| {"formal": "theorem exercise_1_1b\n(x : \u211d)\n(y : \u211a)\n(h : y \u2260 0)\n: ( irrational x ) -> irrational ( x * y ) :=", "informal": "If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $rx$ is irrational."} | |
| {"formal": "theorem exercise_1_4\n(\u03b1 : Type*) [partial_order \u03b1]\n(s : set \u03b1)\n(x y : \u03b1)\n(h\u2080 : set.nonempty s)\n(h\u2081 : x \u2208 lower_bounds s)\n(h\u2082 : y \u2208 upper_bounds s)\n: x \u2264 y :=", "informal": "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 $\\alpha \\leq \\beta$."} | |
| {"formal": "theorem exercise_1_8 : \u00ac \u2203 (r : \u2102 \u2192 \u2102 \u2192 Prop), is_linear_order \u2102 r :=", "informal": "Prove that no order can be defined in the complex field that turns it into an ordered field."} | |
| {"formal": "theorem exercise_1_12 (n : \u2115) (f : \u2115 \u2192 \u2102) : \n abs (\u2211 i in finset.range n, f i) \u2264 \u2211 i in finset.range n, abs (f i) :=", "informal": "If $z_1, \\ldots, z_n$ are complex, prove that $|z_1 + z_2 + \\ldots + z_n| \\leq |z_1| + |z_2| + \\cdots + |z_n|$."} | |
| {"formal": "theorem exercise_1_14\n (z : \u2102) (h : abs z = 1)\n : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=", "informal": "If $z$ is a complex number such that $|z|=1$, that is, such that $z \\bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$."} | |
| {"formal": "theorem exercise_1_17\n (n : \u2115)\n (x y : euclidean_space \u211d (fin n)) -- R^n\n : \u2016x + y\u2016^2 + \u2016x - y\u2016^2 = 2*\u2016x\u2016^2 + 2*\u2016y\u2016^2 :=", "informal": "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 $\\mathbf{y} \\in R^{k}$."} | |
| {"formal": "theorem exercise_1_18b\n : \u00ac \u2200 (x : \u211d), \u2203 (y : \u211d), y \u2260 0 \u2227 x * y = 0 :=", "informal": "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 $\\mathbf{x} \\cdot \\mathbf{y}=0$"} | |
| {"formal": "theorem exercise_2_19a {X : Type*} [metric_space X]\n (A B : set X) (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) :\n separated_nhds A B :=", "informal": "If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated."} | |
| {"formal": "theorem exercise_2_25 {K : Type*} [metric_space K] [compact_space K] :\n \u2203 (B : set (set K)), set.countable B \u2227 is_topological_basis B :=", "informal": "Prove that every compact metric space $K$ has a countable base."} | |
| {"formal": "theorem exercise_2_27b (k : \u2115) (E P : set (euclidean_space \u211d (fin k)))\n (hE : E.nonempty \u2227 \u00ac set.countable E)\n (hP : P = {x | \u2200 U \u2208 \ud835\udcdd x, (P \u2229 E).nonempty \u2227 \u00ac set.countable (P \u2229 E)}) :\n set.countable (E \\ P) :=", "informal": "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 $P$."} | |
| {"formal": "theorem exercise_2_29 (U : set \u211d) (hU : is_open U) :\n \u2203 (f : \u2115 \u2192 set \u211d), (\u2200 n, \u2203 a b : \u211d, f n = {x | a < x \u2227 x < b}) \u2227 (\u2200 n, f n \u2286 U) \u2227\n (\u2200 n m, n \u2260 m \u2192 f n \u2229 f m = \u2205) \u2227\n U = \u22c3 n, f n :=", "informal": "Prove that every open set in $\\mathbb{R}$ is the union of an at most countable collection of disjoint segments."} | |
| {"formal": "theorem exercise_3_2a\n : tendsto (\u03bb (n : \u211d), (sqrt (n^2 + n) - n)) at_top (\ud835\udcdd (1/2)) :=", "informal": "Prove that $\\lim_{n \\rightarrow \\infty}\\sqrt{n^2 + n} -n = 1/2$."} | |
| {"formal": "theorem exercise_3_5 -- TODO fix\n (a b : \u2115 \u2192 \u211d)\n (h : limsup a + limsup b \u2260 0) :\n limsup (\u03bb n, a n + b n) \u2264 limsup a + limsup b :=", "informal": "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 $\\infty-\\infty$."} | |
| {"formal": "theorem exercise_3_7\n (a : \u2115 \u2192 \u211d)\n (h : \u2203 y, (tendsto (\u03bb n, (\u2211 i in (finset.range n), a i)) at_top (\ud835\udcdd y))) :\n \u2203 y, tendsto (\u03bb n, (\u2211 i in (finset.range n), sqrt (a i) / n)) at_top (\ud835\udcdd y) :=", "informal": "Prove that the convergence of $\\Sigma a_{n}$ implies the convergence of $\\sum \\frac{\\sqrt{a_{n}}}{n}$ if $a_n\\geq 0$."} | |
| {"formal": "theorem exercise_3_13\n (a b : \u2115 \u2192 \u211d)\n (ha : \u2203 y, (tendsto (\u03bb n, (\u2211 i in (finset.range n), |a i|)) at_top (\ud835\udcdd y)))\n (hb : \u2203 y, (tendsto (\u03bb n, (\u2211 i in (finset.range n), |b i|)) at_top (\ud835\udcdd y))) :\n \u2203 y, (tendsto (\u03bb n, (\u2211 i in (finset.range n),\n \u03bb i, (\u2211 j in finset.range (i + 1), a j * b (i - j)))) at_top (\ud835\udcdd y)) :=", "informal": "Prove that the Cauchy product of two absolutely convergent series converges absolutely."} | |
| {"formal": "theorem exercise_3_21\n {X : Type*} [metric_space X] [complete_space X]\n (E : \u2115 \u2192 set X)\n (hE : \u2200 n, E n \u2283 E (n + 1))\n (hE' : tendsto (\u03bb n, metric.diam (E n)) at_top (\ud835\udcdd 0)) :\n \u2203 a, set.Inter E = {a} :=", "informal": "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."} | |
| {"formal": "theorem exercise_4_1a\n : \u2203 (f : \u211d \u2192 \u211d), (\u2200 (x : \u211d), tendsto (\u03bb y, f(x + y) - f(x - y)) (\ud835\udcdd 0) (\ud835\udcdd 0)) \u2227 \u00ac continuous f :=", "informal": "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."} | |
| {"formal": "theorem exercise_4_3\n {\u03b1 : Type} [metric_space \u03b1]\n (f : \u03b1 \u2192 \u211d) (h : continuous f) (z : set \u03b1) (g : z = f\u207b\u00b9' {0})\n : is_closed z :=", "informal": "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."} | |
| {"formal": "theorem exercise_4_4b\n {\u03b1 : Type} [metric_space \u03b1]\n {\u03b2 : Type} [metric_space \u03b2]\n (f g : \u03b1 \u2192 \u03b2)\n (s : set \u03b1)\n (h\u2081 : continuous f)\n (h\u2082 : continuous g)\n (h\u2083 : dense s)\n (h\u2084 : \u2200 x \u2208 s, f x = g x)\n : f = g :=", "informal": "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 $p \\in X$."} | |
| {"formal": "theorem exercise_4_5b\n : \u2203 (E : set \u211d) (f : \u211d \u2192 \u211d), (continuous_on f E) \u2227\n (\u00ac \u2203 (g : \u211d \u2192 \u211d), continuous g \u2227 \u2200 x \u2208 E, f x = g x) :=", "informal": "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 $x \\in E$."} | |
| {"formal": "theorem exercise_4_8a\n (E : set \u211d) (f : \u211d \u2192 \u211d) (hf : uniform_continuous_on f E)\n (hE : metric.bounded E) : metric.bounded (set.image f E) :=", "informal": "Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$."} | |
| {"formal": "theorem exercise_4_11a\n {X : Type*} [metric_space X]\n {Y : Type*} [metric_space Y]\n (f : X \u2192 Y) (hf : uniform_continuous f)\n (x : \u2115 \u2192 X) (hx : cauchy_seq x) :\n cauchy_seq (\u03bb n, f (x n)) :=", "informal": "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 $X$."} | |
| {"formal": "theorem exercise_4_15 {f : \u211d \u2192 \u211d}\n (hf : continuous f) (hof : is_open_map f) :\n monotone f :=", "informal": "Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic."} | |
| {"formal": "theorem exercise_4_21a {X : Type*} [metric_space X]\n (K F : set X) (hK : is_compact K) (hF : is_closed F) (hKF : disjoint K F) :\n \u2203 (\u03b4 : \u211d), \u03b4 > 0 \u2227 \u2200 (p q : X), p \u2208 K \u2192 q \u2208 F \u2192 dist p q \u2265 \u03b4 :=", "informal": "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 $p \\in K, q \\in F$."} | |
| {"formal": "theorem exercise_5_1\n {f : \u211d \u2192 \u211d} (hf : \u2200 x y : \u211d, | (f x - f y) | \u2264 (x - y) ^ 2) :\n \u2203 c, f = \u03bb x, c :=", "informal": "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."} | |
| {"formal": "theorem exercise_5_3 {g : \u211d \u2192 \u211d} (hg : continuous g)\n (hg' : \u2203 M : \u211d, \u2200 x : \u211d, | deriv g x | \u2264 M) :\n \u2203 N, \u2200 \u03b5 > 0, \u03b5 < N \u2192 function.injective (\u03bb x : \u211d, x + \u03b5 * g x) :=", "informal": "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."} | |
| {"formal": "theorem exercise_5_5\n {f : \u211d \u2192 \u211d}\n (hfd : differentiable \u211d f)\n (hf : tendsto (deriv f) at_top (\ud835\udcdd 0)) :\n tendsto (\u03bb x, f (x + 1) - f x) at_top at_top :=", "informal": "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 $x \\rightarrow+\\infty$."} | |
| {"formal": "theorem exercise_5_7\n {f g : \u211d \u2192 \u211d} {x : \u211d}\n (hf' : differentiable_at \u211d f 0)\n (hg' : differentiable_at \u211d g 0)\n (hg'_ne_0 : deriv g 0 \u2260 0)\n (f0 : f 0 = 0) (g0 : g 0 = 0) :\n tendsto (\u03bb x, f x / g x) (\ud835\udcdd x) (\ud835\udcdd (deriv f x / deriv g x)) :=", "informal": "Suppose $f^{\\prime}(x), g^{\\prime}(x)$ exist, $g^{\\prime}(x) \\neq 0$, and $f(x)=g(x)=0$. Prove that $\\lim _{t \\rightarrow x} \\frac{f(t)}{g(t)}=\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}.$"} | |
| {"formal": "theorem exercise_5_17\n {f : \u211d \u2192 \u211d}\n (hf' : differentiable_on \u211d f (set.Icc (-1) 1))\n (hf'' : differentiable_on \u211d (deriv f) (set.Icc 1 1))\n (hf''' : differentiable_on \u211d (deriv (deriv f)) (set.Icc 1 1))\n (hf0 : f (-1) = 0)\n (hf1 : f 0 = 0)\n (hf2 : f 1 = 1)\n (hf3 : deriv f 0 = 0) :\n \u2203 x, x \u2208 set.Ioo (-1 : \u211d) 1 \u2227 deriv (deriv (deriv f)) x \u2265 3 :=", "informal": "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 $x \\in(-1,1)$."} | |
| {"formal": "theorem exercise_13_3b : \u00ac \u2200 X : Type, \u2200s : set (set X),\n (\u2200 t : set X, t \u2208 s \u2192 (set.infinite t\u1d9c \u2228 t = \u2205 \u2228 t = \u22a4)) \u2192 \n (set.infinite (\u22c3\u2080 s)\u1d9c \u2228 (\u22c3\u2080 s) = \u2205 \u2228 (\u22c3\u2080 s) = \u22a4) :=", "informal": "Show that the collection $$\\mathcal{T}_\\infty = \\{U | X - U \\text{ is infinite or empty or all of X}\\}$$ does not need to be a topology on the set $X$."} | |
| {"formal": "theorem exercise_13_4a2 :\n \u2203 (X I : Type*) (T : I \u2192 set (set X)),\n (\u2200 i, is_topology X (T i)) \u2227 \u00ac is_topology X (\u22c2 i : I, T i) :=", "informal": "If $\\mathcal{T}_\\alpha$ is a family of topologies on $X$, show that $\\bigcup \\mathcal{T}_\\alpha$ does not need to be a topology on $X$."} | |
| {"formal": "theorem exercise_13_4b2 (X I : Type*) (T : I \u2192 set (set X)) (h : \u2200 i, is_topology X (T i)) :\n \u2203! T', is_topology X T' \u2227 (\u2200 i, T' \u2286 T i) \u2227\n \u2200 T'', is_topology X T'' \u2192 (\u2200 i, T'' \u2286 T i) \u2192 T' \u2286 T'' :=", "informal": "Let $\\mathcal{T}_\\alpha$ be a family of topologies on $X$. Show that there is a unique largest topology on $X$ contained in all the collections $\\mathcal{T}_\\alpha$."} | |
| {"formal": "theorem exercise_13_5b {X : Type*}\n [t : topological_space X] (A : set (set X)) (hA : t = generate_from A) :\n generate_from A = generate_from (sInter {T | is_topology X T \u2227 A \u2286 T}) :=", "informal": "Show that if $\\mathcal{A}$ is a subbasis for a topology on $X$, then the topology generated by $\\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\\mathcal{A}$."} | |
| {"formal": "theorem exercise_13_8a :\n topological_space.is_topological_basis {S : set \u211d | \u2203 a b : \u211a, a < b \u2227 S = Ioo a b} :=", "informal": "Show that the collection $\\{(a,b) \\mid a < b, a \\text{ and } b \\text{ rational}\\}$ is a basis that generates the standard topology on $\\mathbb{R}$."} | |
| {"formal": "theorem exercise_16_1 {X : Type*} [topological_space X]\n (Y : set X)\n (A : set Y) :\n \u2200 U : set A, is_open U \u2194 is_open (subtype.val '' U) :=", "informal": "Show that if $Y$ is a subspace of $X$, and $A$ is a subset of $Y$, then the topology $A$ inherits as a subspace of $Y$ is the same as the topology it inherits as a subspace of $X$."} | |
| {"formal": "theorem exercise_16_6\n (S : set (set (\u211d \u00d7 \u211d)))\n (hS : \u2200 s, s \u2208 S \u2192 \u2203 a b c d, (rational a \u2227 rational b \u2227 rational c \u2227 rational d\n \u2227 s = {x | \u2203 x\u2081 x\u2082, x = (x\u2081, x\u2082) \u2227 a < x\u2081 \u2227 x\u2081 < b \u2227 c < x\u2082 \u2227 x\u2082 < d})) :\n is_topological_basis S :=", "informal": "Show that the countable collection \\[\\{(a, b) \\times (c, d) \\mid a < b \\text{ and } c < d, \\text{ and } a, b, c, d \\text{ are rational}\\}\\] is a basis for $\\mathbb{R}^2$."} | |
| {"formal": "theorem exercise_18_8a {X Y : Type*} [topological_space X] [topological_space Y]\n [linear_order Y] [order_topology Y] {f g : X \u2192 Y}\n (hf : continuous f) (hg : continuous g) :\n is_closed {x | f x \u2264 g x} :=", "informal": "Let $Y$ be an ordered set in the order topology. Let $f, g: X \\rightarrow Y$ be continuous. Show that the set $\\{x \\mid f(x) \\leq g(x)\\}$ is closed in $X$."} | |
| {"formal": "theorem exercise_18_13\n {X : Type*} [topological_space X] {Y : Type*} [topological_space Y]\n [t2_space Y] {A : set X} {f : A \u2192 Y} (hf : continuous f)\n (g : closure A \u2192 Y)\n (g_con : continuous g) :\n \u2200 (g' : closure A \u2192 Y), continuous g' \u2192 (\u2200 (x : closure A), g x = g' x) :=", "informal": "Let $A \\subset X$; let $f: A \\rightarrow Y$ be continuous; let $Y$ be Hausdorff. Show that if $f$ may be extended to a continuous function $g: \\bar{A} \\rightarrow Y$, then $g$ is uniquely determined by $f$."} | |
| {"formal": "theorem exercise_20_2\n [topological_space (\u211d \u00d7\u2097 \u211d)] [order_topology (\u211d \u00d7\u2097 \u211d)]\n : metrizable_space (\u211d \u00d7\u2097 \u211d) :=", "informal": "Show that $\\mathbb{R} \\times \\mathbb{R}$ in the dictionary order topology is metrizable."} | |
| {"formal": "theorem exercise_21_6b\n (f : \u2115 \u2192 I \u2192 \u211d )\n (h : \u2200 x n, f n x = x ^ n) :\n \u00ac \u2203 f\u2080, tendsto_uniformly f f\u2080 at_top :=", "informal": "Define $f_{n}:[0,1] \\rightarrow \\mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\\left(f_{n}\\right)$ does not converge uniformly."} | |
| {"formal": "theorem exercise_22_2a {X Y : Type*} [topological_space X]\n [topological_space Y] (p : X \u2192 Y) (h : continuous p) :\n quotient_map p \u2194 \u2203 (f : Y \u2192 X), continuous f \u2227 p \u2218 f = id :=", "informal": "Let $p: X \\rightarrow Y$ be a continuous map. Show that if there is a continuous map $f: Y \\rightarrow X$ such that $p \\circ f$ equals the identity map of $Y$, then $p$ is a quotient map."} | |
| {"formal": "theorem exercise_22_5 {X Y : Type*} [topological_space X]\n [topological_space Y] (p : X \u2192 Y) (hp : is_open_map p)\n (A : set X) (hA : is_open A) : is_open_map (p \u2218 subtype.val : A \u2192 Y) :=", "informal": "Let $p \\colon X \\rightarrow Y$ be an open map. Show that if $A$ is open in $X$, then the map $q \\colon A \\rightarrow p(A)$ obtained by restricting $p$ is an open map."} | |
| {"formal": "theorem exercise_23_3 {X : Type*} [topological_space X]\n [topological_space X] {A : \u2115 \u2192 set X}\n (hAn : \u2200 n, is_connected (A n))\n (A\u2080 : set X)\n (hA : is_connected A\u2080)\n (h : \u2200 n, A\u2080 \u2229 A n \u2260 \u2205) :\n is_connected (A\u2080 \u222a (\u22c3 n, A n)) :=", "informal": "Let $\\left\\{A_{\\alpha}\\right\\}$ be a collection of connected subspaces of $X$; let $A$ be a connected subset of $X$. Show that if $A \\cap A_{\\alpha} \\neq \\varnothing$ for all $\\alpha$, then $A \\cup\\left(\\bigcup A_{\\alpha}\\right)$ is connected."} | |
| {"formal": "theorem exercise_23_6 {X : Type*}\n [topological_space X] {A C : set X} (hc : is_connected C)\n (hCA : C \u2229 A \u2260 \u2205) (hCXA : C \u2229 A\u1d9c \u2260 \u2205) :\n C \u2229 (frontier A) \u2260 \u2205 :=", "informal": "Let $A \\subset X$. Show that if $C$ is a connected subspace of $X$ that intersects both $A$ and $X-A$, then $C$ intersects $\\operatorname{Bd} A$."} | |
| {"formal": "theorem exercise_23_11 {X Y : Type*} [topological_space X] [topological_space Y]\n (p : X \u2192 Y) (hq : quotient_map p)\n (hY : connected_space Y) (hX : \u2200 y : Y, is_connected (p \u207b\u00b9' {y})) :\n connected_space X :=", "informal": "Let $p: X \\rightarrow Y$ be a quotient map. Show that if each set $p^{-1}(\\{y\\})$ is connected, and if $Y$ is connected, then $X$ is connected."} | |
| {"formal": "theorem exercise_24_3a [topological_space I] [compact_space I]\n (f : I \u2192 I) (hf : continuous f) :\n \u2203 (x : I), f x = x :=", "informal": "Let $f \\colon X \\rightarrow X$ be continuous. Show that if $X = [0, 1]$, there is a point $x$ such that $f(x) = x$. (The point $x$ is called a fixed point of $f$.)"} | |
| {"formal": "theorem exercise_25_9 {G : Type*} [topological_space G] [group G]\n [topological_group G] (C : set G) (h : C = connected_component 1) :\n is_normal_subgroup C :=", "informal": "Let $G$ be a topological group; let $C$ be the component of $G$ containing the identity element $e$. Show that $C$ is a normal subgroup of $G$."} | |
| {"formal": "theorem exercise_26_12 {X Y : Type*} [topological_space X] [topological_space Y]\n (p : X \u2192 Y) (h : function.surjective p) (hc : continuous p) (hp : \u2200 y, is_compact (p \u207b\u00b9' {y}))\n (hY : compact_space Y) : compact_space X :=", "informal": "Let $p: X \\rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(\\{y\\})$ is compact, for each $y \\in Y$. (Such a map is called a perfect map.) Show that if $Y$ is compact, then $X$ is compact."} | |
| {"formal": "theorem exercise_28_4 {X : Type*}\n [topological_space X] (hT1 : t1_space X) :\n countably_compact X \u2194 limit_point_compact X :=", "informal": "A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. Show that for a $T_1$ space $X$, countable compactness is equivalent to limit point compactness."} | |
| {"formal": "theorem exercise_28_6 {X : Type*} [metric_space X]\n [compact_space X] {f : X \u2192 X} (hf : isometry f) :\n function.bijective f :=", "informal": "Let $(X, d)$ be a metric space. If $f: X \\rightarrow X$ satisfies the condition $d(f(x), f(y))=d(x, y)$ for all $x, y \\in X$, then $f$ is called an isometry of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism."} | |
| {"formal": "theorem exercise_29_4 [topological_space (\u2115 \u2192 I)] :\n \u00ac locally_compact_space (\u2115 \u2192 I) :=", "informal": "Show that $[0, 1]^\\omega$ is not locally compact in the uniform topology."} | |
| {"formal": "theorem exercise_30_10\n {X : \u2115 \u2192 Type*} [\u2200 i, topological_space (X i)]\n (h : \u2200 i, \u2203 (s : set (X i)), countable s \u2227 dense s) :\n \u2203 (s : set (\u03a0 i, X i)), countable s \u2227 dense s :=", "informal": "Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset."} | |
| {"formal": "theorem exercise_31_1 {X : Type*} [topological_space X]\n (hX : regular_space X) (x y : X) :\n \u2203 (U V : set X), is_open U \u2227 is_open V \u2227 x \u2208 U \u2227 y \u2208 V \u2227 closure U \u2229 closure V = \u2205 :=", "informal": "Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint."} | |
| {"formal": "theorem exercise_31_3 {\u03b1 : Type*} [partial_order \u03b1]\n [topological_space \u03b1] (h : order_topology \u03b1) : regular_space \u03b1 :=", "informal": "Show that every order topology is regular."} | |
| {"formal": "theorem exercise_32_2a\n {\u03b9 : Type*} {X : \u03b9 \u2192 Type*} [\u2200 i, topological_space (X i)]\n (h : \u2200 i, nonempty (X i)) (h2 : t2_space (\u03a0 i, X i)) :\n \u2200 i, t2_space (X i) :=", "informal": "Show that if $\\prod X_\\alpha$ is Hausdorff, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty."} | |
| {"formal": "theorem exercise_32_2c\n {\u03b9 : Type*} {X : \u03b9 \u2192 Type*} [\u2200 i, topological_space (X i)]\n (h : \u2200 i, nonempty (X i)) (h2 : normal_space (\u03a0 i, X i)) :\n \u2200 i, normal_space (X i) :=", "informal": "Show that if $\\prod X_\\alpha$ is normal, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty."} | |
| {"formal": "theorem exercise_33_7 {X : Type*} [topological_space X]\n (hX : locally_compact_space X) (hX' : t2_space X) :\n \u2200 x A, is_closed A \u2227 \u00ac x \u2208 A \u2192\n \u2203 (f : X \u2192 I), continuous f \u2227 f x = 1 \u2227 f '' A = {0}\n :=", "informal": "Show that every locally compact Hausdorff space is completely regular."} | |
| {"formal": "theorem exercise_34_9\n (X : Type*) [topological_space X] [compact_space X]\n (X1 X2 : set X) (hX1 : is_closed X1) (hX2 : is_closed X2)\n (hX : X1 \u222a X2 = univ) (hX1m : metrizable_space X1)\n (hX2m : metrizable_space X2) : metrizable_space X :=", "informal": "Let $X$ be a compact Hausdorff space that is the union of the closed subspaces $X_1$ and $X_2$. If $X_1$ and $X_2$ are metrizable, show that $X$ is metrizable."} | |
| {"formal": "theorem exercise_43_2 {X : Type*} [metric_space X]\n {Y : Type*} [metric_space Y] [complete_space Y] (A : set X)\n (f : X \u2192 Y) (hf : uniform_continuous_on f A) :\n \u2203! (g : X \u2192 Y), continuous_on g (closure A) \u2227\n uniform_continuous_on g (closure A) \u2227 \u2200 (x : A), g x = f x :=", "informal": "Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \\subset X$. Show that if $f \\colon A \\rightarrow Y$ is uniformly continuous, then $f$ can be uniquely extended to a continuous function $g \\colon \\bar{A} \\rightarrow Y$, and $g$ is uniformly continuous."} | |
| {"formal": "theorem exercise_1_2 :\n (\u27e8-1/2, real.sqrt 3 / 2\u27e9 : \u2102) ^ 3 = -1 :=", "informal": "Show that $\\frac{-1 + \\sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1)."} | |
| {"formal": "theorem exercise_1_4 {F V : Type*} [add_comm_group V] [field F]\n [module F V] (v : V) (a : F): a \u2022 v = 0 \u2194 a = 0 \u2228 v = 0 :=", "informal": "Prove that if $a \\in \\mathbf{F}$, $v \\in V$, and $av = 0$, then $a = 0$ or $v = 0$."} | |
| {"formal": "theorem exercise_1_7 : \u2203 U : set (\u211d \u00d7 \u211d),\n (U \u2260 \u2205) \u2227\n (\u2200 (c : \u211d) (u : \u211d \u00d7 \u211d), u \u2208 U \u2192 c \u2022 u \u2208 U) \u2227\n (\u2200 U' : submodule \u211d (\u211d \u00d7 \u211d), U \u2260 \u2191U') :=", "informal": "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 $\\mathbf{R}^2$."} | |
| {"formal": "theorem exercise_1_9 {F V : Type*} [add_comm_group V] [field F]\n [module F V] (U W : submodule F V):\n \u2203 U' : submodule F V, (U'.carrier = \u2191U \u2229 \u2191W \u2194 (U \u2264 W \u2228 W \u2264 U)) :=", "informal": "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."} | |
| {"formal": "theorem exercise_3_8 {F V W : Type*} [add_comm_group V]\n [add_comm_group W] [field F] [module F V] [module F W]\n (L : V \u2192\u2097[F] W) :\n \u2203 U : submodule F V, U \u2293 L.ker = \u22a5 \u2227\n linear_map.range L = range (dom_restrict L U):=", "informal": "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 $T=\\{T u: u \\in U\\}$."} | |
| {"formal": "theorem exercise_5_1 {F V : Type*} [add_comm_group V] [field F]\n [module F V] {L : V \u2192\u2097[F] V} {n : \u2115} (U : fin n \u2192 submodule F V)\n (hU : \u2200 i : fin n, map L (U i) = U i) :\n map L (\u2211 i : fin n, U i : submodule F V) =\n (\u2211 i : fin n, U i : submodule F V) :=", "informal": "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 $T$."} | |
| {"formal": "theorem exercise_5_11 {F V : Type*} [add_comm_group V] [field F]\n [module F V] (S T : End F V) :\n (S * T).eigenvalues = (T * S).eigenvalues :=", "informal": "Suppose $S, T \\in \\mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues."} | |
| {"formal": "theorem exercise_5_13 {F V : Type*} [add_comm_group V] [field F]\n [module F V] [finite_dimensional F V] {T : End F V}\n (hS : \u2200 U : submodule F V, finrank F U = finrank F V - 1 \u2192\n map T U = U) : \u2203 c : F, T = c \u2022 id :=", "informal": "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."} | |
| {"formal": "theorem exercise_5_24 {V : Type*} [add_comm_group V]\n [module \u211d V] [finite_dimensional \u211d V] {T : End \u211d V}\n (hT : \u2200 c : \u211d, eigenspace T c = \u22a5) {U : submodule \u211d V}\n (hU : map T U = U) : even (finrank U) :=", "informal": "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."} | |
| {"formal": "theorem exercise_6_3 {n : \u2115} (a b : fin n \u2192 \u211d) :\n (\u2211 i, a i * b i) ^ 2 \u2264 (\u2211 i : fin n, i * a i ^ 2) * (\u2211 i, b i ^ 2 / i) :=", "informal": "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 $b_{1}, \\ldots, b_{n}$."} | |
| {"formal": "theorem exercise_6_13 {V : Type*} [inner_product_space \u2102 V] {n : \u2115}\n {e : fin n \u2192 V} (he : orthonormal \u2102 e) (v : V) :\n \u2016v\u2016^2 = \u2211 i : fin n, \u2016\u27eav, e i\u27eb_\u2102\u2016^2 \u2194 v \u2208 span \u2102 (e '' univ) :=", "informal": "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 $v \\in \\operatorname{span}\\left(e_{1}, \\ldots, e_{m}\\right)$."} | |
| {"formal": "theorem exercise_7_5 {V : Type*} [inner_product_space \u2102 V] \n [finite_dimensional \u2102 V] (hV : finrank V \u2265 2) :\n \u2200 U : submodule \u2102 (End \u2102 V), U.carrier \u2260\n {T | T * T.adjoint = T.adjoint * T} :=", "informal": "Show that if $\\operatorname{dim} V \\geq 2$, then the set of normal operators on $V$ is not a subspace of $\\mathcal{L}(V)$."} | |
| {"formal": "theorem exercise_7_9 {V : Type*} [inner_product_space \u2102 V]\n [finite_dimensional \u2102 V] (T : End \u2102 V)\n (hT : T * T.adjoint = T.adjoint * T) :\n is_self_adjoint T \u2194 \u2200 e : T.eigenvalues, (e : \u2102).im = 0 :=", "informal": "Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real."} | |
| {"formal": "theorem exercise_7_11 {V : Type*} [inner_product_space \u2102 V]\n [finite_dimensional \u2102 V] {T : End \u2102 V} (hT : T*T.adjoint = T.adjoint*T) :\n \u2203 (S : End \u2102 V), S ^ 2 = T :=", "informal": "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 $S^{2}=T$.)"} | |
| {"formal": "theorem exercise_1_30 {n : \u2115} : \n \u00ac \u2203 a : \u2124, \u2211 (i : fin n), (1 : \u211a) / (n+2) = a :=", "informal": "Prove that $\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}$ is not an integer."} | |
| {"formal": "theorem exercise_2_4 {a : \u2124} (ha : a \u2260 0) \n (f_a :=", "informal": "If $a$ is a nonzero integer, then for $n>m$ show that $\\left(a^{2^{n}}+1, a^{2^{m}}+1\\right)=1$ or 2 depending on whether $a$ is odd or even."} | |
| {"formal": "theorem exercise_2_27a : \n \u00ac summable (\u03bb i : {p : \u2124 // squarefree p}, (1 : \u211a) / i) :=", "informal": "Show that $\\sum^{\\prime} 1 / n$, the sum being over square free integers, diverges."} | |
| {"formal": "theorem exercise_3_4 : \u00ac \u2203 x y : \u2124, 3*x^2 + 2 = y^2 :=", "informal": "Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers."} | |
| {"formal": "theorem exercise_3_10 {n : \u2115} (hn0 : \u00ac n.prime) (hn1 : n \u2260 4) : \n factorial (n-1) \u2261 0 [MOD n] :=", "informal": "If $n$ is not a prime, show that $(n-1) ! \\equiv 0(n)$, except when $n=4$."} | |
| {"formal": "theorem exercise_4_4 {p t: \u2115} (hp0 : p.prime) (hp1 : p = 4*t + 1) \n (a : zmod p) : \n is_primitive_root a p \u2194 is_primitive_root (-a) p :=", "informal": "Consider a prime $p$ of the form $4 t+1$. Show that $a$ is a primitive root modulo $p$ iff $-a$ is a primitive root modulo $p$."} | |
| {"formal": "theorem exercise_4_6 {p n : \u2115} (hp : p.prime) (hpn : p = 2^n + 1) : \n is_primitive_root 3 p :=", "informal": "If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$."} | |
| {"formal": "theorem exercise_4_11 {p : \u2115} (hp : p.prime) (k s: \u2115) \n (s :=", "informal": "Prove that $1^{k}+2^{k}+\\cdots+(p-1)^{k} \\equiv 0(p)$ if $p-1 \\nmid k$ and $-1(p)$ if $p-1 \\mid k$."} | |
| {"formal": "theorem exercise_5_28 {p : \u2115} (hp : p.prime) (hp1 : p \u2261 1 [MOD 4]): \n \u2203 x, x^4 \u2261 2 [MOD p] \u2194 \u2203 A B, p = A^2 + 64*B^2 :=", "informal": "Show that $x^{4} \\equiv 2(p)$ has a solution for $p \\equiv 1(4)$ iff $p$ is of the form $A^{2}+64 B^{2}$."} | |
| {"formal": "theorem exercise_12_12 : is_algebraic \u211a (sin (real.pi/12)) :=", "informal": "Show that $\\sin (\\pi / 12)$ is an algebraic number."} | |
| {"formal": "theorem exercise_1_13b {f : \u2102 \u2192 \u2102} (\u03a9 : set \u2102) (a b : \u03a9) (h : is_open \u03a9)\n (hf : differentiable_on \u2102 f \u03a9) (hc : \u2203 (c : \u211d), \u2200 z \u2208 \u03a9, (f z).im = c) :\n f a = f b :=", "informal": "Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Im}(f)$ is constant, then $f$ is constant."} | |
| {"formal": "theorem exercise_1_19a (z : \u2102) (hz : abs z = 1) (s : \u2115 \u2192 \u2102)\n (h : s = (\u03bb n, \u2211 i in (finset.range n), i * z ^ i)) :\n \u00ac \u2203 y, tendsto s at_top (\ud835\udcdd y) :=", "informal": "Prove that the power series $\\sum nz^n$ does not converge on any point of the unit circle."} | |
| {"formal": "theorem exercise_1_19c (z : \u2102) (hz : abs z = 1) (hz2 : z \u2260 1) (s : \u2115 \u2192 \u2102)\n (h : s = (\u03bb n, \u2211 i in (finset.range n), i * z / i)) :\n \u2203 z, tendsto s at_top (\ud835\udcdd z) :=", "informal": "Prove that the power series $\\sum zn/n$ converges at every point of the unit circle except $z = 1$."} | |
| {"formal": "theorem exercise_2_2 :\n tendsto (\u03bb y, \u222b x in 0..y, real.sin x / x) at_top (\ud835\udcdd (real.pi / 2)) :=", "informal": "Show that $\\int_{0}^{\\infty} \\frac{\\sin x}{x} d x=\\frac{\\pi}{2}$."} | |
| {"formal": "theorem exercise_2_13 {f : \u2102 \u2192 \u2102}\n (hf : \u2200 z\u2080 : \u2102, \u2203 (s : set \u2102) (c : \u2115 \u2192 \u2102), is_open s \u2227 z\u2080 \u2208 s \u2227\n \u2200 z \u2208 s, tendsto (\u03bb n, \u2211 i in finset.range n, (c i) * (z - z\u2080)^i) at_top (\ud835\udcdd (f z\u2080))\n \u2227 \u2203 i, c i = 0) :\n \u2203 (c : \u2115 \u2192 \u2102) (n : \u2115), f = \u03bb z, \u2211 i in finset.range n, (c i) * z ^ n :=", "informal": "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."} | |
| {"formal": "theorem exercise_3_4 (a : \u211d) (ha : 0 < a) :\n tendsto (\u03bb y, \u222b x in -y..y, x * real.sin x / (x ^ 2 + a ^ 2))\n at_top (\ud835\udcdd (real.pi * (real.exp (-a)))) :=", "informal": "Show that $ \\int_{-\\infty}^{\\infty} \\frac{x \\sin x}{x^2 + a^2} dx = \\pi e^{-a}$ for $a > 0$."} | |
| {"formal": "theorem exercise_3_14 {f : \u2102 \u2192 \u2102} (hf : differentiable \u2102 f)\n (hf_inj : function.injective f) :\n \u2203 (a b : \u2102), f = (\u03bb z, a * z + b) \u2227 a \u2260 0 :=", "informal": "Prove that all entire functions that are also injective take the form $f(z) = az + b$, $a, b \\in \\mathbb{C}$ and $a \\neq 0$."} | |
| {"formal": "theorem exercise_5_1 (f : \u2102 \u2192 \u2102) (hf : differentiable_on \u2102 f (ball 0 1))\n (hb : bounded (set.range f)) (h0 : f \u2260 0) (zeros : \u2115 \u2192 \u2102) (hz : \u2200 n, f (zeros n) = 0)\n (hzz : set.range zeros = {z | f z = 0 \u2227 z \u2208 (ball (0 : \u2102) 1)}) :\n \u2203 (z : \u2102), tendsto (\u03bb n, (\u2211 i in finset.range n, (1 - zeros i))) at_top (\ud835\udcdd z) :=", "informal": "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 $\\sum_{n}\\left(1-\\left|z_{n}\\right|\\right)<\\infty$."} | |
| {"formal": "theorem exercise_2018_a5 (f : \u211d \u2192 \u211d) (hf : cont_diff \u211d \u22a4 f)\n (hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : \u2200 x, f x \u2265 0) :\n \u2203 (n : \u2115) (x : \u211d), iterated_deriv n f x = 0 :=", "informal": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be an infinitely differentiable function satisfying $f(0)=0, f(1)=1$, and $f(x) \\geq 0$ for all $x \\in$ $\\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x)<0$."} | |
| {"formal": "theorem exercise_2018_b4 (a : \u211d) (x : \u2115 \u2192 \u211d) (hx0 : x 0 = a)\n (hx1 : x 1 = a) \n (hxn : \u2200 n : \u2115, n \u2265 2 \u2192 x (n+1) = 2*(x n)*(x (n-1)) - x (n-2))\n (h : \u2203 n, x n = 0) : \n \u2203 c, function.periodic x c :=", "informal": "Given a real number $a$, we define a sequence by $x_{0}=1$, $x_{1}=x_{2}=a$, and $x_{n+1}=2 x_{n} x_{n-1}-x_{n-2}$ for $n \\geq 2$. Prove that if $x_{n}=0$ for some $n$, then the sequence is periodic."} | |
| {"formal": "theorem exercise_2014_a5 (P : \u2115 \u2192 polynomial \u2124) \n (hP : \u2200 n, P n = \u2211 (i : fin n), (n+1) * X ^ n) : \n \u2200 (j k : \u2115), j \u2260 k \u2192 is_coprime (P j) (P k) :=", "informal": "Let"} | |
| {"formal": "theorem exercise_2001_a5 : \n \u2203! a n : \u2115, a > 0 \u2227 n > 0 \u2227 a^(n+1) - (a+1)^n = 2001 :=", "informal": "Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$."} | |
| {"formal": "theorem exercise_1999_b4 (f : \u211d \u2192 \u211d) (hf: cont_diff \u211d 3 f) \n (hf1 : \u2200 (n \u2264 3) (x : \u211d), iterated_deriv n f x > 0) \n (hf2 : \u2200 x : \u211d, iterated_deriv 3 f x \u2264 f x) : \n \u2200 x : \u211d, deriv f x < 2 * f x :=", "informal": "Let $f$ be a real function with a continuous third derivative such that $f(x), f^{\\prime}(x), f^{\\prime \\prime}(x), f^{\\prime \\prime \\prime}(x)$ are positive for all $x$. Suppose that $f^{\\prime \\prime \\prime}(x) \\leq f(x)$ for all $x$. Show that $f^{\\prime}(x)<2 f(x)$ for all $x$."} | |
| {"formal": "theorem exercise_1998_b6 (a b c : \u2124) : \n \u2203 n : \u2124, n > 0 \u2227 \u00ac \u2203 m : \u2124, sqrt (n^3 + a*n^2 + b*n + c) = m :=", "informal": "Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\\sqrt{n^3+a n^2+b n+c}$ is not an integer."} | |
| {"formal": "theorem exercise_2_26 {M : Type*} [topological_space M]\n (U : set M) : is_open U \u2194 \u2200 x \u2208 U, \u00ac cluster_pt x (\ud835\udcdf U\u1d9c) :=", "informal": "Prove that a set $U \\subset M$ is open if and only if none of its points are limits of its complement."} | |
| {"formal": "theorem exercise_2_32a (A : set \u2115) : is_clopen A :=", "informal": "Show that every subset of $\\mathbb{N}$ is clopen."} | |
| {"formal": "theorem exercise_2_46 {M : Type*} [metric_space M]\n {A B : set M} (hA : is_compact A) (hB : is_compact B)\n (hAB : disjoint A B) (hA\u2080 : A \u2260 \u2205) (hB\u2080 : B \u2260 \u2205) :\n \u2203 a\u2080 b\u2080, a\u2080 \u2208 A \u2227 b\u2080 \u2208 B \u2227 \u2200 (a : M) (b : M),\n a \u2208 A \u2192 b \u2208 B \u2192 dist a\u2080 b\u2080 \u2264 dist a b :=", "informal": "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 $d(a_0, b_0) \\leq d(a, b)$."} | |
| {"formal": "theorem exercise_2_92 {\u03b1 : Type*} [topological_space \u03b1]\n {s : \u2115 \u2192 set \u03b1}\n (hs : \u2200 i, is_compact (s i))\n (hs : \u2200 i, (s i).nonempty)\n (hs : \u2200 i, (s i) \u2283 (s (i + 1))) :\n (\u22c2 i, s i).nonempty :=", "informal": "Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty."} | |
| {"formal": "theorem exercise_3_1 {f : \u211d \u2192 \u211d}\n (hf : \u2200 x y, |f x - f y| \u2264 |x - y| ^ 2) :\n \u2203 c, f = \u03bb x, c :=", "informal": "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."} | |
| {"formal": "theorem exercise_3_63a (p : \u211d) (f : \u2115 \u2192 \u211d) (hp : p > 1)\n (h : f = \u03bb k, (1 : \u211d) / (k * (log k) ^ p)) :\n \u2203 l, tendsto f at_top (\ud835\udcdd l) :=", "informal": "Prove that $\\sum 1/k(\\log(k))^p$ converges when $p > 1$."} | |
| {"formal": "theorem exercise_4_15a {\u03b1 : Type*}\n (a b : \u211d) (F : set (\u211d \u2192 \u211d)) :\n (\u2200 (x : \u211d) (\u03b5 > 0), \u2203 (U \u2208 (\ud835\udcdd x)),\n (\u2200 (y z \u2208 U) (f : \u211d \u2192 \u211d), f \u2208 F \u2192 (dist (f y) (f z) < \u03b5)))\n \u2194\n \u2203 (\u03bc : \u211d \u2192 \u211d), \u2200 (x : \u211d), (0 : \u211d) \u2264 \u03bc x \u2227 tendsto \u03bc (\ud835\udcdd 0) (\ud835\udcdd 0) \u2227\n (\u2200 (s t : \u211d) (f : \u211d \u2192 \u211d), f \u2208 F \u2192 |(f s) - (f t)| \u2264 \u03bc (|s - t|)) :=", "informal": "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."} | |
| {"formal": "theorem exercise_2_1_18 {G : Type*} [group G] \n [fintype G] (hG2 : even (fintype.card G)) :\n \u2203 (a : G), a \u2260 1 \u2227 a = a\u207b\u00b9 :=", "informal": "If $G$ is a finite group of even order, show that there must be an element $a \\neq e$ such that $a=a^{-1}$."} | |
| {"formal": "theorem exercise_2_1_26 {G : Type*} [group G] \n [fintype G] (a : G) : \u2203 (n : \u2115), a ^ n = 1 :=", "informal": "If $G$ is a finite group, prove that, given $a \\in G$, there is a positive integer $n$, depending on $a$, such that $a^n = e$."} | |
| {"formal": "theorem exercise_2_2_3 {G : Type*} [group G]\n {P : \u2115 \u2192 Prop} {hP : P = \u03bb i, \u2200 a b : G, (a*b)^i = a^i * b^i}\n (hP1 : \u2203 n : \u2115, P n \u2227 P (n+1) \u2227 P (n+2)) : comm_group G :=", "informal": "If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian."} | |
| {"formal": "theorem exercise_2_2_6c {G : Type*} [group G] {n : \u2115} (hn : n > 1) \n (h : \u2200 (a b : G), (a * b) ^ n = a ^ n * b ^ n) :\n \u2200 (a b : G), (a * b * a\u207b\u00b9 * b\u207b\u00b9) ^ (n * (n - 1)) = 1 :=", "informal": "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 $\\left(a b a^{-1} b^{-1}\\right)^{n(n-1)}=e$."} | |
| {"formal": "theorem exercise_2_3_16 {G : Type*} [group G]\n (hG : \u2200 H : subgroup G, H = \u22a4 \u2228 H = \u22a5) :\n is_cyclic G \u2227 \u2203 (p : \u2115) (fin : fintype G), nat.prime p \u2227 @card G fin = p :=", "informal": "If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number."} | |
| {"formal": "theorem exercise_2_5_23 {G : Type*} [group G] \n (hG : \u2200 (H : subgroup G), H.normal) (a b : G) :\n \u2203 (j : \u2124) , b*a = a^j * b:=", "informal": "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 $j$."} | |
| {"formal": "theorem exercise_2_5_31 {G : Type*} [comm_group G] [fintype G]\n {p m n : \u2115} (hp : nat.prime p) (hp1 : \u00ac p \u2223 m) (hG : card G = p^n*m)\n {H : subgroup G} [fintype H] (hH : card H = p^n) : \n characteristic H :=", "informal": "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 $G$."} | |
| {"formal": "theorem exercise_2_5_43 (G : Type*) [group G] [fintype G]\n (hG : card G = 9) :\n comm_group G :=", "informal": "Prove that a group of order 9 must be abelian."} | |
| {"formal": "theorem exercise_2_5_52 {G : Type*} [group G] [fintype G]\n (\u03c6 : G \u2243* G) {I : finset G} (hI : \u2200 x \u2208 I, \u03c6 x = x\u207b\u00b9)\n (hI1 : (0.75 : \u211a) * card G \u2264 card I) : \n \u2200 x : G, \u03c6 x = x\u207b\u00b9 \u2227 \u2200 x y : G, x*y = y*x :=", "informal": "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."} | |
| {"formal": "theorem exercise_2_7_7 {G : Type*} [group G] {G' : Type*} [group G']\n (\u03c6 : G \u2192* G') (N : subgroup G) [N.normal] : \n (map \u03c6 N).normal :=", "informal": "If $\\varphi$ is a homomorphism of $G$ onto $G'$ and $N \\triangleleft G$, show that $\\varphi(N) \\triangleleft G'$."} | |
| {"formal": "theorem exercise_2_8_15 {G H: Type*} [fintype G] [group G] [fintype H]\n [group H] {p q : \u2115} (hp : nat.prime p) (hq : nat.prime q) \n (h : p > q) (h1 : q \u2223 p - 1) (hG : card G = p*q) (hH : card G = p*q) :\n G \u2243* H :=", "informal": "Prove that if $p > q$ are two primes such that $q \\mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic."} | |
| {"formal": "theorem exercise_2_10_1 {G : Type*} [group G] (A : subgroup G) \n [A.normal] {b : G} (hp : nat.prime (order_of b)) :\n A \u2293 (closure {b}) = \u22a5 :=", "informal": "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 $A \\cap (b) = (e)$."} | |
| {"formal": "theorem exercise_2_11_7 {G : Type*} [group G] {p : \u2115} (hp : nat.prime p)\n {P : sylow p G} (hP : P.normal) : \n characteristic (P : subgroup G) :=", "informal": "If $P \\triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\\varphi(P) = P$ for every automorphism $\\varphi$ of $G$."} | |
| {"formal": "theorem exercise_3_2_21 {\u03b1 : Type*} [fintype \u03b1] {\u03c3 \u03c4: equiv.perm \u03b1} \n (h1 : \u2200 a : \u03b1, \u03c3 a = a \u2194 \u03c4 a \u2260 a) (h2 : \u03c4 \u2218 \u03c3 = id) : \n \u03c3 = 1 \u2227 \u03c4 = 1 :=", "informal": "If $\\sigma, \\tau$ are two permutations that disturb no common element and $\\sigma \\tau = e$, prove that $\\sigma = \\tau = e$."} | |
| {"formal": "theorem exercise_4_1_34 : equiv.perm (fin 3) \u2243* general_linear_group (fin 2) (zmod 2) :=", "informal": "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."} | |
| {"formal": "theorem exercise_4_2_6 {R : Type*} [ring R] (a x : R) \n (h : a ^ 2 = 0) : a * (a * x + x * a) = (x + x * a) * a :=", "informal": "If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$."} | |
| {"formal": "theorem exercise_4_3_1 {R : Type*} [comm_ring R] (a : R) :\n \u2203 I : ideal R, {x : R | x*a=0} = I :=", "informal": "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 $R$."} | |
| {"formal": "theorem exercise_4_4_9 (p : \u2115) (hp : nat.prime p) :\n (\u2203 S : finset (zmod p), S.card = (p-1)/2 \u2227 \u2203 x : zmod p, x^2 = p) \u2227 \n (\u2203 S : finset (zmod p), S.card = (p-1)/2 \u2227 \u00ac \u2203 x : zmod p, x^2 = p) :=", "informal": "Show that $(p - 1)/2$ of the numbers $1, 2, \\ldots, p - 1$ are quadratic residues and $(p - 1)/2$ are quadratic nonresidues $\\mod p$."} | |
| {"formal": "theorem exercise_4_5_23 {p q: polynomial (zmod 7)} \n (hp : p = X^3 - 2) (hq : q = X^3 + 2) : \n irreducible p \u2227 irreducible q \u2227 \n (nonempty $ polynomial (zmod 7) \u29f8 ideal.span ({p} : set $ polynomial $ zmod 7) \u2243+*\n polynomial (zmod 7) \u29f8 ideal.span ({q} : set $ polynomial $ zmod 7)) :=", "informal": "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."} | |
| {"formal": "theorem exercise_4_6_2 : irreducible (X^3 + 3*X + 2 : polynomial \u211a) :=", "informal": "Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$."} | |
| {"formal": "theorem exercise_5_1_8 {p m n: \u2115} {F : Type*} [field F] \n (hp : nat.prime p) (hF : char_p F p) (a b : F) (hm : m = p ^ n) : \n (a + b) ^ m = a^m + b^m :=", "informal": "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 $n$."} | |
| {"formal": "theorem exercise_5_3_7 {K : Type*} [field K] {F : subfield K} \n {a : K} (ha : is_algebraic F (a ^ 2)) : is_algebraic F a :=", "informal": "If $a \\in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$."} | |
| {"formal": "theorem exercise_5_4_3 {a : \u2102} {p : \u2102 \u2192 \u2102} \n (hp : p = \u03bb x, x^5 + real.sqrt 2 * x^3 + real.sqrt 5 * x^2 + \n real.sqrt 7 * x + 11)\n (ha : p a = 0) : \n \u2203 p : polynomial \u2102 , p.degree < 80 \u2227 a \u2208 p.roots \u2227 \n \u2200 n : p.support, \u2203 a b : \u2124, p.coeff n = a / b :=", "informal": "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."} | |
| {"formal": "theorem exercise_5_6_14 {p m n: \u2115} (hp : nat.prime p) {F : Type*} \n [field F] [char_p F p] (hm : m = p ^ n) : \n card (root_set (X ^ m - X : polynomial F) F) = m :=", "informal": "If $F$ is of characteristic $p \\neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct."} | |
| {"formal": "theorem exercise_2_3_2 {G : Type*} [group G] (a b : G) :\n \u2203 g : G, b* a = g * a * b * g\u207b\u00b9 :=", "informal": "Prove that the products $a b$ and $b a$ are conjugate elements in a group."} | |
| {"formal": "theorem exercise_2_8_6 {G H : Type*} [group G] [group H] :\n center (G \u00d7 H) \u2243* (center G) \u00d7 (center H) :=", "informal": "Prove that the center of the product of two groups is the product of their centers."} | |
| {"formal": "theorem exercise_3_2_7 {F : Type*} [field F] {G : Type*} [field G]\n (\u03c6 : F \u2192+* G) : injective \u03c6 :=", "informal": "Prove that every homomorphism of fields is injective."} | |
| {"formal": "theorem exercise_3_7_2 {K V : Type*} [field K] [add_comm_group V]\n [module K V] {\u03b9 : Type*} [fintype \u03b9] (\u03b3 : \u03b9 \u2192 submodule K V)\n (h : \u2200 i : \u03b9, \u03b3 i \u2260 \u22a4) :\n (\u22c2 (i : \u03b9), (\u03b3 i : set V)) \u2260 \u22a4 :=", "informal": "Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces."} | |
| {"formal": "theorem exercise_6_4_2 {G : Type*} [group G] [fintype G] {p q : \u2115}\n (hp : prime p) (hq : prime q) (hG : card G = p*q) :\n is_simple_group G \u2192 false :=", "informal": "Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple."} | |
| {"formal": "theorem exercise_6_4_12 {G : Type*} [group G] [fintype G]\n (hG : card G = 224) :\n is_simple_group G \u2192 false :=", "informal": "Prove that no group of order 224 is simple."} | |
| {"formal": "theorem exercise_10_1_13 {R : Type*} [ring R] {x : R}\n (hx : is_nilpotent x) : is_unit (1 + x) :=", "informal": "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 $R$."} | |
| {"formal": "theorem exercise_10_6_7 {I : ideal gaussian_int}\n (hI : I \u2260 \u22a5) : \u2203 (z : I), z \u2260 0 \u2227 (z : gaussian_int).im = 0 :=", "informal": "Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer."} | |
| {"formal": "theorem exercise_10_4_7a {R : Type*} [comm_ring R] [no_zero_divisors R]\n (I J : ideal R) (hIJ : I + J = \u22a4) : I * J = I \u2293 J :=", "informal": "Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \\cap J$."} | |
| {"formal": "theorem exercise_11_2_13 (a b : \u2124) :\n (of_int a : gaussian_int) \u2223 of_int b \u2192 a \u2223 b :=", "informal": "If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\\mathbb{Z}$."} | |
| {"formal": "theorem exercise_11_4_6a {F : Type*} [field F] [fintype F] (hF : card F = 7) :\n irreducible (X ^ 2 + 1 : polynomial F) :=", "informal": "Prove that $x^2+x+1$ is irreducible in the field $\\mathbb{F}_2$."} | |
| {"formal": "theorem exercise_11_4_6c : irreducible (X^3 - 9 : polynomial (zmod 31)) :=", "informal": "Prove that $x^3 - 9$ is irreducible in $\\mathbb{F}_{31}$."} | |
| {"formal": "theorem exercise_11_13_3 (N : \u2115):\n \u2203 p \u2265 N, nat.prime p \u2227 p + 1 \u2261 0 [MOD 4] :=", "informal": "Prove that there are infinitely many primes congruent to $-1$ (modulo $4$)."} | |
| {"formal": "theorem exercise_13_6_10 {K : Type*} [field K] [fintype K\u02e3] :\n \u220f (x : K\u02e3), x = -1 :=", "informal": "Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$."} | |
| {"formal": "theorem exercise_1_1_2a : \u2203 a b : \u2124, a - b \u2260 b - a :=", "informal": "Prove the the operation $\\star$ on $\\mathbb{Z}$ defined by $a\\star b=a-b$ is not commutative."} | |
| {"formal": "theorem exercise_1_1_4 (n : \u2115) : \n \u2200 (a b c : \u2115), (a * b) * c \u2261 a * (b * c) [ZMOD n] :=", "informal": "Prove that the multiplication of residue class $\\mathbb{Z}/n\\mathbb{Z}$ is associative."} | |
| {"formal": "theorem exercise_1_1_15 {G : Type*} [group G] (as : list G) :\n as.prod\u207b\u00b9 = (as.reverse.map (\u03bb x, x\u207b\u00b9)).prod :=", "informal": "Prove that $(a_1a_2\\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\\dots a_1^{-1}$ for all $a_1, a_2, \\dots, a_n\\in G$."} | |
| {"formal": "theorem exercise_1_1_17 {G : Type*} [group G] {x : G} {n : \u2115}\n (hxn: order_of x = n) :\n x\u207b\u00b9 = x ^ (n - 1 : \u2124) :=", "informal": "Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$."} | |
| {"formal": "theorem exercise_1_1_20 {G : Type*} [group G] {x : G} :\n order_of x = order_of x\u207b\u00b9 :=", "informal": "For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order."} | |
| {"formal": "theorem exercise_1_1_22b {G: Type*} [group G] (a b : G) : \n order_of (a * b) = order_of (b * a) :=", "informal": "Deduce that $|a b|=|b a|$ for all $a, b \\in G$."} | |
| {"formal": "theorem exercise_1_1_29 {A B : Type*} [group A] [group B] :\n \u2200 x y : A \u00d7 B, x*y = y*x \u2194 (\u2200 x y : A, x*y = y*x) \u2227 \n (\u2200 x y : B, x*y = y*x) :=", "informal": "Prove that $A \\times B$ is an abelian group if and only if both $A$ and $B$ are abelian."} | |
| {"formal": "theorem exercise_1_3_8 : infinite (equiv.perm \u2115) :=", "informal": "Prove that if $\\Omega=\\{1,2,3, \\ldots\\}$ then $S_{\\Omega}$ is an infinite group"} | |
| {"formal": "theorem exercise_1_6_11 {A B : Type*} [group A] [group B] : \n A \u00d7 B \u2243* B \u00d7 A :=", "informal": "Let $A$ and $B$ be groups. Prove that $A \\times B \\cong B \\times A$."} | |
| {"formal": "theorem exercise_1_6_23 {G : Type*} \n [group G] (\u03c3 : mul_aut G) (hs : \u2200 g : G, \u03c3 g = 1 \u2192 g = 1) \n (hs2 : \u2200 g : G, \u03c3 (\u03c3 g) = g) :\n \u2200 x y : G, x*y = y*x :=", "informal": "Let $G$ be a finite group which possesses an automorphism $\\sigma$ such that $\\sigma(g)=g$ if and only if $g=1$. If $\\sigma^{2}$ is the identity map from $G$ to $G$, prove that $G$ is abelian."} | |
| {"formal": "theorem exercise_2_1_13 (H : add_subgroup \u211a) {x : \u211a} \n (hH : x \u2208 H \u2192 (1 / x) \u2208 H):\n H = \u22a5 \u2228 H = \u22a4 :=", "informal": "Let $H$ be a subgroup of the additive group of rational numbers with the property that $1 / x \\in H$ for every nonzero element $x$ of $H$. Prove that $H=0$ or $\\mathbb{Q}$."} | |
| {"formal": "theorem exercise_2_4_16a {G : Type*} [group G] {H : subgroup G} \n (hH : H \u2260 \u22a4) : \n \u2203 M : subgroup G, M \u2260 \u22a4 \u2227\n \u2200 K : subgroup G, M \u2264 K \u2192 K = M \u2228 K = \u22a4 \u2227 \n H \u2264 M :=", "informal": "A subgroup $M$ of a group $G$ is called a maximal subgroup if $M \\neq G$ and the only subgroups of $G$ which contain $M$ are $M$ and $G$. Prove that if $H$ is a proper subgroup of the finite group $G$ then there is a maximal subgroup of $G$ containing $H$."} | |
| {"formal": "theorem exercise_2_4_16c {n : \u2115} (H : add_subgroup (zmod n)) : \n \u2203 p : \u2115, nat.prime p \u2227 H = add_subgroup.closure {p} \u2194 \n H \u2260 \u22a4 \u2227 \u2200 K : add_subgroup (zmod n), H \u2264 K \u2192 K = H \u2228 K = \u22a4 :=", "informal": "Show that if $G=\\langle x\\rangle$ is a cyclic group of order $n \\geq 1$ then a subgroup $H$ is maximal if and only $H=\\left\\langle x^{p}\\right\\rangle$ for some prime $p$ dividing $n$."} | |
| {"formal": "theorem exercise_3_1_22a (G : Type*) [group G] (H K : subgroup G) \n [subgroup.normal H] [subgroup.normal K] :\n subgroup.normal (H \u2293 K) :=", "informal": "Prove that if $H$ and $K$ are normal subgroups of a group $G$ then their intersection $H \\cap K$ is also a normal subgroup of $G$."} | |
| {"formal": "theorem exercise_3_2_8 {G : Type*} [group G] (H K : subgroup G)\n [fintype H] [fintype K] \n (hHK : nat.coprime (fintype.card H) (fintype.card K)) : \n H \u2293 K = \u22a5 :=", "informal": "Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \\cap K=1$."} | |
| {"formal": "theorem exercise_3_2_16 (p : \u2115) (hp : nat.prime p) (a : \u2115) :\n nat.coprime a p \u2192 a ^ p \u2261 a [ZMOD p] :=", "informal": "Use Lagrange's Theorem in the multiplicative group $(\\mathbb{Z} / p \\mathbb{Z})^{\\times}$to prove Fermat's Little Theorem: if $p$ is a prime then $a^{p} \\equiv a(\\bmod p)$ for all $a \\in \\mathbb{Z}$."} | |
| {"formal": "theorem exercise_3_3_3 {p : primes} {G : Type*} [group G] \n {H : subgroup G} [hH : H.normal] (hH1 : H.index = p) : \n \u2200 K : subgroup G, K \u2264 H \u2228 H \u2294 K = \u22a4 \u2228 (K \u2293 H).relindex K = p :=", "informal": "Prove that if $H$ is a normal subgroup of $G$ of prime index $p$ then for all $K \\leq G$ either $K \\leq H$, or $G=H K$ and $|K: K \\cap H|=p$."} | |
| {"formal": "theorem exercise_3_4_4 {G : Type*} [comm_group G] [fintype G] {n : \u2115}\n (hn : n \u2223 (fintype.card G)) :\n \u2203 (H : subgroup G) (H_fin : fintype H), @card H H_fin = n :=", "informal": "Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each positive divisor $n$ of its order."} | |
| {"formal": "theorem exercise_3_4_5b {G : Type*} [group G] [is_solvable G] \n (H : subgroup G) [subgroup.normal H] : \n is_solvable (G \u29f8 H) :=", "informal": "Prove that quotient groups of a solvable group are solvable."} | |
| {"formal": "theorem exercise_4_2_8 {G : Type*} [group G] {H : subgroup G} \n {n : \u2115} (hn : n > 0) (hH : H.index = n) : \n \u2203 K \u2264 H, K.normal \u2227 K.index \u2264 n.factorial :=", "informal": "Prove that if $H$ has finite index $n$ then there is a normal subgroup $K$ of $G$ with $K \\leq H$ and $|G: K| \\leq n!$."} | |
| {"formal": "theorem exercise_4_2_9a {G : Type*} [fintype G] [group G] {p \u03b1 : \u2115} \n (hp : p.prime) (ha : \u03b1 > 0) (hG : card G = p ^ \u03b1) : \n \u2200 H : subgroup G, H.index = p \u2192 H.normal :=", "informal": "Prove that if $p$ is a prime and $G$ is a group of order $p^{\\alpha}$ for some $\\alpha \\in \\mathbb{Z}^{+}$, then every subgroup of index $p$ is normal in $G$."} | |
| {"formal": "theorem exercise_4_4_2 {G : Type*} [fintype G] [group G] \n {p q : nat.primes} (hpq : p \u2260 q) (hG : card G = p*q) : \n is_cyclic G :=", "informal": "Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic."} | |
| {"formal": "theorem exercise_4_4_6b : \n \u2203 (G : Type*) (hG : group G) (H : @subgroup G hG), @characteristic G hG H \u2227 \u00ac @subgroup.normal G hG H :=", "informal": "Prove that there exists a normal subgroup that is not characteristic."} | |
| {"formal": "theorem exercise_4_4_8a {G : Type*} [group G] (H K : subgroup G) \n (hHK : H \u2264 K) [hHK1 : (H.subgroup_of K).normal] [hK : K.normal] : \n H.normal :=", "informal": "Let $G$ be a group with subgroups $H$ and $K$ with $H \\leq K$. Prove that if $H$ is characteristic in $K$ and $K$ is normal in $G$ then $H$ is normal in $G$."} | |
| {"formal": "theorem exercise_4_5_13 {G : Type*} [group G] [fintype G]\n (hG : card G = 56) :\n \u2203 (p : \u2115) (P : sylow p G), P.normal :=", "informal": "Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order."} | |
| {"formal": "theorem exercise_4_5_15 {G : Type*} [group G] [fintype G] \n (hG : card G = 351) : \n \u2203 (p : \u2115) (P : sylow p G), P.normal :=", "informal": "Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order."} | |
| {"formal": "theorem exercise_4_5_17 {G : Type*} [fintype G] [group G] \n (hG : card G = 105) : \n nonempty(sylow 5 G) \u2227 nonempty(sylow 7 G) :=", "informal": "Prove that if $|G|=105$ then $G$ has a normal Sylow 5 -subgroup and a normal Sylow 7-subgroup."} | |
| {"formal": "theorem exercise_4_5_19 {G : Type*} [fintype G] [group G] \n (hG : card G = 6545) : \u00ac is_simple_group G :=", "informal": "Prove that if $|G|=6545$ then $G$ is not simple."} | |
| {"formal": "theorem exercise_4_5_21 {G : Type*} [fintype G] [group G]\n (hG : card G = 2907) : \u00ac is_simple_group G :=", "informal": "Prove that if $|G|=2907$ then $G$ is not simple."} | |
| {"formal": "theorem exercise_4_5_23 {G : Type*} [fintype G] [group G]\n (hG : card G = 462) : \u00ac is_simple_group G :=", "informal": "Prove that if $|G|=462$ then $G$ is not simple."} | |
| {"formal": "theorem exercise_4_5_33 {G : Type*} [group G] [fintype G] {p : \u2115} \n (P : sylow p G) [hP : P.normal] (H : subgroup G) [fintype H] : \n \u2200 R : sylow p H, R.to_subgroup = (H \u2293 P.to_subgroup).subgroup_of H \u2227\n nonempty (sylow p H) :=", "informal": "Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P \\cap H$ is the unique Sylow $p$-subgroup of $H$."} | |
| {"formal": "theorem exercise_7_1_2 {R : Type*} [ring R] {u : R}\n (hu : is_unit u) : is_unit (-u) :=", "informal": "Prove that if $u$ is a unit in $R$ then so is $-u$."} | |
| {"formal": "theorem exercise_7_1_12 {F : Type*} [field F] {K : subring F}\n (hK : (1 : F) \u2208 K) : is_domain K :=", "informal": "Prove that any subring of a field which contains the identity is an integral domain."} | |
| {"formal": "theorem exercise_7_2_2 {R : Type*} [ring R] (p : polynomial R) :\n p \u2223 0 \u2194 \u2203 b : R, b \u2260 0 \u2227 b \u2022 p = 0 :=", "informal": "Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\\cdots+a_{1} x+a_{0}$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in $R[x]$ if and only if there is a nonzero $b \\in R$ such that $b p(x)=0$."} | |
| {"formal": "theorem exercise_7_3_16 {R S : Type*} [ring R] [ring S] \n {\u03c6 : R \u2192+* S} (hf : surjective \u03c6) : \n \u03c6 '' (center R) \u2282 center S :=", "informal": "Let $\\varphi: R \\rightarrow S$ be a surjective homomorphism of rings. Prove that the image of the center of $R$ is contained in the center of $S$."} | |
| {"formal": "theorem exercise_7_4_27 {R : Type*} [comm_ring R] (hR : (0 : R) \u2260 1) \n {a : R} (ha : is_nilpotent a) (b : R) : \n is_unit (1-a*b) :=", "informal": "Let $R$ be a commutative ring with $1 \\neq 0$. Prove that if $a$ is a nilpotent element of $R$ then $1-a b$ is a unit for all $b \\in R$."} | |
| {"formal": "theorem exercise_8_2_4 {R : Type*} [ring R][no_zero_divisors R] \n [cancel_comm_monoid_with_zero R] [gcd_monoid R]\n (h1 : \u2200 a b : R, a \u2260 0 \u2192 b \u2260 0 \u2192 \u2203 r s : R, gcd a b = r*a + s*b)\n (h2 : \u2200 a : \u2115 \u2192 R, (\u2200 i j : \u2115, i < j \u2192 a i \u2223 a j) \u2192 \n \u2203 N : \u2115, \u2200 n \u2265 N, \u2203 u : R, is_unit u \u2227 a n = u * a N) : \n is_principal_ideal_ring R :=", "informal": "Let $R$ be an integral domain. Prove that if the following two conditions hold then $R$ is a Principal Ideal Domain: (i) any two nonzero elements $a$ and $b$ in $R$ have a greatest common divisor which can be written in the form $r a+s b$ for some $r, s \\in R$, and (ii) if $a_{1}, a_{2}, a_{3}, \\ldots$ are nonzero elements of $R$ such that $a_{i+1} \\mid a_{i}$ for all $i$, then there is a positive integer $N$ such that $a_{n}$ is a unit times $a_{N}$ for all $n \\geq N$."} | |
| {"formal": "theorem exercise_8_3_5a {n : \u2124} (hn0 : n > 3) (hn1 : squarefree n) : \n irreducible (2 :zsqrtd $ -n) \u2227 \n irreducible (\u27e80, 1\u27e9 : zsqrtd $ -n) \u2227 \n irreducible (1 + \u27e80, 1\u27e9 : zsqrtd $ -n) :=", "informal": "Let $R=\\mathbb{Z}[\\sqrt{-n}]$ where $n$ is a squarefree integer greater than 3. Prove that $2, \\sqrt{-n}$ and $1+\\sqrt{-n}$ are irreducibles in $R$."} | |
| {"formal": "theorem exercise_8_3_6b {q : \u2115} (hq0 : q.prime) \n (hq1 : q \u2261 3 [ZMOD 4]) {R : Type*} [ring R]\n (hR : R = (gaussian_int \u29f8 ideal.span ({q} : set gaussian_int))) : \n is_field R \u2227 \u2203 finR : fintype R, @card R finR = q^2 :=", "informal": "Let $q \\in \\mathbb{Z}$ be a prime with $q \\equiv 3 \\bmod 4$. Prove that the quotient ring $\\mathbb{Z}[i] /(q)$ is a field with $q^{2}$ elements."} | |
| {"formal": "theorem exercise_9_1_10 {f : \u2115 \u2192 mv_polynomial \u2115 \u2124} \n (hf : f = \u03bb i, X i * X (i+1)): \n infinite (minimal_primes (mv_polynomial \u2115 \u2124 \u29f8 ideal.span (range f))) :=", "informal": "Prove that the ring $\\mathbb{Z}\\left[x_{1}, x_{2}, x_{3}, \\ldots\\right] /\\left(x_{1} x_{2}, x_{3} x_{4}, x_{5} x_{6}, \\ldots\\right)$ contains infinitely many minimal prime ideals."} | |
| {"formal": "theorem exercise_9_4_2a : irreducible (X^4 - 4*X^3 + 6 : polynomial \u2124) :=", "informal": "Prove that $x^4-4x^3+6$ is irreducible in $\\mathbb{Z}[x]$."} | |
| {"formal": "theorem exercise_9_4_2c : irreducible \n (X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : polynomial \u2124) :=", "informal": "Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\\mathbb{Z}[x]$."} | |
| {"formal": "theorem exercise_9_4_9 : \n irreducible (X^2 - C sqrtd : polynomial (zsqrtd 2)) :=", "informal": "Prove that the polynomial $x^{2}-\\sqrt{2}$ is irreducible over $\\mathbb{Z}[\\sqrt{2}]$. You may assume that $\\mathbb{Z}[\\sqrt{2}]$ is a U.F.D."} | |
| {"formal": "theorem exercise_11_1_13 {\u03b9 : Type*} [fintype \u03b9] : \n (\u03b9 \u2192 \u211d) \u2243\u2097[\u211a] \u211d :=", "informal": "Prove that as vector spaces over $\\mathbb{Q}, \\mathbb{R}^n \\cong \\mathbb{R}$, for all $n \\in \\mathbb{Z}^{+}$."} | |