \mathcal{L}(V)$ has $\operatorname{dim} V$ distinct eigenvalues and that $S \in \mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $S T=T S$. |
exercise_6_2 | valid | /-- Suppose $u, v \in V$. Prove that $\langle u, v\rangle=0$ if and only if $\|u\| \leq\|u+a v\|$ for all $a \in \mathbf{F}$.-/
| theorem exercise_6_2 {V : Type*} [NormedAddCommGroup V] [Module ℂ V]
[InnerProductSpace ℂ V] (u v : V) :
⟪u, v⟫_ℂ = 0 ↔ ∀ (a : ℂ), ‖u‖ ≤ ‖u + a • v‖:= sorry | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : Module ℂ V
inst✝ : InnerProductSpace ℂ V
u v : V
⊢ ⟪u, v⟫_ℂ = 0 ↔ ∀ (a : ℂ), ‖u‖ ≤ ‖u + a • v‖ | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $u, v \in V$. Prove that $\langle u, v\rangle=0$ if and only if $\|u\| \leq\|u+a v\|$ for all $a \in \mathbf{F}$. |
exercise_6_7 | valid | /-- Prove that if $V$ is a complex inner-product space, then $\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$ for all $u, v \in V$.-/
| theorem exercise_6_7 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] (u v : V) :
⟪u, v⟫_ℂ = (‖u + v‖^2 - ‖u - v‖^2 + I*‖u + I•v‖^2 - I*‖u-I•v‖^2) / 4:= sorry | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℂ V
u v : V
⊢ ⟪u, v⟫_ℂ = (↑‖u + v‖ ^ 2 - ↑‖u - v‖ ^ 2 + I * ↑‖u + I • v‖ ^ 2 - I * ↑‖u - I • v‖ ^ 2) / 4 | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Prove that if $V$ is a complex inner-product space, then $\langle u, v\rangle=\frac{\|u+v\|^{2}-\|u-v\|^{2}+\|u+i v\|^{2} i-\|u-i v\|^{2} i}{4}$ for all $u, v \in V$. |
exercise_6_16 | valid | /-- Suppose $U$ is a subspace of $V$. Prove that $U^{\perp}=\{0\}$ if and only if $U=V$-/
| theorem exercise_6_16 {K V : Type*} [RCLike K] [NormedAddCommGroup V] [InnerProductSpace K V]
{U : Submodule K V} :
U.orthogonal = ⊥ ↔ U = ⊤:= sorry | K : Type u_1
V : Type u_2
inst✝² : RCLike K
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace K V
U : Submodule K V
⊢ Uᗮ = ⊥ ↔ U = ⊤ | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $U$ is a subspace of $V$. Prove that $U^{\perp}=\{0\}$ if and only if $U=V$ |
exercise_7_6 | valid | /-- Prove that if $T \in \mathcal{L}(V)$ is normal, then $\operatorname{range} T=\operatorname{range} T^{*}.$-/
| theorem exercise_7_6 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (T : End ℂ V)
(hT : T * adjoint T = adjoint T * T) :
range T = range (adjoint T):= sorry | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℂ V
inst✝ : FiniteDimensional ℂ V
T : End ℂ V
hT : T * adjoint T = adjoint T * T
⊢ range T = range (adjoint T) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Prove that if $T \in \mathcal{L}(V)$ is normal, then $\operatorname{range} T=\operatorname{range} T^{*}.$ |
exercise_7_10 | valid | /-- Suppose $V$ is a complex inner-product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$.-/
| theorem exercise_7_10 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (T : End ℂ V)
(hT : T * adjoint T = adjoint T * T) (hT1 : T^9 = T^8) :
IsSelfAdjoint T ∧ T^2 = T:= sorry | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℂ V
inst✝ : FiniteDimensional ℂ V
T : End ℂ V
hT : T * adjoint T = adjoint T * T
hT1 : T ^ 9 = T ^ 8
⊢ IsSelfAdjoint T ∧ T ^ 2 = T | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $V$ is a complex inner-product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$. |
exercise_7_14 | valid | /-- Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon>0$. Prove that if there exists $v \in V$ such that $\|v\|=1$ and $\|T v-\lambda v\|<\epsilon,$ then $T$ has an eigenvalue $\lambda^{\prime}$ such that $\left|\lambda-\lambda^{\prime}\right|<\epsilon$.-/
| theorem exercise_7_14 {𝕜 V : Type*} [RCLike 𝕜] [NormedAddCommGroup V]
[InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V]
{T : Module.End 𝕜 V} (hT : IsSelfAdjoint T)
{l : 𝕜} {ε : ℝ} (he : ε > 0) : ∃ v : V, ‖v‖= 1 ∧ (‖T v - l • v‖ < ε →
(∃ l' : T.Eigenvalues, ‖l - l'‖ < ε)):= sorry | 𝕜 : Type u_1
V : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : FiniteDimensional 𝕜 V
T : End 𝕜 V
hT : IsSelfAdjoint T
l : 𝕜
ε : ℝ
he : ε > 0
⊢ ∃ v, ‖v‖ = 1 ∧ (‖T v - l • v‖ < ε → ∃ l', ‖l - ↑T l'‖ < ε) | import Mathlib
open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End
open scoped BigOperators
| Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon>0$. Prove that if there exists $v \in V$ such that $\|v\|=1$ and $\|T v-\lambda v\|<\epsilon,$ then $T$ has an eigenvalue $\lambda^{\prime}$ such that $\left|\lambda-\lambda^{\prime}\right|<\epsilon$. |
exercise_1_1_3 | valid | /-- Prove that the addition of residue classes $\mathbb{Z}/n\mathbb{Z}$ is associative.-/
| theorem exercise_1_1_3 (n : ℤ) :
∀ (a b c : ℤ), (a+b)+c ≡ a+(b+c) [ZMOD n]:= sorry | n : ℤ
⊢ ∀ (a b c : ℤ), a + b + c ≡ a + (b + c) [ZMOD n] | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| Prove that the addition of residue classes $\mathbb{Z}/n\mathbb{Z}$ is associative. |
exercise_1_1_5 | valid | /-- Prove that for all $n>1$ that $\mathbb{Z}/n\mathbb{Z}$ is not a group under multiplication of residue classes.-/
| theorem exercise_1_1_5 (n : ℕ) (hn : 1 < n) :
IsEmpty (Group (ZMod n)):= sorry | n : ℕ
hn : 1 < n
⊢ IsEmpty (Group (ZMod n)) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| Prove that for all $n>1$ that $\mathbb{Z}/n\mathbb{Z}$ is not a group under multiplication of residue classes. |
exercise_1_1_16 | valid | /-- Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.-/
| theorem exercise_1_1_16 {G : Type*} [Group G]
(x : G) (hx : x ^ 2 = 1) :
orderOf x = 1 ∨ orderOf x = 2:= sorry | G : Type u_1
inst✝ : Group G
x : G
hx : x ^ 2 = 1
⊢ orderOf x = 1 ∨ orderOf x = 2 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$. |
exercise_1_1_18 | valid | /-- Let $x$ and $y$ be elements of $G$. Prove that $xy=yx$ if and only if $y^{-1}xy=x$ if and only if $x^{-1}y^{-1}xy=1$.-/
| theorem exercise_1_1_18 {G : Type*} [Group G]
(x y : G) : (x * y = y * x ↔ y⁻¹ * x * y = x) ↔ (x⁻¹ * y⁻¹ * x * y = 1):= sorry | G : Type u_1
inst✝ : Group G
x y : G
⊢ (x * y = y * x ↔ y⁻¹ * x * y = x) ↔ x⁻¹ * y⁻¹ * x * y = 1 | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| Let $x$ and $y$ be elements of $G$. Prove that $xy=yx$ if and only if $y^{-1}xy=x$ if and only if $x^{-1}y^{-1}xy=1$. |
exercise_1_1_22a | valid | /-- If $x$ and $g$ are elements of the group $G$, prove that $|x|=\left|g^{-1} x g\right|$.-/
| theorem exercise_1_1_22a {G : Type*} [Group G] (x g : G) :
orderOf x = orderOf (g⁻¹ * x * g):= sorry | G : Type u_1
inst✝ : Group G
x g : G
⊢ orderOf x = orderOf (g⁻¹ * x * g) | import Mathlib
open Fintype Subgroup Set Polynomial Ideal
open scoped BigOperators
| If $x$ and $g$ are elements of the group $G$, prove that $|x|=\left|g^{-1} x g\right|$. |
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