diff --git "a/data.json" "b/data.json" deleted file mode 100644--- "a/data.json" +++ /dev/null @@ -1,2620 +0,0 @@ -[ - { - "id": "Rudin|exercise_1_1a", - "formal_statement": "theorem exercise_1_1a\n (x : \u211d) (y : \u211a) :\n ( irrational x ) -> irrational ( x + y ) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational.\n", - "nl_proof": "\\begin{proof}\n\n If $r$ and $r+x$ were both rational, then $x=r+x-r$ would also be rational.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_1b", - "formal_statement": "theorem exercise_1_1b\n(x : \u211d)\n(y : \u211a)\n(h : y \u2260 0)\n: ( irrational x ) -> irrational ( x * y ) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.\n", - "nl_proof": "\\begin{proof}\n\n If $r x$ were rational, then $x=\\frac{r x}{r}$ would also be rational.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_2", - "formal_statement": "theorem exercise_1_2\n: \u00ac \u2203 (x : \u211a), ( x ^ 2 = 12 ) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that there is no rational number whose square is $12$.\n", - "nl_proof": "\\begin{proof}\n\n Suppose $m^2=12 n^2$, where $m$ and $n$ have no common factor. It follows that $m$ must be even, and therefore $n$ must be odd. Let $m=2 r$. Then we have $r^2=3 n^2$, so that $r$ is also odd. Let $r=2 s+1$ and $n=2 t+1$. Then\n\n$$\n\n4 s^2+4 s+1=3\\left(4 t^2+4 t+1\\right)=12 t^2+12 t+3,\n\n$$\n\nso that\n\n$$\n\n4\\left(s^2+s-3 t^2-3 t\\right)=2 .\n\n$$\n\nBut this is absurd, since 2 cannot be a multiple of 4 .\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_4", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $\\alpha \\leq \\beta$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since $E$ is nonempty, there exists $x \\in E$. Then by definition of lower and upper bounds we have $\\alpha \\leq x \\leq \\beta$, and hence by property $i i$ in the definition of an ordering, we have $\\alpha<\\beta$ unless $\\alpha=x=\\beta$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_5", - "formal_statement": "theorem exercise_1_5\n (A minus_A : set \u211d) (hA : A.nonempty) (hA_bdd_below : bdd_below A)\n (hminus_A : minus_A = {x | -x \u2208 A}) :\n Inf A = Sup minus_A :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \\in A$. Prove that $\\inf A=-\\sup (-A)$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: We need to prove that $-\\sup (-A)$ is the greatest lower bound of $A$. For brevity, let $\\alpha=-\\sup (-A)$. We need to show that $\\alpha \\leq x$ for all $x \\in A$ and $\\alpha \\geq \\beta$ if $\\beta$ is any lower bound of $A$.\n\n\n\nSuppose $x \\in A$. Then, $-x \\in-A$, and, hence $-x \\leq \\sup (-A)$. It follows that $x \\geq-\\sup (-A)$, i.e., $\\alpha \\leq x$. Thus $\\alpha$ is a lower bound of $A$.\n\n\n\nNow let $\\beta$ be any lower bound of $A$. This means $\\beta \\leq x$ for all $x$ in $A$. Hence $-x \\leq-\\beta$ for all $x \\in A$, which says $y \\leq-\\beta$ for all $y \\in-A$. This means $-\\beta$ is an upper bound of $-A$. Hence $-\\beta \\geq \\sup (-A)$ by definition of sup, i.e., $\\beta \\leq-\\sup (-A)$, and so $-\\sup (-A)$ is the greatest lower bound of $A$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_8", - "formal_statement": "theorem exercise_1_8\n : \u00ac \u2203 (r : \u2102 \u2192 \u2102 \u2192 Prop), is_linear_order \u2102 r :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that no order can be defined in the complex field that turns it into an ordered field.\n", - "nl_proof": "\\begin{proof}\n\n Solution. By Part (a) of Proposition $1.18$, either $i$ or $-i$ must be positive. Hence $-1=i^2=(-i)^2$ must be positive. But then $1=(-1)^2$, must also be positive, and this contradicts Part $(a)$ of Proposition 1.18, since 1 and $-1$ cannot both be positive.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_11a", - "formal_statement": "theorem exercise_1_11a\n (z : \u2102) : \u2203 (r : \u211d) (w : \u2102), abs w = 1 \u2227 z = r * w :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $z$ is a complex number, prove that there exists an $r\\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. If $z=0$, we take $r=0, w=1$. (In this case $w$ is not unique.) Otherwise we take $r=|z|$ and $w=z /|z|$, and these choices are unique, since if $z=r w$, we must have $r=r|w|=|r w|=|z|, z / r$\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_12", - "formal_statement": "theorem exercise_1_12\n (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) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $z_1, \\ldots, z_n$ are complex, prove that $|z_1 + z_2 + \\ldots + z_n| \\leq |z_1| + |z_2| + \\cdots + |z_n|$.\n", - "nl_proof": "\\begin{proof}\n\n We can apply the case $n=2$ and induction on $n$ to get\n\n$$\n\n\\begin{aligned}\n\n\\left|z_1+z_2+\\cdots z_n\\right| &=\\left|\\left(z_1+z_2+\\cdots+z_{n-1}\\right)+z_n\\right| \\\\\n\n& \\leq\\left|z_1+z_2+\\cdots+z_{n-1}\\right|+\\left|z_n\\right| \\\\\n\n& \\leq\\left|z_1\\right|+\\left|z_2\\right|+\\cdots+\\left|z_{n-1}\\right|+\\left|z_n\\right|\n\n\\end{aligned}\n\n$$\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_13", - "formal_statement": "theorem exercise_1_13\n (x y : \u2102)\n : |(abs x) - (abs y)| \u2264 abs (x - y) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $x, y$ are complex, prove that $||x|-|y|| \\leq |x-y|$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since $x=x-y+y$, the triangle inequality gives\n\n$$\n\n|x| \\leq|x-y|+|y|\n\n$$\n\nso that $|x|-|y| \\leq|x-y|$. Similarly $|y|-|x| \\leq|x-y|$. Since $|x|-|y|$ is a real number we have either ||$x|-| y||=|x|-|y|$ or ||$x|-| y||=|y|-|x|$. In either case, we have shown that ||$x|-| y|| \\leq|x-y|$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_14", - "formal_statement": "theorem exercise_1_14\n (z : \u2102) (h : abs z = 1)\n : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $z$ is a complex number such that $|z|=1$, that is, such that $z \\bar{z}=1$, compute $|1+z|^{2}+|1-z|^{2}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. $|1+z|^2=(1+z)(1+\\bar{z})=1+\\bar{z}+z+z \\bar{z}=2+z+\\bar{z}$. Similarly $|1-z|^2=(1-z)(1-\\bar{z})=1-z-\\bar{z}+z \\bar{z}=2-z-\\bar{z}$. Hence\n\n$$\n\n|1+z|^2+|1-z|^2=4 \\text {. }\n\n$$\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_16a", - "formal_statement": "theorem exercise_1_16a\n (n : \u2115)\n (d r : \u211d)\n (x y z : euclidean_space \u211d (fin n)) -- R^n\n (h\u2081 : n \u2265 3)\n (h\u2082 : \u2225x - y\u2225 = d)\n (h\u2083 : d > 0)\n (h\u2084 : r > 0)\n (h\u2085 : 2 * r > d)\n : set.infinite {z : euclidean_space \u211d (fin n) | \u2225z - x\u2225 = r \u2227 \u2225z - y\u2225 = r} :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $k \\geq 3, x, y \\in \\mathbb{R}^k, |x - y| = d > 0$, and $r > 0$. Prove that if $2r > d$, there are infinitely many $z \\in \\mathbb{R}^k$ such that $|z-x|=|z-y|=r$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. (a) Let w be any vector satisfying the following two equations:\n\n$$\n\n\\begin{aligned}\n\n\\mathbf{w} \\cdot(\\mathbf{x}-\\mathbf{y}) &=0, \\\\\n\n|\\mathbf{w}|^2 &=r^2-\\frac{d^2}{4} .\n\n\\end{aligned}\n\n$$\n\nFrom linear algebra it is known that all but one of the components of a solution $\\mathbf{w}$ of the first equation can be arbitrary. The remaining component is then uniquely determined. Also, if $w$ is any non-zero solution of the first equation, there is a unique positive number $t$ such that $t$ w satisfies both equations. (For example, if $x_1 \\neq y_1$, the first equation is satisfied whenever\n\n$$\n\nz_1=\\frac{z_2\\left(x_2-y_2\\right)+\\cdots+z_k\\left(x_k-y_k\\right)}{y_1-x_1} .\n\n$$\n\nIf $\\left(z_1, z_2, \\ldots, z_k\\right)$ satisfies this equation, so does $\\left(t z_1, t z_2, \\ldots, t z_k\\right)$ for any real number $t$.) Since at least two of these components can vary independently, we can find a solution with these components having any prescribed ratio. This ratio does not change when we multiply by the positive number $t$ to obtain a solution of both equations. Since there are infinitely many ratios, there are infinitely many distinct solutions. For each such solution $\\mathbf{w}$ the vector $\\mathbf{z}=$ $\\frac{1}{2} \\mathrm{x}+\\frac{1}{2} \\mathrm{y}+\\mathrm{w}$ is a solution of the required equation. For\n\n$$\n\n\\begin{aligned}\n\n|\\mathrm{z}-\\mathbf{x}|^2 &=\\left|\\frac{\\mathbf{y}-\\mathbf{x}}{2}+\\mathbf{w}\\right|^2 \\\\\n\n&=\\left|\\frac{\\mathbf{y}-\\mathbf{x}}{2}\\right|^2+2 \\mathbf{w} \\cdot \\frac{\\mathbf{x}-\\mathbf{y}}{2}+|\\mathbf{w}|^2 \\\\\n\n&=\\frac{d^2}{4}+0+r^2-\\frac{d^2}{4} \\\\\n\n&=r^2\n\n\\end{aligned}\n\n$$\n\nand a similar relation holds for $|z-y|^2$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_17", - "formal_statement": "theorem exercise_1_17\n (n : \u2115)\n (x y : euclidean_space \u211d (fin n)) -- R^n\n : \u2225x + y\u2225^2 + \u2225x - y\u2225^2 = 2*\u2225x\u2225^2 + 2*\u2225y\u2225^2 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $\\mathbf{y} \\in R^{k}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. The proof is a routine computation, using the relation\n\n$$\n\n|x \\pm y|^2=(x \\pm y) \\cdot(x \\pm y)=|x|^2 \\pm 2 x \\cdot y+|y|^2 .\n\n$$\n\nIf $\\mathrm{x}$ and $\\mathrm{y}$ are the sides of a parallelogram, then $\\mathrm{x}+\\mathrm{y}$ and $\\mathbf{x}-\\mathrm{y}$ are its diagonals. Hence this result says that the sum of the squares on the diagonals of a parallelogram equals the sum of the squares on the sides.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_18a", - "formal_statement": "theorem exercise_1_18a\n (n : \u2115)\n (h : n > 1)\n (x : euclidean_space \u211d (fin n)) -- R^n\n : \u2203 (y : euclidean_space \u211d (fin n)), y \u2260 0 \u2227 (inner x y) = (0 : \u211d) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $k \\geq 2$ and $\\mathbf{x} \\in R^{k}$, prove that there exists $\\mathbf{y} \\in R^{k}$ such that $\\mathbf{y} \\neq 0$ but $\\mathbf{x} \\cdot \\mathbf{y}=0$\n", - "nl_proof": "\\begin{proof}\n\n Solution. If $\\mathbf{x}$ has any components equal to 0 , then $\\mathbf{y}$ can be taken to have the corresponding components equal to 1 and all others equal to 0 . If all the components of $\\mathbf{x}$ are nonzero, $\\mathbf{y}$ can be taken as $\\left(-x_2, x_1, 0, \\ldots, 0\\right)$. This is, of course, not true when $k=1$, since the product of two nonzero real numbers is nonzero.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_18b", - "formal_statement": "theorem exercise_1_18b\n : \u00ac \u2200 (x : \u211d), \u2203 (y : \u211d), y \u2260 0 \u2227 x * y = 0 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $\\mathbf{x} \\cdot \\mathbf{y}=0$\n", - "nl_proof": "\\begin{proof}\n\n Not true when $k=1$, since the product of two nonzero real numbers is nonzero.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_1_19", - "formal_statement": "theorem exercise_1_19\n (n : \u2115)\n (a b c x : euclidean_space \u211d (fin n))\n (r : \u211d)\n (h\u2081 : r > 0)\n (h\u2082 : 3 \u2022 c = 4 \u2022 b - a)\n (h\u2083 : 3 * r = 2 * \u2225x - b\u2225)\n : \u2225x - a\u2225 = 2 * \u2225x - b\u2225 \u2194 \u2225x - c\u2225 = r :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $a, b \\in R^k$. Find $c \\in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since the solution is given to us, all we have to do is verify it, i.e., we need to show that the equation\n\n$$\n\n|\\mathrm{x}-\\mathrm{a}|=2|\\mathrm{x}-\\mathrm{b}|\n\n$$\n\nis equivalent to $|\\mathrm{x}-\\mathbf{c}|=r$, which says\n\n$$\n\n\\left|\\mathbf{x}-\\frac{4}{3} \\mathbf{b}+\\frac{1}{3} \\mathbf{a}\\right|=\\frac{2}{3}|\\mathbf{b}-\\mathbf{a}| .\n\n$$\n\nIf we square both sides of both equations, we an equivalent pair of equations, the first of which reduces to\n\n$$\n\n3|\\mathbf{x}|^2+2 \\mathbf{a} \\cdot \\mathbf{x}-8 \\mathbf{b} \\cdot \\mathbf{x}-|\\mathbf{a}|^2+4|\\mathbf{b}|^2=0,\n\n$$\n\nand the second of which reduces to this equation divided by 3 . Hence these equations are indeed equivalent.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_19a", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.\n", - "nl_proof": "\\begin{proof}\n\n Solution. We are given that $A \\cap B=\\varnothing$. Since $A$ and $B$ are closed, this means $A \\cap \\bar{B}=\\varnothing=\\bar{A} \\cap B$, which says that $A$ and $B$ are separated.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_24", - "formal_statement": "theorem exercise_2_24 {X : Type*} [metric_space X]\n (hX : \u2200 (A : set X), infinite A \u2192 \u2203 (x : X), x \u2208 closure A) :\n separable_space X :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_2_25", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that every compact metric space $K$ has a countable base.\n", - "nl_proof": "\\begin{proof}\n\n Solution. It is easier simply to refer to the previous problem. The hint shows that $K$ can be covered by a finite union of neighborhoods of radius $1 / n$, and the previous problem shows that this implies that $K$ is separable.\n\n\n\nIt is not entirely obvious that a metric space with a countable base is separable. To prove this, let $\\left\\{V_n\\right\\}_{n=1}^{\\infty}$ be a countable base, and let $x_n \\in V_n$. The points $V_n$ must be dense in $X$. For if $G$ is any non-empty open set, then $G$ contains $V_n$ for some $n$, and hence $x_n \\in G$. (Thus for a metric space, having a countable base and being separable are equivalent.)\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_27a", - "formal_statement": "theorem exercise_2_27a (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, \u00ac set.countable (P \u2229 E)}) :\n is_closed P \u2227 P = {x | cluster_pt x (\ud835\udcdf P)} :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $E\\subset\\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect.\n", - "nl_proof": "\\begin{proof}\n\n Solution. We see that $E \\cap W$ is at most countable, being a countable union of at-most-countable sets. It remains to show that $P=W^c$, and that $P$ is perfect.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_27b", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $P$.\n", - "nl_proof": "\\begin{proof}\n\n If $x \\in W^c$, and $O$ is any neighborhood of $x$, then $x \\in V_n \\subseteq O$ for some n. Since $x \\notin W, V_n \\cap E$ is uncountable. Hence $O$ contains uncountably many points of $E$, and so $x$ is a condensation point of $E$. Thus $x \\in P$, i.e., $W^c \\subseteq P$.\n\nConversely if $x \\in W$, then $x \\in V_n$ for some $V_n$ such that $V_n \\cap E$ is countable. Hence $x$ has a neighborhood (any neighborhood contained in $V_n$ ) containing at most a countable set of points of $E$, and so $x \\notin P$, i.e., $W \\subseteq P^c$. Hence $P=W^c$.\n\nIt is clear that $P$ is closed (since its complement $W$ is open), so that we need only show that $P \\subseteq P^{\\prime}$. Hence suppose $x \\in P$, and $O$ is any neighborhood of $x$. (By definition of $P$ this means $O \\cap E$ is uncountable.) We need to show that there is a point $y \\in P \\cap(O \\backslash\\{x\\})$. If this is not the case, i.e., if every point $y$ in $O \\backslash\\{x\\}$ is in $P^c$, then for each such point $y$ there is a set $V_n$ containing $y$ such that $V_n \\cap E$ is at most countable. That would mean that $y \\in W$, i.e., that $O \\backslash\\{x\\}$ is contained in $W$. It would follow that $O \\cap E \\subseteq\\{x\\} \\cup(W \\cap E)$, and so $O \\cap E$ contains at most a countable set of points, contrary to the hypothesis that $x \\in P$. Hence $O$ contains a point of $P$ different from $x$, and so $P \\subseteq P^{\\prime}$. Thus $P$ is perfect.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_28", - "formal_statement": "theorem exercise_2_28 (X : Type*) [metric_space X] [separable_space X]\n (A : set X) (hA : is_closed A) :\n \u2203 P\u2081 P\u2082 : set X, A = P\u2081 \u222a P\u2082 \u2227\n is_closed P\u2081 \u2227 P\u2081 = {x | cluster_pt x (\ud835\udcdf P\u2081)} \u2227\n set.countable P\u2082 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.\n", - "nl_proof": "\\begin{proof}\n\n Solution. If $E$ is closed, it contains all its limit points, and hence certainly all its condensation points. Thus $E=P \\cup(E \\backslash P)$, where $P$ is perfect (the set of all condensation points of $E$ ), and $E \\backslash P$ is at most countable.\n\n\n\nSince a perfect set in a separable metric space has the same cardinality as the real numbers, the set $P$ must be empty if $E$ is countable. The at-mostcountable set $E \\backslash P$ cannot be perfect, hence must have isolated points if it is nonempty.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_2_29", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that every open set in $\\mathbb{R}$ is the union of an at most countable collection of disjoint segments.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $O$ be open. For each pair of points $x \\in O, y \\in O$, we define an equivalence relation $x \\sim y$ by saying $x \\sim y$ if and only if $[\\min (x, y), \\max (x, y)] \\subset$ 0 . This is an equivalence relation, since $x \\sim x([x, x] \\subset O$ if $x \\in O)$; if $x \\sim y$, then $y \\sim x$ (since $\\min (x, y)=\\min (y, x)$ and $\\max (x, y)=\\max (y, x))$; and if $x \\sim y$ and $y \\sim z$, then $x \\sim z([\\min (x, z), \\max (x, z)] \\subseteq[\\min (x, y), \\max (x, y)] \\cup$ $[\\min (y, z), \\max (y, z)] \\subseteq O)$. In fact it is easy to prove that\n\n$$\n\n\\min (x, z) \\geq \\min (\\min (x, y), \\min (y, z))\n\n$$\n\nand\n\n$$\n\n\\max (x, z) \\leq \\max (\\max (x, y), \\max (y, z))\n\n$$\n\nIt follows that $O$ can be written as a disjoint union of pairwise disjoint equivalence classes. We claim that each equivalence class is an open interval.\n\n\n\nTo show this, for each $x \\in O$; let $A=\\{z:[z, x] \\subseteq O\\}$ and $B=\\{z:[x, z] \\subseteq$ $O\\}$, and let $a=\\inf A, b=\\sup B$. We claim that $(a, b) \\subset O$. Indeed if $az$. Then $z \\in[c, x] \\cup[x, d] \\subseteq O$. We now claim that $(a, b)$ is the equivalence class containing $x$. It is clear that each element of $(a, b)$ is equivalent to $x$ by the way in which $a$ and $b$ were chosen. We need to show that if $z \\notin(a, b)$, then $z$ is not equivalent to $x$. Suppose that $zb$ and $z \\sim x$, then $b$ could not be an upper bound for $B$.\n\n\n\nWe have now established that $O$ is a union of pairwise disjoint open intervals. Such a union must be at most countable, since each open interval contains a rational number not in any other interval.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_1a", - "formal_statement": "theorem exercise_3_1a\n (f : \u2115 \u2192 \u211d)\n (h : \u2203 (a : \u211d), tendsto (\u03bb (n : \u2115), f n) at_top (\ud835\udcdd a))\n : \u2203 (a : \u211d), tendsto (\u03bb (n : \u2115), |f n|) at_top (\ud835\udcdd a) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that convergence of $\\left\\{s_{n}\\right\\}$ implies convergence of $\\left\\{\\left|s_{n}\\right|\\right\\}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $\\varepsilon>0$. Since the sequence $\\left\\{s_n\\right\\}$ is a Cauchy sequence, there exists $N$ such that $\\left|s_m-s_n\\right|<\\varepsilon$ for all $m>N$ and $n>N$. We then have ||$s_m|-| s_n|| \\leq\\left|s_m-s_n\\right|<\\varepsilon$ for all $m>N$ and $n>N$. Hence the sequence $\\left\\{\\left|s_n\\right|\\right\\}$ is also a Cauchy sequence, and therefore must converge.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_2a", - "formal_statement": "theorem exercise_3_2a\n : tendsto (\u03bb (n : \u211d), (sqrt (n^2 + n) - n)) at_top (\ud835\udcdd (1/2)) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\lim_{n \\rightarrow \\infty}\\sqrt{n^2 + n} -n = 1/2$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Multiplying and dividing by $\\sqrt{n^2+n}+n$ yields\n\n$$\n\n\\sqrt{n^2+n}-n=\\frac{n}{\\sqrt{n^2+n}+n}=\\frac{1}{\\sqrt{1+\\frac{1}{n}}+1} .\n\n$$\n\nIt follows that the limit is $\\frac{1}{2}$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_3", - "formal_statement": "theorem exercise_3_3\n : \u2203 (x : \u211d), tendsto f at_top (\ud835\udcdd x) \u2227 \u2200 n, f n < 2 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $s_{1}=\\sqrt{2}$, and $s_{n+1}=\\sqrt{2+\\sqrt{s_{n}}} \\quad(n=1,2,3, \\ldots),$ prove that $\\left\\{s_{n}\\right\\}$ converges, and that $s_{n}<2$ for $n=1,2,3, \\ldots$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since $\\sqrt{2}<2$, it is manifest that if $s_n<2$, then $s_{n+1}<\\sqrt{2+2}=2$. Hence it follows by induction that $\\sqrt{2}1$, i.e., $s_n>s_n^2-2=s_{n-1}$. Hence the sequence is an increasing sequence that is bounded above (by 2 ) and so converges. Since the limit $s$ satisfies $s^2-s-2=0$, it follows that the limit is 2.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_5", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $\\infty-\\infty$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since the case when $\\limsup _{n \\rightarrow \\infty} a_n=+\\infty$ and $\\limsup _{n \\rightarrow \\infty} b_n=-\\infty$ has been excluded from consideration, we note that the inequality is obvious if $\\limsup _{n \\rightarrow \\infty} a_n=+\\infty$. Hence we shall assume that $\\left\\{a_n\\right\\}$ is bounded above.\n\n\n\nLet $\\left\\{n_k\\right\\}$ be a subsequence of the positive integers such that $\\lim _{k \\rightarrow \\infty}\\left(a_{n_k}+\\right.$ $\\left.b_{n_k}\\right)=\\limsup _{n \\rightarrow \\infty}\\left(a_n+b_n\\right)$. Then choose a subsequence of the positive integers $\\left\\{k_m\\right\\}$ such that\n\n$$\n\n\\lim _{m \\rightarrow \\infty} a_{n_{k_m}}=\\limsup _{k \\rightarrow \\infty} a_{n_k} .\n\n$$\n\nThe subsequence $a_{n_{k_m}}+b_{n_{k_m}}$ still converges to the same limit as $a_{n_k}+b_{n_k}$, i.e., to $\\limsup _{n \\rightarrow \\infty}\\left(a_n+b_n\\right)$. Hence, since $a_{n_k}$ is bounded above (so that $\\limsup _{k \\rightarrow \\infty} a_{n_k}$ is finite), it follows that $b_{n_{k_m}}$ converges to the difference\n\n$$\n\n\\lim _{m \\rightarrow \\infty} b_{n_{k_m}}=\\lim _{m \\rightarrow \\infty}\\left(a_{n_{k_m}}+b_{n_{k_m}}\\right)-\\lim _{m \\rightarrow \\infty} a_{n_{k_m}} .\n\n$$\n\nThus we have proved that there exist subsequences $\\left\\{a_{n_{k_m}}\\right\\}$ and $\\left\\{b_{n_{k_m}}\\right\\}$ which converge to limits $a$ and $b$ respectively such that $a+b=\\limsup _{n \\rightarrow \\infty}\\left(a_n+b_n^*\\right)$. Since $a$ is the limit of a subsequence of $\\left\\{a_n\\right\\}$ and $b$ is the limit of a subsequence of $\\left\\{b_n\\right\\}$, it follows that $a \\leq \\limsup _{n \\rightarrow \\infty} a_n$ and $b \\leq \\limsup _{n \\rightarrow \\infty} b_n$, from which the desired inequality follows.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_6a", - "formal_statement": "theorem exercise_3_6a\n: tendsto (\u03bb (n : \u2115), (\u2211 i in finset.range n, g i)) at_top at_top :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\lim_{n \\rightarrow \\infty} \\sum_{i0$ there exists $N$ such that $\\left|\\sum_{k=m+1}^n a_k b_k\\right|\\langle\\varepsilon$ if $n\\rangle$ $m \\geq N$. To do this, let $S_n=\\sum_{k=1}^n a_k\\left(S_0=0\\right)$, so that $a_k=S_k-S_{k-1}$ for $k=1,2, \\ldots$ Let $M$ be an uppper bound for both $\\left|b_n\\right|$ and $\\left|S_n\\right|$, and let $S=\\sum a_n$ and $b=\\lim b_n$. Choose $N$ so large that the following three inequalities hold for all $m>N$ and $n>N$ :\n\n$$\n\n\\left|b_n S_n-b S\\right|<\\frac{\\varepsilon}{3} ; \\quad\\left|b_m S_m-b S\\right|<\\frac{\\varepsilon}{3} ; \\quad\\left|b_m-b_n\\right|<\\frac{\\varepsilon}{3 M} .\n\n$$\n\nThen if $n>m>N$, we have, from the formula for summation by parts,\n\n$$\n\n\\sum_{k=m+1}^n a_n b_n=b_n S_n-b_m S_m+\\sum_{k=m}^{n-1}\\left(b_k-b_{k+1}\\right) S_k\n\n$$\n\nOur assumptions yield immediately that $\\left|b_n S_n-b_m S_m\\right|<\\frac{2 \\varepsilon}{3}$, and\n\n$$\n\n\\left|\\sum_{k=m}^{n-1}\\left(b_k-b_{k+1}\\right) S_k\\right| \\leq M \\sum_{k=m}^{n-1}\\left|b_k-b_{k+1}\\right| .\n\n$$\n\nSince the sequence $\\left\\{b_n\\right\\}$ is monotonic, we have\n\n$$\n\n\\sum_{k=m}^{n-1}\\left|b_k-b_{k+1}\\right|=\\left|\\sum_{k=m}^{n-1}\\left(b_k-b_{k+1}\\right)\\right|=\\left|b_m-b_n\\right|<\\frac{\\varepsilon}{3 M},\n\n$$\n\nfrom which the desired inequality follows.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_13", - "formal_statement": "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)) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Prove that the Cauchy product of two absolutely convergent series converges absolutely.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since both the hypothesis and conclusion refer to absolute convergence, we may assume both series consist of nonnegative terms. We let $S_n=\\sum_{k=0}^n a_n, T_n=\\sum_{k=0}^n b_n$, and $U_n=\\sum_{k=0}^n \\sum_{l=0}^k a_l b_{k-l}$. We need to show that $U_n$ remains bounded, given that $S_n$ and $T_n$ are bounded. To do this we make the convention that $a_{-1}=T_{-1}=0$, in order to save ourselves from having to separate off the first and last terms when we sum by parts. We then have\n\n$$\n\n\\begin{aligned}\n\nU_n &=\\sum_{k=0}^n \\sum_{l=0}^k a_l b_{k-l} \\\\\n\n&=\\sum_{k=0}^n \\sum_{l=0}^k a_l\\left(T_{k-l}-T_{k-l-1}\\right) \\\\\n\n&=\\sum_{k=0}^n \\sum_{j=0}^k a_{k-j}\\left(T_j-T_{j-1}\\right) \\\\\n\n&=\\sum_{k=0}^n \\sum_{j=0}^k\\left(a_{k-j}-a_{k-j-1}\\right) T_j \\\\\n\n&=\\sum_{j=0}^n \\sum_{k=j}^n\\left(a_{k-j}-a_{k-j-1}\\right) T_j\n\n&=\\sum_{j=0}^n a_{n-j} T_j \\\\\n\n&\\leq T \\sum_{m=0}^n a_m \\\\\n\n&=T S_n \\\\\n\n&\\leq S T .\n\n\\end{aligned}\n\n$$\n\nThus $U_n$ is bounded, and hence approaches a finite limit.\n\n\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_20", - "formal_statement": "theorem exercise_3_20 {X : Type*} [metric_space X]\n (p : \u2115 \u2192 X) (l : \u2115) (r : X)\n (hp : cauchy_seq p)\n (hpl : tendsto (\u03bb n, p (l * n)) at_top (\ud835\udcdd r)) :\n tendsto p at_top (\ud835\udcdd r) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $\\left\\{p_{n}\\right\\}$ is a Cauchy sequence in a metric space $X$, and some sequence $\\left\\{p_{n l}\\right\\}$ converges to a point $p \\in X$. Prove that the full sequence $\\left\\{p_{n}\\right\\}$ converges to $p$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $\\varepsilon>0$. Choose $N_1$ so large that $d\\left(p_m, p_n\\right)<\\frac{\\varepsilon}{2}$ if $m>N_1$ and $n>N_1$. Then choose $N \\geq N_1$ so large that $d\\left(p_{n_k}, p\\right)<\\frac{\\varepsilon}{2}$ if $k>N$. Then if $n>N$, we have\n\n$$\n\nd\\left(p_n, p\\right) \\leq d\\left(p_n, p_{n_{N+1}}\\right)+d\\left(p_{n_{N+1}}, p\\right)<\\varepsilon\n\n$$\n\nFor the first term on the right is less than $\\frac{\\varepsilon}{2}$ since $n>N_1$ and $n_{N+1}>N+1>$ $N_1$. The second term is less than $\\frac{\\varepsilon}{2}$ by the choice of $N$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_21", - "formal_statement": "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} :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Choose $x_n \\in E_n$. (We use the axiom of choice here.) The sequence $\\left\\{x_n\\right\\}$ is a Cauchy sequence, since the diameter of $E_n$ tends to zero as $n$ tends to infinity and $E_n$ contains $E_{n+1}$. Since the metric space $X$ is complete, the sequence $x_n$ converges to a point $x$, which must belong to $E_n$ for all $n$, since $E_n$ is closed and contains $x_m$ for all $m \\geq n$. There cannot be a second point $y$ in all of the $E_n$, since for any point $y \\neq x$ the diameter of $E_n$ is less $\\operatorname{than} d(x, y)$ for large $n$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_3_22", - "formal_statement": "theorem exercise_3_22 (X : Type*) [metric_space X] [complete_space X]\n (G : \u2115 \u2192 set X) (hG : \u2200 n, is_open (G n) \u2227 dense (G n)) :\n \u2203 x, \u2200 n, x \u2208 G n :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $X$ is a nonempty complete metric space, and $\\left\\{G_{n}\\right\\}$ is a sequence of dense open sets of $X$. Prove Baire's theorem, namely, that $\\bigcap_{1}^{\\infty} G_{n}$ is not empty.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $F_n$ be the complement of $G_n$, so that $F_n$ is closed and contains no open sets. We shall prove that any nonempty open set $U$ contains a point not in any $F_n$, hence in all $G_n$. To this end, we note that $U$ is not contained in $F_1$, so that there is a point $x_1 \\in U \\backslash F_1$. Since $U \\backslash F_1$ is open, there exists $r_1>0$ such that $B_1$, defined as the open ball of radius $r_1$ about $x_1$, is contained in $U \\backslash F_1$. Let $E_1$ be the open ball of radius $\\frac{r_1}{2}$ about $x_1$, so that the closure of $E_1$ is contained in $B_1$. Now $F_2$ does not contain $E_1$, and so we can find a point $x_2 \\in E_1 \\backslash F_2$. Since $E_1 \\backslash F_2$ is an open set, there exists a positive number $r_2$ such that $B_2$, the open ball of radius $R_2$ about $x_2$, is contained in $E_1 \\backslash F_2$, which in turn is contained in $U \\backslash\\left(F_1 \\cup F_2\\right)$. We let $E_2$ be the open ball of radius $\\frac{r_2}{2}$ about $x_2$, so that $\\bar{E}_2 \\subseteq B_2$. Proceeding in this way, we construct a sequence of open balls $E_j$, such that $E_j \\supseteq \\bar{E}_{j+1}$, and the diameter of $E_j$ tends to zero. By the previous exercise, there is a point $x$ belonging to all the sets $\\bar{E}_j$, hence to all the sets $U \\backslash\\left(F_1 \\cup F_2 \\cup \\cdots \\cup F_n\\right)$. Thus the point $x$ belongs to $U \\cap\\left(\\cap_1^{\\infty} G_n\\right)$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_1a", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "\\begin{proof}\n\n $$\n\nf(x)= \\begin{cases}1 & \\text { if } x \\text { is an integer } \\\\ 0 & \\text { if } x \\text { is not an integer. }\\end{cases}\n\n$$\n\n(If $x$ is an integer, then $f(x+h)-f(x-h) \\equiv 0$ for all $h$; while if $x$ is not an integer, $f(x+h)-f(x-h)=0$ for $|h|<\\min (x-[x], 1+[x]-x)$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_2a", - "formal_statement": "theorem exercise_4_2a\n {\u03b1 : Type} [metric_space \u03b1]\n {\u03b2 : Type} [metric_space \u03b2]\n (f : \u03b1 \u2192 \u03b2)\n (h\u2081 : continuous f)\n : \u2200 (x : set \u03b1), f '' (closure x) \u2286 closure (f '' x) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that $f(\\overline{E}) \\subset \\overline{f(E)}$ for every set $E \\subset X$. ($\\overline{E}$ denotes the closure of $E$).\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $x \\in \\bar{E}$. We need to show that $f(x) \\in \\overline{f(E)}$. To this end, let $O$ be any neighborhood of $f(x)$. Since $f$ is continuous, $f^{-1}(O)$ contains (is) a neighborhood of $x$. Since $x \\in \\bar{E}$, there is a point $u$ of $E$ in $f^{-1}(O)$. Hence $\\frac{f(u)}{f(E)} \\in O \\cap f(E)$. Since $O$ was any neighborhood of $f(x)$, it follows that $f(x) \\in \\overline{f(E)}$\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_3", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "\\begin{proof}\n\n Solution. $Z(f)=f^{-1}(\\{0\\})$, which is the inverse image of a closed set. Hence $Z(f)$ is closed.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_4a", - "formal_statement": "theorem exercise_4_4a\n {\u03b1 : Type} [metric_space \u03b1]\n {\u03b2 : Type} [metric_space \u03b2]\n (f : \u03b1 \u2192 \u03b2)\n (s : set \u03b1)\n (h\u2081 : continuous f)\n (h\u2082 : dense s)\n : f '' set.univ \u2286 closure (f '' s) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $f$ and $g$ be continuous mappings of a metric space $X$ into a metric space $Y$, and let $E$ be a dense subset of $X$. Prove that $f(E)$ is dense in $f(X)$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. To prove that $f(E)$ is dense in $f(X)$, simply use that $f(X)=f(\\bar{E}) \\subseteq \\overline{f(E)}$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_4b", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $p \\in X$.\n", - "nl_proof": "\\begin{proof}\n\n The function $\\varphi: X \\rightarrow R^1$ given by\n\n$$\n\n\\varphi(p)=d_Y(f(p), g(p))\n\n$$\n\nis continuous, since\n\n$$\n\n\\left|d_Y(f(p), g(p))-d_Y(f(q), g(q))\\right| \\leq d_Y(f(p), f(q))+d_Y(g(p), g(q))\n\n$$\n\n(This inequality follows from the triangle inequality, since\n\n$$\n\nd_Y(f(p), g(p)) \\leq d_Y(f(p), f(q))+d_Y(f(q), g(q))+d_Y(g(q), g(p)),\n\n$$\n\nand the same inequality holds with $p$ and $q$ interchanged. The absolute value $\\left|d_Y(f(p), g(p))-d_Y(f(q), g(q))\\right|$ must be either $d_Y(f(p), g(p))-d_Y(f(q), g(q))$ or $d_Y(f(q), g(q))-d_Y(f(p), g(p))$, and the triangle inequality shows that both of these numbers are at most $d_Y(f(p), f(q))+d_Y(g(p), g(q))$.)\n\nBy the previous problem, the zero set of $\\varphi$ is closed. But by definition\n\n$$\n\nZ(\\varphi)=\\{p: f(p)=g(p)\\} .\n\n$$\n\nHence the set of $p$ for which $f(p)=g(p)$ is closed. Since by hypothesis it is dense, it must be $X$.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_5a", - "formal_statement": "theorem exercise_4_5a\n (f : \u211d \u2192 \u211d)\n (E : set \u211d)\n (h\u2081 : is_closed E)\n (h\u2082 : continuous_on f E)\n : \u2203 (g : \u211d \u2192 \u211d), continuous g \u2227 \u2200 x \u2208 E, f x = g x :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $f$ is a real continuous function defined on a closed set $E \\subset \\mathbb{R}$, prove that there exist continuous real functions $g$ on $\\mathbb{R}$ such that $g(x)=f(x)$ for all $x \\in E$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_5b", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $x \\in E$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_6", - "formal_statement": "theorem exercise_4_6\n (f : \u211d \u2192 \u211d)\n (E : set \u211d)\n (G : set (\u211d \u00d7 \u211d))\n (h\u2081 : is_compact E)\n (h\u2082 : G = {(x, f x) | x \u2208 E})\n : continuous_on f E \u2194 is_compact G :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $f$ is defined on $E$, the graph of $f$ is the set of points $(x, f(x))$, for $x \\in E$. In particular, if $E$ is a set of real numbers, and $f$ is real-valued, the graph of $f$ is a subset of the plane. Suppose $E$ is compact, and prove that $f$ is continuous on $E$ if and only if its graph is compact.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $Y$ be the co-domain of the function $f$. We invent a new metric space $E \\times Y$ as the set of pairs of points $(x, y), x \\in E, y \\in Y$, with the metric $\\rho\\left(\\left(x_1, y_1\\right),\\left(x_2, y_2\\right)\\right)=d_E\\left(x_1, x_2\\right)+d_Y\\left(y_1, y_2\\right)$. The function $\\varphi(x)=(x, f(x))$ is then a mapping of $E$ into $E \\times Y$.\n\n\n\nWe claim that the mapping $\\varphi$ is continuous if $f$ is continuous. Indeed, let $x \\in X$ and $\\varepsilon>0$ be given. Choose $\\eta>0$ so that $d_Y(f(x), f(u))<\\frac{\\varepsilon}{2}$ if $d_E(x, y)<\\eta$. Then let $\\delta=\\min \\left(\\eta, \\frac{\\varepsilon}{2}\\right)$. It is easy to see that $\\rho(\\varphi(x), \\varphi(u))<\\varepsilon$ if $d_E(x, u)<\\delta$. Conversely if $\\varphi$ is continuous, it is obvious from the inequality $\\rho(\\varphi(x), \\varphi(u)) \\geq d_Y(f(x), f(u))$ that $f$ is continuous.\n\n\n\nFrom these facts we deduce immediately that the graph of a continuous function $f$ on a compact set $E$ is compact, being the image of $E$ under the continuous mapping $\\varphi$. Conversely, if $f$ is not continuous at some point $x$, there is a sequence of points $x_n$ converging to $x$ such that $f\\left(x_n\\right)$ does not converge to $f(x)$. If no subsequence of $f\\left(x_n\\right)$ converges, then the sequence $\\left\\{\\left(x_n, f\\left(x_n\\right)\\right\\}_{n=1}^{\\infty}\\right.$ has no convergent subsequence, and so the graph is not compact. If some subsequence of $f\\left(x_n\\right)$ converges, say $f\\left(x_{n_k}\\right) \\rightarrow z$, but $z \\neq f(x)$, then the graph of $f$ fails to contain the limit point $(x, z)$, and hence is not closed. A fortiori it is not compact.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_8a", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_8b", - "formal_statement": "theorem exercise_4_8b\n (E : set \u211d) :\n \u2203 f : \u211d \u2192 \u211d, uniform_continuous_on f E \u2227 \u00ac metric.bounded (set.image f E) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $E$ be a bounded set in $R^{1}$. Prove that there exists a real function $f$ such that $f$ is uniformly continuous and is not bounded on $E$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_11a", - "formal_statement": "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)) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $X$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_12", - "formal_statement": "theorem exercise_4_12\n {\u03b1 \u03b2 \u03b3 : Type*} [uniform_space \u03b1] [uniform_space \u03b2] [uniform_space \u03b3]\n {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n (hf : uniform_continuous f) (hg : uniform_continuous g) :\n uniform_continuous (g \u2218 f) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "A uniformly continuous function of a uniformly continuous function is uniformly continuous.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $f: X \\rightarrow Y$ and $g: Y \\rightarrow Z$ be uniformly continuous. Then $g \\circ f: X \\rightarrow Z$ is uniformly continuous, where $g \\circ f(x)=g(f(x))$ for all $x \\in X$.\n\nTo prove this fact, let $\\varepsilon>0$ be given. Then, since $g$ is uniformly continuous, there exists $\\eta>0$ such that $d_Z(g(u), g(v))<\\varepsilon$ if $d_Y(u, v)<\\eta$. Since $f$ is uniformly continuous, there exists $\\delta>0$ such that $d_Y(f(x), f(y))<\\eta$ if $d_X(x, y)<\\delta$\n\n\n\nIt is then obvious that $d_Z(g(f(x)), g(f(y)))<\\varepsilon$ if $d_X(x, y)<\\delta$, so that $g \\circ f$ is uniformly continuous.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_4_14", - "formal_statement": "theorem exercise_4_14 [topological_space I]\n [linear_order I] (f : I \u2192 I) (hf : continuous f) :\n \u2203 (x : I), f x = x :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Let $I=[0,1]$ be the closed unit interval. Suppose $f$ is a continuous mapping of $I$ into $I$. Prove that $f(x)=x$ for at least one $x \\in I$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. If $f(0)=0$ or $f(1)=1$, we are done. If not, then $0 0 \u2227 \u2200 (p q : X), p \u2208 K \u2192 q \u2208 F \u2192 dist p q \u2265 \u03b4 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $p \\in K, q \\in F$.\n", - "nl_proof": "" - }, - { - "id": "Rudin|exercise_4_24", - "formal_statement": "theorem exercise_4_24 {f : \u211d \u2192 \u211d}\n (hf : continuous f) (a b : \u211d) (hab : a < b)\n (h : \u2200 x y : \u211d, a < x \u2192 x < b \u2192 a < y \u2192 y < b \u2192 f ((x + y) / 2) \u2264 (f x + f y) / 2) :\n convex_on \u211d (set.Ioo a b) f :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\\left(\\frac{x+y}{2}\\right) \\leq \\frac{f(x)+f(y)}{2}$ for all $x, y \\in(a, b)$. Prove that $f$ is convex.\n", - "nl_proof": "\\begin{proof}\n\n Solution. We shall prove that\n\n$$\n\nf(\\lambda x+(1-\\lambda) y) \\leq \\lambda f(x)+(1-\\lambda) f(y)\n\n$$\n\nfor all \"dyadic rational\" numbers, i.e., all numbers of the form $\\lambda=\\frac{k}{2^n}$, where $k$ is a nonnegative integer not larger than $2^n$. We do this by induction on $n$. The case $n=0$ is trivial (since $\\lambda=0$ or $\\lambda=1$ ). In the case $n=1$ we have $\\lambda=0$ or $\\lambda=1$ or $\\lambda=\\frac{1}{2}$. The first two cases are again trivial, and the third is precisely the hypothesis of the theorem. Suppose the result is proved for $n \\leq r$, and consider $\\lambda=\\frac{k}{2^{r+1}}$. If $k$ is even, say $k=2 l$, then $\\frac{k}{2^{r+1}}=\\frac{l}{2^r}$, and we can appeal to the induction hypothesis. Now suppose $k$ is odd. Then $1 \\leq k \\leq 2^{r+1}-1$, and so the numbers $l=\\frac{k-1}{2}$ and $m=\\frac{k+1}{2}$ are integers with $0 \\leq l 0)\n (hg : g = f\u207b\u00b9)\n (hg_diff : differentiable_on \u211d g (set.Ioo a b)) :\n differentiable_on \u211d g (set.Ioo a b) \u2227\n \u2200 x \u2208 set.Ioo a b, deriv g x = 1 / deriv f x :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $f^{\\prime}(x)>0$ in $(a, b)$. Prove that $f$ is strictly increasing in $(a, b)$, and let $g$ be its inverse function. Prove that $g$ is differentiable, and that $g^{\\prime}(f(x))=\\frac{1}{f^{\\prime}(x)} \\quad(a0$. Hence $f(c)0$ there exists $\\eta>0$ such that\n\n$$\n\n\\left|\\frac{1}{\\frac{f(x+h)-f(x)}{h}}-\\frac{1}{f^{\\prime}(x)}\\right|<\\varepsilon\n\n$$\n\nif $0<|h|<\\eta$. Since $h=g(y+k)-g(y)$, there exists $\\delta>0$ such that $0<|h|<\\eta$ if $0<|k|<\\delta$. The proof is now complete. \n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_3", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "\\begin{proof}\n\n Solution. If $0<\\varepsilon<\\frac{1}{M}$, we certainly have\n\n$$\n\nf^{\\prime}(x) \\geq 1-\\varepsilon M>0,\n\n$$\n\nand this implies that $f(x)$ is one-to-one, by the preceding problem.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_4", - "formal_statement": "theorem exercise_5_4 {n : \u2115}\n (C : \u2115 \u2192 \u211d)\n (hC : \u2211 i in (finset.range (n + 1)), (C i) / (i + 1) = 0) :\n \u2203 x, x \u2208 (set.Icc (0 : \u211d) 1) \u2227 \u2211 i in finset.range (n + 1), (C i) * (x^i) = 0 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "If $C_{0}+\\frac{C_{1}}{2}+\\cdots+\\frac{C_{n-1}}{n}+\\frac{C_{n}}{n+1}=0,$ where $C_{0}, \\ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Consider the polynomial\n\n$$\n\np(x)=C_0 x+\\frac{C_1}{2} x^2+\\cdots+\\frac{C_{n-1}}{n} x^n+\\frac{C_n}{n+1} x^{n+1},\n\n$$\n\nwhose derivative is\n\n$$\n\np^{\\prime}(x)=C_0+C_1 x+\\cdots+C_{n-1} x^{n-1}+C_n x^n .\n\n$$\n\nIt is obvious that $p(0)=0$, and the hypothesis of the problem is that $p(1)=0$. Hence Rolle's theorem implies that $p^{\\prime}(x)=0$ for some $x$ between 0 and 1 .\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_5", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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 $x \\rightarrow+\\infty$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Let $\\varepsilon>0$. Choose $x_0$ such that $\\left|f^{\\prime}(x)\\right|<\\varepsilon$ if $x>x_0$. Then for any $x \\geq x_0$ there exists $x_1 \\in(x, x+1)$ such that\n\n$$\n\nf(x+1)-f(x)=f^{\\prime}\\left(x_1\\right) .\n\n$$\n\nSince $\\left|f^{\\prime}\\left(x_1\\right)\\right|<\\varepsilon$, it follows that $|f(x+1)-f(x)|<\\varepsilon$, as required.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_6", - "formal_statement": "theorem exercise_5_6\n {f : \u211d \u2192 \u211d}\n (hf1 : continuous f)\n (hf2 : \u2200 x, differentiable_at \u211d f x)\n (hf3 : f 0 = 0)\n (hf4 : monotone (deriv f)) :\n monotone_on (\u03bb x, f x / x) (set.Ioi 0) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose (a) $f$ is continuous for $x \\geq 0$, (b) $f^{\\prime}(x)$ exists for $x>0$, (c) $f(0)=0$, (d) $f^{\\prime}$ is monotonically increasing. Put $g(x)=\\frac{f(x)}{x} \\quad(x>0)$ and prove that $g$ is monotonically increasing.\n", - "nl_proof": "\\begin{proof}\n\n Put\n\n$$\n\ng(x)=\\frac{f(x)}{x} \\quad(x>0)\n\n$$\n\nand prove that $g$ is monotonically increasing.\n\nSolution. By the mean-value theorem\n\n$$\n\nf(x)=f(x)-f(0)=f^{\\prime}(c) x\n\n$$\n\nfor some $c \\in(0, x)$. Since $f^{\\prime}$ is monotonically increasing, this result implies that $f(x)0,\n\n$$\n\nso that $g$ is also monotonically increasing.\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_7", - "formal_statement": "theorem exercise_5_7\n {f g : \u211d \u2192 \u211d} {x : \u211d}\n (hf' : differentiable_at \u211d f 0)\n (hg' : differentiable_at \u211d g 0)\n (hg'_ne_0 : deriv g 0 \u2260 0)\n (f0 : f 0 = 0) (g0 : g 0 = 0) :\n tendsto (\u03bb x, f x / g x) (\ud835\udcdd x) (\ud835\udcdd (deriv f x / deriv g x)) :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $f^{\\prime}(x), g^{\\prime}(x)$ exist, $g^{\\prime}(x) \\neq 0$, and $f(x)=g(x)=0$. Prove that $\\lim _{t \\rightarrow x} \\frac{f(t)}{g(t)}=\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}.$\n", - "nl_proof": "\\begin{proof}\n\n Solution. Since $f(x)=g(x)=0$, we have\n\n$$\n\n\\begin{aligned}\n\n\\lim _{t \\rightarrow x} \\frac{f(t)}{g(t)} &=\\lim _{t \\rightarrow x} \\frac{\\frac{f(t)-f(x)}{t-x}}{\\frac{g(t)-g(x)}{t-x}} \\\\\n\n&=\\frac{\\lim _{t \\rightarrow x} \\frac{f(t)-f(x)}{t-x}}{\\lim _{t \\rightarrow x} \\frac{g(t)-g(x)}{t-x}} \\\\\n\n&=\\frac{f^{\\prime}(x)}{g^{\\prime}(x)}\n\n\\end{aligned}\n\n$$\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_15", - "formal_statement": "theorem exercise_5_15 {f : \u211d \u2192 \u211d} (a M0 M1 M2 : \u211d)\n (hf' : differentiable_on \u211d f (set.Ici a))\n (hf'' : differentiable_on \u211d (deriv f) (set.Ici a))\n (hM0 : M0 = Sup {(| f x | )| x \u2208 (set.Ici a)})\n (hM1 : M1 = Sup {(| deriv f x | )| x \u2208 (set.Ici a)})\n (hM2 : M2 = Sup {(| deriv (deriv f) x | )| x \u2208 (set.Ici a)}) :\n (M1 ^ 2) \u2264 4 * M0 * M2 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "Suppose $a \\in R^{1}, f$ is a twice-differentiable real function on $(a, \\infty)$, and $M_{0}, M_{1}, M_{2}$ are the least upper bounds of $|f(x)|,\\left|f^{\\prime}(x)\\right|,\\left|f^{\\prime \\prime}(x)\\right|$, respectively, on $(a, \\infty)$. Prove that $M_{1}^{2} \\leq 4 M_{0} M_{2} .$\n", - "nl_proof": "\\begin{proof}\n\n Solution. The inequality is obvious if $M_0=+\\infty$ or $M_2=+\\infty$, so we shall assume that $M_0$ and $M_2$ are both finite. We need to show that\n\n$$\n\n\\left|f^{\\prime}(x)\\right| \\leq 2 \\sqrt{M_0 M_2}\n\n$$\n\nfor all $x>a$. We note that this is obvious if $M_2=0$, since in that case $f^{\\prime}(x)$ is constant, $f(x)$ is a linear function, and the only bounded linear function is a constant, whose derivative is zero. Hence we shall assume from now on that $0-1$. Now $f^{\\prime}(x)=\\frac{4 x}{\\left(x^2+1\\right)^2}$ for $x>0$ and $f^{\\prime}(x)=4 x$ for $x<0$. It thus follows from Exercise 9 above that $f^{\\prime}(0)=0$, and that $f^{\\prime}(x)$ is continuous. Likewise $f^{\\prime \\prime}(x)=4$ for $x<0$ and $f^{\\prime \\prime}(x)=\\frac{4-4 x^2}{\\left(x^2+1\\right)^3}=-4 \\frac{x^2-1}{\\left(x^2+1\\right)^3}$. This shows that $\\left|f^{\\prime \\prime}(x)\\right|<4$ for $x>0$ and also that $\\lim _{x \\rightarrow 0} f^{\\prime \\prime}(x)=4$. Hence Exercise 9 again implies that $f^{\\prime \\prime}(x)$ is continuous and $f^{\\prime \\prime}(0)=4$.\n\n\n\nOn $n$-dimensional space let $\\mathbf{f}(x)=\\left(f_1(x), \\ldots, f_n(x)\\right), M_0=\\sup |\\mathbf{f}(x)|$, $M_1=\\sup \\left|\\mathbf{f}^{\\prime}(x)\\right|$, and $M_2=\\sup \\left|\\mathbf{f}^{\\prime \\prime}(x)\\right|$. Just as in the numerical case, there is nothing to prove if $M_2=0$ or $M_0=+\\infty$ or $M_2=+\\infty$, and so we assume $0a$, and let $\\mathbf{u}=\\frac{1}{\\left|\\mathbf{f}^{\\prime}\\left(x_0\\right)\\right|} \\mathbf{f}^{\\prime}\\left(x_0\\right)$. Consider the real-valued function $\\varphi(x)=\\mathrm{u} \\cdot \\mathrm{f}(x)$. Let $N_0, N_1$, and $N_2$ be the suprema of $|\\varphi(x)|,\\left|\\varphi^{\\prime}(x)\\right|$, and $\\left|\\varphi^{\\prime \\prime}(x)\\right|$ respectively. By the Schwarz inequality we have (since $|\\mathbf{u}|=1) N_0 \\leq M_0$ and $N_2 \\leq M_2$, while $N_1 \\geq \\varphi\\left(x_0\\right)=\\left|\\mathbf{f}^{\\prime}\\left(x_0\\right)\\right|>a$. We therefore have $a^2<4 N_0 N_2 \\leq 4 M_0 M_2$. Since $a$ was any positive number less than $M_1$, we have $M_1^2 \\leq 4 M_0 M_2$, i.e., the result holds also for vector-valued functions.\n\n\n\nEquality can hold on any $R^n$, as we see by taking $\\mathbf{f}(x)=(f(x), 0, \\ldots, 0)$ or $\\mathbf{f}(x)=(f(x), f(x), \\ldots, f(x))$, where $f(x)$ is a real-valued function for which equality holds.\n\n\n\n\\end{proof}" - }, - { - "id": "Rudin|exercise_5_17", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen real\nopen topological_space\nopen filter\nopen_locale topological_space\nopen_locale big_operators\nopen_locale complex_conjugate\nopen_locale filter\n\n\nnoncomputable theory\n\n", - "nl_statement": "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)$.\n", - "nl_proof": "\\begin{proof}\n\n Solution. Following the hint, we observe that Theorem $5.15$ (Taylor's formula with remainder) implies that\n\n$$\n\n\\begin{aligned}\n\nf(1) &=f(0)+f^{\\prime}(0)+\\frac{1}{2} f^{\\prime \\prime}(0)+\\frac{1}{6} f^{(3)}(s) \\\\\n\nf(-1) &=f(0)-f^{\\prime}(0)+\\frac{1}{2} f^{\\prime \\prime}(0)-\\frac{1}{6} f^{(3)}(t)\n\n\\end{aligned}\n\n$$\n\nfor some $s \\in(0,1), t \\in(-1,0)$. By subtracting the second equation from the first and using the given values of $f(1), f(-1)$, and $f^{\\prime}(0)$, we obtain\n\n$$\n\n1=\\frac{1}{6}\\left(f^{(3)}(s)+f^{(3)}(t)\\right),\n\n$$\n\nwhich is the desired result. Note that we made no use of the hypothesis $f(0)=0$.\n\n\\end{proof}" - }, - { - "id": "Munkres|exercise_13_1", - "formal_statement": "theorem exercise_13_1 (X : Type*) [topological_space X] (A : set X)\n (h1 : \u2200 x \u2208 A, \u2203 U : set X, x \u2208 U \u2227 is_open U \u2227 U \u2286 A) :\n is_open A :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be a topological space; let $A$ be a subset of $X$. Suppose that for each $x \\in A$ there is an open set $U$ containing $x$ such that $U \\subset A$. Show that $A$ is open in $X$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_3b", - "formal_statement": "theorem exercise_13_3b : \u00ac \u2200 X : Type, \u2200s : set (set X),\n (\u2200 t : set X, t \u2208 s \u2192 (set.infinite t\u1d9c \u2228 t = \u2205 \u2228 t = \u22a4)) \u2192 \n (set.infinite (\u22c3\u2080 s)\u1d9c \u2228 (\u22c3\u2080 s) = \u2205 \u2228 (\u22c3\u2080 s) = \u22a4) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that the collection $$\\mathcal{T}_\\infty = \\{U | X - U \\text{ is infinite or empty or all of X}\\}$$ is does not need to be a topology on the set $X$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_4a1", - "formal_statement": "theorem exercise_13_4a1 (X I : Type*) (T : I \u2192 set (set X)) (h : \u2200 i, is_topology X (T i)) :\n is_topology X (\u22c2 i : I, T i) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "If $\\mathcal{T}_\\alpha$ is a family of topologies on $X$, show that $\\bigcap \\mathcal{T}_\\alpha$ is a topology on $X$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_4a2", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_4b1", - "formal_statement": "theorem exercise_13_4b1 (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 i \u2286 T') \u2227\n \u2200 T'', is_topology X T'' \u2192 (\u2200 i, T i \u2286 T'') \u2192 T'' \u2286 T' :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $\\mathcal{T}_\\alpha$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $\\mathcal{T}_\\alpha$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_4b2", - "formal_statement": "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'' :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_5a", - "formal_statement": "theorem exercise_13_5a {X : Type*}\n [topological_space X] (A : set (set X)) (hA : is_topological_basis A) :\n generate_from A = generate_from (sInter {T | is_topology X T \u2227 A \u2286 T}) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $\\mathcal{A}$ is a basis for a topology on $X$, then the topology generated by $\\mathcal{A}$ equals the intersection of all topologies on $X$ that contain $\\mathcal{A}$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_5b", - "formal_statement": "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}) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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}$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_6", - "formal_statement": "theorem exercise_13_6 :\n \u00ac (\u2200 U, Rl.is_open U \u2192 K_topology.is_open U) \u2227 \u00ac (\u2200 U, K_topology.is_open U \u2192 Rl.is_open U) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that the lower limit topology $\\mathbb{R}_l$ and $K$-topology $\\mathbb{R}_K$ are not comparable.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_8a", - "formal_statement": "theorem exercise_13_8a :\n topological_space.is_topological_basis {S : set \u211d | \u2203 a b : \u211a, a < b \u2227 S = Ioo a b} :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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}$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_13_8b", - "formal_statement": "theorem exercise_13_8b :\n (topological_space.generate_from {S : set \u211d | \u2203 a b : \u211a, a < b \u2227 S = Ico a b}).is_open \u2260\n (lower_limit_topology \u211d).is_open :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that the collection $\\{(a,b) \\mid a < b, a \\text{ and } b \\text{ rational}\\}$ is a basis that generates a topology different from the lower limit topology on $\\mathbb{R}$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_16_1", - "formal_statement": "theorem exercise_16_1 {X : Type*} [topological_space X]\n (Y : set X)\n (A : set Y)\n :\n \u2200 U : set A, is_open U \u2194 is_open (subtype.val '' U) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_16_4", - "formal_statement": "theorem exercise_16_4 {X Y : Type*} [topological_space X] [topological_space Y]\n (\u03c0\u2081 : X \u00d7 Y \u2192 X)\n (\u03c0\u2082 : X \u00d7 Y \u2192 Y)\n (h\u2081 : \u03c0\u2081 = prod.fst)\n (h\u2082 : \u03c0\u2082 = prod.snd) :\n is_open_map \u03c0\u2081 \u2227 is_open_map \u03c0\u2082 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "A map $f: X \\rightarrow Y$ is said to be an open map if for every open set $U$ of $X$, the set $f(U)$ is open in $Y$. Show that $\\pi_{1}: X \\times Y \\rightarrow X$ and $\\pi_{2}: X \\times Y \\rightarrow Y$ are open maps.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_16_6", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_17_4", - "formal_statement": "theorem exercise_17_4 {X : Type*} [topological_space X]\n (U A : set X) (hU : is_open U) (hA : is_closed A) :\n is_open (U \\ A) \u2227 is_closed (A \\ U) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $U$ is open in $X$ and $A$ is closed in $X$, then $U-A$ is open in $X$, and $A-U$ is closed in $X$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_18_8a", - "formal_statement": "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} :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_18_8b", - "formal_statement": "theorem exercise_18_8b {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 continuous (\u03bb x, min (f x) (g x)) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $Y$ be an ordered set in the order topology. Let $f, g: X \\rightarrow Y$ be continuous. Let $h: X \\rightarrow Y$ be the function $h(x)=\\min \\{f(x), g(x)\\}.$ Show that $h$ is continuous.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_18_13", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_19_6a", - "formal_statement": "theorem exercise_19_6a\n {n : \u2115}\n {f : fin n \u2192 Type*} {x : \u2115 \u2192 \u03a0a, f a}\n (y : \u03a0i, f i)\n [\u03a0a, topological_space (f a)] :\n tendsto x at_top (\ud835\udcdd y) \u2194 \u2200 i, tendsto (\u03bb j, (x j) i) at_top (\ud835\udcdd (y i)) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $\\mathbf{x}_1, \\mathbf{x}_2, \\ldots$ be a sequence of the points of the product space $\\prod X_\\alpha$. Show that this sequence converges to the point $\\mathbf{x}$ if and only if the sequence $\\pi_\\alpha(\\mathbf{x}_i)$ converges to $\\pi_\\alpha(\\mathbf{x})$ for each $\\alpha$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_20_2", - "formal_statement": "theorem exercise_20_2\n [topological_space (\u211d \u00d7\u2097 \u211d)] [order_topology (\u211d \u00d7\u2097 \u211d)]\n : metrizable_space (\u211d \u00d7\u2097 \u211d) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that $\\mathbb{R} \\times \\mathbb{R}$ in the dictionary order topology is metrizable.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_21_6a", - "formal_statement": "theorem exercise_21_6a\n (f : \u2115 \u2192 I \u2192 \u211d )\n (h : \u2200 x n, f n x = x ^ n) :\n \u2200 x, \u2203 y, tendsto (\u03bb n, f n x) at_top (\ud835\udcdd y) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Define $f_{n}:[0,1] \\rightarrow \\mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\\left(f_{n}(x)\\right)$ converges for each $x \\in[0,1]$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_21_6b", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_21_8", - "formal_statement": "theorem exercise_21_8\n {X : Type*} [topological_space X] {Y : Type*} [metric_space Y]\n {f : \u2115 \u2192 X \u2192 Y} {x : \u2115 \u2192 X}\n (hf : \u2200 n, continuous (f n))\n (x\u2080 : X)\n (hx : tendsto x at_top (\ud835\udcdd x\u2080))\n (f\u2080 : X \u2192 Y)\n (hh : tendsto_uniformly f f\u2080 at_top) :\n tendsto (\u03bb n, f n (x n)) at_top (\ud835\udcdd (f\u2080 x\u2080)) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be a topological space and let $Y$ be a metric space. Let $f_{n}: X \\rightarrow Y$ be a sequence of continuous functions. Let $x_{n}$ be a sequence of points of $X$ converging to $x$. Show that if the sequence $\\left(f_{n}\\right)$ converges uniformly to $f$, then $\\left(f_{n}\\left(x_{n}\\right)\\right)$ converges to $f(x)$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_22_2a", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_22_2b", - "formal_statement": "theorem exercise_22_2b {X : Type*} [topological_space X]\n {A : set X} (r : X \u2192 A) (hr : continuous r) (h : \u2200 x : A, r x = x) :\n quotient_map r :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "If $A \\subset X$, a retraction of $X$ onto $A$ is a continuous map $r: X \\rightarrow A$ such that $r(a)=a$ for each $a \\in A$. Show that a retraction is a quotient map.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_22_5", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_2", - "formal_statement": "theorem exercise_23_2 {X : Type*}\n [topological_space X] {A : \u2115 \u2192 set X} (hA : \u2200 n, is_connected (A n))\n (hAn : \u2200 n, A n \u2229 A (n + 1) \u2260 \u2205) :\n is_connected (\u22c3 n, A n) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $\\left\\{A_{n}\\right\\}$ be a sequence of connected subspaces of $X$, such that $A_{n} \\cap A_{n+1} \\neq \\varnothing$ for all $n$. Show that $\\bigcup A_{n}$ is connected.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_3", - "formal_statement": "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)) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_4", - "formal_statement": "theorem exercise_23_4 {X : Type*} [topological_space X] [cofinite_topology X]\n (s : set X) : set.infinite s \u2192 is_connected s :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ is an infinite set, it is connected in the finite complement topology.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_6", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_9", - "formal_statement": "theorem exercise_23_9 {X Y : Type*}\n [topological_space X] [topological_space Y]\n (A\u2081 A\u2082 : set X)\n (B\u2081 B\u2082 : set Y)\n (hA : A\u2081 \u2282 A\u2082)\n (hB : B\u2081 \u2282 B\u2082)\n (hA : is_connected A\u2082)\n (hB : is_connected B\u2082) :\n is_connected ({x | \u2203 a b, x = (a, b) \u2227 a \u2208 A\u2082 \u2227 b \u2208 B\u2082} \\\n {x | \u2203 a b, x = (a, b) \u2227 a \u2208 A\u2081 \u2227 b \u2208 B\u2081}) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X \\times Y)-(A \\times B)$ is connected.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_23_11", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_24_2", - "formal_statement": "theorem exercise_24_2 {f : (metric.sphere 0 1 : set \u211d) \u2192 \u211d}\n (hf : continuous f) : \u2203 x, f x = f (-x) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $f: S^{1} \\rightarrow \\mathbb{R}$ be a continuous map. Show there exists a point $x$ of $S^{1}$ such that $f(x)=f(-x)$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_24_3a", - "formal_statement": "theorem exercise_24_3a [topological_space I]\n (f : I \u2192 I) (hf : continuous f) :\n \u2203 (x : I), f x = x :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.)\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_25_4", - "formal_statement": "theorem exercise_25_4 {X : Type*} [topological_space X]\n [loc_path_connected_space X] (U : set X) (hU : is_open U)\n (hcU : is_connected U) : is_path_connected U :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_25_9", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_26_11", - "formal_statement": "theorem exercise_26_11\n {X : Type*} [topological_space X] [compact_space X] [t2_space X]\n (A : set (set X)) (hA : \u2200 (a b : set X), a \u2208 A \u2192 b \u2208 A \u2192 a \u2286 b \u2228 b \u2286 a)\n (hA' : \u2200 a \u2208 A, is_closed a) (hA'' : \u2200 a \u2208 A, is_connected a) :\n is_connected (\u22c2\u2080 A) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be a compact Hausdorff space. Let $\\mathcal{A}$ be a collection of closed connected subsets of $X$ that is simply ordered by proper inclusion. Then $Y=\\bigcap_{A \\in \\mathcal{A}} A$ is connected.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_26_12", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_27_4", - "formal_statement": "theorem exercise_27_4\n {X : Type*} [metric_space X] [connected_space X] (hX : \u2203 x y : X, x \u2260 y) :\n \u00ac countable (univ : set X) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that a connected metric space having more than one point is uncountable.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_28_4", - "formal_statement": "theorem exercise_28_4 {X : Type*}\n [topological_space X] (hT1 : t1_space X) :\n countably_compact X \u2194 limit_point_compact X :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_28_5", - "formal_statement": "theorem exercise_28_5\n (X : Type*) [topological_space X] :\n countably_compact X \u2194 \u2200 (C : \u2115 \u2192 set X), (\u2200 n, is_closed (C n)) \u2227\n (\u2200 n, C n \u2260 \u2205) \u2227 (\u2200 n, C n \u2286 C (n + 1)) \u2192 \u2203 x, \u2200 n, x \u2208 C n :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that X is countably compact if and only if every nested sequence $C_1 \\supset C_2 \\supset \\cdots$ of closed nonempty sets of X has a nonempty intersection.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_28_6", - "formal_statement": "theorem exercise_28_6 {X : Type*} [metric_space X]\n [compact_space X] {f : X \u2192 X} (hf : isometry f) :\n function.bijective f :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_29_1", - "formal_statement": "theorem exercise_29_1 : \u00ac locally_compact_space \u211a :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that the rationals $\\mathbb{Q}$ are not locally compact.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_29_4", - "formal_statement": "theorem exercise_29_4 [topological_space (\u2115 \u2192 I)] :\n \u00ac locally_compact_space (\u2115 \u2192 I) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that $[0, 1]^\\omega$ is not locally compact in the uniform topology.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_29_10", - "formal_statement": "theorem exercise_29_10 {X : Type*}\n [topological_space X] [t2_space X] (x : X)\n (hx : \u2203 U : set X, x \u2208 U \u2227 is_open U \u2227 (\u2203 K : set X, U \u2282 K \u2227 is_compact K))\n (U : set X) (hU : is_open U) (hxU : x \u2208 U) :\n \u2203 (V : set X), is_open V \u2227 x \u2208 V \u2227 is_compact (closure V) \u2227 closure V \u2286 U :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each neighborhood $U$ of $x$, there is a neighborhood $V$ of $x$ such that $\\bar{V}$ is compact and $\\bar{V} \\subset U$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_30_10", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_30_13", - "formal_statement": "theorem exercise_30_13 {X : Type*} [topological_space X]\n (h : \u2203 (s : set X), countable s \u2227 dense s) (U : set (set X))\n (hU : \u2200 (x y : set X), x \u2208 U \u2192 y \u2208 U \u2192 x \u2260 y \u2192 x \u2229 y = \u2205) :\n countable U :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_31_1", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_31_2", - "formal_statement": "theorem exercise_31_2 {X : Type*}\n [topological_space X] [normal_space X] {A B : set X}\n (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) :\n \u2203 (U V : set X), is_open U \u2227 is_open V \u2227 A \u2286 U \u2227 B \u2286 V \u2227 closure U \u2229 closure V = \u2205 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_31_3", - "formal_statement": "theorem exercise_31_3 {\u03b1 : Type*} [partial_order \u03b1]\n [topological_space \u03b1] (h : order_topology \u03b1) : regular_space \u03b1 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that every order topology is regular.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_32_1", - "formal_statement": "theorem exercise_32_1 {X : Type*} [topological_space X]\n (hX : normal_space X) (A : set X) (hA : is_closed A) :\n normal_space {x // x \u2208 A} :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that a closed subspace of a normal space is normal.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_32_2a", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $\\prod X_\\alpha$ is Hausdorff, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_32_2b", - "formal_statement": "theorem exercise_32_2b\n {\u03b9 : Type*} {X : \u03b9 \u2192 Type*} [\u2200 i, topological_space (X i)]\n (h : \u2200 i, nonempty (X i)) (h2 : regular_space (\u03a0 i, X i)) :\n \u2200 i, regular_space (X i) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $\\prod X_\\alpha$ is regular, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_32_2c", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that if $\\prod X_\\alpha$ is normal, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_32_3", - "formal_statement": "theorem exercise_32_3 {X : Type*} [topological_space X]\n (hX : locally_compact_space X) (hX' : t2_space X) :\n regular_space X :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that every locally compact Hausdorff space is regular.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_33_7", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Show that every locally compact Hausdorff space is completely regular.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_33_8", - "formal_statement": "theorem exercise_33_8\n (X : Type*) [topological_space X] [regular_space X]\n (h : \u2200 x A, is_closed A \u2227 \u00ac x \u2208 A \u2192\n \u2203 (f : X \u2192 I), continuous f \u2227 f x = (1 : I) \u2227 f '' A = {0})\n (A B : set X) (hA : is_closed A) (hB : is_closed B)\n (hAB : disjoint A B)\n (hAc : is_compact A) :\n \u2203 (f : X \u2192 I), continuous f \u2227 f '' A = {0} \u2227 f '' B = {1} :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be completely regular, let $A$ and $B$ be disjoint closed subsets of $X$. Show that if $A$ is compact, there is a continuous function $f \\colon X \\rightarrow [0, 1]$ such that $f(A) = \\{0\\}$ and $f(B) = \\{1\\}$.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_34_9", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_38_6", - "formal_statement": "theorem exercise_38_6 {X : Type*}\n (X : Type*) [topological_space X] [regular_space X]\n (h : \u2200 x A, is_closed A \u2227 \u00ac x \u2208 A \u2192\n \u2203 (f : X \u2192 I), continuous f \u2227 f x = (1 : I) \u2227 f '' A = {0}) :\n is_connected (univ : set X) \u2194 is_connected (univ : set (stone_cech X)) :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "Let $X$ be completely regular. Show that $X$ is connected if and only if the Stone-\u010cech compactification of $X$ is connected.\n", - "nl_proof": "" - }, - { - "id": "Munkres|exercise_43_2", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set topological_space \nopen_locale classical\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_1_2", - "formal_statement": "theorem exercise_1_2 :\n (\u27e8-1/2, real.sqrt 3 / 2\u27e9 : \u2102) ^ 3 = -1 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Show that $\\frac{-1 + \\sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).\n", - "nl_proof": "\\begin{proof}\n\n$$\n\n\\left(\\frac{-1+\\sqrt{3} i}{2}\\right)^2=\\frac{-1-\\sqrt{3} i}{2},\n\n$$\n\nhence\n\n$$\n\n\\left(\\frac{-1+\\sqrt{3} i}{2}\\right)^3=\\frac{-1-\\sqrt{3} i}{2} \\cdot \\frac{-1+\\sqrt{3} i}{2}=1\n\n$$\n\nThis means $\\frac{-1+\\sqrt{3} i}{2}$ is a cube root of 1.\n\n\\end{proof}" - }, - { - "id": "Axler|exercise_1_3", - "formal_statement": "theorem exercise_1_3 {F V : Type*} [add_comm_group V] [field F]\n [module F V] {v : V} : -(-v) = v :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that $-(-v) = v$ for every $v \\in V$.\n", - "nl_proof": "\\begin{proof}\n\n By definition, we have\n\n$$\n\n(-v)+(-(-v))=0 \\quad \\text { and } \\quad v+(-v)=0 .\n\n$$\n\nThis implies both $v$ and $-(-v)$ are additive inverses of $-v$, by the uniqueness of additive inverse, it follows that $-(-v)=v$.\n\n\\end{proof}" - }, - { - "id": "Axler|exercise_1_4", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that if $a \\in \\mathbf{F}$, $v \\in V$, and $av = 0$, then $a = 0$ or $v = 0$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_1_6", - "formal_statement": "theorem exercise_1_6 : \u2203 U : set (\u211d \u00d7 \u211d),\n (U \u2260 \u2205) \u2227\n (\u2200 (u v : \u211d \u00d7 \u211d), u \u2208 U \u2227 v \u2208 U \u2192 u + v \u2208 U) \u2227\n (\u2200 (u : \u211d \u00d7 \u211d), u \u2208 U \u2192 -u \u2208 U) \u2227\n (\u2200 U' : submodule \u211d (\u211d \u00d7 \u211d), U \u2260 \u2191U') :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Give an example of a nonempty subset $U$ of $\\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \\in U$ whenever $u \\in U$), but $U$ is not a subspace of $\\mathbf{R}^2$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_1_7", - "formal_statement": "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') :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_1_8", - "formal_statement": "theorem exercise_1_8 {F V : Type*} [add_comm_group V] [field F]\n [module F V] {\u03b9 : Type*} (u : \u03b9 \u2192 submodule F V) :\n \u2203 U : submodule F V, (\u22c2 (i : \u03b9), (u i).carrier) = \u2191U :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_1_9", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_3_1", - "formal_statement": "theorem exercise_3_1 {F V : Type*} \n [add_comm_group V] [field F] [module F V] [finite_dimensional F V]\n (T : V \u2192\u2097[F] V) (hT : finrank F V = 1) :\n \u2203 c : F, \u2200 v : V, T v = c \u2022 v:=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\\operatorname{dim} V=1$ and $T \\in \\mathcal{L}(V, V)$, then there exists $a \\in \\mathbf{F}$ such that $T v=a v$ for all $v \\in V$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_3_8", - "formal_statement": "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):=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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\\}$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_4_4", - "formal_statement": "theorem exercise_4_4 (p : polynomial \u2102) :\n p.degree = @card (root_set p \u2102) (polynomial.root_set_fintype p \u2102) \u2194\n disjoint\n (@card (root_set p.derivative \u2102) (polynomial.root_set_fintype p.derivative \u2102))\n (@card (root_set p \u2102) (polynomial.root_set_fintype p \u2102)) :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $p \\in \\mathcal{P}(\\mathbf{C})$ has degree $m$. Prove that $p$ has $m$ distinct roots if and only if $p$ and its derivative $p^{\\prime}$ have no roots in common.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_1", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_4", - "formal_statement": "theorem exercise_5_4 {F V : Type*} [add_comm_group V] [field F]\n [module F V] (S T : V \u2192\u2097[F] V) (hST : S \u2218 T = T \u2218 S) (c : F):\n map S (T - c \u2022 id).ker = (T - c \u2022 id).ker :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose that $S, T \\in \\mathcal{L}(V)$ are such that $S T=T S$. Prove that $\\operatorname{null} (T-\\lambda I)$ is invariant under $S$ for every $\\lambda \\in \\mathbf{F}$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_11", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $S, T \\in \\mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_12", - "formal_statement": "theorem exercise_5_12 {F V : Type*} [add_comm_group V] [field F]\n [module F V] {S : End F V}\n (hS : \u2200 v : V, \u2203 c : F, v \u2208 eigenspace S c) :\n \u2203 c : F, S = c \u2022 id :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $T \\in \\mathcal{L}(V)$ is such that every vector in $V$ is an eigenvector of $T$. Prove that $T$ is a scalar multiple of the identity operator.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_13", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_20", - "formal_statement": "theorem exercise_5_20 {F V : Type*} [add_comm_group V] [field F]\n [module F V] [finite_dimensional F V] {S T : End F V}\n (h1 : @card T.eigenvalues (eigenvalues.fintype T) = finrank F V)\n (h2 : \u2200 v : V, \u2203 c : F, v \u2208 eigenspace S c \u2194 \u2203 c : F, v \u2208 eigenspace T c) :\n S * T = T * S :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose that $T \\in \\mathcal{L}(V)$ has $\\operatorname{dim} V$ distinct eigenvalues and that $S \\in \\mathcal{L}(V)$ has the same eigenvectors as $T$ (not necessarily with the same eigenvalues). Prove that $S T=T S$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_5_24", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_6_2", - "formal_statement": "theorem exercise_6_2 {V : Type*} [add_comm_group V] [module \u2102 V]\n [inner_product_space \u2102 V] (u v : V) :\n \u27eau, v\u27eb_\u2102 = 0 \u2194 \u2200 (a : \u2102), \u2225u\u2225 \u2264 \u2225u + a \u2022 v\u2225 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $u, v \\in V$. Prove that $\\langle u, v\\rangle=0$ if and only if $\\|u\\| \\leq\\|u+a v\\|$ for all $a \\in \\mathbf{F}$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_6_3", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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}$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_6_7", - "formal_statement": "theorem exercise_6_7 {V : Type*} [inner_product_space \u2102 V] (u v : V) :\n \u27eau, v\u27eb_\u2102 = (\u2225u + v\u2225^2 - \u2225u - v\u2225^2 + I*\u2225u + I\u2022v\u2225^2 - I*\u2225u-I\u2022v\u2225^2) / 4 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that if $V$ is a complex inner-product space, then $\\langle u, v\\rangle=\\frac{\\|u+v\\|^{2}-\\|u-v\\|^{2}+\\|u+i v\\|^{2} i-\\|u-i v\\|^{2} i}{4}$ for all $u, v \\in V$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_6_13", - "formal_statement": "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 \u2225v\u2225^2 = \u2211 i : fin n, \u2225\u27eav, e i\u27eb_\u2102\u2225^2 \u2194 v \u2208 span \u2102 (e '' univ) :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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)$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_6_16", - "formal_statement": "theorem exercise_6_16 {K V : Type*} [is_R_or_C K] [inner_product_space K V]\n {U : submodule K V} : \n U.orthogonal = \u22a5 \u2194 U = \u22a4 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $U$ is a subspace of $V$. Prove that $U^{\\perp}=\\{0\\}$ if and only if $U=V$\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_5", - "formal_statement": "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} :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Show that if $\\operatorname{dim} V \\geq 2$, then the set of normal operators on $V$ is not a subspace of $\\mathcal{L}(V)$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_6", - "formal_statement": "theorem exercise_7_6 {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 T.range = T.adjoint.range :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that if $T \\in \\mathcal{L}(V)$ is normal, then $\\operatorname{range} T=\\operatorname{range} T^{*}.$\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_9", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_10", - "formal_statement": "theorem exercise_7_10 {V : Type*} [inner_product_space \u2102 V]\n [finite_dimensional \u2102 V] (T : End \u2102 V)\n (hT : T * T.adjoint = T.adjoint * T) (hT1 : T^9 = T^8) :\n is_self_adjoint T \u2227 T^2 = T :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $V$ is a complex inner-product space and $T \\in \\mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$.\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_11", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "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$.)\n", - "nl_proof": "" - }, - { - "id": "Axler|exercise_7_14", - "formal_statement": "theorem exercise_7_14 {\ud835\udd5c V : Type*} [is_R_or_C \ud835\udd5c]\n [inner_product_space \ud835\udd5c V] [finite_dimensional \ud835\udd5c V]\n {T : End \ud835\udd5c V} (hT : is_self_adjoint T)\n {l : \ud835\udd5c} {\u03b5 : \u211d} (he : \u03b5 > 0) : \u2203 v : V, \u2225v\u2225 = 1 \u2227 \u2225T v - l \u2022 v\u2225 < \u03b5 \u2192\n \u2203 l' : T.eigenvalues, \u2225l - l'\u2225 < \u03b5 :=", - "src_header": "import .common \n\nopen set fintype complex polynomial submodule linear_map finite_dimensional\nopen module module.End inner_product_space\n\nopen_locale big_operators\n\n", - "nl_statement": "Suppose $T \\in \\mathcal{L}(V)$ is self-adjoint, $\\lambda \\in \\mathbf{F}$, and $\\epsilon>0$. Prove that if there exists $v \\in V$ such that $\\|v\\|=1$ and $\\|T v-\\lambda v\\|<\\epsilon,$ then $T$ has an eigenvalue $\\lambda^{\\prime}$ such that $\\left|\\lambda-\\lambda^{\\prime}\\right|<\\epsilon$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_1_27", - "formal_statement": "theorem exercise_1_27 {n : \u2115} (hn : odd n) : 8 \u2223 (n^2 - 1) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "For all odd $n$ show that $8 \\mid n^{2}-1$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_1_30", - "formal_statement": "theorem exercise_1_30 {n : \u2115} : \n \u00ac \u2203 a : \u2124, \u2211 (i : fin n), (1 : \u211a) / (n+2) = a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}$ is not an integer.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_1_31", - "formal_statement": "theorem exercise_1_31 : (\u27e81, 1\u27e9 : gaussian_int) ^ 2 \u2223 2 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that 2 is divisible by $(1+i)^{2}$ in $\\mathbb{Z}[i]$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_2_4", - "formal_statement": "theorem exercise_2_4 {a : \u2124} (ha : a \u2260 0) \n (f_a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_2_21", - "formal_statement": "theorem exercise_2_21 {l : \u2115 \u2192 \u211d} \n (hl : \u2200 p n : \u2115, p.prime \u2192 l (p^n) = log p )\n (hl1 : \u2200 m : \u2115, \u00ac is_prime_pow m \u2192 l m = 0) :\n l = \u03bb n, \u2211 d : divisors n, moebius (n/d) * log d :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Define $\\wedge(n)=\\log p$ if $n$ is a power of $p$ and zero otherwise. Prove that $\\sum_{A \\mid n} \\mu(n / d) \\log d$ $=\\wedge(n)$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_2_27a", - "formal_statement": "theorem exercise_2_27a : \n \u00ac summable (\u03bb i : {p : \u2124 // squarefree p}, (1 : \u211a) / i) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that $\\sum^{\\prime} 1 / n$, the sum being over square free integers, diverges.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_3_1", - "formal_statement": "theorem exercise_3_1 : infinite {p : primes // p \u2261 -1 [ZMOD 6]} :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that there are infinitely many primes congruent to $-1$ modulo 6 .\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_3_4", - "formal_statement": "theorem exercise_3_4 : \u00ac \u2203 x y : \u2124, 3*x^2 + 2 = y^2 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_3_5", - "formal_statement": "theorem exercise_3_5 : \u00ac \u2203 x y : \u2124, 7*x^3 + 2 = y^3 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that the equation $7 x^{3}+2=y^{3}$ has no solution in integers.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_3_10", - "formal_statement": "theorem exercise_3_10 {n : \u2115} (hn0 : \u00ac n.prime) (hn1 : n \u2260 4) : \n factorial (n-1) \u2261 0 [MOD n] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $n$ is not a prime, show that $(n-1) ! \\equiv 0(n)$, except when $n=4$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_3_14", - "formal_statement": "theorem exercise_3_14 {p q n : \u2115} (hp0 : p.prime \u2227 p > 2) \n (hq0 : q.prime \u2227 q > 2) (hpq0 : p \u2260 q) (hpq1 : p - 1 \u2223 q - 1)\n (hn : n.gcd (p*q) = 1) : \n n^(q-1) \u2261 1 [MOD p*q] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $p$ and $q$ be distinct odd primes such that $p-1$ divides $q-1$. If $(n, p q)=1$, show that $n^{q-1} \\equiv 1(p q)$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_4_4", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_5_13", - "formal_statement": "theorem exercise_5_13 {p x: \u2124} (hp : prime p) \n (hpx : p \u2223 (x^4 - x^2 + 1)) : p \u2261 1 [ZMOD 12] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that any prime divisor of $x^{4}-x^{2}+1$ is congruent to 1 modulo 12 .\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_5_28", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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}$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_5_37", - "formal_statement": "theorem exercise_5_37 {p q : \u2115} [fact(p.prime)] [fact(q.prime)] {a : \u2124}\n (ha : a < 0) (h0 : p \u2261 q [ZMOD 4*a]) (h1 : \u00ac ((p : \u2124) \u2223 a)) :\n legendre_sym p a = legendre_sym q a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that if $a$ is negative then $p \\equiv q(4 a) together with p\\not | a$ imply $(a / p)=(a / q)$.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_18_1", - "formal_statement": "theorem exercise_18_1 : \u00ac \u2203 x y : \u2124, 165*x^2 - 21*y^2 = 19 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that $165 x^{2}-21 y^{2}=19$ has no integral solution.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_12_12", - "formal_statement": "theorem exercise_12_12 : is_algebraic \u211a (sin (real.pi/12)) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that $\\sin (\\pi / 12)$ is an algebraic number.\n", - "nl_proof": "" - }, - { - "id": "Ireland-Rosen|exercise_18_4", - "formal_statement": "theorem exercise_18_4 {n : \u2115} (hn : \u2203 x y z w : \u2124, \n x^3 + y^3 = n \u2227 z^3 + w^3 = n \u2227 x \u2260 z \u2227 x \u2260 w \u2227 y \u2260 z \u2227 y \u2260 w) : \n n \u2265 1729 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen zsqrtd gaussian_int char_p nat.arithmetic_function \n\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_13a", - "formal_statement": "theorem exercise_1_13a {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).re = c) :\n f a = f b :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Re}(f)$ is constant, then $f$ is constant.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_13b", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Im}(f)$ is constant, then $f$ is constant.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_13c", - "formal_statement": "theorem exercise_1_13c {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, abs (f z) = c) :\n f a = f b :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $|f|$ is constant, then $f$ is constant.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_19a", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Prove that the power series $\\sum nz^n$ does not converge on any point of the unit circle.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_19b", - "formal_statement": "theorem exercise_1_19b (z : \u2102) (hz : abs z = 1) (s : \u2115 \u2192 \u2102)\n (h : s = (\u03bb n, \u2211 i in (finset.range n), i * z / i ^ 2)) :\n \u2203 y, tendsto s at_top (\ud835\udcdd y) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Prove that the power series $\\sum zn/n^2$ converges at every point of the unit circle.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_19c", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Prove that the power series $\\sum zn/n$ converges at every point of the unit circle except $z = 1$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_22", - "formal_statement": "theorem exercise_1_22 (n : \u2115) (S : fin n \u2192 set \u2115) (f : fin n \u2192 \u2115 \u00d7 \u2115)\n (h : \u2200 i, S i = set.range (\u03bb j, (f i).fst + j * (f i).snd)) :\n \u00ac (\u22c3 i, S i) = (set.univ : set \u2115) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Let $\\mathbb{N} = {1, 2, 3, \\ldots}$ denote the set of positive integers. A subset $S \\subset \\mathbb{N}$ is said to be in arithmetic progression if $S = {a, a + d, a + 2d, a + 3d, \\ldots}$ where $a, d \\in \\mathbb{N}$. Here $d$ is called the step of $S$. Show that $\\mathbb{N}$ cannot be partitioned into a finite number of subsets that are in arithmetic progression with distinct steps (except for the trivial case $a = d = 1$).\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_1_26", - "formal_statement": "theorem exercise_1_26\n (f F\u2081 F\u2082 : \u2102 \u2192 \u2102) (\u03a9 : set \u2102) (h1 : is_open \u03a9) (h2 : is_connected \u03a9)\n (hF\u2081 : differentiable_on \u2102 F\u2081 \u03a9) (hF\u2082 : differentiable_on \u2102 F\u2082 \u03a9)\n (hdF\u2081 : \u2200 x \u2208 \u03a9, deriv F\u2081 x = f x) (hdF\u2082 : \u2200 x \u2208 \u03a9, deriv F\u2082 x = f x)\n : \u2203 c : \u2102, \u2200 x, F\u2081 x = F\u2082 x + c :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Suppose $f$ is continuous in a region $\\Omega$. Prove that any two primitives of $f$ (if they exist) differ by a constant.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_2_2", - "formal_statement": "theorem exercise_2_2 :\n tendsto (\u03bb y, \u222b x in 0..y, real.sin x / x) at_top (\ud835\udcdd (real.pi / 2)) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Show that $\\int_{0}^{\\infty} \\frac{\\sin x}{x} d x=\\frac{\\pi}{2}$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_2_9", - "formal_statement": "theorem exercise_2_9\n {f : \u2102 \u2192 \u2102} (\u03a9 : set \u2102) (b : metric.bounded \u03a9) (h : is_open \u03a9)\n (hf : differentiable_on \u2102 f \u03a9) (z \u2208 \u03a9) (hz : f z = z) (h'z : deriv f z = 1) :\n \u2203 (f_lin : \u2102 \u2192L[\u2102] \u2102), \u2200 x \u2208 \u03a9, f x = f_lin x :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Let $\\Omega$ be a bounded open subset of $\\mathbb{C}$, and $\\varphi: \\Omega \\rightarrow \\Omega$ a holomorphic function. Prove that if there exists a point $z_{0} \\in \\Omega$ such that $\\varphi\\left(z_{0}\\right)=z_{0} \\quad \\text { and } \\quad \\varphi^{\\prime}\\left(z_{0}\\right)=1$ then $\\varphi$ is linear.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_2_13", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_3_3", - "formal_statement": "theorem exercise_3_3 (a : \u211d) (ha : 0 < a) :\n tendsto (\u03bb y, \u222b x in -y..y, real.cos x / (x ^ 2 + a ^ 2))\n at_top (\ud835\udcdd (real.pi * (real.exp (-a) / a))) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Show that $ \\int_{-\\infty}^{\\infty} \\frac{\\cos x}{x^2 + a^2} dx = \\pi \\frac{e^{-a}}{a}$ for $a > 0$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_3_4", - "formal_statement": "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)))) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Show that $ \\int_{-\\infty}^{\\infty} \\frac{x \\sin x}{x^2 + a^2} dx = \\pi e^{-a}$ for $a > 0$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_3_9", - "formal_statement": "theorem exercise_3_9 : \u222b x in 0..1, real.log (real.sin (real.pi * x)) = - real.log 2 :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Show that $\\int_0^1 \\log(\\sin \\pi x) dx = - \\log 2$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_3_14", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_3_22", - "formal_statement": "theorem exercise_3_22 (D : set \u2102) (hD : D = ball 0 1) (f : \u2102 \u2192 \u2102)\n (hf : differentiable_on \u2102 f D) (hfc : continuous_on f (closure D)) :\n \u00ac \u2200 z \u2208 (sphere (0 : \u2102) 1), f z = 1 / z :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "Show that there is no holomorphic function $f$ in the unit disc $D$ that extends continuously to $\\partial D$ such that $f(z) = 1/z$ for $z \\in \\partial D$.\n", - "nl_proof": "" - }, - { - "id": "Shakarchi|exercise_5_1", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen complex filter function interval_integral\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_12a", - "formal_statement": "theorem exercise_2_12a (f : \u2115 \u2192 \u2115) (p : \u2115 \u2192 \u211d) (a : \u211d)\n (hf : injective f) (hp : tendsto p at_top (\ud835\udcdd a)) :\n tendsto (\u03bb n, p (f n)) at_top (\ud835\udcdd a) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $(p_n)$ be a sequence and $f:\\mathbb{N}\\to\\mathbb{N}$. The sequence $(q_k)_{k\\in\\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_12b", - "formal_statement": "theorem exercise_2_12b (f : \u2115 \u2192 \u2115) (p : \u2115 \u2192 \u211d) (a : \u211d)\n (hf : surjective f) (hp : tendsto p at_top (\ud835\udcdd a)) :\n tendsto (\u03bb n, p (f n)) at_top (\ud835\udcdd a) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $(p_n)$ be a sequence and $f:\\mathbb{N}\\to\\mathbb{N}$. The sequence $(q_k)_{k\\in\\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is a surjection, the limit of a sequence is unaffected by rearrangement.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_26", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Prove that a set $U \\subset M$ is open if and only if none of its points are limits of its complement.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_29", - "formal_statement": "theorem exercise_2_29 (M : Type*) [metric_space M]\n (O C : set (set M))\n (hO : O = {s | is_open s})\n (hC : C = {s | is_closed s}) :\n \u2203 f : O \u2192 C, bijective f :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $\\mathcal{T}$ be the collection of open subsets of a metric space $\\mathrm{M}$, and $\\mathcal{K}$ the collection of closed subsets. Show that there is a bijection from $\\mathcal{T}$ onto $\\mathcal{K}$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_32a", - "formal_statement": "theorem exercise_2_32a (A : set \u2115) : is_clopen A :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Show that every subset of $\\mathbb{N}$ is clopen.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_41", - "formal_statement": "theorem exercise_2_41 (m : \u2115) {X : Type*} [normed_space \u211d ((fin m) \u2192 \u211d)] :\n is_compact (metric.closed_ball 0 1) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $\\|\\cdot\\|$ be any norm on $\\mathbb{R}^{m}$ and let $B=\\left\\{x \\in \\mathbb{R}^{m}:\\|x\\| \\leq 1\\right\\}$. Prove that $B$ is compact.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_46", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "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)$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_57", - "formal_statement": "theorem exercise_2_57 {X : Type*} [topological_space X]\n : \u2203 (S : set X), is_connected S \u2227 \u00ac is_connected (interior S) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Show that if $S$ is connected, it is not true in general that its interior is connected.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_79", - "formal_statement": "theorem exercise_2_79\n {M : Type*} [topological_space M] [compact_space M]\n [loc_path_connected_space M] (hM : nonempty M)\n (hM : connected_space M) : path_connected_space M :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Prove that if $M$ is nonempty compact, locally path-connected and connected then it is path-connected.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_85", - "formal_statement": "theorem exercise_2_85\n (M : Type*) [topological_space M] [compact_space M]\n (U : set (set M)) (hU : \u2200 p, \u2203 (U\u2081 U\u2082 \u2208 U), p \u2208 U\u2081 \u2227 p \u2208 U\u2082 \u2227 U\u2081 \u2260 U\u2082) :\n \u2203 (V : set (set M)), set.finite V \u2227\n \u2200 p, \u2203 (V\u2081 V\u2082 \u2208 V), p \u2208 V\u2081 \u2227 p \u2208 V\u2082 \u2227 V\u2081 \u2260 V\u2082 :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Suppose that $M$ is compact and that $\\mathcal{U}$ is an open covering of $M$ which is redundant in the sense that each $p \\in M$ is contained in at least two members of $\\mathcal{U}$. Show that $\\mathcal{U}$ reduces to a finite subcovering with the same property.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_92", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_109", - "formal_statement": "theorem exercise_2_109\n {M : Type*} [metric_space M]\n (h : \u2200 x y z : M, dist x z = max (dist x y) (dist y z)) :\n totally_disconnected_space M :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "A metric on $M$ is an ultrametric if for all $x, y, z \\in M$, $d(x, z) \\leq \\max \\{d(x, y), d(y, z)\\} .$ Show that a metric space with an ultrametric is totally disconnected.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_126", - "formal_statement": "theorem exercise_2_126 {E : set \u211d}\n (hE : \u00ac set.countable E) : \u2203 (p : \u211d), cluster_pt p (\ud835\udcdf E) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Suppose that $E$ is an uncountable subset of $\\mathbb{R}$. Prove that there exists a point $p \\in \\mathbb{R}$ at which $E$ condenses.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_2_137", - "formal_statement": "theorem exercise_2_137\n {M : Type*} [metric_space M] [separable_space M] [complete_space M]\n {P : set M} (hP : is_closed P)\n (hP' : is_closed P \u2227 P = {x | cluster_pt x (\ud835\udcdf P)}) :\n \u2200 x \u2208 P, \u2200 n \u2208 (\ud835\udcdd x), \u00ac set.countable n :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $P$ be a closed perfect subset of a separable complete metric space $M$. Prove that each point of $P$ is a condensation point of $P$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_3_1", - "formal_statement": "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 :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_3_4", - "formal_statement": "theorem exercise_3_4 (n : \u2115) :\n tendsto (\u03bb n, (sqrt (n + 1) - sqrt n)) at_top (\ud835\udcdd 0) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Prove that $\\sqrt{n+1}-\\sqrt{n} \\rightarrow 0$ as $n \\rightarrow \\infty$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_3_11a", - "formal_statement": "theorem exercise_3_11a\n {f : \u211d \u2192 \u211d} {a b x : \u211d}\n (h1 : differentiable_within_at \u211d f (set.Ioo a b) x)\n (h2 : differentiable_within_at \u211d (deriv f) (set.Ioo a b) x) :\n \u2203 l, tendsto (\u03bb h, (f (x - h) - 2 * f x + f (x + h)) / h ^ 2) (\ud835\udcdd 0) (\ud835\udcdd l)\n \u2227 deriv (deriv f) x = l :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $f \\colon (a, b) \\rightarrow \\mathbb{R}$ be given. If $f''(x)$ exists, prove that \\[\\lim_{h \\rightarrow 0} \\frac{f(x - h) - 2f(x) + f(x + h)}{h^2} = f''(x).\\]\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_3_63a", - "formal_statement": "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) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Prove that $\\sum 1/k(\\log(k))^p$ converges when $p > 1$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_3_63b", - "formal_statement": "theorem exercise_3_63b (p : \u211d) (f : \u2115 \u2192 \u211d) (hp : p \u2264 1)\n (h : f = \u03bb k, (1 : \u211d) / (k * (log k) ^ p)) :\n \u00ac \u2203 l, tendsto f at_top (\ud835\udcdd l) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Prove that $\\sum 1/k(\\log(k))^p$ diverges when $p \\leq 1$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_4_15a", - "formal_statement": "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|)) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_4_19", - "formal_statement": "theorem exercise_4_19 {M : Type*} [metric_space M]\n [compact_space M] (A : set M) (hA : dense A) (\u03b4 : \u211d) (h\u03b4 : \u03b4 > 0) :\n \u2203 (A_fin : set M), A_fin \u2282 A \u2227 set.finite A_fin \u2227 \u2200 (x : M), \u2203 i \u2208 A_fin, dist x i < \u03b4 :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "If $M$ is compact and $A$ is dense in $M$, prove that for each $\\delta > 0$ there is a finite subset $\\{a_1, \\ldots , a_k\\} \\subset A$ which is $\\delta$-dense in $M$ in the sense that each $x \\in M$ lies within distance $\\delta$ of at least one of the points $a_1,\\ldots, a_k$.\n", - "nl_proof": "" - }, - { - "id": "Pugh|exercise_5_2", - "formal_statement": "theorem exercise_5_2 {V : Type*} [normed_add_comm_group V]\n [normed_space \u2102 V] {W : Type*} [normed_add_comm_group W] [normed_space \u2102 W] :\n normed_space \u2102 (continuous_linear_map (id \u2102) V W) :=", - "src_header": "import .common\n\nopen set real filter function ring_hom\nopen_locale big_operators\nopen_locale filter\nopen_locale topological_space\nnoncomputable theory \n\n", - "nl_statement": "Let $L$ be the vector space of continuous linear transformations from a normed space $V$ to a normed space $W$. Show that the operator norm makes $L$ a normed space.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_1_18", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is a finite group of even order, show that there must be an element $a \\neq e$ such that $a=a^{-1}$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_1_21", - "formal_statement": "theorem exercise_2_1_21 (G : Type*) [group G] [fintype G]\n (hG : card G = 5) :\n comm_group G :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that a group of order 5 must be abelian.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_1_26", - "formal_statement": "theorem exercise_2_1_26 {G : Type*} [group G] \n [fintype G] (a : G) : \u2203 (n : \u2115), a ^ n = 1 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_1_27", - "formal_statement": "theorem exercise_2_1_27 {G : Type*} [group G] \n [fintype G] : \u2203 (m : \u2115), \u2200 (a : G), a ^ m = 1 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is a finite group, prove that there is an integer $m > 0$ such that $a^m = e$ for all $a \\in G$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_2_3", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is a group in which $(a b)^{i}=a^{i} b^{i}$ for three consecutive integers $i$, prove that $G$ is abelian.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_2_5", - "formal_statement": "theorem exercise_2_2_5 {G : Type*} [group G] \n (h : \u2200 (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 \u2227 (a * b) ^ 5 = a ^ 5 * b ^ 5) :\n comm_group G :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G$ be a group in which $(a b)^{3}=a^{3} b^{3}$ and $(a b)^{5}=a^{5} b^{5}$ for all $a, b \\in G$. Show that $G$ is abelian.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_2_6c", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$. \n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_3_17", - "formal_statement": "theorem exercise_2_3_17 {G : Type*} [has_mul G] [group G] (a x : G) : \n set.centralizer {x\u207b\u00b9*a*x} = \n (\u03bb g : G, x\u207b\u00b9*g*x) '' (set.centralizer {a}) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is a group and $a, x \\in G$, prove that $C\\left(x^{-1} a x\\right)=x^{-1} C(a) x$\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_3_19", - "formal_statement": "theorem exercise_2_3_19 {G : Type*} [group G] {M : subgroup G}\n (hM : \u2200 (x : G), (\u03bb g : G, x\u207b\u00b9*g*x) '' M \u2282 M) :\n \u2200 x : G, (\u03bb g : G, x\u207b\u00b9*g*x) '' M = M :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $M$ is a subgroup of $G$ such that $x^{-1} M x \\subset M$ for all $x \\in G$, prove that actually $x^{-1} M x=M$. \n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_3_16", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_4_36", - "formal_statement": "theorem exercise_2_4_36 {a n : \u2115} (h : a > 1) :\n n \u2223 (a ^ n - 1).totient :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $a > 1$ is an integer, show that $n \\mid \\varphi(a^n - 1)$, where $\\phi$ is the Euler $\\varphi$-function.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_23", - "formal_statement": "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:=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_30", - "formal_statement": "theorem exercise_2_5_30 {G : Type*} [group G] [fintype G]\n {p m : \u2115} (hp : nat.prime p) (hp1 : \u00ac p \u2223 m) (hG : card G = p*m) \n {H : subgroup G} [fintype H] [H.normal] (hH : card H = p):\n characteristic H :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Suppose that $|G| = pm$, where $p \\nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_31", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_37", - "formal_statement": "theorem exercise_2_5_37 (G : Type*) [group G] [fintype G]\n (hG : card G = 6) (hG' : is_empty (comm_group G)) :\n G \u2245 equiv.perm (fin 3) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is a nonabelian group of order 6, prove that $G \\simeq S_3$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_43", - "formal_statement": "theorem exercise_2_5_43 (G : Type*) [group G] [fintype G]\n (hG : card G = 9) :\n comm_group G :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of order 9 must be abelian.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_44", - "formal_statement": "theorem exercise_2_5_44 {G : Type*} [group G] [fintype G] {p : \u2115}\n (hp : nat.prime p) (hG : card G = p^2) :\n \u2203 (N : subgroup G) (fin : fintype N), @card N fin = p \u2227 N.normal :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_5_52", - "formal_statement": "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 * card G \u2264 card I) : \n \u2200 x : G, \u03c6 x = x\u207b\u00b9 \u2227 \u2200 x y : G, x*y = y*x :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_6_15", - "formal_statement": "theorem exercise_2_6_15 {G : Type*} [comm_group G] {m n : \u2115} \n (hm : \u2203 (g : G), order_of g = m) \n (hn : \u2203 (g : G), order_of g = n) \n (hmn : m.coprime n) :\n \u2203 (g : G), order_of g = m * n :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G$ is an abelian group and if $G$ has an element of order $m$ and one of order $n$, where $m$ and $n$ are relatively prime, prove that $G$ has an element of order $mn$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_7_7", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $\\varphi$ is a homomorphism of $G$ onto $G'$ and $N \\triangleleft G$, show that $\\varphi(N) \\triangleleft G'$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_8_12", - "formal_statement": "theorem exercise_2_8_12 {G H : Type*} [fintype G] [fintype H] \n [group G] [group H] (hG : card G = 21) (hH : card H = 21) \n (hG1 : is_empty(comm_group G)) (hH1 : is_empty (comm_group H)) :\n G \u2243* H :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that any two nonabelian groups of order 21 are isomorphic.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_8_15", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $p > q$ are two primes such that $q \\mid p - 1$, then any two nonabelian groups of order $pq$ are isomorphic.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_9_2", - "formal_statement": "theorem exercise_2_9_2 {G H : Type*} [fintype G] [fintype H] [group G] \n [group H] (hG : is_cyclic G) (hH : is_cyclic H) :\n is_cyclic (G \u00d7 H) \u2194 (card G).coprime (card H) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $G_1$ and $G_2$ are cyclic groups of orders $m$ and $n$, respectively, prove that $G_1 \\times G_2$ is cyclic if and only if $m$ and $n$ are relatively prime.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_10_1", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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)$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_11_6", - "formal_statement": "theorem exercise_2_11_6 {G : Type*} [group G] {p : \u2115} (hp : nat.prime p) \n {P : sylow p G} (hP : P.normal) :\n \u2200 (Q : sylow p G), P = Q :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $P$ is a $p$-Sylow subgroup of $G$ and $P \\triangleleft G$, prove that $P$ is the only $p$-Sylow subgroup of $G$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_11_7", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $P \\triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\\varphi(P) = P$ for every automorphism $\\varphi$ of $G$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_2_11_22", - "formal_statement": "theorem exercise_2_11_22 {p : \u2115} {n : \u2115} {G : Type*} [fintype G] \n [group G] (hp : nat.prime p) (hG : card G = p ^ n) {K : subgroup G}\n [fintype K] (hK : card K = p ^ (n-1)) : \n K.normal :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that any subgroup of order $p^{n-1}$ in a group $G$ of order $p^n$ is normal in $G$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_3_2_21", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $\\sigma, \\tau$ are two permutations that disturb no common element and $\\sigma \\tau = e$, prove that $\\sigma = \\tau = e$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_1_19", - "formal_statement": "theorem exercise_4_1_19 : infinite {x : quaternion \u211d | x^2 = -1} :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that there is an infinite number of solutions to $x^2 = -1$ in the quaternions.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_1_34", - "formal_statement": "theorem exercise_4_1_34 : equiv.perm (fin 3) \u2243* general_linear_group (fin 2) (zmod 2) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_2_5", - "formal_statement": "theorem exercise_4_2_5 {R : Type*} [ring R] \n (h : \u2200 x : R, x ^ 3 = x) : comm_ring R :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $R$ be a ring in which $x^3 = x$ for every $x \\in R$. Prove that $R$ is commutative.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_2_6", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_2_9", - "formal_statement": "theorem exercise_4_2_9 {p : \u2115} (hp : nat.prime p) (hp1 : odd p) :\n \u2203 (a b : \u2124), a / b = \u2211 i in finset.range p, 1 / (i + 1) \u2192 \u2191p \u2223 a :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $p$ be an odd prime and let $1 + \\frac{1}{2} + ... + \\frac{1}{p - 1} = \\frac{a}{b}$, where $a, b$ are integers. Show that $p \\mid a$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_3_1", - "formal_statement": "theorem exercise_4_3_1 {R : Type*} [comm_ring R] (a : R) :\n \u2203 I : ideal R, {x : R | x*a=0} = I :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_3_25", - "formal_statement": "theorem exercise_4_3_25 (I : ideal (matrix (fin 2) (fin 2) \u211d)) : \n I = \u22a5 \u2228 I = \u22a4 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $R$ be the ring of $2 \\times 2$ matrices over the real numbers; suppose that $I$ is an ideal of $R$. Show that $I = (0)$ or $I = R$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_4_9", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_5_16", - "formal_statement": "theorem exercise_4_5_16 {p n: \u2115} (hp : nat.prime p) \n {q : polynomial (zmod p)} (hq : irreducible q) (hn : q.degree = n) :\n \u2203 is_fin : fintype $ polynomial (zmod p) \u29f8 ideal.span ({q} : set (polynomial $ zmod p)), \n @card (polynomial (zmod p) \u29f8 ideal.span {q}) is_fin = p ^ n \u2227 \n is_field (polynomial $ zmod p):=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $F = \\mathbb{Z}_p$ be the field of integers $\\mod p$, where $p$ is a prime, and let $q(x) \\in F[x]$ be irreducible of degree $n$. Show that $F[x]/(q(x))$ is a field having at exactly $p^n$ elements.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_5_23", - "formal_statement": "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)) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_5_25", - "formal_statement": "theorem exercise_4_5_25 {p : \u2115} (hp : nat.prime p) :\n irreducible (\u2211 i : finset.range p, X ^ p : polynomial \u211a) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $p$ is a prime, show that $q(x) = 1 + x + x^2 + \\cdots x^{p - 1}$ is irreducible in $Q[x]$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_6_2", - "formal_statement": "theorem exercise_4_6_2 : irreducible (X^3 + 3*X + 2 : polynomial \u211a) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_4_6_3", - "formal_statement": "theorem exercise_4_6_3 :\n infinite {a : \u2124 | irreducible (X^7 + 15*X^2 - 30*X + a : polynomial \u211a)} :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_1_8", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_2_20", - "formal_statement": "theorem exercise_5_2_20 {F V \u03b9: Type*} [infinite F] [field F] \n [add_comm_group V] [module F V] {u : \u03b9 \u2192 submodule F V} \n (hu : \u2200 i : \u03b9, u i \u2260 \u22a4) : \n (\u22c3 i : \u03b9, (u i : set V)) \u2260 \u22a4 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $V$ be a vector space over an infinite field $F$. Show that $V$ cannot be the set-theoretic union of a finite number of proper subspaces of $V$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_3_7", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $a \\in K$ is such that $a^2$ is algebraic over the subfield $F$ of $K$, show that a is algebraic over $F$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_3_10", - "formal_statement": "theorem exercise_5_3_10 : is_algebraic \u211a (cos (real.pi / 180)) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\cos 1^{\\circ}$ is algebraic over $\\mathbb{Q}$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_4_3", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_5_2", - "formal_statement": "theorem exercise_5_5_2 : irreducible (X^3 - 3*X - 1 : polynomial \u211a) :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^3 - 3x - 1$ is irreducible over $\\mathbb{Q}$.\n", - "nl_proof": "" - }, - { - "id": "Herstein|exercise_5_6_14", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat fintype real subgroup ideal polynomial submodule zsqrtd \nopen char_p mul_aut matrix\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $F$ is of characteristic $p \\neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_2_2_9", - "formal_statement": "theorem exercise_2_2_9 {G : Type*} [group G] {a b : G}\n (h : a * b = b * a) :\n \u2200 x y : closure {x | x = a \u2228 x = b}, x*y = y*x :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $H$ be the subgroup generated by two elements $a, b$ of a group $G$. Prove that if $a b=b a$, then $H$ is an abelian group.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_2_3_2", - "formal_statement": "theorem exercise_2_3_2 {G : Type*} [group G] (a b : G) :\n \u2203 g : G, b* a = g * a * b * g\u207b\u00b9 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the products $a b$ and $b a$ are conjugate elements in a group.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_2_4_19", - "formal_statement": "theorem exercise_2_4_19 {G : Type*} [group G] {x : G}\n (hx : order_of x = 2) (hx1 : \u2200 y, order_of y = 2 \u2192 y = x) :\n x \u2208 center G :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if a group contains exactly one element of order 2 , then that element is in the center of the group.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_2_8_6", - "formal_statement": "theorem exercise_2_8_6 {G H : Type*} [group G] [group H] :\n center (G \u00d7 H) \u2243* (center G) \u00d7 (center H) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the center of the product of two groups is the product of their centers.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_2_11_3", - "formal_statement": "theorem exercise_2_11_3 {G : Type*} [group G] [fintype G]\n (hG : even (card G)) : \u2203 x : G, order_of x = 2 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of even order contains an element of order $2 .$\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_3_2_7", - "formal_statement": "theorem exercise_3_2_7 {F : Type*} [field F] {G : Type*} [field G]\n (\u03c6 : F \u2192+* G) : injective \u03c6 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that every homomorphism of fields is injective.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_3_5_6", - "formal_statement": "theorem exercise_3_5_6 {K V : Type*} [field K] [add_comm_group V]\n [module K V] {S : set V} (hS : set.countable S)\n (hS1 : span K S = \u22a4) {\u03b9 : Type*} (R : \u03b9 \u2192 V)\n (hR : linear_independent K R) : countable \u03b9 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_3_7_2", - "formal_statement": "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 (\u22c2 (i : \u03b9), (\u03b3 i : set V)) \u2260 \u22a4 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_1_14", - "formal_statement": "theorem exercise_6_1_14 (G : Type*) [group G]\n (hG : is_cyclic $ G \u29f8 (center G)) :\n center G = \u22a4 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $Z$ be the center of a group $G$. Prove that if $G / Z$ is a cyclic group, then $G$ is abelian and hence $G=Z$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_4_2", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_4_3", - "formal_statement": "theorem exercise_6_4_3 {G : Type*} [group G] [fintype G] {p q : \u2115}\n (hp : prime p) (hq : prime q) (hG : card G = p^2 *q) :\n is_simple_group G \u2192 false :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_4_12", - "formal_statement": "theorem exercise_6_4_12 {G : Type*} [group G] [fintype G]\n (hG : card G = 224) :\n is_simple_group G \u2192 false :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that no group of order 224 is simple.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_8_1", - "formal_statement": "theorem exercise_6_8_1 {G : Type*} [group G]\n (a b : G) : closure ({a, b} : set G) = closure {b*a*b^2, b*a*b^3} :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that two elements $a, b$ of a group generate the same subgroup as $b a b^2, b a b^3$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_8_4", - "formal_statement": "theorem exercise_6_8_4 {\u03b1 : Type*} [group \u03b1] [free_group \u03b1] (x y z : \u03b1):\n closure ({x,y,z} : set \u03b1) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the group generated by $x, y, z$ with the single relation $y x y z^{-2}=1$ is actually a free group.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_6_8_6", - "formal_statement": "theorem exercise_6_8_6 {G : Type*} [group G] (N : subgroup G)\n [N.normal] (hG : is_cyclic G) (hGN : is_cyclic (G \u29f8 N)) :\n \u2203 (g h : G), closure ({g,h} : set G) = \u22a4 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G$ be a group with a normal subgroup $N$. Assume that $G$ and $G / N$ are both cyclic groups. Prove that $G$ can be generated by two elements.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_1_13", - "formal_statement": "theorem exercise_10_1_13 {R : Type*} [ring R] {x : R}\n (hx : is_nilpotent x) : is_unit (1 + x) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_2_4", - "formal_statement": "theorem exercise_10_2_4 :\n span ({2} : set $ polynomial \u2124) \u2293 (span {X}) =\n span ({2 * X} : set $ polynomial \u2124) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that in the ring $\\mathbb{Z}[x],(2) \\cap(x)=(2 x)$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_6_7", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_4_6", - "formal_statement": "theorem exercise_10_4_6 {R : Type*} [comm_ring R] \n [no_zero_divisors R] {I J : ideal R} (x : I \u2293 J) : \n is_nilpotent ((ideal.quotient.mk (I*J)) x) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \\cap J$ in $R / I J$ is nilpotent.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_4_7a", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \\cap J$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_5_16", - "formal_statement": "theorem exercise_10_5_16 {F : Type*} [fintype F] [field F] :\n is_empty ((polynomial F) \u29f8 ideal.span ({X^2} : set (polynomial F)) \u2243\n (polynomial F) \u29f8 ideal.span ({X^2 - 1} : set (polynomial F))) \u2194\n ring_char F \u2260 2 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $F$ be a field. Prove that the rings $F[x] /\\left(x^2\\right)$ and $F[x] /\\left(x^2-1\\right)$ are isomorphic if and only if $F$ has characteristic $2 .$\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_7_6", - "formal_statement": "theorem exercise_10_7_6 {F : Type*} [fintype F] [field F]\n (hF : card F = 5) :\n field $ (polynomial F) \u29f8 ideal.span ({X^2 + X + 1} : set (polynomial F)) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the ring $\\mathbb{F}_5[x] /\\left(x^2+x+1\\right)$ is a field.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_10_7_10", - "formal_statement": "theorem exercise_10_7_10 {R : Type*} [ring R]\n (M : ideal R) (hM : \u2200 (x : R), x \u2209 M \u2192 is_unit x) :\n is_maximal M \u2227 \u2200 (N : ideal R), is_maximal N \u2192 N = M :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_2_13", - "formal_statement": "theorem exercise_11_2_13 (a b : \u2124) :\n (of_int a : gaussian_int) \u2223 of_int b \u2192 a \u2223 b :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\\mathbb{Z}$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_3_1", - "formal_statement": "theorem exercise_11_3_1 {F : Type*} [field F] (a b : F) (ha : a \u2260 0) (p : polynomial F) :\n irreducible p \u2194 irreducible (\u2211 n in p.support, p.coeff n \u2022 (a \u2022 X + b \u2022 1)^n : polynomial F) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $a, b$ be elements of a field $F$, with $a \\neq 0$. Prove that a polynomial $f(x) \\in F[x]$ is irreducible if and only if $f(a x+b)$ is irreducible.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_3_4", - "formal_statement": "theorem exercise_11_3_4 : irreducible (X^3 + 6*X + 12 : polynomial \u211a) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that two integer polynomials are relatively prime in $\\mathbb{Q}[x]$ if and only if the ideal they generate in $\\mathbb{Z}[x]$ contains an integer.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_4_1b", - "formal_statement": "theorem exercise_11_4_1b {F : Type*} [field F] [fintype F] (hF : card F = 2) :\n irreducible (12 + 6 * X + X ^ 3 : polynomial F) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^3 + 6x + 12$ is irreducible in $\\mathbb{Q}$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_4_6a", - "formal_statement": "theorem exercise_11_4_6a {F : Type*} [field F] [fintype F] (hF : card F = 7) :\n irreducible (X ^ 2 + 1 : polynomial F) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^2+x+1$ is irreducible in the field $\\mathbb{F}_2$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_4_6b", - "formal_statement": "theorem exercise_11_4_6b {F : Type*} [field F] [fintype F] (hF : card F = 31) :\n irreducible (X ^ 3 - 9 : polynomial F) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^2+1$ is irreducible in $\\mathbb{F}_7$\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_4_6c", - "formal_statement": "theorem exercise_11_4_6c : irreducible (X^3 - 9 : polynomial (zmod 31)) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^3 - 9$ is irreducible in $\\mathbb{F}_{31}$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_4_8", - "formal_statement": "theorem exercise_11_4_8 {p : \u2115} (hp : prime p) (n : \u2115) :\n irreducible (X ^ n - p : polynomial \u211a) :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\\mathbb{Q}[x]$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_12_3", - "formal_statement": "theorem exercise_11_12_3 (p : \u2115) (hp : nat.prime p) {a : zmod p} \n (ha : a^2 = -5) :\n \u2203 (x y : \u2124), x ^ 2 + 5 * y ^ 2 = p \u2228 2 * x ^ 2 + 2 * x * y + 3 * y ^ 2 = p :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $x^2 \\equiv-5$ (modulo $p$) has a solution, then there is an integer point on one of the two ellipses $x^2+5 y^2=p$ or $2 x^2+2 x y+3 y^2=p$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_11_13_3", - "formal_statement": "theorem exercise_11_13_3 (N : \u2115):\n \u2203 p \u2265 N, nat.prime p \u2227 p + 1 \u2261 0 [MOD 4] :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that there are infinitely many primes congruent to $-1$ (modulo $4$).\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_13_4_10", - "formal_statement": "theorem exercise_13_4_10 \n {p : \u2115} {hp : nat.prime p} (h : \u2203 r : \u2115, p = 2 ^ r + 1) :\n \u2203 (k : \u2115), p = 2 ^ (2 ^ k) + 1 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.\n", - "nl_proof": "" - }, - { - "id": "Artin|exercise_13_6_10", - "formal_statement": "theorem exercise_13_6_10 {K : Type*} [field K] [fintype K\u02e3] :\n \u220f (x : K\u02e3), x = -1 :=", - "src_header": "import .common \n\nopen function fintype subgroup ideal polynomial submodule zsqrtd char_p\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_1_1_2a", - "formal_statement": "theorem exercise_1_1_2a : \u2203 a b : \u2124, a - b \u2260 b - a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove the the operation $\\star$ on $\\mathbb{Z}$ defined by $a\\star b=a-b$ is not commutative.\n", - "nl_proof": "\\begin{proof}\n\n Not commutative since\n\n$$\n\n1 \\star(-1)=1-(-1)=2\n\n$$\n\n$$\n\n(-1) \\star 1=-1-1=-2 .\n\n$$\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_3", - "formal_statement": "theorem exercise_1_1_3 (n : \u2124) : \n \u2200 (a b c : \u2124), (a+b)+c \u2261 a+(b+c) [ZMOD n] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the addition of residue classes $\\mathbb{Z}/n\\mathbb{Z}$ is associative.\n", - "nl_proof": "\\begin{proof}\n\n We have\n\n$$\n\n\\begin{aligned}\n\n(\\bar{a}+\\bar{b})+\\bar{c} &=\\overline{a+b}+\\bar{c} \\\\\n\n&=\\overline{(a+b)+c} \\\\\n\n&=\\overline{a+(b+c)} \\\\\n\n&=\\bar{a}+\\overline{b+c} \\\\\n\n&=\\bar{a}+(\\bar{b}+\\bar{c})\n\n\\end{aligned}\n\n$$\n\nsince integer addition is associative.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_4", - "formal_statement": "theorem exercise_1_1_4 (n : \u2115) : \n \u2200 (a b c : \u2115), (a * b) * c \u2261 a * (b * c) [ZMOD n] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the multiplication of residue class $\\mathbb{Z}/n\\mathbb{Z}$ is associative.\n", - "nl_proof": "\\begin{proof}\n\n We have\n\n$$\n\n\\begin{aligned}\n\n(\\bar{a} \\cdot \\bar{b}) \\cdot \\bar{c} &=\\overline{a \\cdot b} \\cdot \\bar{c} \\\\\n\n&=\\overline{(a \\cdot b) \\cdot c} \\\\\n\n&=\\overline{a \\cdot(b \\cdot c)} \\\\\n\n&=\\bar{a} \\cdot \\overline{b \\cdot c} \\\\\n\n&=\\bar{a} \\cdot(\\bar{b} \\cdot \\bar{c})\n\n\\end{aligned}\n\n$$\n\nsince integer multiplication is associative.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_5", - "formal_statement": "theorem exercise_1_1_5 (n : \u2115) (hn : 1 < n) : \n is_empty (group (zmod n)) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that for all $n>1$ that $\\mathbb{Z}/n\\mathbb{Z}$ is not a group under multiplication of residue classes.\n", - "nl_proof": "\\begin{proof}\n\n Note that since $n>1, \\overline{1} \\neq \\overline{0}$. Now suppose $\\mathbb{Z} /(n)$ contains a multiplicative identity element $\\bar{e}$. Then in particular,\n\n$$\n\n\\bar{e} \\cdot \\overline{1}=\\overline{1}\n\n$$\n\nso that $\\bar{e}=\\overline{1}$. Note, however, that\n\n$$\n\n\\overline{0} \\cdot \\bar{k}=\\overline{0}\n\n$$\n\nfor all k, so that $\\overline{0}$ does not have a multiplicative inverse. Hence $\\mathbb{Z} /(n)$ is not a group under multiplication.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_15", - "formal_statement": "theorem exercise_1_1_15 {G : Type*} [group G] (as : list G) :\n as.prod\u207b\u00b9 = (as.reverse.map (\u03bb x, x\u207b\u00b9)).prod :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $(a_1a_2\\dots a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}\\dots a_1^{-1}$ for all $a_1, a_2, \\dots, a_n\\in G$.\n", - "nl_proof": "\\begin{proof}\n\n For $n=1$, note that for all $a_1 \\in G$ we have $a_1^{-1}=a_1^{-1}$.\n\nNow for $n \\geq 2$ we proceed by induction on $n$. For the base case, note that for all $a_1, a_2 \\in G$ we have\n\n$$\n\n\\left(a_1 \\cdot a_2\\right)^{-1}=a_2^{-1} \\cdot a_1^{-1}\n\n$$\n\nsince\n\n$$\n\na_1 \\cdot a_2 \\cdot a_2^{-1} a_1^{-1}=1 .\n\n$$\n\nFor the inductive step, suppose that for some $n \\geq 2$, for all $a_i \\in G$ we have\n\n$$\n\n\\left(a_1 \\cdot \\ldots \\cdot a_n\\right)^{-1}=a_n^{-1} \\cdot \\ldots \\cdot a_1^{-1} .\n\n$$\n\nThen given some $a_{n+1} \\in G$, we have\n\n$$\n\n\\begin{aligned}\n\n\\left(a_1 \\cdot \\ldots \\cdot a_n \\cdot a_{n+1}\\right)^{-1} &=\\left(\\left(a_1 \\cdot \\ldots \\cdot a_n\\right) \\cdot a_{n+1}\\right)^{-1} \\\\\n\n&=a_{n+1}^{-1} \\cdot\\left(a_1 \\cdot \\ldots \\cdot a_n\\right)^{-1} \\\\\n\n&=a_{n+1}^{-1} \\cdot a_n^{-1} \\cdot \\ldots \\cdot a_1^{-1},\n\n\\end{aligned}\n\n$$\n\nusing associativity and the base case where necessary.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_16", - "formal_statement": "theorem exercise_1_1_16 {G : Type*} [group G] \n (x : G) (hx : x ^ 2 = 1) :\n order_of x = 1 \u2228 order_of x = 2 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.\n", - "nl_proof": "\\begin{proof}\n\n $(\\Rightarrow)$ Suppose $x^2=1$. Then we have $0<|x| \\leq 2$, i.e., $|x|$ is either 1 or 2 .\n\n( $\\Leftarrow$ ) If $|x|=1$, then we have $x=1$ so that $x^2=1$. If $|x|=2$ then $x^2=1$ by definition. So if $|x|$ is 1 or 2 , we have $x^2=1$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_17", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.\n", - "nl_proof": "\\begin{proof}\n\n We have $x \\cdot x^{n-1}=x^n=1$, so by the uniqueness of inverses $x^{-1}=x^{n-1}$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_18", - "formal_statement": "theorem exercise_1_1_18 {G : Type*} [group G]\n (x y : G) : x * y = y * x \u2194 y\u207b\u00b9 * x * y = x \u2194 x\u207b\u00b9 * y\u207b\u00b9 * x * y = 1 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $x$ and $y$ be elements of $G$. Prove that $xy=yx$ if and only if $y^{-1}xy=x$ if and only if $x^{-1}y^{-1}xy=1$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_1_1_20", - "formal_statement": "theorem exercise_1_1_20 {G : Type*} [group G] {x : G} :\n order_of x = order_of x\u207b\u00b9 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order.\n", - "nl_proof": "\\begin{proof}\n\n Recall that the order of a group element is either a positive integer or infinity.\n\nSuppose $|x|$ is infinite and that $\\left|x^{-1}\\right|=n$ for some $n$. Then\n\n$$\n\nx^n=x^{(-1) \\cdot n \\cdot(-1)}=\\left(\\left(x^{-1}\\right)^n\\right)^{-1}=1^{-1}=1,\n\n$$\n\na contradiction. So if $|x|$ is infinite, $\\left|x^{-1}\\right|$ must also be infinite. Likewise, if $\\left|x^{-1}\\right|$ is infinite, then $\\left|\\left(x^{-1}\\right)^{-1}\\right|=|x|$ is also infinite.\n\nSuppose now that $|x|=n$ and $\\left|x^{-1}\\right|=m$ are both finite. Then we have\n\n$$\n\n\\left(x^{-1}\\right)^n=\\left(x^n\\right)^{-1}=1^{-1}=1,\n\n$$\n\nso that $m \\leq n$. Likewise, $n \\leq m$. Hence $m=n$ and $x$ and $x^{-1}$ have the same order.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_22a", - "formal_statement": "theorem exercise_1_1_22a {G : Type*} [group G] (x g : G) :\n order_of x = order_of (g\u207b\u00b9 * x * g) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $x$ and $g$ are elements of the group $G$, prove that $|x|=\\left|g^{-1} x g\\right|$.\n", - "nl_proof": "\\begin{proof}\n\n First we prove a technical lemma:\n\n\n\n {\\bf Lemma.} For all $a, b \\in G$ and $n \\in \\mathbb{Z},\\left(b^{-1} a b\\right)^n=b^{-1} a^n b$.\n\nThe statement is clear for $n=0$. We prove the case $n>0$ by induction; the base case $n=1$ is clear. Now suppose $\\left(b^{-1} a b\\right)^n=b^{-1} a^n b$ for some $n \\geq 1$; then\n\n$$\n\n\\left(b^{-1} a b\\right)^{n+1}=\\left(b^{-1} a b\\right)\\left(b^{-1} a b\\right)^n=b^{-1} a b b^{-1} a^n b=b^{-1} a^{n+1} b .\n\n$$\n\nBy induction the statement holds for all positive $n$.\n\nNow suppose $n<0$; we have\n\n$$\n\n\\left(b^{-1} a b\\right)^n=\\left(\\left(b^{-1} a b\\right)^{-n}\\right)^{-1}=\\left(b^{-1} a^{-n} b\\right)^{-1}=b^{-1} a^n b .\n\n$$\n\nHence, the statement holds for all integers $n$.\n\nNow to the main result. Suppose first that $|x|$ is infinity and that $\\left|g^{-1} x g\\right|=n$ for some positive integer $n$. Then we have\n\n$$\n\n\\left(g^{-1} x g\\right)^n=g^{-1} x^n g=1,\n\n$$\n\nand multiplying on the left by $g$ and on the right by $g^{-1}$ gives us that $x^n=1$, a contradiction. Thus if $|x|$ is infinity, so is $\\left|g^{-1} x g\\right|$. Similarly, if $\\left|g^{-1} x g\\right|$ is infinite and $|x|=n$, we have\n\n$$\n\n\\left(g^{-1} x g\\right)^n=g^{-1} x^n g=g^{-1} g=1,\n\n$$\n\na contradiction. Hence if $\\left|g^{-1} x g\\right|$ is infinite, so is $|x|$.\n\nSuppose now that $|x|=n$ and $\\left|g^{-1} x g\\right|=m$ for some positive integers $n$ and $m$. We have\n\n$$\n\n\\left(g^{-1} x g\\right)^n=g^{-1} x^n g=g^{-1} g=1,\n\n$$\n\nSo that $m \\leq n$, and\n\n$$\n\n\\left(g^{-1} x g\\right)^m=g^{-1} x^m g=1,\n\n$$\n\nso that $x^m=1$ and $n \\leq m$. Thus $n=m$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_22b", - "formal_statement": "theorem exercise_1_1_22b {G: Type*} [group G] (a b : G) : \n order_of (a * b) = order_of (b * a) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Deduce that $|a b|=|b a|$ for all $a, b \\in G$.\n", - "nl_proof": "\\begin{proof}\n\n Let $a$ and $b$ be arbitrary group elements. Letting $x=a b$ and $g=a$, we see that\n\n$$\n\n|a b|=\\left|a^{-1} a b a\\right|=|b a| .\n\n$$\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_25", - "formal_statement": "theorem exercise_1_1_25 {G : Type*} [group G] \n (h : \u2200 x : G, x ^ 2 = 1) : \u2200 a b : G, a*b = b*a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $x^{2}=1$ for all $x \\in G$ then $G$ is abelian.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Note that since $x^2=1$ for all $x \\in G$, we have $x^{-1}=x$. Now let $a, b \\in G$. We have\n\n$$\n\na b=(a b)^{-1}=b^{-1} a^{-1}=b a .\n\n$$\n\nThus $G$ is abelian.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_29", - "formal_statement": "theorem exercise_1_1_29 {A B : Type*} [group A] [group B] :\n \u2200 x y : A \u00d7 B, x*y = y*x \u2194 (\u2200 x y : A, x*y = y*x) \u2227 \n (\u2200 x y : B, x*y = y*x) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $A \\times B$ is an abelian group if and only if both $A$ and $B$ are abelian.\n", - "nl_proof": "\\begin{proof}\n\n $(\\Rightarrow)$ Suppose $a_1, a_2 \\in A$ and $b_1, b_2 \\in B$. Then\n\n$$\n\n\\left(a_1 a_2, b_1 b_2\\right)=\\left(a_1, b_1\\right) \\cdot\\left(a_2, b_2\\right)=\\left(a_2, b_2\\right) \\cdot\\left(a_1, b_1\\right)=\\left(a_2 a_1, b_2 b_1\\right) .\n\n$$\n\nSince two pairs are equal precisely when their corresponding entries are equal, we have $a_1 a_2=a_2 a_1$ and $b_1 b_2=b_2 b_1$. Hence $A$ and $B$ are abelian.\n\n$(\\Leftarrow)$ Suppose $\\left(a_1, b_1\\right),\\left(a_2, b_2\\right) \\in A \\times B$. Then we have\n\n$$\n\n\\left(a_1, b_1\\right) \\cdot\\left(a_2, b_2\\right)=\\left(a_1 a_2, b_1 b_2\\right)=\\left(a_2 a_1, b_2 b_1\\right)=\\left(a_2, b_2\\right) \\cdot\\left(a_1, b_1\\right) .\n\n$$\n\nHence $A \\times B$ is abelian.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_1_34", - "formal_statement": "theorem exercise_1_1_34 {G : Type*} [group G] {x : G} \n (hx_inf : order_of x = 0) (n m : \u2124) :\n x ^ n \u2260 x ^ m :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $x$ is an element of infinite order in $G$, prove that the elements $x^{n}, n \\in \\mathbb{Z}$ are all distinct.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Suppose to the contrary that $x^a=x^b$ for some $0 \\leq a|x|>\\left|x^2\\right|>\\cdots,\n\n$$\n\nand in particular, $1>\\left|x^n\\right|$ for all $n$. So $x$ has infinite order in $\\mathbb{R}^{\\times}$.\n\nSimilarly, if $|x|>1$ (absolute value) then $x$ has infinite order in $\\mathbb{R}^{\\times}$. So $\\mathbb{R}^{\\times}$has 1 element of order 1,1 element of order 2 , and all other elements have infinite order.\n\nIn $\\mathbb{C}^{\\times}$, on the other hand, $i$ has order 4 . Thus $\\mathbb{R}^{\\times}$and $\\mathbb{C}^{\\times}$are not isomorphic.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_6_11", - "formal_statement": "theorem exercise_1_6_11 {A B : Type*} [group A] [group B] : \n A \u00d7 B \u2243* B \u00d7 A :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $A$ and $B$ be groups. Prove that $A \\times B \\cong B \\times A$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: We know from set theory that the mapping $\\varphi: A \\times B \\rightarrow B \\times A$ given by $\\varphi((a, b))=(b, a)$ is a bijection with inverse $\\psi: B \\times A \\rightarrow A \\times B$ given by $\\psi((b, a))=(a, b)$. Also $\\varphi$ is a homomorphism, as we show below.\n\nLet $a_1, a_2 \\in A$ and $b_1, b_2 \\in B$. Then\n\n$$\n\n\\begin{aligned}\n\n\\varphi\\left(\\left(a_1, b_1\\right) \\cdot\\left(a_2, b_2\\right)\\right) &=\\varphi\\left(\\left(a_1 a_2, b_1 b_2\\right)\\right) \\\\\n\n&=\\left(b_1 b_2, a_1 a_2\\right) \\\\\n\n&=\\left(b_1, a_1\\right) \\cdot\\left(b_2, a_2\\right) \\\\\n\n&=\\varphi\\left(\\left(a_1, b_1\\right)\\right) \\cdot \\varphi\\left(\\left(a_2, b_2\\right)\\right)\n\n\\end{aligned}\n\n$$\n\nHence $A \\times B \\cong B \\times A$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_6_17", - "formal_statement": "theorem exercise_1_6_17 {G : Type*} [group G] (f : G \u2192 G) \n (hf : f = \u03bb g, g\u207b\u00b9) :\n \u2200 x y : G, f x * f y = f (x*y) \u2194 \u2200 x y : G, x*y = y*x :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G$ be any group. Prove that the map from $G$ to itself defined by $g \\mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.\n", - "nl_proof": "\\begin{proof}\n\n $(\\Rightarrow)$ Suppose $G$ is abelian. Then\n\n$$\n\n\\varphi(a b)=(a b)^{-1}=b^{-1} a^{-1}=a^{-1} b^{-1}=\\varphi(a) \\varphi(b),\n\n$$\n\nso that $\\varphi$ is a homomorphism.\n\n$(\\Leftarrow)$ Suppose $\\varphi$ is a homomorphism, and let $a, b \\in G$. Then\n\n$$\n\na b=\\left(b^{-1} a^{-1}\\right)^{-1}=\\varphi\\left(b^{-1} a^{-1}\\right)=\\varphi\\left(b^{-1}\\right) \\varphi\\left(a^{-1}\\right)=\\left(b^{-1}\\right)^{-1}\\left(a^{-1}\\right)^{-1}=b a,\n\n$$\n\nso that $G$ is abelian.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_1_6_23", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "\\begin{proof}\n\n Solution: We define a mapping $f: G \\rightarrow G$ by $f(x)=x^{-1} \\sigma(x)$.\n\nClaim: $f$ is injective.\n\nProof of claim: Suppose $f(x)=f(y)$. Then $y^{-1} \\sigma(y)=x^{-1} \\sigma(x)$, so that $x y^{-1}=\\sigma(x) \\sigma\\left(y^{-1}\\right)$, and $x y^{-1}=\\sigma\\left(x y^{-1}\\right)$. Then we have $x y^{-1}=1$, hence $x=y$. So $f$ is injective.\n\n\n\nSince $G$ is finite and $f$ is injective, $f$ is also surjective. Then every $z \\in G$ is of the form $x^{-1} \\sigma(x)$ for some $x$. Now let $z \\in G$ with $z=x^{-1} \\sigma(x)$. We have\n\n$$\n\n\\sigma(z)=\\sigma\\left(x^{-1} \\sigma(x)\\right)=\\sigma(x)^{-1} x=\\left(x^{-1} \\sigma(x)\\right)^{-1}=z^{-1} .\n\n$$\n\nThus $\\sigma$ is in fact the inversion mapping, and we assumed that $\\sigma$ is a homomorphism. By a previous example, then, $G$ is abelian.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_2_1_5", - "formal_statement": "theorem exercise_2_1_5 {G : Type*} [group G] [fintype G] \n (hG : card G > 2) (H : subgroup G) [fintype H] : \n card H \u2260 card G - 1 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $G$ cannot have a subgroup $H$ with $|H|=n-1$, where $n=|G|>2$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Under these conditions, there exists a nonidentity element $x \\in H$ and an element $y \\notin H$. Consider the product $x y$. If $x y \\in H$, then since $x^{-1} \\in H$ and $H$ is a subgroup, $y \\in H$, a contradiction. If $x y \\notin H$, then we have $x y=y$. Thus $x=1$, a contradiction. Thus no such subgroup exists.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_2_1_13", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $\\mathbb{Q}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: First, suppose there does not exist a nonzero element in $H$. Then $H=0$.\n\nNow suppose there does exist a nonzero element $a \\in H$; without loss of generality, say $a=p / q$ in lowest terms for some integers $p$ and $q$ - that is, $\\operatorname{gcd}(p, q)=1$. Now $q \\cdot \\frac{p}{q}=p \\in H$, and since $q / p \\in H$, we have $p \\cdot \\frac{q}{p} \\in H$. There exist integers $x, y$ such that $q x+p y=1$; note that $q x \\in H$ and $p y \\in H$, so that $1 \\in H$. Thus $n \\in H$ for all $n \\in \\mathbb{Z}$. Moreover, if $n \\neq 0,1 / n \\in H$. Then $m / n \\in H$ for all integers $m, n$ with $n \\neq 0$; hence $H=\\mathbb{Q}$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_2_4_4", - "formal_statement": "theorem exercise_2_4_4 {G : Type*} [group G] (H : subgroup G) : \n subgroup.closure ((H : set G) \\ {1}) = \u22a4 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $H$ is a subgroup of $G$ then $H$ is generated by the set $H-\\{1\\}$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_2_4_16a", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $H$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_2_4_16b", - "formal_statement": "theorem exercise_2_4_16b {n : \u2115} {hn : n \u2260 0} \n {R : subgroup (dihedral_group n)} \n (hR : R = subgroup.closure {dihedral_group.r 1}) : \n R \u2260 \u22a4 \u2227 \n \u2200 K : subgroup (dihedral_group n), R \u2264 K \u2192 K = R \u2228 K = \u22a4 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_2_4_16c", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $n$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_3_1_3a", - "formal_statement": "theorem exercise_3_1_3a {A : Type*} [comm_group A] (B : subgroup A) :\n \u2200 a b : A \u29f8 B, a*b = b*a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A / B$ is abelian.\n", - "nl_proof": "\\begin{proof}\n\n Lemma: Let $G$ be a group. If $|G|=2$, then $G \\cong Z_2$.\n\nProof: Since $G=\\{e a\\}$ has an identity element, say $e$, we know that $e e=e, e a=a$, and $a e=a$. If $a^2=a$, we have $a=e$, a contradiction. Thus $a^2=e$. We can easily see that $G \\cong Z_2$.\n\n\n\nIf $A$ is abelian, every subgroup of $A$ is normal; in particular, $B$ is normal, so $A / B$ is a group. Now let $x B, y B \\in A / B$. Then\n\n$$\n\n(x B)(y B)=(x y) B=(y x) B=(y B)(x B) .\n\n$$\n\nHence $A / B$ is abelian.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_3_1_22a", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $G$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_3_1_22b", - "formal_statement": "theorem exercise_3_1_22b {G : Type*} [group G] (I : Type*)\n (H : I \u2192 subgroup G) (hH : \u2200 i : I, subgroup.normal (H i)) : \n subgroup.normal (\u2a05 (i : I), H i):=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the intersection of an arbitrary nonempty collection of normal subgroups of a group is a normal subgroup (do not assume the collection is countable).\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_3_2_8", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \\cap K=1$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Let $|H|=p$ and $|K|=q$. We saw in a previous exercise that $H \\cap K$ is a subgroup of both $H$ and $K$; by Lagrange's Theorem, then, $|H \\cap K|$ divides $p$ and $q$. Since $\\operatorname{gcd}(p, q)=1$, then, $|H \\cap K|=1$. Thus $H \\cap K=1$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_3_2_11", - "formal_statement": "theorem exercise_3_2_11 {G : Type*} [group G] {H K : subgroup G}\n (hHK : H \u2264 K) : \n H.index = K.index * H.relindex K :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $H \\leq K \\leq G$. Prove that $|G: H|=|G: K| \\cdot|K: H|$ (do not assume $G$ is finite).\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_3_2_16", - "formal_statement": "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] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $a \\in \\mathbb{Z}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: If $p$ is prime, then $\\varphi(p)=p-1$ (where $\\varphi$ denotes the Euler totient). Thus\n\n$$\n\n\\mid\\left((\\mathbb{Z} /(p))^{\\times} \\mid=p-1 .\\right.\n\n$$\n\nSo for all $a \\in(\\mathbb{Z} /(p))^{\\times}$, we have $|a|$ divides $p-1$. Hence\n\n$$\n\na=1 \\cdot a=a^{p-1} a=a^p \\quad(\\bmod p) .\n\n$$\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_3_2_21a", - "formal_statement": "theorem exercise_3_2_21a (H : add_subgroup \u211a) (hH : H \u2260 \u22a4) : H.index = 0 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\mathbb{Q}$ has no proper subgroups of finite index.\n", - "nl_proof": "\\begin{proof}\n\n Solution: We begin with a lemma.\n\nLemma: If $D$ is a divisible abelian group, then no proper subgroup of $D$ has finite index.\n\nProof: We saw previously that no finite group is divisible and that every proper quotient $D / A$ of a divisible group is divisible; thus no proper quotient of a divisible group is finite. Equivalently, $[D: A]$ is not finite.\n\nBecause $\\mathbb{Q}$ and $\\mathbb{Q} / \\mathbb{Z}$ are divisible, the conclusion follows.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_3_3_3", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $|K: K \\cap H|=p$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Suppose $K \\backslash N \\neq \\emptyset$; say $k \\in K \\backslash N$. Now $G / N \\cong \\mathbb{Z} /(p)$ is cyclic, and moreover is generated by any nonidentity- in particular by $\\bar{k}$\n\n\n\nNow $K N \\leq G$ since $N$ is normal. Let $g \\in G$. We have $g N=k^a N$ for some integer a. In particular, $g=k^a n$ for some $n \\in N$, hence $g \\in K N$. We have $[K: K \\cap N]=p$ by the Second Isomorphism Theorem.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_3_4_1", - "formal_statement": "theorem exercise_3_4_1 (G : Type*) [comm_group G] [is_simple_group G] :\n is_cyclic G \u2227 \u2203 G_fin : fintype G, nat.prime (@card G G_fin) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $G$ is an abelian simple group then $G \\cong Z_{p}$ for some prime $p$ (do not assume $G$ is a finite group).\n", - "nl_proof": "\\begin{proof}\n\n Solution: Let $G$ be an abelian simple group.\n\nSuppose $G$ is infinite. If $x \\in G$ is a nonidentity element of finite order, then $\\langle x\\rangle 0) (hH : H.index = n) : \n \u2203 K \u2264 H, K.normal \u2227 K.index \u2264 n.factorial :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $H$ has finite index $n$ then there is a normal subgroup $K$ of $G$ with $K \\leq H$ and $|G: K| \\leq n!$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: $G$ acts on the cosets $G / H$ by left multiplication. Let $\\lambda: G \\rightarrow S_{G / H}$ be the permutation representation induced by this action, and let $K$ be the kernel of the representation.\n\nNow $K$ is normal in $G$, and $K \\leq \\operatorname{stab}_G(H)=H$. By the First Isomorphism Theorem, we have an injective group homomorphism $\\bar{\\lambda}: G / K \\rightarrow S_{G / H}$. Since $\\left|S_{G / H}\\right|=n !$, we have $[G: K] \\leq n !$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_4_3_26", - "formal_statement": "theorem exercise_4_3_26 {\u03b1 : Type*} [fintype \u03b1] (ha : fintype.card \u03b1 > 1)\n (h_tran : \u2200 a b: \u03b1, \u2203 \u03c3 : equiv.perm \u03b1, \u03c3 a = b) : \n \u2203 \u03c3 : equiv.perm \u03b1, \u2200 a : \u03b1, \u03c3 a \u2260 a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G$ be a transitive permutation group on the finite set $A$ with $|A|>1$. Show that there is some $\\sigma \\in G$ such that $\\sigma(a) \\neq a$ for all $a \\in A$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_2_9a", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $G$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Let $G$ be a group of order $p^k$ and $H \\leq G$ a subgroup with $[G: H]=p$. Now $G$ acts on the conjugates $g H g^{-1}$ by conjugation, since\n\n$$\n\ng_1 g_2 \\cdot H=\\left(g_1 g_2\\right) H\\left(g_1 g_2\\right)^{-1}=g_1\\left(g_2 H g_2^{-1}\\right) g_1^{-1}=g_1 \\cdot\\left(g_2 \\cdot H\\right)\n\n$$\n\nand $1 \\cdot H=1 H 1=H$. Moreover, under this action we have $H \\leq \\operatorname{stab}(H)$. By Exercise 3.2.11, we have\n\n$$\n\n[G: \\operatorname{stab}(H)][\\operatorname{stab}(H): H]=[G: H]=p,\n\n$$\n\na prime.\n\nIf $[G: \\operatorname{stab}(H)]=p$, then $[\\operatorname{stab}(H): H]=1$ and we have $H=\\operatorname{stab}(H)$; moreover, $H$ has exactly $p$ conjugates in $G$. Let $\\varphi: G \\rightarrow S_p$ be the permutation representation induced by the action of $G$ on the conjugates of $H$, and let $K$ be the kernel of this representation. Now $K \\leq \\operatorname{stab}(H)=H$. By the first isomorphism theorem, the induced map $\\bar{\\varphi}: G / K \\rightarrow S_p$ is injective, so that $|G / K|$ divides $p$ !. Note, however, that $|G / K|$ is a power of $p$ and that the only powers of $p$ that divide $p$ ! are 1 and $p$. So $[G: K]$ is 1 or $p$. If $[G: K]=1$, then $G=K$ so that $g H g^{-1}=H$ for all $g \\in G$; then $\\operatorname{stab}(H)=G$ and we have $[G: \\operatorname{stab}(H)]=1$, a contradiction. Now suppose $[G: K]=p$. Again by Exercise $3.2$.11 we have $[G: K]=[G: H][H: K]$, so that $[H: K]=1$, hence $H=K$. Again, this implies that $H$ is normal so that $g H g^{-1}=H$ for all $g \\in G$, and we have $[G: \\operatorname{stab}(H)]=1$, a contradiction. Thus $[G: \\operatorname{stab}(H)] \\neq p$\n\nIf $[G: \\operatorname{stab}(H)]=1$, then $G=\\operatorname{stab}(H)$. That is, $g H g^{-1}=H$ for all $g \\in G$; thus $H \\leq G$ is normal.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_4_2_14", - "formal_statement": "theorem exercise_4_2_14 {G : Type*} [fintype G] [group G] \n (hG : \u00ac (card G).prime) (hG1 : \u2200 k \u2223 card G, \n \u2203 (H : subgroup G) (fH : fintype H), @card H fH = k) : \n \u00ac is_simple_group G :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G$ be a finite group of composite order $n$ with the property that $G$ has a subgroup of order $k$ for each positive integer $k$ dividing $n$. Prove that $G$ is not simple.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Let $p$ be the smallest prime dividing $n$, and write $n=p m$. Now $G$ has a subgroup $H$ of order $m$, and $H$ has index $p$. By Corollary 5 in the text, $H$ is normal in $G$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_4_4_2", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_4_6a", - "formal_statement": "theorem exercise_4_4_6a {G : Type*} [group G] (H : subgroup G)\n [subgroup.characteristic H] : subgroup.normal H :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that characteristic subgroups are normal.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_4_6b", - "formal_statement": "theorem exercise_4_4_6b {G : Type*} [group G] : \n \u2203 H : subgroup G, H.characteristic \u2227 \u00ac H.normal :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that there exists a normal subgroup that is not characteristic.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_4_7", - "formal_statement": "theorem exercise_4_4_7 {G : Type*} [group G] {H : subgroup G} [fintype H]\n (hH : \u2200 (K : subgroup G) (fK : fintype K), card H = @card K fK \u2192 H = K) : \n H.characteristic :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "If $H$ is the unique subgroup of a given order in a group $G$ prove $H$ is characteristic in $G$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_4_8a", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $G$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_5_1a", - "formal_statement": "theorem exercise_4_5_1a {p : \u2115} {G : Type*} [group G] \n {P : subgroup G} (hP : is_p_group p P) (H : subgroup G) \n (hH : P \u2264 H) : is_p_group p H :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $P \\in \\operatorname{Syl}_{p}(G)$ and $H$ is a subgroup of $G$ containing $P$ then $P \\in \\operatorname{Syl}_{p}(H)$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: If $P \\leq H \\leq G$ is a Sylow $p$-subgroup of $G$, then $p$ does not divide $[G: P]$. Now $[G: P]=[G: H][H: P]$, so that $p$ does not divide $[H: P]$; hence $P$ is a Sylow $p$-subgroup of $H$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_4_5_13", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_5_14", - "formal_statement": "theorem exercise_4_5_14 {G : Type*} [group G] [fintype G]\n (hG : card G = 312) :\n \u2203 (p : \u2115) (P : sylow p G), P.normal :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_5_15", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_4_5_16", - "formal_statement": "theorem exercise_4_5_16 {p q r : \u2115} {G : Type*} [group G] \n [fintype G] (hpqr : p < q \u2227 q < r) \n (hpqr1 : p.prime \u2227 q.prime \u2227 r.prime)(hG : card G = p*q*r) : \n nonempty (sylow p G) \u2228 nonempty(sylow q G) \u2228 nonempty(sylow r G) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $|G|=p q r$, where $p, q$ and $r$ are primes with $pm$, a contradiction. Thus we have $a_t b_m=0$. As in the base case,\n\n$$\n\na_t q(x) p(x)=0\n\n$$\n\nand $a_t q(x)$ has degree strictly less than that of $q(x)$, so that by minimality, $a_t q(x)=0$.\n\nBy induction, $a_i q(x)=0$ for all $0 \\leq i \\leq n$. In particular, $a_i b_m=0$. Thus $b_m p(x)=0$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_7_2_12", - "formal_statement": "theorem exercise_7_2_12 {R G : Type*} [ring R] [group G] [fintype G] : \n \u2211 g : G, monoid_algebra.of R G g \u2208 center (monoid_algebra R G) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $G=\\left\\{g_{1}, \\ldots, g_{n}\\right\\}$ be a finite group. Prove that the element $N=g_{1}+g_{2}+\\ldots+g_{n}$ is in the center of the group ring $R G$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Let $M=\\sum_{i=1}^n r_i g_i$ be an element of $R[G]$. Note that for each $g_i \\in G$, the action of $g_i$ on $G$ by conjugation permutes the subscripts. Then we have the following.\n\n$$\n\n\\begin{aligned}\n\nN M &=\\left(\\sum_{i=1}^n g_i\\right)\\left(\\sum_{j=1}^n r_j g_j\\right) \\\\\n\n&=\\sum_{j=1}^n \\sum_{i=1}^n r_j g_i g_j \\\\\n\n&=\\sum_{j=1}^n \\sum_{i=1}^n r_j g_j g_j^{-1} g_i g_j \\\\\n\n&=\\sum_{j=1}^n r_j g_j\\left(\\sum_{i=1}^n g_j^{-1} g_i g_j\\right) \\\\\n\n&=\\sum_{j=1}^n r_j g_j\\left(\\sum_{i=1}^n g_i\\right) \\\\\n\n&=\\left(\\sum_{j=1}^n r_j g_j\\right)\\left(\\sum_{i=1}^n g_i\\right) \\\\\n\n&=M N .\n\n\\end{aligned}\n\n$$\n\nThus $N \\in Z(R[G])$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_7_3_16", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $S$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: Suppose $r \\in \\varphi[Z(R)]$. Then $r=\\varphi(z)$ for some $z \\in Z(R)$. Now let $x \\in S$. Since $\\varphi$ is surjective, we have $x=\\varphi y$ for some $y \\in R$. Now\n\n$$\n\nx r=\\varphi(y) \\varphi(z)=\\varphi(y z)=\\varphi(z y)=\\varphi(z) \\varphi(y)=r x .\n\n$$\n\nThus $r \\in Z(S)$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_7_3_37", - "formal_statement": "theorem exercise_7_3_37 {R : Type*} {p m : \u2115} (hp : p.prime) \n (N : ideal $ zmod $ p^m) : \n is_nilpotent N \u2194 is_nilpotent (ideal.span ({p} : set $ zmod $ p^m)) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "An ideal $N$ is called nilpotent if $N^{n}$ is the zero ideal for some $n \\geq 1$. Prove that the ideal $p \\mathbb{Z} / p^{m} \\mathbb{Z}$ is a nilpotent ideal in the ring $\\mathbb{Z} / p^{m} \\mathbb{Z}$.\n", - "nl_proof": "\\begin{proof}\n\n Solution: First we prove a lemma.\n\nLemma: Let $R$ be a ring, and let $I_1, I_2, J \\subseteq R$ be ideals such that $J \\subseteq I_1, I_2$. Then $\\left(I_1 / J\\right)\\left(I_2 / J\\right)=I_1 I_2 / J$.\n\nProof: ( $\\subseteq$ ) Let\n\n$$\n\n\\alpha=\\sum\\left(x_i+J\\right)\\left(y_i+J\\right) \\in\\left(I_1 / J\\right)\\left(I_2 / J\\right) .\n\n$$\n\nThen\n\n$$\n\n\\alpha=\\sum\\left(x_i y_i+J\\right)=\\left(\\sum x_i y_i\\right)+J \\in\\left(I_1 I_2\\right) / J .\n\n$$\n\nNow let $\\alpha=\\left(\\sum x_i y_i\\right)+J \\in\\left(I_1 I_2\\right) / J$. Then\n\n$$\n\n\\alpha=\\sum\\left(x_i+J\\right)\\left(y_i+J\\right) \\in\\left(I_1 / J\\right)\\left(I_2 / J\\right) .\n\n$$\n\nFrom this lemma and the lemma to Exercise 7.3.36, it follows by an easy induction that\n\n$$\n\n\\left(p \\mathbb{Z} / p^m \\mathbb{Z}\\right)^m=(p \\mathbb{Z})^m / p^m \\mathbb{Z}=p^m \\mathbb{Z} / p^m \\mathbb{Z} \\cong 0 .\n\n$$\n\nThus $p \\mathbb{Z} / p^m \\mathbb{Z}$ is nilpotent in $\\mathbb{Z} / p^m \\mathbb{Z}$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_7_4_27", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $b \\in R$.\n", - "nl_proof": "\\begin{proof}\n\n $\\mathfrak{N}(R)$ is an ideal of $R$. Thus for all $b \\in R,-a b$ is nilpotent. Hence $1-a b$ is a unit in $R$.\n\n\\end{proof}" - }, - { - "id": "Dummit-Foote|exercise_8_1_12", - "formal_statement": "theorem exercise_8_1_12 {N : \u2115} (hN : N > 0) {M M': \u2124} {d : \u2115}\n (hMN : M.gcd N = 1) (hMd : d.gcd N.totient = 1) \n (hM' : M' \u2261 M^d [ZMOD N]) : \n \u2203 d' : \u2115, d' * d \u2261 1 [ZMOD N.totient] \u2227 \n M \u2261 M'^d' [ZMOD N] :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Let $N$ be a positive integer. Let $M$ be an integer relatively prime to $N$ and let $d$ be an integer relatively prime to $\\varphi(N)$, where $\\varphi$ denotes Euler's $\\varphi$-function. Prove that if $M_{1} \\equiv M^{d} \\pmod N$ then $M \\equiv M_{1}^{d^{\\prime}} \\pmod N$ where $d^{\\prime}$ is the inverse of $d \\bmod \\varphi(N)$: $d d^{\\prime} \\equiv 1 \\pmod {\\varphi(N)}$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_8_2_4", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $n \\geq N$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_8_3_4", - "formal_statement": "theorem exercise_8_3_4 {R : Type*} {n : \u2124} {r s : \u211a} \n (h : r^2 + s^2 = n) : \n \u2203 a b : \u2124, a^2 + b^2 = n :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if an integer is the sum of two rational squares, then it is the sum of two integer squares.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_8_3_5a", - "formal_statement": "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) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 $R$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_8_3_6a", - "formal_statement": "theorem exercise_8_3_6a {R : Type*} [ring R]\n (hR : R = (gaussian_int \u29f8 ideal.span ({\u27e80, 1\u27e9} : set gaussian_int))) :\n is_field R \u2227 \u2203 finR : fintype R, @card R finR = 2 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that the quotient ring $\\mathbb{Z}[i] /(1+i)$ is a field of order 2.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_8_3_6b", - "formal_statement": "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 :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_1_6", - "formal_statement": "theorem exercise_9_1_6 : \u00ac is_principal \n (ideal.span ({X 0, X 1} : set (mv_polynomial (fin 2) \u211a))) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $(x, y)$ is not a principal ideal in $\\mathbb{Q}[x, y]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_1_10", - "formal_statement": "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))) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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 (cf. exercise.9.1.36 of Section 7.4).\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_3_2", - "formal_statement": "theorem exercise_9_3_2 {f g : polynomial \u211a} (i j : \u2115)\n (hfg : \u2200 n : \u2115, \u2203 a : \u2124, (f*g).coeff = a) :\n \u2203 a : \u2124, f.coeff i * g.coeff j = a :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that if $f(x)$ and $g(x)$ are polynomials with rational coefficients whose product $f(x) g(x)$ has integer coefficients, then the product of any coefficient of $g(x)$ with any coefficient of $f(x)$ is an integer.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_2a", - "formal_statement": "theorem exercise_9_4_2a : irreducible (X^4 - 4*X^3 + 6 : polynomial \u2124) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^4-4x^3+6$ is irreducible in $\\mathbb{Z}[x]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_2b", - "formal_statement": "theorem exercise_9_4_2b : irreducible \n (X^6 + 30*X^5 - 15*X^3 + 6*X - 120 : polynomial \u2124) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^6+30x^5-15x^3 + 6x-120$ is irreducible in $\\mathbb{Z}[x]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_2c", - "formal_statement": "theorem exercise_9_4_2c : irreducible \n (X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : polynomial \u2124) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\\mathbb{Z}[x]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_2d", - "formal_statement": "theorem exercise_9_4_2d {p : \u2115} (hp : p.prime \u2227 p > 2) \n {f : polynomial \u2124} (hf : f = (X + 2)^p): \n irreducible (\u2211 n in (f.support - {0}), (f.coeff n) * X ^ (n-1) : \n polynomial \u2124) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $\\frac{(x+2)^p-2^p}{x}$, where $p$ is an odd prime, is irreducible in $\\mathbb{Z}[x]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_9", - "formal_statement": "theorem exercise_9_4_9 : \n irreducible (X^2 - C sqrtd : polynomial (zsqrtd 2)) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "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.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_9_4_11", - "formal_statement": "theorem exercise_9_4_11 : \n irreducible ((X 0)^2 + (X 1)^2 - 1 : mv_polynomial (fin 2) \u211a) :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that $x^2+y^2-1$ is irreducible in $\\mathbb{Q}[x,y]$.\n", - "nl_proof": "" - }, - { - "id": "Dummit-Foote|exercise_11_1_13", - "formal_statement": "theorem exercise_11_1_13 {\u03b9 : Type*} [fintype \u03b9] : \n (\u03b9 \u2192 \u211d) \u2243\u2097[\u211a] \u211d :=", - "src_header": "import .common \n\nopen set function nat int fintype real polynomial mv_polynomial\nopen subgroup ideal submodule zsqrtd gaussian_int char_p mul_aut matrix\n\nopen_locale pointwise\nopen_locale big_operators\nnoncomputable theory\n\n", - "nl_statement": "Prove that as vector spaces over $\\mathbb{Q}, \\mathbb{R}^n \\cong \\mathbb{R}$, for all $n \\in \\mathbb{Z}^{+}$.\n", - "nl_proof": "" - } -] \ No newline at end of file