Datasets:

HaimingW

/

proofnet-lean4

like 0

exercise_1_13b

test

/-- Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Im}(f)$ is constant, then $f$ is constant.-/

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 :=

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

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_1_19a

test

/-- Prove that the power series $\sum nz^n$ does not converge on any point of the unit circle.-/

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) :=

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)

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_1_19c

test

/-- Prove that the power series $\sum zn/n$ converges at every point of the unit circle except $z = 1$.-/

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) :=

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)

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_2_2

test

/-- Show that $\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$.-/

theorem exercise_2_2 : Tendsto (λ y => ∫ x in (0 : ℝ)..y, Real.sin x / x) atTop (𝓝 (Real.pi / 2)) :=

⊢ Tendsto (fun y => ∫ (x : ℝ) in 0 ..y, x.sin / x) atTop (𝓝 (Real.pi / 2))

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_2_13

test

/-- Suppose $f$ is an analytic function defined everywhere in $\mathbb{C}$ and such that for each $z_0 \in \mathbb{C}$ at least one coefficient in the expansion $f(z) = \sum_{n=0}^\infty c_n(z - z_0)^n$ is equal to 0. Prove that $f$ is a polynomial.-/

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 :=

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

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_3_4

test

/-- Show that $ \int_{-\infty}^{\infty} \frac{x \sin x}{x^2 + a^2} dx = \pi e^{-a}$ for $a > 0$.-/

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)))) :=

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))

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_3_14

test

/-- Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \in \mathbb{C}$ and $a \neq 0$.-/

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 :=

f : ℂ → ℂ hf : Differentiable ℂ f hf_inj : Injective f ⊢ ∃ a b, (f = fun z => a * z + b) ∧ a ≠ 0

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_5_1

test

/-- Prove that if $f$ is holomorphic in the unit disc, bounded and not identically zero, and $z_{1}, z_{2}, \ldots, z_{n}, \ldots$ are its zeros $\left(\left|z_{k}\right|<1\right)$, then $\sum_{n}\left(1-\left|z_{n}\right|\right)<\infty$.-/

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))) atTop (𝓝 z) :=

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)

import Mathlib open Complex Filter Function Metric Finset open scoped BigOperators Topology

exercise_1_1b

test

/-- If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.-/

theorem exercise_1_1b (x : ℝ) (y : ℚ) (h : y ≠ 0) : ( Irrational x ) -> Irrational ( x * y ) :=

x : ℝ y : ℚ h : y ≠ 0 ⊢ Irrational x → Irrational (x * ↑y)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_4

test

/-- Let $E$ be a nonempty subset of an ordered set; suppose $\alpha$ is a lower bound of $E$ and $\beta$ is an upper bound of $E$. Prove that $\alpha \leq \beta$.-/

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 :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_8

test

/-- Prove that no order can be defined in the complex field that turns it into an ordered field.-/

theorem exercise_1_8 : ¬ ∃ (r : ℂ → ℂ → Prop), IsLinearOrder ℂ r :=

⊢ ¬∃ r, IsLinearOrder ℂ r

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_12

test

/-- If $z_1, \ldots, z_n$ are complex, prove that $|z_1 + z_2 + \ldots + z_n| \leq |z_1| + |z_2| + \cdots + |z_n|$.-/

theorem exercise_1_12 (n : ℕ) (f : ℕ → ℂ) : abs (∑ i in range n, f i) ≤ ∑ i in range n, abs (f i) :=

n : ℕ f : ℕ → ℂ ⊢ Complex.abs (∑ i ∈ range n, f i) ≤ ∑ i ∈ range n, Complex.abs (f i)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_14

test

/-- If $z$ is a complex number such that $|z|=1$, that is, such that $z \bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$.-/

theorem exercise_1_14 (z : ℂ) (h : abs z = 1) : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=

z : ℂ h : Complex.abs z = 1 ⊢ Complex.abs (1 + z) ^ 2 + Complex.abs (1 - z) ^ 2 = 4

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_17

test

/-- Prove that $|\mathbf{x}+\mathbf{y}|^{2}+|\mathbf{x}-\mathbf{y}|^{2}=2|\mathbf{x}|^{2}+2|\mathbf{y}|^{2}$ if $\mathbf{x} \in R^{k}$ and $\mathbf{y} \in R^{k}$.-/

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 :=

n : ℕ x y : EuclideanSpace ℝ (Fin n) ⊢ ‖x + y‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * ‖x‖ ^ 2 + 2 * ‖y‖ ^ 2

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_1_18b

test

/-- If $k = 1$ and $\mathbf{x} \in R^{k}$, prove that there does not exist $\mathbf{y} \in R^{k}$ such that $\mathbf{y} \neq 0$ but $\mathbf{x} \cdot \mathbf{y}=0$-/

theorem exercise_1_18b : ¬ ∀ (x : ℝ), ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0 :=

⊢ ¬∀ (x : ℝ), ∃ y, y ≠ 0 ∧ x * y = 0

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_2_19a

test

/-- If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.-/

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 :=

X : Type u_1 inst✝ : MetricSpace X A B : Set X hA : IsClosed A hB : IsClosed B hAB : Disjoint A B ⊢ SeparatedNhds A B

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_2_25

test

/-- Prove that every compact metric space $K$ has a countable base.-/

theorem exercise_2_25 {K : Type*} [MetricSpace K] [CompactSpace K] : ∃ (B : Set (Set K)), Set.Countable B ∧ IsTopologicalBasis B :=

K : Type u_1 inst✝¹ : MetricSpace K inst✝ : CompactSpace K ⊢ ∃ B, B.Countable ∧ IsTopologicalBasis B

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_2_27b

test

/-- Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that at most countably many points of $E$ are not in $P$.-/

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) :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_2_29

test

/-- Prove that every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments.-/

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 :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_3_2a

test

/-- Prove that $\lim_{n \rightarrow \infty}\sqrt{n^2 + n} -n = 1/2$.-/

theorem exercise_3_2a : Tendsto (λ (n : ℝ) => (sqrt (n^2 + n) - n)) atTop (𝓝 (1/2)) :=

⊢ Tendsto (fun n => √(n ^ 2 + n) - n) atTop (𝓝 (1 / 2))

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_3_5

test

/-- For any two real sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$, prove that $\limsup _{n \rightarrow \infty}\left(a_{n}+b_{n}\right) \leq \limsup _{n \rightarrow \infty} a_{n}+\limsup _{n \rightarrow \infty} b_{n},$ provided the sum on the right is not of the form $\infty-\infty$.-/

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 :=

a b : ℕ → ℝ h : limsup a + limsup b ≠ 0 ⊢ (limsup fun n => a n + b n) ≤ limsup a + limsup b

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_3_7

test

/-- Prove that the convergence of $\Sigma a_{n}$ implies the convergence of $\sum \frac{\sqrt{a_{n}}}{n}$ if $a_n\geq 0$.-/

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) :=

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)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_3_13

test

/-- Prove that the Cauchy product of two absolutely convergent series converges absolutely.-/

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)) :=

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)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_3_21

test

/-- If $\left\{E_{n}\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \supset E_{n+1}$, and if $\lim _{n \rightarrow \infty} \operatorname{diam} E_{n}=0,$ then $\bigcap_{1}^{\infty} E_{n}$ consists of exactly one point.-/

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} :=

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}

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_1a

test

/-- Suppose $f$ is a real function defined on $\mathbb{R}$ which satisfies $\lim_{h \rightarrow 0} f(x + h) - f(x - h) = 0$ for every $x \in \mathbb{R}$. Show that $f$ does not need to be continuous.-/

theorem exercise_4_1a : ∃ (f : ℝ → ℝ), (∀ (x : ℝ), Tendsto (λ y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬ Continuous f :=

⊢ ∃ f, (∀ (x : ℝ), Tendsto (fun y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬Continuous f

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_3

test

/-- Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$ ) be the set of all $p \in X$ at which $f(p)=0$. Prove that $Z(f)$ is closed.-/

theorem exercise_4_3 {α : Type} [MetricSpace α] (f : α → ℝ) (h : Continuous f) (z : Set α) (g : z = f⁻¹' {0}) : IsClosed z :=

α : Type inst✝ : MetricSpace α f : α → ℝ h : Continuous f z : Set α g : z = f ⁻¹' {0} ⊢ IsClosed z

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_4b

test

/-- 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 if $g(p) = f(p)$ for all $p \in P$ then $g(p) = f(p)$ for all $p \in X$.-/

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 :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_5b

test

/-- Show that there exist a set $E \subset \mathbb{R}$ and a real continuous function $f$ defined on $E$, such that there does not exist a continuous real function $g$ on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x \in E$.-/

theorem exercise_4_5b : ∃ (E : Set ℝ) (f : ℝ → ℝ), (ContinuousOn f E) ∧ (¬ ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x) :=

⊢ ∃ E f, ContinuousOn f E ∧ ¬∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_8a

test

/-- Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.-/

theorem exercise_4_8a (E : Set ℝ) (f : ℝ → ℝ) (hf : UniformContinuousOn f E) (hE : Bornology.IsBounded E) : Bornology.IsBounded (Set.image f E) :=

E : Set ℝ f : ℝ → ℝ hf : UniformContinuousOn f E hE : Bornology.IsBounded E ⊢ Bornology.IsBounded (f '' E)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_11a

test

/-- Suppose $f$ is a uniformly continuous mapping of a metric space $X$ into a metric space $Y$ and prove that $\left\{f\left(x_{n}\right)\right\}$ is a Cauchy sequence in $Y$ for every Cauchy sequence $\{x_n\}$ in $X$.-/

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)) :=

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)

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_15

test

/-- Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic.-/

theorem exercise_4_15 {f : ℝ → ℝ} (hf : Continuous f) (hof : IsOpenMap f) : Monotone f :=

f : ℝ → ℝ hf : Continuous f hof : IsOpenMap f ⊢ Monotone f

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_4_21a

test

/-- Suppose $K$ and $F$ are disjoint sets in a metric space $X, K$ is compact, $F$ is closed. Prove that there exists $\delta>0$ such that $d(p, q)>\delta$ if $p \in K, q \in F$.-/

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 :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_5_1

test

/-- Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant.-/

theorem exercise_5_1 {f : ℝ → ℝ} (hf : ∀ x y : ℝ, |(f x - f y)| ≤ (x - y) ^ 2) : ∃ c, f = λ x => c :=

f : ℝ → ℝ hf : ∀ (x y : ℝ), |f x - f y| ≤ (x - y) ^ 2 ⊢ ∃ c, f = fun x => c

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_5_3

test

/-- Suppose $g$ is a real function on $R^{1}$, with bounded derivative (say $\left|g^{\prime}\right| \leq M$ ). Fix $\varepsilon>0$, and define $f(x)=x+\varepsilon g(x)$. Prove that $f$ is one-to-one if $\varepsilon$ is small enough.-/

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) :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_5_5

test

/-- Suppose $f$ is defined and differentiable for every $x>0$, and $f^{\prime}(x) \rightarrow 0$ as $x \rightarrow+\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x) \rightarrow 0$ as $x \rightarrow+\infty$.-/

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 :=

f : ℝ → ℝ hfd : Differentiable ℝ f hf : Tendsto (deriv f) atTop (𝓝 0) ⊢ Tendsto (fun x => f (x + 1) - f x) atTop atTop

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_5_7

test

/-- Suppose $f^{\prime}(x), g^{\prime}(x)$ exist, $g^{\prime}(x) \neq 0$, and $f(x)=g(x)=0$. Prove that $\lim _{t \rightarrow x} \frac{f(t)}{g(t)}=\frac{f^{\prime}(x)}{g^{\prime}(x)}.$-/

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)) :=

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))

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_5_17

test

/-- Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that $f(-1)=0, \quad f(0)=0, \quad f(1)=1, \quad f^{\prime}(0)=0 .$ Prove that $f^{(3)}(x) \geq 3$ for some $x \in(-1,1)$.-/

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 (-1 : ℝ) 1 ∧ deriv (deriv (deriv f)) x ≥ 3 :=

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

import Mathlib open Topology Filter Real Complex TopologicalSpace Finset open scoped BigOperators

exercise_2_1_18

test

/-- If $G$ is a finite group of even order, show that there must be an element $a \neq e$ such that $a=a^{-1}$.-/

theorem exercise_2_1_18 {G : Type*} [Group G] [Fintype G] (hG2 : Even (card G)) : ∃ (a : G), a ≠ 1 ∧ a = a⁻¹ :=

G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG2 : Even (card G) ⊢ ∃ a, a ≠ 1 ∧ a = a⁻¹

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_1_26

test

/-- If $G$ is a finite group, prove that, given $a \in G$, there is a positive integer $n$, depending on $a$, such that $a^n = e$.-/

theorem exercise_2_1_26 {G : Type*} [Group G] [Fintype G] (a : G) : ∃ (n : ℕ), a ^ n = 1 :=

G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G a : G ⊢ ∃ n, a ^ n = 1

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_2_3

test

/-- If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_2_6c

test

/-- Let $G$ be a group in which $(a b)^{n}=a^{n} b^{n}$ for some fixed integer $n>1$ for all $a, b \in G$. For all $a, b \in G$, prove that $\left(a b a^{-1} b^{-1}\right)^{n(n-1)}=e$.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_3_16

test

/-- If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.-/

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 :=

G : Type u_1 inst✝ : Group G hG : ∀ (H : Subgroup G), H = ⊤ ∨ H = ⊥ ⊢ IsCyclic G ∧ ∃ p Fin, p.Prime ∧ card G = p

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_5_23

test

/-- Let $G$ be a group such that all subgroups of $G$ are normal in $G$. If $a, b \in G$, prove that $ba = a^jb$ for some $j$.-/

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 :=

G : Type u_1 inst✝ : Group G hG : ∀ (H : Subgroup G), H.Normal a b : G ⊢ ∃ j, b * a = a ^ j * b

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_5_31

test

/-- Suppose that $G$ is an abelian group of order $p^nm$ where $p \nmid m$ is a prime. If $H$ is a subgroup of $G$ of order $p^n$, prove that $H$ is a characteristic subgroup of $G$.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_5_43

test

/-- Prove that a group of order 9 must be abelian.-/

def exercise_2_5_43 (G : Type*) [Group G] [Fintype G] (hG : card G = 9) : CommGroup G :=

G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG : card G = 9 ⊢ CommGroup G

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_5_52

test

/-- Let $G$ be a finite group and $\varphi$ an automorphism of $G$ such that $\varphi(x) = x^{-1}$ for more than three-fourths of the elements of $G$. Prove that $\varphi(y) = y^{-1}$ for all $y \in G$, and so $G$ is abelian.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_7_7

test

/-- If $\varphi$ is a homomorphism of $G$ onto $G'$ and $N \triangleleft G$, show that $\varphi(N) \triangleleft G'$.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_8_15

test

/-- Prove that if $p > q$ are two primes such that $q \mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic.-/

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 :=

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

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_10_1

test

/-- Let $A$ be a normal subgroup of a group $G$, and suppose that $b \in G$ is an element of prime order $p$, and that $b \not\in A$. Show that $A \cap (b) = (e)$.-/

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}) = ⊥ :=

G : Type u_1 inst✝¹ : Group G A : Subgroup G inst✝ : A.Normal b : G hp : (orderOf b).Prime ⊢ A ⊓ Subgroup.closure {b} = ⊥

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_11_7

test

/-- If $P \triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\varphi(P) = P$ for every automorphism $\varphi$ of $G$.-/

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) :=

G : Type u_1 inst✝ : Group G p : ℕ hp : p.Prime P : Sylow p G hP : (↑P).Normal ⊢ (↑P).Characteristic

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_3_2_21

test

/-- If $\sigma, \tau$ are two permutations that disturb no common element and $\sigma \tau = e$, prove that $\sigma = \tau = e$.-/

theorem exercise_3_2_21 {α : Type*} [Fintype α] {σ τ: Equiv.Perm α} (h1 : ∀ a : α, σ a = a ↔ τ a ≠ a) (h2 : τ ∘ σ = id) : σ = 1 ∧ τ = 1 :=

α : Type u_1 inst✝ : Fintype α σ τ : Equiv.Perm α h1 : ∀ (a : α), σ a = a ↔ τ a ≠ a h2 : ⇑τ ∘ ⇑σ = id ⊢ σ = 1 ∧ τ = 1

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_1_34

test

/-- Let $T$ be the group of $2\times 2$ matrices $A$ with entries in the field $\mathbb{Z}_2$ such that $\det A$ is not equal to 0. Prove that $T$ is isomorphic to $S_3$, the symmetric group of degree 3.-/

def exercise_4_1_34 : Equiv.Perm (Fin 3) ≃* Matrix.GeneralLinearGroup (Fin 2) (ZMod 2) :=

⊢ Equiv.Perm (Fin 3) ≃* GL (Fin 2) (ZMod 2)

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_2_6

test

/-- If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$.-/

theorem exercise_4_2_6 {R : Type*} [Ring R] (a x : R) (h : a ^ 2 = 0) : a * (a * x + x * a) = (x + x * a) * a :=

R : Type u_1 inst✝ : Ring R a x : R h : a ^ 2 = 0 ⊢ a * (a * x + x * a) = (x + x * a) * a

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_3_1

test

/-- If $R$ is a commutative ring and $a \in R$, let $L(a) = \{x \in R \mid xa = 0\}$. Prove that $L(a)$ is an ideal of $R$.-/

theorem exercise_4_3_1 {R : Type*} [CommRing R] (a : R) : ∃ I : Ideal R, {x : R | x*a=0} = I :=

R : Type u_1 inst✝ : CommRing R a : R ⊢ ∃ I, {x | x * a = 0} = ↑I

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_4_9

test

/-- Show that $(p - 1)/2$ of the numbers $1, 2, \ldots, p - 1$ are quadratic residues and $(p - 1)/2$ are quadratic nonresidues $\mod p$.-/

theorem exercise_4_4_9 (p : ℕ) (hp : Nat.Prime p) : (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ∃ x : ZMod p, x^2 = p) ∧ (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ¬ ∃ x : ZMod p, x^2 = p) :=

p : ℕ hp : p.Prime ⊢ (∃ S, S.card = (p - 1) / 2 ∧ ∃ x, x ^ 2 = ↑p) ∧ ∃ S, S.card = (p - 1) / 2 ∧ ¬∃ x, x ^ 2 = ↑p

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_5_23

test

/-- Let $F = \mathbb{Z}_7$ and let $p(x) = x^3 - 2$ and $q(x) = x^3 + 2$ be in $F[x]$. Show that $p(x)$ and $q(x)$ are irreducible in $F[x]$ and that the fields $F[x]/(p(x))$ and $F[x]/(q(x))$ are isomorphic.-/

theorem exercise_4_5_23 {p q: Polynomial (ZMod 7)} (hp : p = X^3 - 2) (hq : q = X^3 + 2) : Irreducible p ∧ Irreducible q ∧ (Nonempty $ Polynomial (ZMod 7) ⧸ span ({p} : Set $ Polynomial $ ZMod 7) ≃+* Polynomial (ZMod 7) ⧸ span ({q} : Set $ Polynomial $ ZMod 7)) :=

p q : (ZMod 7)[X] hp : p = X ^ 3 - 2 hq : q = X ^ 3 + 2 ⊢ Irreducible p ∧ Irreducible q ∧ Nonempty ((ZMod 7)[X] ⧸ span {p} ≃+* (ZMod 7)[X] ⧸ span {q})

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_4_6_2

test

/-- Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$.-/

theorem exercise_4_6_2 : Irreducible (X^3 + 3*X + 2 : Polynomial ℚ) :=

⊢ Irreducible (X ^ 3 + 3 * X + 2)

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_5_1_8

test

/-- If $F$ is a field of characteristic $p \neq 0$, show that $(a + b)^m = a^m + b^m$, where $m = p^n$, for all $a, b \in F$ and any positive integer $n$.-/

theorem exercise_5_1_8 {p m n: ℕ} {F : Type*} [Field F] (hp : Nat.Prime p) (hF : CharP F p) (a b : F) (hm : m = p ^ n) : (a + b) ^ m = a^m + b^m :=

p m n : ℕ F : Type u_1 inst✝ : Field F hp : p.Prime hF : CharP F p a b : F hm : m = p ^ n ⊢ (a + b) ^ m = a ^ m + b ^ m

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_5_3_7

test

/-- If $a \in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$.-/

theorem exercise_5_3_7 {K : Type*} [Field K] {F : Subfield K} {a : K} (ha : IsAlgebraic F (a ^ 2)) : IsAlgebraic F a :=

K : Type u_1 inst✝ : Field K F : Subfield K a : K ha : IsAlgebraic (↥F) (a ^ 2) ⊢ IsAlgebraic (↥F) a

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_5_4_3

test

/-- If $a \in C$ is such that $p(a) = 0$, where $p(x) = x^5 + \sqrt{2}x^3 + \sqrt{5}x^2 + \sqrt{7}x + \sqrt{11}$, show that $a$ is algebraic over $\mathbb{Q}$ of degree at most 80.-/

theorem exercise_5_4_3 {a : ℂ} {p : ℂ → ℂ} (hp : p = λ (x : ℂ) => x^5 + sqrt 2 * x^3 + sqrt 5 * x^2 + sqrt 7 * x + 11) (ha : p a = 0) : ∃ p : Polynomial ℂ , p.degree < 80 ∧ a ∈ p.roots ∧ ∀ n : p.support, ∃ a b : ℤ, p.coeff n = a / b :=

a : ℂ p : ℂ → ℂ hp : p = fun x => x ^ 5 + ↑√2 * x ^ 3 + ↑√5 * x ^ 2 + ↑√7 * x + 11 ha : p a = 0 ⊢ ∃ p, p.degree < 80 ∧ a ∈ p.roots ∧ ∀ (n : { x // x ∈ p.support }), ∃ a b, p.coeff ↑n = ↑a / ↑b

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_5_6_14

test

/-- If $F$ is of characteristic $p \neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct.-/

theorem exercise_5_6_14 {p m n: ℕ} (hp : Nat.Prime p) {F : Type*} [Field F] [CharP F p] (hm : m = p ^ n) : card (rootSet (X ^ m - X : Polynomial F) F) = m :=

p m n : ℕ hp : p.Prime F : Type u_1 inst✝¹ : Field F inst✝ : CharP F p hm : m = p ^ n ⊢ card ↑((X ^ m - X).rootSet F) = m

import Mathlib open Fintype Set Real Ideal Polynomial open scoped BigOperators

exercise_2_26

test

/-- Prove that a set $U \subset M$ is open if and only if none of its points are limits of its complement.-/

theorem exercise_2_26 {M : Type*} [TopologicalSpace M] (U : Set M) : IsOpen U ↔ ∀ x ∈ U, ¬ ClusterPt x (𝓟 Uᶜ) :=

M : Type u_1 inst✝ : TopologicalSpace M U : Set M ⊢ IsOpen U ↔ ∀ x ∈ U, ¬ClusterPt x (𝓟 Uᶜ)

import Mathlib open Filter Real Function open scoped Topology

exercise_2_32a

test

/-- Show that every subset of $\mathbb{N}$ is clopen.-/

theorem exercise_2_32a (A : Set ℕ) : IsClopen A :=

A : Set ℕ ⊢ IsClopen A

import Mathlib open Filter Real Function open scoped Topology

exercise_2_46

test

/-- Assume that $A, B$ are compact, disjoint, nonempty subsets of $M$. Prove that there are $a_0 \in A$ and $b_0 \in B$ such that for all $a \in A$ and $b \in B$ we have $d(a_0, b_0) \leq d(a, b)$.-/

theorem exercise_2_46 {M : Type*} [MetricSpace M] {A B : Set M} (hA : IsCompact A) (hB : IsCompact B) (hAB : Disjoint A B) (hA₀ : A ≠ ∅) (hB₀ : B ≠ ∅) : ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a : M) (b : M), a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b :=

M : Type u_1 inst✝ : MetricSpace M A B : Set M hA : IsCompact A hB : IsCompact B hAB : Disjoint A B hA₀ : A ≠ ∅ hB₀ : B ≠ ∅ ⊢ ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a b : M), a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b

import Mathlib open Filter Real Function open scoped Topology

exercise_2_92

test

/-- Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty.-/

theorem exercise_2_92 {α : Type*} [TopologicalSpace α] {s : ℕ → Set α} (hs : ∀ i, IsCompact (s i)) (hs : ∀ i, (s i).Nonempty) (hs : ∀ i, (s i) ⊃ (s (i + 1))) : (⋂ i, s i).Nonempty :=

α : Type u_1 inst✝ : TopologicalSpace α s : ℕ → Set α hs✝¹ : ∀ (i : ℕ), IsCompact (s i) hs✝ : ∀ (i : ℕ), (s i).Nonempty hs : ∀ (i : ℕ), s i ⊃ s (i + 1) ⊢ (⋂ i, s i).Nonempty

import Mathlib open Filter Real Function open scoped Topology

exercise_3_1

test

/-- Assume that $f \colon \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t)-f(x)| \leq|t-x|^{2}$ for all $t, x$. Prove that $f$ is constant.-/

theorem exercise_3_1 {f : ℝ → ℝ} (hf : ∀ x y, |f x - f y| ≤ |x - y| ^ 2) : ∃ c, f = λ x => c :=

f : ℝ → ℝ hf : ∀ (x y : ℝ), |f x - f y| ≤ |x - y| ^ 2 ⊢ ∃ c, f = fun x => c

import Mathlib open Filter Real Function open scoped Topology

exercise_3_63a

test

/-- Prove that $\sum 1/k(\log(k))^p$ converges when $p > 1$.-/

theorem exercise_3_63a (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) :=

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

exercise_4_15a

test

/-- A continuous, strictly increasing function $\mu \colon (0, \infty) \rightarrow (0, \infty)$ is a modulus of continuity if $\mu(s) \rightarrow 0$ as $s \rightarrow 0$. A function $f \colon [a, b] \rightarrow \mathbb{R}$ has modulus of continuity $\mu$ if $|f(s) - f(t)| \leq \mu(|s - t|)$ for all $s, t \in [a, b]$. Prove that a function is uniformly continuous if and only if it has a modulus of continuity.-/

theorem exercise_4_15a {α : Type*} (a b : ℝ) (F : Set (ℝ → ℝ)) : (∀ x : ℝ, ∀ ε > 0, ∃ U ∈ (𝓝 x), (∀ y z : U, ∀ f : ℝ → ℝ, f ∈ F → (dist (f y) (f z) < ε))) ↔ ∃ (μ : ℝ → ℝ), ∀ (x : ℝ), (0 : ℝ) ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧ (∀ (s t : ℝ) (f : ℝ → ℝ), f ∈ F → |(f s) - (f t)| ≤ μ (|s - t|)) :=

α : Type u_1 a b : ℝ F : Set (ℝ → ℝ) ⊢ (∀ (x ε : ℝ), ε > 0 → ∃ U ∈ 𝓝 x, ∀ (y z : ↑U), ∀ f ∈ F, dist (f ↑y) (f ↑z) < ε) ↔ ∃ μ, ∀ (x : ℝ), 0 ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧ ∀ (s t : ℝ), ∀ f ∈ F, |f s - f t| ≤ μ |s - t|

import Mathlib open Filter Real Function open scoped Topology

exercise_2_3_2

test

/-- Prove that the products $a b$ and $b a$ are conjugate elements in a group.-/

theorem exercise_2_3_2 {G : Type*} [Group G] (a b : G) : ∃ g : G, b* a = g * a * b * g⁻¹ :=

G : Type u_1 inst✝ : Group G a b : G ⊢ ∃ g, b * a = g * a * b * g⁻¹

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_2_8_6

test

/-- Prove that the center of the product of two groups is the product of their centers.-/

noncomputable def exercise_2_8_6 {G H : Type*} [Group G] [Group H] : center (G × H) ≃* (center G) × (center H) :=

G : Type u_1 H : Type u_2 inst✝¹ : Group G inst✝ : Group H ⊢ ↥(center (G × H)) ≃* ↥(center G) × ↥(center H)

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators --center of (G × H) equivalent, preserves multiplication with (center G) × (center H)

exercise_3_2_7

test

/-- Prove that every homomorphism of fields is injective.-/

theorem exercise_3_2_7 {F : Type*} [Field F] {G : Type*} [Field G] (φ : F →+* G) : Injective φ :=

F : Type u_1 inst✝¹ : Field F G : Type u_2 inst✝ : Field G φ : F →+* G ⊢ Injective ⇑φ

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators open RingHom

exercise_3_7_2

test

/-- Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces.-/

theorem exercise_3_7_2 {K V : Type*} [Field K] [AddCommGroup V] [Module K V] {ι : Type*} [Fintype ι] (γ : ι → Submodule K V) (h : ∀ i : ι, γ i ≠ ⊤) : (⋂ (i : ι), (γ i : Set V)) ≠ ⊤ :=

K : Type u_1 V : Type u_2 inst✝³ : Field K inst✝² : AddCommGroup V inst✝¹ : Module K V ι : Type u_3 inst✝ : Fintype ι γ : ι → Submodule K V h : ∀ (i : ι), γ i ≠ ⊤ ⊢ ⋂ i, ↑(γ i) ≠ ⊤

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_6_4_2

test

/-- Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple.-/

theorem exercise_6_4_2 {G : Type*} [Group G] [Fintype G] {p q : ℕ} (hp : Prime p) (hq : Prime q) (hG : card G = p*q) : IsSimpleGroup G → false :=

G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G p q : ℕ hp : Prime p hq : Prime q hG : card G = p * q ⊢ IsSimpleGroup G → false = true

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_6_4_12

test

/-- Prove that no group of order 224 is simple.-/

theorem exercise_6_4_12 {G : Type*} [Group G] [Fintype G] (hG : card G = 224) : IsSimpleGroup G → false :=

G : Type u_1 inst✝¹ : Group G inst✝ : Fintype G hG : card G = 224 ⊢ IsSimpleGroup G → false = true

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_10_1_13

test

/-- An element $x$ of a ring $R$ is called nilpotent if some power of $x$ is zero. Prove that if $x$ is nilpotent, then $1+x$ is a unit in $R$.-/

theorem exercise_10_1_13 {R : Type*} [Ring R] {x : R} (hx : IsNilpotent x) : IsUnit (1 + x) :=

R : Type u_1 inst✝ : Ring R x : R hx : IsNilpotent x ⊢ IsUnit (1 + x)

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_10_6_7

test

/-- Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer.-/

theorem exercise_10_6_7 {I : Ideal GaussianInt} (hI : I ≠ ⊥) : ∃ (z : I), z ≠ 0 ∧ (z : GaussianInt).im = 0 :=

I : Ideal GaussianInt hI : I ≠ ⊥ ⊢ ∃ z, z ≠ 0 ∧ (↑z).im = 0

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_10_4_7a

test

/-- Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \cap J$.-/

theorem exercise_10_4_7a {R : Type*} [CommRing R] [NoZeroDivisors R] (I J : Ideal R) (hIJ : I + J = ⊤) : I * J = I ⊓ J :=

R : Type u_1 inst✝¹ : CommRing R inst✝ : NoZeroDivisors R I J : Ideal R hIJ : I + J = ⊤ ⊢ I * J = I ⊓ J

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_11_2_13

test

/-- If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\mathbb{Z}$.-/

theorem exercise_11_2_13 (a b : ℤ) : (ofInt a : GaussianInt) ∣ ofInt b → a ∣ b :=

a b : ℤ ⊢ ofInt a ∣ ofInt b → a ∣ b

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_11_4_6a

test

/-- Prove that $x^2+x+1$ is irreducible in the field $\mathbb{F}_2$.-/

theorem exercise_11_4_6a {F : Type*} [Field F] [Fintype F] (hF : card F = 7) : Irreducible (X ^ 2 + 1 : Polynomial F) :=

F : Type u_1 inst✝¹ : Field F inst✝ : Fintype F hF : card F = 7 ⊢ Irreducible (X ^ 2 + 1)

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_11_4_6c

test

/-- Prove that $x^3 - 9$ is irreducible in $\mathbb{F}_{31}$.-/

theorem exercise_11_4_6c : Irreducible (X^3 - 9 : Polynomial (ZMod 31)) :=

⊢ Irreducible (X ^ 3 - 9)

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_11_13_3

test

/-- Prove that there are infinitely many primes congruent to $-1$ (modulo $4$).-/

theorem exercise_11_13_3 (N : ℕ): ∃ p ≥ N, Nat.Prime p ∧ p + 1 ≡ 0 [MOD 4] :=

N : ℕ ⊢ ∃ p ≥ N, p.Prime ∧ p + 1 ≡ 0 [MOD 4]

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_13_6_10

test

/-- Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$.-/

theorem exercise_13_6_10 {K : Type*} [Field K] [Fintype Kˣ] : (∏ x : Kˣ, x) = -1 :=

K : Type u_1 inst✝¹ : Field K inst✝ : Fintype Kˣ ⊢ ∏ x : Kˣ, x = -1

import Mathlib open Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd open scoped BigOperators

exercise_1_2

test

/-- Show that $\frac{-1 + \sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).-/

theorem exercise_1_2 : (⟨-1/2, Real.sqrt 3 / 2⟩ : ℂ) ^ 3 = -1 :=

⊢ { re := -1 / 2, im := √3 / 2 } ^ 3 = -1

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators open scoped BigOperators

exercise_1_4

test

/-- Prove that if $a \in \mathbf{F}$, $v \in V$, and $av = 0$, then $a = 0$ or $v = 0$.-/

theorem exercise_1_4 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] (v : V) (a : F): a • v = 0 ↔ a = 0 ∨ v = 0 :=

F : Type u_1 V : Type u_2 inst✝² : AddCommGroup V inst✝¹ : Field F inst✝ : Module F V v : V a : F ⊢ a • v = 0 ↔ a = 0 ∨ v = 0

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_1_7

test

/-- Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbf{R}^2$.-/

theorem exercise_1_7 : ∃ U : Set (ℝ × ℝ), (U ≠ ∅) ∧ (∀ (c : ℝ) (u : ℝ × ℝ), u ∈ U → c • u ∈ U) ∧ (∀ U' : Submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=

⊢ ∃ U, U ≠ ∅ ∧ (∀ (c : ℝ), ∀ u ∈ U, c • u ∈ U) ∧ ∀ (U' : Submodule ℝ (ℝ × ℝ)), U ≠ ↑U'

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_1_9

test

/-- Prove that the union of two subspaces of $V$ is a subspace of $V$ if and only if one of the subspaces is contained in the other.-/

theorem exercise_1_9 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] (U W : Submodule F V): ∃ U' : Submodule F V, (U'.carrier = ↑U ∩ ↑W ↔ (U ≤ W ∨ W ≤ U)) :=

F : Type u_1 V : Type u_2 inst✝² : AddCommGroup V inst✝¹ : Field F inst✝ : Module F V U W : Submodule F V ⊢ ∃ U', U'.carrier = ↑U ∩ ↑W ↔ U ≤ W ∨ W ≤ U

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_3_8

test

/-- Suppose that $V$ is finite dimensional and that $T \in \mathcal{L}(V, W)$. Prove that there exists a subspace $U$ of $V$ such that $U \cap \operatorname{null} T=\{0\}$ and range $T=\{T u: u \in U\}$.-/

theorem exercise_3_8 {F V W : Type*} [AddCommGroup V] [AddCommGroup W] [Field F] [Module F V] [Module F W] (L : V →ₗ[F] W) : ∃ U : Submodule F V, U ⊓ (ker L) = ⊥ ∧ (range L = range (domRestrict L U)) :=

F : Type u_1 V : Type u_2 W : Type u_3 inst✝⁴ : AddCommGroup V inst✝³ : AddCommGroup W inst✝² : Field F inst✝¹ : Module F V inst✝ : Module F W L : V →ₗ[F] W ⊢ ∃ U, U ⊓ ker L = ⊥ ∧ range L = range (L.domRestrict U)

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_5_1

test

/-- Suppose $T \in \mathcal{L}(V)$. Prove that if $U_{1}, \ldots, U_{m}$ are subspaces of $V$ invariant under $T$, then $U_{1}+\cdots+U_{m}$ is invariant under $T$.-/

theorem exercise_5_1 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] {L : V →ₗ[F] V} {n : ℕ} (U : Fin n → Submodule F V) (hU : ∀ i : Fin n, Submodule.map L (U i) = U i) : Submodule.map L (∑ i : Fin n, U i : Submodule F V) = (∑ i : Fin n, U i : Submodule F V) :=

F : Type u_1 V : Type u_2 inst✝² : AddCommGroup V inst✝¹ : Field F inst✝ : Module F V L : V →ₗ[F] V n : ℕ U : Fin n → Submodule F V hU : ∀ (i : Fin n), Submodule.map L (U i) = U i ⊢ Submodule.map L (∑ i : Fin n, U i) = ∑ i : Fin n, U i

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_5_11

test

/-- Suppose $S, T \in \mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.-/

theorem exercise_5_11 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] (S T : End F V) : (S * T).Eigenvalues = (T * S).Eigenvalues :=

F : Type u_1 V : Type u_2 inst✝² : AddCommGroup V inst✝¹ : Field F inst✝ : Module F V S T : End F V ⊢ (S * T).Eigenvalues = (T * S).Eigenvalues

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_5_13

test

/-- Suppose $T \in \mathcal{L}(V)$ is such that every subspace of $V$ with dimension $\operatorname{dim} V-1$ is invariant under $T$. Prove that $T$ is a scalar multiple of the identity operator.-/

theorem exercise_5_13 {F V : Type*} [AddCommGroup V] [Field F] [Module F V] [FiniteDimensional F V] {T : End F V} (hS : ∀ U : Submodule F V, finrank F U = finrank F V - 1 → Submodule.map T U = U) : ∃ c : F, T = c • LinearMap.id :=

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 : End F V hS : ∀ (U : Submodule F V), finrank F ↥U = finrank F V - 1 → Submodule.map T U = U ⊢ ∃ c, T = c • LinearMap.id

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_5_24

test

/-- Suppose $V$ is a real vector space and $T \in \mathcal{L}(V)$ has no eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.-/

theorem exercise_5_24 {V : Type*} [AddCommGroup V] [Module ℝ V] [FiniteDimensional ℝ V] {T : End ℝ V} (hT : ∀ c : ℝ, eigenspace T c = ⊥) {U : Submodule ℝ V} (hU : Submodule.map T U = U) : Even (finrank U) :=

V : Type u_1 inst✝² : AddCommGroup V inst✝¹ : Module ℝ V inst✝ : FiniteDimensional ℝ V T : End ℝ V hT : ∀ (c : ℝ), T.eigenspace c = ⊥ U : Submodule ℝ V hU : Submodule.map T U = U ⊢ Even (finrank ↥U)

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_6_3

test

/-- Prove that $\left(\sum_{j=1}^{n} a_{j} b_{j}\right)^{2} \leq\left(\sum_{j=1}^{n} j a_{j}{ }^{2}\right)\left(\sum_{j=1}^{n} \frac{b_{j}{ }^{2}}{j}\right)$ for all real numbers $a_{1}, \ldots, a_{n}$ and $b_{1}, \ldots, b_{n}$.-/

theorem exercise_6_3 {n : ℕ} (a b : Fin n → ℝ) : (∑ i, a i * b i) ^ 2 ≤ (∑ i : Fin n, i * a i ^ 2) * (∑ i, b i ^ 2 / i) :=

n : ℕ a b : Fin n → ℝ ⊢ (∑ i : Fin n, a i * b i) ^ 2 ≤ (∑ i : Fin n, ↑↑i * a i ^ 2) * ∑ i : Fin n, b i ^ 2 / ↑↑i

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_6_13

test

/-- Suppose $\left(e_{1}, \ldots, e_{m}\right)$ is an or thonormal list of vectors in $V$. Let $v \in V$. Prove that $\|v\|^{2}=\left|\left\langle v, e_{1}\right\rangle\right|^{2}+\cdots+\left|\left\langle v, e_{m}\right\rangle\right|^{2}$ if and only if $v \in \operatorname{span}\left(e_{1}, \ldots, e_{m}\right)$.-/

theorem exercise_6_13 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] {n : ℕ} {e : Fin n → V} (he : Orthonormal ℂ e) (v : V) : ‖v‖^2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖^2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ) :=

V : Type u_1 inst✝¹ : NormedAddCommGroup V inst✝ : InnerProductSpace ℂ V n : ℕ e : Fin n → V he : Orthonormal ℂ e v : V ⊢ ‖v‖ ^ 2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖ ^ 2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ)

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_7_5

test

/-- Show that if $\operatorname{dim} V \geq 2$, then the set of normal operators on $V$ is not a subspace of $\mathcal{L}(V)$.-/

theorem exercise_7_5 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (hV : finrank V ≥ 2) : ∀ U : Submodule ℂ (End ℂ V), U.carrier ≠ {T | T * adjoint T = adjoint T * T} :=

V : Type u_1 inst✝² : NormedAddCommGroup V inst✝¹ : InnerProductSpace ℂ V inst✝ : FiniteDimensional ℂ V hV : finrank V ≥ 2 ⊢ ∀ (U : Submodule ℂ (End ℂ V)), U.carrier ≠ {T | T * adjoint T = adjoint T * T}

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_7_9

test

/-- Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.-/

theorem exercise_7_9 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] (T : End ℂ V) (hT : T * adjoint T = adjoint T * T) : IsSelfAdjoint T ↔ ∀ e : T.Eigenvalues, (e : ℂ).im = 0 :=

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 ⊢ IsSelfAdjoint T ↔ ∀ (e : T.Eigenvalues), (↑T e).im = 0

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_7_11

test

/-- Suppose $V$ is a complex inner-product space. Prove that every normal operator on $V$ has a square root. (An operator $S \in \mathcal{L}(V)$ is called a square root of $T \in \mathcal{L}(V)$ if $S^{2}=T$.)-/

theorem exercise_7_11 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V] {T : End ℂ V} (hT : T*adjoint T = adjoint T*T) : ∃ (S : End ℂ V), S ^ 2 = T :=

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 ⊢ ∃ S, S ^ 2 = T

import Mathlib open Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End open scoped BigOperators

exercise_1_1_2a

test

/-- Prove the the operation $\star$ on $\mathbb{Z}$ defined by $a\star b=a-b$ is not commutative.-/

theorem exercise_1_1_2a : ∃ a b : ℤ, a - b ≠ b - a :=

⊢ ∃ a b, a - b ≠ b - a

import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators

exercise_1_1_4

test

/-- Prove that the multiplication of residue class $\mathbb{Z}/n\mathbb{Z}$ is associative.-/

theorem exercise_1_1_4 (n : ℕ) : ∀ (a b c : ℕ), (a * b) * c ≡ a * (b * c) [ZMOD n] :=

n : ℕ ⊢ ∀ (a b c : ℕ), ↑a * ↑b * ↑c ≡ ↑a * (↑b * ↑c) [ZMOD ↑n]

import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators

exercise_1_1_15

test

/-- Prove that $(a_1a_2\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\dots a_1^{-1}$ for all $a_1, a_2, \dots, a_n\in G$.-/

theorem exercise_1_1_15 {G : Type*} [Group G] (as : List G) : as.prod⁻¹ = (as.reverse.map (λ x => x⁻¹)).prod :=

G : Type u_1 inst✝ : Group G as : List G ⊢ as.prod⁻¹ = (List.map (fun x => x⁻¹) as.reverse).prod

import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators

exercise_1_1_17

test

/-- Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.-/

theorem exercise_1_1_17 {G : Type*} [Group G] {x : G} {n : ℕ} (hxn: orderOf x = n) : x⁻¹ = x ^ (n - 1 : ℤ) :=

G : Type u_1 inst✝ : Group G x : G n : ℕ hxn : orderOf x = n ⊢ x⁻¹ = x ^ (↑n - 1)

import Mathlib open Fintype Subgroup Set Polynomial Ideal open scoped BigOperators