proofnet/test.jsonl

{"name": "exercise_1_13b", "split": "test", "informal_prefix": "/-- Suppose that $f$ is holomorphic in an open set $\\Omega$. Prove that if $\\text{Im}(f)$ is constant, then $f$ is constant.-/\n", "formal_statement": "theorem exercise_1_13b {f : ℂ → ℂ} (Ω : Set ℂ) (a b : Ω) (h : IsOpen Ω)\n  (hf : DifferentiableOn ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).im = c) :\n  f a = f b :=", "goal": "f : ℂ → ℂ\nΩ : Set ℂ\na b : ↑Ω\nh : IsOpen Ω\nhf : DifferentiableOn ℂ f Ω\nhc : ∃ c, ∀ z ∈ Ω, (f z).im = c\n⊢ f ↑a = f ↑b", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_19a", "split": "test", "informal_prefix": "/-- Prove that the power series $\\sum nz^n$ does not converge on any point of the unit circle.-/\n", "formal_statement": "theorem exercise_1_19a (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ)\n    (h : s = (λ n => ∑ i in (range n), i * z ^ i)) :\n    ¬ ∃ y, Tendsto s atTop (𝓝 y) :=", "goal": "z : ℂ\nhz : Complex.abs z = 1\ns : ℕ → ℂ\nh : s = fun n => ∑ i ∈ range n, ↑i * z ^ i\n⊢ ¬∃ y, Tendsto s atTop (𝓝 y)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_19c", "split": "test", "informal_prefix": "/-- Prove that the power series $\\sum zn/n$ converges at every point of the unit circle except $z = 1$.-/\n", "formal_statement": "theorem exercise_1_19c (z : ℂ) (hz : abs z = 1) (hz2 : z ≠ 1) (s : ℕ → ℂ)\n    (h : s = (λ n => ∑ i in (range n), i * z / i)) :\n    ∃ z, Tendsto s atTop (𝓝 z) :=", "goal": "z : ℂ\nhz : Complex.abs z = 1\nhz2 : z ≠ 1\ns : ℕ → ℂ\nh : s = fun n => ∑ i ∈ range n, ↑i * z / ↑i\n⊢ ∃ z, Tendsto s atTop (𝓝 z)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_2_2", "split": "test", "informal_prefix": "/-- Show that $\\int_{0}^{\\infty} \\frac{\\sin x}{x} d x=\\frac{\\pi}{2}$.-/\n", "formal_statement": "theorem exercise_2_2 :\n  Tendsto (λ y => ∫ x in (0 : ℝ)..y, Real.sin x / x) atTop (𝓝 (Real.pi / 2)) :=", "goal": "⊢ Tendsto (fun y => ∫ (x : ℝ) in 0 ..y, x.sin / x) atTop (𝓝 (Real.pi / 2))", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_2_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_13 {f : ℂ → ℂ}\n    (hf : ∀ z₀ : ℂ, ∃ (s : Set ℂ) (c : ℕ → ℂ), IsOpen s ∧ z₀ ∈ s ∧\n      ∀ z ∈ s, Tendsto (λ n => ∑ i in range n, (c i) * (z - z₀)^i) atTop (𝓝 (f z₀))\n      ∧ ∃ i, c i = 0) :\n    ∃ (c : ℕ → ℂ) (n : ℕ), f = λ z => ∑ i in range n, (c i) * z ^ n :=", "goal": "f : ℂ → ℂ\nhf :\n  ∀ (z₀ : ℂ),\n    ∃ s c,\n      IsOpen s ∧ z₀ ∈ s ∧ ∀ z ∈ s, Tendsto (fun n => ∑ i ∈ range n, c i * (z - z₀) ^ i) atTop (𝓝 (f z₀)) ∧ ∃ i, c i = 0\n⊢ ∃ c n, f = fun z => ∑ i ∈ range n, c i * z ^ n", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_4", "split": "test", "informal_prefix": "/-- Show that $ \\int_{-\\infty}^{\\infty} \\frac{x \\sin x}{x^2 + a^2} dx = \\pi e^{-a}$ for $a > 0$.-/\n", "formal_statement": "theorem exercise_3_4 (a : ℝ) (ha : 0 < a) :\n    Tendsto (λ y => ∫ x in -y..y, x * Real.sin x / (x ^ 2 + a ^ 2))\n    atTop (𝓝 (Real.pi * (Real.exp (-a)))) :=", "goal": "a : ℝ\nha : 0 < a\n⊢ Tendsto (fun y => ∫ (x : ℝ) in -y..y, x * x.sin / (x ^ 2 + a ^ 2)) atTop (𝓝 (Real.pi * (-a).exp))", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_3_14", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_14 {f : ℂ → ℂ} (hf : Differentiable ℂ f)\n    (hf_inj : Function.Injective f) :\n    ∃ (a b : ℂ), f = (λ z => a * z + b) ∧ a ≠ 0 :=", "goal": "f : ℂ → ℂ\nhf : Differentiable ℂ f\nhf_inj : Injective f\n⊢ ∃ a b, (f = fun z => a * z + b) ∧ a ≠ 0", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_1 (f : ℂ → ℂ) (hf : DifferentiableOn ℂ f (ball 0 1))\n  (hb : Bornology.IsBounded (Set.range f)) (h0 : f ≠ 0) (zeros : ℕ → ℂ) (hz : ∀ n, f (zeros n) = 0)\n  (hzz : Set.range zeros = {z | f z = 0 ∧ z ∈ (ball (0 : ℂ) 1)}) :\n  ∃ (z : ℂ), Tendsto (λ n => (∑ i in range n, (1 - zeros i))) atTop (𝓝 z) :=", "goal": "f : ℂ → ℂ\nhf : DifferentiableOn ℂ f (ball 0 1)\nhb : Bornology.IsBounded (Set.range f)\nh0 : f ≠ 0\nzeros : ℕ → ℂ\nhz : ∀ (n : ℕ), f (zeros n) = 0\nhzz : Set.range zeros = {z | f z = 0 ∧ z ∈ ball 0 1}\n⊢ ∃ z, Tendsto (fun n => ∑ i ∈ range n, (1 - zeros i)) atTop (𝓝 z)", "header": "import Mathlib\n\nopen Complex Filter Function Metric Finset\nopen scoped BigOperators Topology\n\n"}
{"name": "exercise_1_1b", "split": "test", "informal_prefix": "/-- If $r$ is rational $(r \\neq 0)$ and $x$ is irrational, prove that $rx$ is irrational.-/\n", "formal_statement": "theorem exercise_1_1b\n(x : ℝ)\n(y : ℚ)\n(h : y ≠ 0)\n: ( Irrational x ) -> Irrational ( x * y ) :=", "goal": "x : ℝ\ny : ℚ\nh : y ≠ 0\n⊢ Irrational x → Irrational (x * ↑y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_4", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_4\n(α : Type*) [PartialOrder α]\n(s : Set α)\n(x y : α)\n(h₀ : Set.Nonempty s)\n(h₁ : x ∈ lowerBounds s)\n(h₂ : y ∈ upperBounds s)\n: x ≤ y :=", "goal": "α : Type u_1\ninst✝ : PartialOrder α\ns : Set α\nx y : α\nh₀ : s.Nonempty\nh₁ : x ∈ lowerBounds s\nh₂ : y ∈ upperBounds s\n⊢ x ≤ y", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_8", "split": "test", "informal_prefix": "/-- Prove that no order can be defined in the complex field that turns it into an ordered field.-/\n", "formal_statement": "theorem exercise_1_8 : ¬ ∃ (r : ℂ → ℂ → Prop), IsLinearOrder ℂ r :=", "goal": "⊢ ¬∃ r, IsLinearOrder ℂ r", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_12", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_12 (n : ℕ) (f : ℕ → ℂ) :\n  abs (∑ i in range n, f i) ≤ ∑ i in range n, abs (f i) :=", "goal": "n : ℕ\nf : ℕ → ℂ\n⊢ Complex.abs (∑ i ∈ range n, f i) ≤ ∑ i ∈ range n, Complex.abs (f i)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_14", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_14\n  (z : ℂ) (h : abs z = 1)\n  : (abs (1 + z)) ^ 2 + (abs (1 - z)) ^ 2 = 4 :=", "goal": "z : ℂ\nh : Complex.abs z = 1\n⊢ Complex.abs (1 + z) ^ 2 + Complex.abs (1 - z) ^ 2 = 4", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_17", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_17\n  (n : ℕ)\n  (x y : EuclideanSpace ℝ (Fin n)) -- R^n\n  : ‖x + y‖^2 + ‖x - y‖^2 = 2*‖x‖^2 + 2*‖y‖^2 :=", "goal": "n : ℕ\nx y : EuclideanSpace ℝ (Fin n)\n⊢ ‖x + y‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * ‖x‖ ^ 2 + 2 * ‖y‖ ^ 2", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_18b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_18b\n  : ¬ ∀ (x : ℝ), ∃ (y : ℝ), y ≠ 0 ∧ x * y = 0 :=", "goal": "⊢ ¬∀ (x : ℝ), ∃ y, y ≠ 0 ∧ x * y = 0", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_19a", "split": "test", "informal_prefix": "/-- If $A$ and $B$ are disjoint closed sets in some metric space $X$, prove that they are separated.-/\n", "formal_statement": "theorem exercise_2_19a {X : Type*} [MetricSpace X]\n  (A B : Set X) (hA : IsClosed A) (hB : IsClosed B) (hAB : Disjoint A B) :\n  SeparatedNhds A B :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\nA B : Set X\nhA : IsClosed A\nhB : IsClosed B\nhAB : Disjoint A B\n⊢ SeparatedNhds A B", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_25", "split": "test", "informal_prefix": "/-- Prove that every compact metric space $K$ has a countable base.-/\n", "formal_statement": "theorem exercise_2_25 {K : Type*} [MetricSpace K] [CompactSpace K] :\n  ∃ (B : Set (Set K)), Set.Countable B ∧ IsTopologicalBasis B :=", "goal": "K : Type u_1\ninst✝¹ : MetricSpace K\ninst✝ : CompactSpace K\n⊢ ∃ B, B.Countable ∧ IsTopologicalBasis B", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_27b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_27b (k : ℕ) (E P : Set (EuclideanSpace ℝ (Fin k)))\n  (hE : E.Nonempty ∧ ¬ Set.Countable E)\n  (hP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬ Set.Countable (P ∩ E)}) :\n  Set.Countable (E \\ P) :=", "goal": "k : ℕ\nE P : Set (EuclideanSpace ℝ (Fin k))\nhE : E.Nonempty ∧ ¬E.Countable\nhP : P = {x | ∀ U ∈ 𝓝 x, (P ∩ E).Nonempty ∧ ¬(P ∩ E).Countable}\n⊢ (E \\ P).Countable", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_29", "split": "test", "informal_prefix": "/-- Prove that every open set in $\\mathbb{R}$ is the union of an at most countable collection of disjoint segments.-/\n", "formal_statement": "theorem exercise_2_29 (U : Set ℝ) (hU : IsOpen U) :\n  ∃ (f : ℕ → Set ℝ), (∀ n, ∃ a b : ℝ, f n = {x | a < x ∧ x < b}) ∧ (∀ n, f n ⊆ U) ∧\n  (∀ n m, n ≠ m → f n ∩ f m = ∅) ∧\n  U = ⋃ n, f n :=", "goal": "U : Set ℝ\nhU : IsOpen U\n⊢ ∃ f,\n    (∀ (n : ℕ), ∃ a b, f n = {x | a < x ∧ x < b}) ∧\n      (∀ (n : ℕ), f n ⊆ U) ∧ (∀ (n m : ℕ), n ≠ m → f n ∩ f m = ∅) ∧ U = ⋃ n, f n", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2a", "split": "test", "informal_prefix": "/-- Prove that $\\lim_{n \\rightarrow \\infty}\\sqrt{n^2 + n} -n = 1/2$.-/\n", "formal_statement": "theorem exercise_3_2a\n  : Tendsto (λ (n : ℝ) => (sqrt (n^2 + n) - n)) atTop (𝓝 (1/2)) :=", "goal": "⊢ Tendsto (fun n => √(n ^ 2 + n) - n) atTop (𝓝 (1 / 2))", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_5", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_5\n  (a b : ℕ → ℝ)\n  (h : limsup a + limsup b ≠ 0) :\n  limsup (λ n => a n + b n) ≤ limsup a + limsup b :=", "goal": "a b : ℕ → ℝ\nh : limsup a + limsup b ≠ 0\n⊢ (limsup fun n => a n + b n) ≤ limsup a + limsup b", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_7", "split": "test", "informal_prefix": "/-- Prove that the convergence of $\\Sigma a_{n}$ implies the convergence of $\\sum \\frac{\\sqrt{a_{n}}}{n}$ if $a_n\\geq 0$.-/\n", "formal_statement": "theorem exercise_3_7\n  (a : ℕ → ℝ)\n  (h : ∃ y, (Tendsto (λ n => (∑ i in (range n), a i)) atTop (𝓝 y))) :\n  ∃ y, Tendsto (λ n => (∑ i in (range n), sqrt (a i) / n)) atTop (𝓝 y) :=", "goal": "a : ℕ → ℝ\nh : ∃ y, Tendsto (fun n => ∑ i ∈ range n, a i) atTop (𝓝 y)\n⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, √(a i) / ↑n) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_13", "split": "test", "informal_prefix": "/-- Prove that the Cauchy product of two absolutely convergent series converges absolutely.-/\n", "formal_statement": "theorem exercise_3_13\n  (a b : ℕ → ℝ)\n  (ha : ∃ y, (Tendsto (λ n => (∑ i in (range n), |a i|)) atTop (𝓝 y)))\n  (hb : ∃ y, (Tendsto (λ n => (∑ i in (range n), |b i|)) atTop (𝓝 y))) :\n  ∃ y, (Tendsto (λ n => (∑ i in (range n),\n  λ i => (∑ j in range (i + 1), a j * b (i - j)))) atTop (𝓝 y)) :=", "goal": "a b : ℕ → ℝ\nha : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |a i|) atTop (𝓝 y)\nhb : ∃ y, Tendsto (fun n => ∑ i ∈ range n, |b i|) atTop (𝓝 y)\n⊢ ∃ y, Tendsto (fun n => ∑ i ∈ range n, fun i => ∑ j ∈ range (i + 1), a j * b (i - j)) atTop (𝓝 y)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_21", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_21\n  {X : Type*} [MetricSpace X] [CompleteSpace X]\n  (E : ℕ → Set X)\n  (hE : ∀ n, E n ⊃ E (n + 1))\n  (hE' : Tendsto (λ n => Metric.diam (E n)) atTop (𝓝 0)) :\n  ∃ a, Set.iInter E = {a} :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nE : ℕ → Set X\nhE : ∀ (n : ℕ), E n ⊃ E (n + 1)\nhE' : Tendsto (fun n => Metric.diam (E n)) atTop (𝓝 0)\n⊢ ∃ a, Set.iInter E = {a}", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_1a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_1a\n  : ∃ (f : ℝ → ℝ), (∀ (x : ℝ), Tendsto (λ y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬ Continuous f :=", "goal": "⊢ ∃ f, (∀ (x : ℝ), Tendsto (fun y => f (x + y) - f (x - y)) (𝓝 0) (𝓝 0)) ∧ ¬Continuous f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_3\n  {α : Type} [MetricSpace α]\n  (f : α → ℝ) (h : Continuous f) (z : Set α) (g : z = f⁻¹' {0})\n  : IsClosed z :=", "goal": "α : Type\ninst✝ : MetricSpace α\nf : α → ℝ\nh : Continuous f\nz : Set α\ng : z = f ⁻¹' {0}\n⊢ IsClosed z", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_4b\n  {α : Type} [MetricSpace α]\n  {β : Type} [MetricSpace β]\n  (f g : α → β)\n  (s : Set α)\n  (h₁ : Continuous f)\n  (h₂ : Continuous g)\n  (h₃ : Dense s)\n  (h₄ : ∀ x ∈ s, f x = g x)\n  : f = g :=", "goal": "α : Type\ninst✝¹ : MetricSpace α\nβ : Type\ninst✝ : MetricSpace β\nf g : α → β\ns : Set α\nh₁ : Continuous f\nh₂ : Continuous g\nh₃ : Dense s\nh₄ : ∀ x ∈ s, f x = g x\n⊢ f = g", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_5b\n  : ∃ (E : Set ℝ) (f : ℝ → ℝ), (ContinuousOn f E) ∧\n  (¬ ∃ (g : ℝ → ℝ), Continuous g ∧ ∀ x ∈ E, f x = g x) :=", "goal": "⊢ ∃ E f, ContinuousOn f E ∧ ¬∃ g, Continuous g ∧ ∀ x ∈ E, f x = g x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_8a", "split": "test", "informal_prefix": "/-- Let $f$ be a real uniformly continuous function on the bounded set $E$ in $R^{1}$. Prove that $f$ is bounded on $E$.-/\n", "formal_statement": "theorem exercise_4_8a\n  (E : Set ℝ) (f : ℝ → ℝ) (hf : UniformContinuousOn f E)\n  (hE : Bornology.IsBounded E) : Bornology.IsBounded (Set.image f E) :=", "goal": "E : Set ℝ\nf : ℝ → ℝ\nhf : UniformContinuousOn f E\nhE : Bornology.IsBounded E\n⊢ Bornology.IsBounded (f '' E)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_11a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_11a\n  {X : Type*} [MetricSpace X]\n  {Y : Type*} [MetricSpace Y]\n  (f : X → Y) (hf : UniformContinuous f)\n  (x : ℕ → X) (hx : CauchySeq x) :\n  CauchySeq (λ n => f (x n)) :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\nY : Type u_2\ninst✝ : MetricSpace Y\nf : X → Y\nhf : UniformContinuous f\nx : ℕ → X\nhx : CauchySeq x\n⊢ CauchySeq fun n => f (x n)", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_15", "split": "test", "informal_prefix": "/-- Prove that every continuous open mapping of $R^{1}$ into $R^{1}$ is monotonic.-/\n", "formal_statement": "theorem exercise_4_15 {f : ℝ → ℝ}\n  (hf : Continuous f) (hof : IsOpenMap f) :\n  Monotone f :=", "goal": "f : ℝ → ℝ\nhf : Continuous f\nhof : IsOpenMap f\n⊢ Monotone f", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_21a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_21a {X : Type*} [MetricSpace X]\n  (K F : Set X) (hK : IsCompact K) (hF : IsClosed F) (hKF : Disjoint K F) :\n  ∃ (δ : ℝ), δ > 0 ∧ ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ :=", "goal": "X : Type u_1\ninst✝ : MetricSpace X\nK F : Set X\nhK : IsCompact K\nhF : IsClosed F\nhKF : Disjoint K F\n⊢ ∃ δ > 0, ∀ (p q : X), p ∈ K → q ∈ F → dist p q ≥ δ", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- Let $f$ be defined for all real $x$, and suppose that $|f(x)-f(y)| \\leq (x-y)^{2}$ for all real $x$ and $y$. Prove that $f$ is constant.-/\n", "formal_statement": "theorem exercise_5_1\n  {f : ℝ → ℝ} (hf : ∀ x y : ℝ, |(f x - f y)| ≤ (x - y) ^ 2) :\n  ∃ c, f = λ x => c :=", "goal": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), |f x - f y| ≤ (x - y) ^ 2\n⊢ ∃ c, f = fun x => c", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_3 {g : ℝ → ℝ} (hg : Continuous g)\n  (hg' : ∃ M : ℝ, ∀ x : ℝ, |deriv g x| ≤ M) :\n  ∃ N, ∀ ε > 0, ε < N → Function.Injective (λ x : ℝ => x + ε * g x) :=", "goal": "g : ℝ → ℝ\nhg : Continuous g\nhg' : ∃ M, ∀ (x : ℝ), |deriv g x| ≤ M\n⊢ ∃ N, ∀ ε > 0, ε < N → Function.Injective fun x => x + ε * g x", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_5", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_5\n  {f : ℝ → ℝ}\n  (hfd : Differentiable ℝ f)\n  (hf : Tendsto (deriv f) atTop (𝓝 0)) :\n  Tendsto (λ x => f (x + 1) - f x) atTop atTop :=", "goal": "f : ℝ → ℝ\nhfd : Differentiable ℝ f\nhf : Tendsto (deriv f) atTop (𝓝 0)\n⊢ Tendsto (fun x => f (x + 1) - f x) atTop atTop", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_7", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_7\n  {f g : ℝ → ℝ} {x : ℝ}\n  (hf' : DifferentiableAt ℝ f 0)\n  (hg' : DifferentiableAt ℝ g 0)\n  (hg'_ne_0 : deriv g 0 ≠ 0)\n  (f0 : f 0 = 0) (g0 : g 0 = 0) :\n  Tendsto (λ x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x)) :=", "goal": "f g : ℝ → ℝ\nx : ℝ\nhf' : DifferentiableAt ℝ f 0\nhg' : DifferentiableAt ℝ g 0\nhg'_ne_0 : deriv g 0 ≠ 0\nf0 : f 0 = 0\ng0 : g 0 = 0\n⊢ Tendsto (fun x => f x / g x) (𝓝 x) (𝓝 (deriv f x / deriv g x))", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_17", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_17\n  {f : ℝ → ℝ}\n  (hf' : DifferentiableOn ℝ f (Set.Icc (-1) 1))\n  (hf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1))\n  (hf''' : DifferentiableOn ℝ (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  ∃ x, x ∈ Set.Ioo (-1 : ℝ) 1 ∧ deriv (deriv (deriv f)) x ≥ 3 :=", "goal": "f : ℝ → ℝ\nhf' : DifferentiableOn ℝ f (Set.Icc (-1) 1)\nhf'' : DifferentiableOn ℝ (deriv f) (Set.Icc 1 1)\nhf''' : DifferentiableOn ℝ (deriv (deriv f)) (Set.Icc 1 1)\nhf0 : f (-1) = 0\nhf1 : f 0 = 0\nhf2 : f 1 = 1\nhf3 : deriv f 0 = 0\n⊢ ∃ x ∈ Set.Ioo (-1) 1, deriv (deriv (deriv f)) x ≥ 3", "header": "import Mathlib\n\nopen Topology Filter Real Complex TopologicalSpace Finset\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_18", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_1_18 {G : Type*} [Group G]\n  [Fintype G] (hG2 : Even (card G)) :\n  ∃ (a : G), a ≠ 1 ∧ a = a⁻¹ :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG2 : Even (card G)\n⊢ ∃ a, a ≠ 1 ∧ a = a⁻¹", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_26", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_1_26 {G : Type*} [Group G]\n  [Fintype G] (a : G) : ∃ (n : ℕ), a ^ n = 1 :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\na : G\n⊢ ∃ n, a ^ n = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_2_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "def exercise_2_2_3 {G : Type*} [Group G]\n  {P : ℕ → Prop} {hP : P = λ i => ∀ a b : G, (a*b)^i = a^i * b^i}\n  (hP1 : ∃ n : ℕ, P n ∧ P (n+1) ∧ P (n+2)) : CommGroup G :=", "goal": "G : Type u_1\ninst✝ : Group G\nP : ℕ → Prop\nhP : P = fun i => ∀ (a b : G), (a * b) ^ i = a ^ i * b ^ i\nhP1 : ∃ n, P n ∧ P (n + 1) ∧ P (n + 2)\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_2_6c", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_2_6c {G : Type*} [Group G] {n : ℕ} (hn : n > 1)\n  (h : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n) :\n  ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1 :=", "goal": "G : Type u_1\ninst✝ : Group G\nn : ℕ\nhn : n > 1\nh : ∀ (a b : G), (a * b) ^ n = a ^ n * b ^ n\n⊢ ∀ (a b : G), (a * b * a⁻¹ * b⁻¹) ^ (n * (n - 1)) = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_3_16", "split": "test", "informal_prefix": "/-- If a group $G$ has no proper subgroups, prove that $G$ is cyclic of order $p$, where $p$ is a prime number.-/\n", "formal_statement": "theorem exercise_2_3_16 {G : Type*} [Group G]\n  (hG : ∀ H : Subgroup G, H = ⊤ ∨ H = ⊥) :\n  IsCyclic G ∧ ∃ (p : ℕ) (Fin : Fintype G), Nat.Prime p ∧ @card G Fin = p :=", "goal": "G : Type u_1\ninst✝ : Group G\nhG : ∀ (H : Subgroup G), H = ⊤ ∨ H = ⊥\n⊢ IsCyclic G ∧ ∃ p Fin, p.Prime ∧ card G = p", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_23", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_5_23 {G : Type*} [Group G]\n  (hG : ∀ (H : Subgroup G), H.Normal) (a b : G) :\n  ∃ (j : ℤ) , b*a = a^j * b :=", "goal": "G : Type u_1\ninst✝ : Group G\nhG : ∀ (H : Subgroup G), H.Normal\na b : G\n⊢ ∃ j, b * a = a ^ j * b", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_31", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_5_31 {G : Type*} [CommGroup G] [Fintype G]\n  {p m n : ℕ} (hp : Nat.Prime p) (hp1 : ¬ p ∣ m) (hG : card G = p^n*m)\n  {H : Subgroup G} [Fintype H] (hH : card H = p^n) :\n  Subgroup.Characteristic H :=", "goal": "G : Type u_1\ninst✝² : CommGroup G\ninst✝¹ : Fintype G\np m n : ℕ\nhp : p.Prime\nhp1 : ¬p ∣ m\nhG : card G = p ^ n * m\nH : Subgroup G\ninst✝ : Fintype ↥H\nhH : card ↥H = p ^ n\n⊢ H.Characteristic", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_43", "split": "test", "informal_prefix": "/-- Prove that a group of order 9 must be abelian.-/\n", "formal_statement": "def exercise_2_5_43 (G : Type*) [Group G] [Fintype G]\n  (hG : card G = 9) :\n  CommGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 9\n⊢ CommGroup G", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_5_52", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_5_52 {G : Type*} [Group G] [Fintype G]\n  (φ : G ≃* G) {I : Finset G} (hI : ∀ x ∈ I, φ x = x⁻¹)\n  (hI1 : (0.75 : ℚ) * card G ≤ card I) :\n  ∀ x : G, φ x = x⁻¹ ∧ ∀ x y : G, x*y = y*x :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nφ : G ≃* G\nI : Finset G\nhI : ∀ x ∈ I, φ x = x⁻¹\nhI1 : 0.75 * ↑(card G) ≤ ↑(card { x // x ∈ I })\n⊢ ∀ (x : G), φ x = x⁻¹ ∧ ∀ (x y : G), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_7_7", "split": "test", "informal_prefix": "/-- If $\\varphi$ is a homomorphism of $G$ onto $G'$ and $N \\triangleleft G$, show that $\\varphi(N) \\triangleleft G'$.-/\n", "formal_statement": "theorem exercise_2_7_7 {G : Type*} [Group G] {G' : Type*} [Group G']\n  (φ : G →* G') (N : Subgroup G) [N.Normal] :\n  (Subgroup.map φ N).Normal :=", "goal": "G : Type u_1\ninst✝² : Group G\nG' : Type u_2\ninst✝¹ : Group G'\nφ : G →* G'\nN : Subgroup G\ninst✝ : N.Normal\n⊢ (Subgroup.map φ N).Normal", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_8_15", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "def exercise_2_8_15 {G H: Type*} [Fintype G] [Group G] [Fintype H]\n  [Group H] {p q : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q)\n  (h : p > q) (h1 : q ∣ p - 1) (hG : card G = p*q) (hH : card G = p*q) :\n  G ≃* H :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝³ : Fintype G\ninst✝² : Group G\ninst✝¹ : Fintype H\ninst✝ : Group H\np q : ℕ\nhp : p.Prime\nhq : q.Prime\nh : p > q\nh1 : q ∣ p - 1\nhG hH : card G = p * q\n⊢ G ≃* H", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_10_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_10_1 {G : Type*} [Group G] (A : Subgroup G)\n  [A.Normal] {b : G} (hp : Nat.Prime (orderOf b)) :\n  A ⊓ (Subgroup.closure {b}) = ⊥ :=", "goal": "G : Type u_1\ninst✝¹ : Group G\nA : Subgroup G\ninst✝ : A.Normal\nb : G\nhp : (orderOf b).Prime\n⊢ A ⊓ Subgroup.closure {b} = ⊥", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_11_7", "split": "test", "informal_prefix": "/-- If $P \\triangleleft G$, $P$ a $p$-Sylow subgroup of $G$, prove that $\\varphi(P) = P$ for every automorphism $\\varphi$ of $G$.-/\n", "formal_statement": "theorem exercise_2_11_7 {G : Type*} [Group G] {p : ℕ} (hp : Nat.Prime p)\n  {P : Sylow p G} (hP : P.Normal) :\n  Subgroup.Characteristic (P : Subgroup G) :=", "goal": "G : Type u_1\ninst✝ : Group G\np : ℕ\nhp : p.Prime\nP : Sylow p G\nhP : (↑P).Normal\n⊢ (↑P).Characteristic", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_21", "split": "test", "informal_prefix": "/-- If $\\sigma, \\tau$ are two permutations that disturb no common element and $\\sigma \\tau = e$, prove that $\\sigma = \\tau = e$.-/\n", "formal_statement": "theorem exercise_3_2_21 {α : Type*} [Fintype α] {σ τ: Equiv.Perm α}\n  (h1 : ∀ a : α, σ a = a ↔ τ a ≠ a) (h2 : τ ∘ σ = id) :\n  σ = 1 ∧ τ = 1 :=", "goal": "α : Type u_1\ninst✝ : Fintype α\nσ τ : Equiv.Perm α\nh1 : ∀ (a : α), σ a = a ↔ τ a ≠ a\nh2 : ⇑τ ∘ ⇑σ = id\n⊢ σ = 1 ∧ τ = 1", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_1_34", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "def exercise_4_1_34 : Equiv.Perm (Fin 3) ≃* Matrix.GeneralLinearGroup (Fin 2) (ZMod 2) :=", "goal": "⊢ Equiv.Perm (Fin 3) ≃* GL (Fin 2) (ZMod 2)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_6", "split": "test", "informal_prefix": "/-- If $a^2 = 0$ in $R$, show that $ax + xa$ commutes with $a$.-/\n", "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 :=", "goal": "R : Type u_1\ninst✝ : Ring R\na x : R\nh : a ^ 2 = 0\n⊢ a * (a * x + x * a) = (x + x * a) * a", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_3_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_3_1 {R : Type*} [CommRing R] (a : R) :\n  ∃ I : Ideal R, {x : R | x*a=0} = I :=", "goal": "R : Type u_1\ninst✝ : CommRing R\na : R\n⊢ ∃ I, {x | x * a = 0} = ↑I", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_9", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_4_9 (p : ℕ) (hp : Nat.Prime p) :\n  (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ∃ x : ZMod p, x^2 = p) ∧\n  (∃ S : Finset (ZMod p), S.card = (p-1)/2 ∧ ¬ ∃ x : ZMod p, x^2 = p) :=", "goal": "p : ℕ\nhp : p.Prime\n⊢ (∃ S, S.card = (p - 1) / 2 ∧ ∃ x, x ^ 2 = ↑p) ∧ ∃ S, S.card = (p - 1) / 2 ∧ ¬∃ x, x ^ 2 = ↑p", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_23", "split": "test", "informal_prefix": "/-- 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", "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 ∧ Irreducible q ∧\n  (Nonempty $ Polynomial (ZMod 7) ⧸ span ({p} : Set $ Polynomial $ ZMod 7) ≃+*\n  Polynomial (ZMod 7) ⧸ span ({q} : Set $ Polynomial $ ZMod 7)) :=", "goal": "p q : (ZMod 7)[X]\nhp : p = X ^ 3 - 2\nhq : q = X ^ 3 + 2\n⊢ Irreducible p ∧ Irreducible q ∧ Nonempty ((ZMod 7)[X] ⧸ span {p} ≃+* (ZMod 7)[X] ⧸ span {q})", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6_2", "split": "test", "informal_prefix": "/-- Prove that $f(x) = x^3 + 3x + 2$ is irreducible in $Q[x]$.-/\n", "formal_statement": "theorem exercise_4_6_2 : Irreducible (X^3 + 3*X + 2 : Polynomial ℚ) :=", "goal": "⊢ Irreducible (X ^ 3 + 3 * X + 2)", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1_8", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_1_8 {p m n: ℕ} {F : Type*} [Field F]\n  (hp : Nat.Prime p) (hF : CharP F p) (a b : F) (hm : m = p ^ n) :\n  (a + b) ^ m = a^m + b^m :=", "goal": "p m n : ℕ\nF : Type u_1\ninst✝ : Field F\nhp : p.Prime\nhF : CharP F p\na b : F\nhm : m = p ^ n\n⊢ (a + b) ^ m = a ^ m + b ^ m", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_3_7", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_3_7 {K : Type*} [Field K] {F : Subfield K}\n  {a : K} (ha : IsAlgebraic F (a ^ 2)) : IsAlgebraic F a :=", "goal": "K : Type u_1\ninst✝ : Field K\nF : Subfield K\na : K\nha : IsAlgebraic (↥F) (a ^ 2)\n⊢ IsAlgebraic (↥F) a", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_4_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_4_3 {a : ℂ} {p : ℂ → ℂ}\n  (hp : p = λ (x : ℂ) => x^5 + sqrt 2 * x^3 + sqrt 5 * x^2 + sqrt 7 * x + 11)\n  (ha : p a = 0) :\n  ∃ p : Polynomial ℂ , p.degree < 80 ∧ a ∈ p.roots ∧\n  ∀ n : p.support, ∃ a b : ℤ, p.coeff n = a / b :=", "goal": "a : ℂ\np : ℂ → ℂ\nhp : p = fun x => x ^ 5 + ↑√2 * x ^ 3 + ↑√5 * x ^ 2 + ↑√7 * x + 11\nha : p a = 0\n⊢ ∃ p, p.degree < 80 ∧ a ∈ p.roots ∧ ∀ (n : { x // x ∈ p.support }), ∃ a b, p.coeff ↑n = ↑a / ↑b", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_6_14", "split": "test", "informal_prefix": "/-- If $F$ is of characteristic $p \\neq 0$, show that all the roots of $x^m - x$, where $m = p^n$, are distinct.-/\n", "formal_statement": "theorem exercise_5_6_14 {p m n: ℕ} (hp : Nat.Prime p) {F : Type*}\n  [Field F] [CharP F p] (hm : m = p ^ n) :\n  card (rootSet (X ^ m - X : Polynomial F) F) = m :=", "goal": "p m n : ℕ\nhp : p.Prime\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharP F p\nhm : m = p ^ n\n⊢ card ↑((X ^ m - X).rootSet F) = m", "header": "import Mathlib\n\nopen Fintype Set Real Ideal Polynomial\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_26", "split": "test", "informal_prefix": "/-- Prove that a set $U \\subset M$ is open if and only if none of its points are limits of its complement.-/\n", "formal_statement": "theorem exercise_2_26 {M : Type*} [TopologicalSpace M]\n  (U : Set M) : IsOpen U ↔ ∀ x ∈ U, ¬ ClusterPt x (𝓟 Uᶜ) :=", "goal": "M : Type u_1\ninst✝ : TopologicalSpace M\nU : Set M\n⊢ IsOpen U ↔ ∀ x ∈ U, ¬ClusterPt x (𝓟 Uᶜ)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_32a", "split": "test", "informal_prefix": "/-- Show that every subset of $\\mathbb{N}$ is clopen.-/\n", "formal_statement": "theorem exercise_2_32a (A : Set ℕ) : IsClopen A :=", "goal": "A : Set ℕ\n⊢ IsClopen A", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_46", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_46 {M : Type*} [MetricSpace M]\n  {A B : Set M} (hA : IsCompact A) (hB : IsCompact B)\n  (hAB : Disjoint A B) (hA₀ : A ≠ ∅) (hB₀ : B ≠ ∅) :\n  ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a : M) (b : M),\n  a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b :=", "goal": "M : Type u_1\ninst✝ : MetricSpace M\nA B : Set M\nhA : IsCompact A\nhB : IsCompact B\nhAB : Disjoint A B\nhA₀ : A ≠ ∅\nhB₀ : B ≠ ∅\n⊢ ∃ a₀ b₀, a₀ ∈ A ∧ b₀ ∈ B ∧ ∀ (a b : M), a ∈ A → b ∈ B → dist a₀ b₀ ≤ dist a b", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_92", "split": "test", "informal_prefix": "/-- Give a direct proof that the nested decreasing intersection of nonempty covering compact sets is nonempty.-/\n", "formal_statement": "theorem exercise_2_92 {α : Type*} [TopologicalSpace α]\n  {s : ℕ → Set α}\n  (hs : ∀ i, IsCompact (s i))\n  (hs : ∀ i, (s i).Nonempty)\n  (hs : ∀ i, (s i) ⊃ (s (i + 1))) :\n  (⋂ i, s i).Nonempty :=", "goal": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : ℕ → Set α\nhs✝¹ : ∀ (i : ℕ), IsCompact (s i)\nhs✝ : ∀ (i : ℕ), (s i).Nonempty\nhs : ∀ (i : ℕ), s i ⊃ s (i + 1)\n⊢ (⋂ i, s i).Nonempty", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_1 {f : ℝ → ℝ}\n  (hf : ∀ x y, |f x - f y| ≤ |x - y| ^ 2) :\n  ∃ c, f = λ x => c :=", "goal": "f : ℝ → ℝ\nhf : ∀ (x y : ℝ), |f x - f y| ≤ |x - y| ^ 2\n⊢ ∃ c, f = fun x => c", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_3_63a", "split": "test", "informal_prefix": "/-- Prove that $\\sum 1/k(\\log(k))^p$ converges when $p > 1$.-/\n", "formal_statement": "theorem exercise_3_63a (p : ℝ) (f : ℕ → ℝ) (hp : p > 1)\n  (h : f = λ (k : ℕ) => (1 : ℝ) / (k * (log k) ^ p)) :\n  ∃ l, Tendsto f atTop (𝓝 l) :=", "goal": "p : ℝ\nf : ℕ → ℝ\nhp : p > 1\nh : f = fun k => 1 / (↑k * (↑k).log ^ p)\n⊢ ∃ l, Tendsto f atTop (𝓝 l)", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_4_15a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_15a {α : Type*}\n  (a b : ℝ) (F : Set (ℝ → ℝ)) :\n  (∀ x : ℝ, ∀ ε > 0, ∃ U ∈ (𝓝 x),\n  (∀ y z : U, ∀ f : ℝ → ℝ, f ∈ F → (dist (f y) (f z) < ε)))\n  ↔\n  ∃ (μ : ℝ → ℝ), ∀ (x : ℝ), (0 : ℝ) ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧\n  (∀ (s t : ℝ) (f : ℝ → ℝ), f ∈ F → |(f s) - (f t)| ≤ μ (|s - t|)) :=", "goal": "α : Type u_1\na b : ℝ\nF : Set (ℝ → ℝ)\n⊢ (∀ (x ε : ℝ), ε > 0 → ∃ U ∈ 𝓝 x, ∀ (y z : ↑U), ∀ f ∈ F, dist (f ↑y) (f ↑z) < ε) ↔\n    ∃ μ, ∀ (x : ℝ), 0 ≤ μ x ∧ Tendsto μ (𝓝 0) (𝓝 0) ∧ ∀ (s t : ℝ), ∀ f ∈ F, |f s - f t| ≤ μ |s - t|", "header": "import Mathlib\n\nopen Filter Real Function\nopen scoped Topology\n\n"}
{"name": "exercise_2_3_2", "split": "test", "informal_prefix": "/-- Prove that the products $a b$ and $b a$ are conjugate elements in a group.-/\n", "formal_statement": "theorem exercise_2_3_2 {G : Type*} [Group G] (a b : G) :\n    ∃ g : G, b* a = g * a * b * g⁻¹ :=", "goal": "G : Type u_1\ninst✝ : Group G\na b : G\n⊢ ∃ g, b * a = g * a * b * g⁻¹", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_8_6", "split": "test", "informal_prefix": "/-- Prove that the center of the product of two groups is the product of their centers.-/\n", "formal_statement": "noncomputable def exercise_2_8_6 {G H : Type*} [Group G] [Group H] :\n    center (G × H) ≃* (center G) × (center H) :=", "goal": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\n⊢ ↥(center (G × H)) ≃* ↥(center G) × ↥(center H)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n--center of (G × H) equivalent, preserves multiplication with (center G) × (center H)\n"}
{"name": "exercise_3_2_7", "split": "test", "informal_prefix": "/-- Prove that every homomorphism of fields is injective.-/\n", "formal_statement": "theorem exercise_3_2_7 {F : Type*} [Field F] {G : Type*} [Field G]\n    (φ : F →+* G) : Injective φ :=", "goal": "F : Type u_1\ninst✝¹ : Field F\nG : Type u_2\ninst✝ : Field G\nφ : F →+* G\n⊢ Injective ⇑φ", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\nopen RingHom\n"}
{"name": "exercise_3_7_2", "split": "test", "informal_prefix": "/-- Let $V$ be a vector space over an infinite field $F$. Prove that $V$ is not the union of finitely many proper subspaces.-/\n", "formal_statement": "theorem exercise_3_7_2 {K V : Type*} [Field K] [AddCommGroup V]\n  [Module K V] {ι : Type*} [Fintype ι] (γ : ι → Submodule K V)\n  (h : ∀ i : ι, γ i ≠ ⊤) :\n  (⋂ (i : ι), (γ i : Set V)) ≠ ⊤ :=", "goal": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\nι : Type u_3\ninst✝ : Fintype ι\nγ : ι → Submodule K V\nh : ∀ (i : ι), γ i ≠ ⊤\n⊢ ⋂ i, ↑(γ i) ≠ ⊤", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_4_2", "split": "test", "informal_prefix": "/-- Prove that no group of order $p q$, where $p$ and $q$ are prime, is simple.-/\n", "formal_statement": "theorem exercise_6_4_2 {G : Type*} [Group G] [Fintype G] {p q : ℕ}\n  (hp : Prime p) (hq : Prime q) (hG : card G = p*q) :\n  IsSimpleGroup G → false :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\np q : ℕ\nhp : Prime p\nhq : Prime q\nhG : card G = p * q\n⊢ IsSimpleGroup G → false = true", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_4_12", "split": "test", "informal_prefix": "/-- Prove that no group of order 224 is simple.-/\n", "formal_statement": "theorem exercise_6_4_12 {G : Type*} [Group G] [Fintype G]\n  (hG : card G = 224) :\n  IsSimpleGroup G → false :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 224\n⊢ IsSimpleGroup G → false = true", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_1_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_10_1_13 {R : Type*} [Ring R] {x : R}\n  (hx : IsNilpotent x) : IsUnit (1 + x) :=", "goal": "R : Type u_1\ninst✝ : Ring R\nx : R\nhx : IsNilpotent x\n⊢ IsUnit (1 + x)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_6_7", "split": "test", "informal_prefix": "/-- Prove that every nonzero ideal in the ring of Gauss integers contains a nonzero integer.-/\n", "formal_statement": "theorem exercise_10_6_7 {I : Ideal GaussianInt}\n  (hI : I ≠ ⊥) : ∃ (z : I), z ≠ 0 ∧ (z : GaussianInt).im = 0 :=", "goal": "I : Ideal GaussianInt\nhI : I ≠ ⊥\n⊢ ∃ z, z ≠ 0 ∧ (↑z).im = 0", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_10_4_7a", "split": "test", "informal_prefix": "/-- Let $I, J$ be ideals of a ring $R$ such that $I+J=R$. Prove that $I J=I \\cap J$.-/\n", "formal_statement": "theorem exercise_10_4_7a {R : Type*} [CommRing R] [NoZeroDivisors R]\n  (I J : Ideal R) (hIJ : I + J = ⊤) : I * J = I ⊓ J :=", "goal": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\nI J : Ideal R\nhIJ : I + J = ⊤\n⊢ I * J = I ⊓ J", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_2_13", "split": "test", "informal_prefix": "/-- If $a, b$ are integers and if $a$ divides $b$ in the ring of Gauss integers, then $a$ divides $b$ in $\\mathbb{Z}$.-/\n", "formal_statement": "theorem exercise_11_2_13 (a b : ℤ) :\n  (ofInt a : GaussianInt) ∣ ofInt b → a ∣ b :=", "goal": "a b : ℤ\n⊢ ofInt a ∣ ofInt b → a ∣ b", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_6a", "split": "test", "informal_prefix": "/-- Prove that $x^2+x+1$ is irreducible in the field $\\mathbb{F}_2$.-/\n", "formal_statement": "theorem exercise_11_4_6a {F : Type*} [Field F] [Fintype F] (hF : card F = 7) :\n  Irreducible (X ^ 2 + 1 : Polynomial F) :=", "goal": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : card F = 7\n⊢ Irreducible (X ^ 2 + 1)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_4_6c", "split": "test", "informal_prefix": "/-- Prove that $x^3 - 9$ is irreducible in $\\mathbb{F}_{31}$.-/\n", "formal_statement": "theorem exercise_11_4_6c : Irreducible (X^3 - 9 : Polynomial (ZMod 31)) :=", "goal": "⊢ Irreducible (X ^ 3 - 9)", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_13_3", "split": "test", "informal_prefix": "/-- Prove that there are infinitely many primes congruent to $-1$ (modulo $4$).-/\n", "formal_statement": "theorem exercise_11_13_3 (N : ℕ):\n  ∃ p ≥ N, Nat.Prime p ∧ p + 1 ≡ 0 [MOD 4] :=", "goal": "N : ℕ\n⊢ ∃ p ≥ N, p.Prime ∧ p + 1 ≡ 0 [MOD 4]", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_13_6_10", "split": "test", "informal_prefix": "/-- Let $K$ be a finite field. Prove that the product of the nonzero elements of $K$ is $-1$.-/\n", "formal_statement": "theorem exercise_13_6_10 {K : Type*} [Field K] [Fintype Kˣ] :\n  (∏ x : Kˣ,  x) = -1 :=", "goal": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype Kˣ\n⊢ ∏ x : Kˣ, x = -1", "header": "import Mathlib\n\nopen Function Fintype Subgroup Ideal Polynomial Submodule Zsqrtd\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_2", "split": "test", "informal_prefix": "/-- Show that $\\frac{-1 + \\sqrt{3}i}{2}$ is a cube root of 1 (meaning that its cube equals 1).-/\n", "formal_statement": "theorem exercise_1_2 :\n  (⟨-1/2, Real.sqrt 3 / 2⟩ : ℂ) ^ 3 = -1 :=", "goal": "⊢ { re := -1 / 2, im := √3 / 2 } ^ 3 = -1", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_4", "split": "test", "informal_prefix": "/-- Prove that if $a \\in \\mathbf{F}$, $v \\in V$, and $av = 0$, then $a = 0$ or $v = 0$.-/\n", "formal_statement": "theorem exercise_1_4 {F V : Type*} [AddCommGroup V] [Field F]\n  [Module F V] (v : V) (a : F): a • v = 0 ↔ a = 0 ∨ v = 0 :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nv : V\na : F\n⊢ a • v = 0 ↔ a = 0 ∨ v = 0", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_7", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_7 : ∃ U : Set (ℝ × ℝ),\n  (U ≠ ∅) ∧\n  (∀ (c : ℝ) (u : ℝ × ℝ), u ∈ U → c • u ∈ U) ∧\n  (∀ U' : Submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=", "goal": "⊢ ∃ U, U ≠ ∅ ∧ (∀ (c : ℝ), ∀ u ∈ U, c • u ∈ U) ∧ ∀ (U' : Submodule ℝ (ℝ × ℝ)), U ≠ ↑U'", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_9", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_9 {F V : Type*} [AddCommGroup V] [Field F]\n  [Module F V] (U W : Submodule F V):\n  ∃ U' : Submodule F V, (U'.carrier = ↑U ∩ ↑W ↔ (U ≤ W ∨ W ≤ U)) :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nU W : Submodule F V\n⊢ ∃ U', U'.carrier = ↑U ∩ ↑W ↔ U ≤ W ∨ W ≤ U", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_8", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_8 {F V W : Type*}  [AddCommGroup V]\n  [AddCommGroup W] [Field F] [Module F V] [Module F W]\n  (L : V →ₗ[F] W) :\n  ∃ U : Submodule F V, U ⊓ (ker L) = ⊥ ∧\n  (range L = range (domRestrict L U)) :=", "goal": "F : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddCommGroup W\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : Module F W\nL : V →ₗ[F] W\n⊢ ∃ U, U ⊓ ker L = ⊥ ∧ range L = range (L.domRestrict U)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_1 {F V : Type*} [AddCommGroup V] [Field F]\n  [Module F V] {L : V →ₗ[F] V} {n : ℕ} (U : Fin n → Submodule F V)\n  (hU : ∀ i : Fin n, Submodule.map L (U i) = U i) :\n  Submodule.map L (∑ i : Fin n, U i : Submodule F V) =\n  (∑ i : Fin n, U i : Submodule F V) :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nL : V →ₗ[F] V\nn : ℕ\nU : Fin n → Submodule F V\nhU : ∀ (i : Fin n), Submodule.map L (U i) = U i\n⊢ Submodule.map L (∑ i : Fin n, U i) = ∑ i : Fin n, U i", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_11", "split": "test", "informal_prefix": "/-- Suppose $S, T \\in \\mathcal{L}(V)$. Prove that $S T$ and $T S$ have the same eigenvalues.-/\n", "formal_statement": "theorem exercise_5_11 {F V : Type*} [AddCommGroup V] [Field F]\n  [Module F V] (S T : End F V) :\n  (S * T).Eigenvalues = (T * S).Eigenvalues :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field F\ninst✝ : Module F V\nS T : End F V\n⊢ (S * T).Eigenvalues = (T * S).Eigenvalues", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_13 {F V : Type*} [AddCommGroup V] [Field F]\n  [Module F V] [FiniteDimensional F V] {T : End F V}\n  (hS : ∀ U : Submodule F V, finrank F U = finrank F V - 1 →\n  Submodule.map T U = U) : ∃ c : F, T = c • LinearMap.id :=", "goal": "F : Type u_1\nV : Type u_2\ninst✝³ : AddCommGroup V\ninst✝² : Field F\ninst✝¹ : Module F V\ninst✝ : FiniteDimensional F V\nT : End F V\nhS : ∀ (U : Submodule F V), finrank F ↥U = finrank F V - 1 → Submodule.map T U = U\n⊢ ∃ c, T = c • LinearMap.id", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_24", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_24 {V : Type*} [AddCommGroup V]\n  [Module ℝ V] [FiniteDimensional ℝ V] {T : End ℝ V}\n  (hT : ∀ c : ℝ, eigenspace T c = ⊥) {U : Submodule ℝ V}\n  (hU : Submodule.map T U = U) : Even (finrank U) :=", "goal": "V : Type u_1\ninst✝² : AddCommGroup V\ninst✝¹ : Module ℝ V\ninst✝ : FiniteDimensional ℝ V\nT : End ℝ V\nhT : ∀ (c : ℝ), T.eigenspace c = ⊥\nU : Submodule ℝ V\nhU : Submodule.map T U = U\n⊢ Even (finrank ↥U)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_6_3 {n : ℕ} (a b : Fin n → ℝ) :\n  (∑ i, a i * b i) ^ 2 ≤ (∑ i : Fin n, i * a i ^ 2) * (∑ i, b i ^ 2 / i) :=", "goal": "n : ℕ\na b : Fin n → ℝ\n⊢ (∑ i : Fin n, a i * b i) ^ 2 ≤ (∑ i : Fin n, ↑↑i * a i ^ 2) * ∑ i : Fin n, b i ^ 2 / ↑↑i", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_6_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_6_13 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] {n : ℕ}\n  {e : Fin n → V} (he : Orthonormal ℂ e) (v : V) :\n  ‖v‖^2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖^2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ) :=", "goal": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nn : ℕ\ne : Fin n → V\nhe : Orthonormal ℂ e\nv : V\n⊢ ‖v‖ ^ 2 = ∑ i : Fin n, ‖⟪v, e i⟫_ℂ‖ ^ 2 ↔ v ∈ Submodule.span ℂ (e '' Set.univ)", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_5", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_7_5 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n  [FiniteDimensional ℂ V] (hV : finrank V ≥ 2) :\n  ∀ U : Submodule ℂ (End ℂ V), U.carrier ≠\n  {T | T * adjoint T = adjoint T * T} :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nhV : finrank V ≥ 2\n⊢ ∀ (U : Submodule ℂ (End ℂ V)), U.carrier ≠ {T | T * adjoint T = adjoint T * T}", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_9", "split": "test", "informal_prefix": "/-- Prove that a normal operator on a complex inner-product space is self-adjoint if and only if all its eigenvalues are real.-/\n", "formal_statement": "theorem exercise_7_9 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n  [FiniteDimensional ℂ V] (T : End ℂ V)\n  (hT : T * adjoint T = adjoint T * T) :\n  IsSelfAdjoint T ↔ ∀ e : T.Eigenvalues, (e : ℂ).im = 0 :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\n⊢ IsSelfAdjoint T ↔ ∀ (e : T.Eigenvalues), (↑T e).im = 0", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_11", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_7_11 {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]\n  [FiniteDimensional ℂ V] {T : End ℂ V} (hT : T*adjoint T = adjoint T*T) :\n  ∃ (S : End ℂ V), S ^ 2 = T :=", "goal": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℂ V\ninst✝ : FiniteDimensional ℂ V\nT : End ℂ V\nhT : T * adjoint T = adjoint T * T\n⊢ ∃ S, S ^ 2 = T", "header": "import Mathlib\n\nopen Fintype Complex Polynomial LinearMap FiniteDimensional Module Module.End\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_2a", "split": "test", "informal_prefix": "/-- Prove the the operation $\\star$ on $\\mathbb{Z}$ defined by $a\\star b=a-b$ is not commutative.-/\n", "formal_statement": "theorem exercise_1_1_2a : ∃ a b : ℤ, a - b ≠ b - a :=", "goal": "⊢ ∃ a b, a - b ≠ b - a", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_4", "split": "test", "informal_prefix": "/-- Prove that the multiplication of residue class $\\mathbb{Z}/n\\mathbb{Z}$ is associative.-/\n", "formal_statement": "theorem exercise_1_1_4 (n : ℕ) :\n  ∀ (a b c : ℕ), (a * b) * c ≡ a * (b * c) [ZMOD n] :=", "goal": "n : ℕ\n⊢ ∀ (a b c : ℕ), ↑a * ↑b * ↑c ≡ ↑a * (↑b * ↑c) [ZMOD ↑n]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_15", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_1_15 {G : Type*} [Group G] (as : List G) :\n  as.prod⁻¹ = (as.reverse.map (λ x => x⁻¹)).prod :=", "goal": "G : Type u_1\ninst✝ : Group G\nas : List G\n⊢ as.prod⁻¹ = (List.map (fun x => x⁻¹) as.reverse).prod", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_17", "split": "test", "informal_prefix": "/-- Let $x$ be an element of $G$. Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.-/\n", "formal_statement": "theorem exercise_1_1_17 {G : Type*} [Group G] {x : G} {n : ℕ}\n  (hxn: orderOf x = n) :\n  x⁻¹ = x ^ (n - 1 : ℤ) :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\nn : ℕ\nhxn : orderOf x = n\n⊢ x⁻¹ = x ^ (↑n - 1)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_20", "split": "test", "informal_prefix": "/-- For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order.-/\n", "formal_statement": "theorem exercise_1_1_20 {G : Type*} [Group G] {x : G} :\n  orderOf x = orderOf x⁻¹ :=", "goal": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ orderOf x = orderOf x⁻¹", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_22b", "split": "test", "informal_prefix": "/-- Deduce that $|a b|=|b a|$ for all $a, b \\in G$.-/\n", "formal_statement": "theorem exercise_1_1_22b {G: Type*} [Group G] (a b : G) :\n  orderOf (a * b) = orderOf (b * a) :=", "goal": "G : Type u_1\ninst✝ : Group G\na b : G\n⊢ orderOf (a * b) = orderOf (b * a)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_1_29", "split": "test", "informal_prefix": "/-- Prove that $A \\times B$ is an abelian group if and only if both $A$ and $B$ are abelian.-/\n", "formal_statement": "theorem exercise_1_1_29 {A B : Type*} [Group A] [Group B] :\n  ∀ x y : A × B, x*y = y*x ↔ (∀ x y : A, x*y = y*x) ∧\n  (∀ x y : B, x*y = y*x) :=", "goal": "A : Type u_1\nB : Type u_2\ninst✝¹ : Group A\ninst✝ : Group B\n⊢ ∀ (x y : A × B), x * y = y * x ↔ (∀ (x y : A), x * y = y * x) ∧ ∀ (x y : B), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_3_8", "split": "test", "informal_prefix": "/-- Prove that if $\\Omega=\\{1,2,3, \\ldots\\}$ then $S_{\\Omega}$ is an infinite group-/\n", "formal_statement": "theorem exercise_1_3_8 : Infinite (Equiv.Perm ℕ) :=", "goal": "⊢ Infinite (Equiv.Perm ℕ)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_11", "split": "test", "informal_prefix": "/-- Let $A$ and $B$ be groups. Prove that $A \\times B \\cong B \\times A$.-/\n", "formal_statement": "noncomputable def exercise_1_6_11 {A B : Type*} [Group A] [Group B] :\n  A × B ≃* B × A :=", "goal": "A : Type u_1\nB : Type u_2\ninst✝¹ : Group A\ninst✝ : Group B\n⊢ A × B ≃* B × A", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_1_6_23", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_1_6_23 {G : Type*}\n  [Group G] (σ : MulAut G) (hs : ∀ g : G, σ g = 1 → g = 1)\n  (hs2 : ∀ g : G, σ (σ g) = g) :\n  ∀ x y : G, x*y = y*x :=", "goal": "G : Type u_1\ninst✝ : Group G\nσ : MulAut G\nhs : ∀ (g : G), σ g = 1 → g = 1\nhs2 : ∀ (g : G), σ (σ g) = g\n⊢ ∀ (x y : G), x * y = y * x", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_1_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_1_13 (H : AddSubgroup ℚ) {x : ℚ}\n  (hH : x ∈ H → (1 / x) ∈ H):\n  H = ⊥ ∨ H = ⊤ :=", "goal": "H : AddSubgroup ℚ\nx : ℚ\nhH : x ∈ H → 1 / x ∈ H\n⊢ H = ⊥ ∨ H = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_16a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_4_16a {G : Type*} [Group G] {H : Subgroup G}\n  (hH : H ≠ ⊤) :\n  ∃ M : Subgroup G, M ≠ ⊤ ∧\n  ∀ K : Subgroup G, M ≤ K → K = M ∨ K = ⊤ ∧\n  H ≤ M :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : H ≠ ⊤\n⊢ ∃ M, M ≠ ⊤ ∧ ∀ (K : Subgroup G), M ≤ K → K = M ∨ K = ⊤ ∧ H ≤ M", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4_16c", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_4_16c {n : ℕ} (H : AddSubgroup (ZMod n)) :\n  ∃ p : (ZMod n), Prime p ∧ H = AddSubgroup.closure {p} ↔\n  (H ≠ ⊤ ∧ ∀ K : AddSubgroup (ZMod n), H ≤ K → K = H ∨ K = ⊤) :=", "goal": "n : ℕ\nH : AddSubgroup (ZMod n)\n⊢ ∃ p, Prime p ∧ H = AddSubgroup.closure {p} ↔ H ≠ ⊤ ∧ ∀ (K : AddSubgroup (ZMod n)), H ≤ K → K = H ∨ K = ⊤", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_1_22a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_1_22a (G : Type*) [Group G] (H K : Subgroup G)\n  [Normal H] [Normal K] :\n  Normal (H ⊓ K) :=", "goal": "G : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : H.Normal\ninst✝ : K.Normal\n⊢ (H ⊓ K).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_8", "split": "test", "informal_prefix": "/-- Prove that if $H$ and $K$ are finite subgroups of $G$ whose orders are relatively prime then $H \\cap K=1$.-/\n", "formal_statement": "theorem exercise_3_2_8 {G : Type*} [Group G] (H K : Subgroup G)\n  [Fintype H] [Fintype K]\n  (hHK : Nat.Coprime (card H) (card K)) :\n  H ⊓ K = ⊥ :=", "goal": "G : Type u_1\ninst✝² : Group G\nH K : Subgroup G\ninst✝¹ : Fintype ↥H\ninst✝ : Fintype ↥K\nhHK : (card ↥H).Coprime (card ↥K)\n⊢ H ⊓ K = ⊥", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_2_16", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_2_16 (p : ℕ) (hp : Nat.Prime p) (a : ℕ) :\n  Nat.Coprime a p → a ^ p ≡ a [ZMOD p] :=", "goal": "p : ℕ\nhp : p.Prime\na : ℕ\n⊢ a.Coprime p → ↑a ^ p ≡ ↑a [ZMOD ↑p]", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_3_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_3_3_3 {p : Nat.Primes} {G : Type*} [Group G]\n  {H : Subgroup G} [hH : H.Normal] (hH1 : H.index = p) :\n  ∀ K : Subgroup G, K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = p :=", "goal": "p : Nat.Primes\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhH : H.Normal\nhH1 : H.index = ↑p\n⊢ ∀ (K : Subgroup G), K ≤ H ∨ H ⊔ K = ⊤ ∨ (K ⊓ H).relindex K = ↑p", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_4", "split": "test", "informal_prefix": "/-- Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each positive divisor $n$ of its order.-/\n", "formal_statement": "theorem exercise_3_4_4 {G : Type*} [CommGroup G] [Fintype G] {n : ℕ}\n    (hn : n ∣ (card G)) :\n    ∃ (H : Subgroup G) (H_fin : Fintype H), @card H H_fin = n :=", "goal": "G : Type u_1\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nn : ℕ\nhn : n ∣ card G\n⊢ ∃ H H_fin, card ↥H = n", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4_5b", "split": "test", "informal_prefix": "/-- Prove that quotient groups of a solvable group are solvable.-/\n", "formal_statement": "theorem exercise_3_4_5b {G : Type*} [Group G] [IsSolvable G]\n  (H : Subgroup G) [Normal H] :\n  IsSolvable (G ⧸ H) :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsSolvable G\nH : Subgroup G\ninst✝ : H.Normal\n⊢ IsSolvable (G ⧸ H)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_8", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_2_8 {G : Type*} [Group G] {H : Subgroup G}\n  {n : ℕ} (hn : n > 0) (hH : H.index = n) :\n  ∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial :=", "goal": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn : ℕ\nhn : n > 0\nhH : H.index = n\n⊢ ∃ K ≤ H, K.Normal ∧ K.index ≤ n.factorial", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_2_9a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_2_9a {G : Type*} [Fintype G] [Group G] {p α : ℕ}\n  (hp : p.Prime) (ha : α > 0) (hG : card G = p ^ α) :\n  ∀ H : Subgroup G, H.index = p → H.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\np α : ℕ\nhp : p.Prime\nha : α > 0\nhG : card G = p ^ α\n⊢ ∀ (H : Subgroup G), H.index = p → H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_2", "split": "test", "informal_prefix": "/-- Prove that if $G$ is an abelian group of order $p q$, where $p$ and $q$ are distinct primes, then $G$ is cyclic.-/\n", "formal_statement": "theorem exercise_4_4_2 {G : Type*} [Fintype G] [Group G]\n  {p q : Nat.Primes} (hpq : p ≠ q) (hG : card G = p*q) :\n  IsCyclic G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\np q : Nat.Primes\nhpq : p ≠ q\nhG : card G = ↑p * ↑q\n⊢ IsCyclic G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_6b", "split": "test", "informal_prefix": "/-- Prove that there exists a normal subgroup that is not characteristic.-/\n", "formal_statement": "theorem exercise_4_4_6b :\n  ∃ (G : Type*) (hG : Group G) (H : @Subgroup G hG), @Characteristic G hG H  ∧ ¬ @Normal G hG H :=", "goal": "⊢ ∃ G hG H, H.Characteristic ∧ ¬H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4_8a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_4_8a {G : Type*} [Group G] (H K : Subgroup G)\n  (hHK : H ≤ K) [hHK1 : (H.subgroupOf K).Normal] [hK : K.Normal] :\n  H.Normal :=", "goal": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhHK : H ≤ K\nhHK1 : (H.subgroupOf K).Normal\nhK : K.Normal\n⊢ H.Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_13", "split": "test", "informal_prefix": "/-- Prove that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/\n", "formal_statement": "theorem exercise_4_5_13 {G : Type*} [Group G] [Fintype G]\n  (hG : card G = 56) :\n  ∃ (p : ℕ) (P : Sylow p G), P.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 56\n⊢ ∃ p P, (↑P).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_15", "split": "test", "informal_prefix": "/-- Prove that a group of order 351 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.-/\n", "formal_statement": "theorem exercise_4_5_15 {G : Type*} [Group G] [Fintype G]\n  (hG : card G = 351) :\n  ∃ (p : ℕ) (P : Sylow p G), P.Normal :=", "goal": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Fintype G\nhG : card G = 351\n⊢ ∃ p P, (↑P).Normal", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_17", "split": "test", "informal_prefix": "/-- Prove that if $|G|=105$ then $G$ has a normal Sylow 5 -subgroup and a normal Sylow 7-subgroup.-/\n", "formal_statement": "theorem exercise_4_5_17 {G : Type*} [Fintype G] [Group G]\n  (hG : card G = 105) :\n  Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G) :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 105\n⊢ Nonempty (Sylow 5 G) ∧ Nonempty (Sylow 7 G)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_19", "split": "test", "informal_prefix": "/-- Prove that if $|G|=6545$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_19 {G : Type*} [Fintype G] [Group G]\n  (hG : card G = 6545) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 6545\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_21", "split": "test", "informal_prefix": "/-- Prove that if $|G|=2907$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_21 {G : Type*} [Fintype G] [Group G]\n  (hG : card G = 2907) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 2907\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_23", "split": "test", "informal_prefix": "/-- Prove that if $|G|=462$ then $G$ is not simple.-/\n", "formal_statement": "theorem exercise_4_5_23 {G : Type*} [Fintype G] [Group G]\n  (hG : card G = 462) : ¬ IsSimpleGroup G :=", "goal": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : Group G\nhG : card G = 462\n⊢ ¬IsSimpleGroup G", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_5_33", "split": "test", "informal_prefix": "/-- Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P \\cap H$ is the unique Sylow $p$-subgroup of $H$.-/\n", "formal_statement": "theorem exercise_4_5_33 {G : Type*} [Group G] [Fintype G] {p : ℕ}\n  (P : Sylow p G) [hP : P.Normal] (H : Subgroup G) [Fintype H] :\n  ∀ R : Sylow p H, R.toSubgroup = (H ⊓ P.toSubgroup).subgroupOf H ∧\n  Nonempty (Sylow p H) :=", "goal": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Fintype G\np : ℕ\nP : Sylow p G\nhP : (↑P).Normal\nH : Subgroup G\ninst✝ : Fintype ↥H\n⊢ ∀ (R : Sylow p ↥H), ↑R = (H ⊓ ↑P).subgroupOf H ∧ Nonempty (Sylow p ↥H)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_2", "split": "test", "informal_prefix": "/-- Prove that if $u$ is a unit in $R$ then so is $-u$.-/\n", "formal_statement": "theorem exercise_7_1_2 {R : Type*} [Ring R] {u : R}\n  (hu : IsUnit u) : IsUnit (-u) :=", "goal": "R : Type u_1\ninst✝ : Ring R\nu : R\nhu : IsUnit u\n⊢ IsUnit (-u)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_1_12", "split": "test", "informal_prefix": "/-- Prove that any subring of a field which contains the identity is an integral domain.-/\n", "formal_statement": "theorem exercise_7_1_12 {F : Type*} [Field F] {K : Subring F}\n  (hK : (1 : F) ∈ K) : IsDomain K :=", "goal": "F : Type u_1\ninst✝ : Field F\nK : Subring F\nhK : 1 ∈ K\n⊢ IsDomain ↥K", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_2_2", "split": "test", "informal_prefix": "/-- Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\\cdots+a_{1} x+a_{0}$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in $R[x]$ if and only if there is a nonzero $b \\in R$ such that $b p(x)=0$.-/\n", "formal_statement": "theorem exercise_7_2_2 {R : Type*} [Ring R] (p : Polynomial R) :\n  p ∣ 0 ↔ ∃ b : R, b ≠ 0 ∧ b • p = 0 :=", "goal": "R : Type u_1\ninst✝ : Ring R\np : R[X]\n⊢ p ∣ 0 ↔ ∃ b, b ≠ 0 ∧ b • p = 0", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_3_16", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_7_3_16 {R S : Type*} [Ring R] [Ring S]\n  {φ : R →+* S} (hf : Function.Surjective φ) :\n  φ '' (center R) ⊂ center S :=", "goal": "R : Type u_1\nS : Type u_2\ninst✝¹ : Ring R\ninst✝ : Ring S\nφ : R →+* S\nhf : Function.Surjective ⇑φ\n⊢ ⇑φ '' Set.center R ⊂ Set.center S", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_7_4_27", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_7_4_27 {R : Type*} [CommRing R] (hR : (0 : R) ≠ 1)\n  {a : R} (ha : IsNilpotent a) (b : R) :\n  IsUnit (1-a*b) :=", "goal": "R : Type u_1\ninst✝ : CommRing R\nhR : 0 ≠ 1\na : R\nha : IsNilpotent a\nb : R\n⊢ IsUnit (1 - a * b)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_2_4", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_8_2_4 {R : Type*} [Ring R][NoZeroDivisors R]\n  [CancelCommMonoidWithZero R] [GCDMonoid R]\n  (h1 : ∀ a b : R, a ≠ 0 → b ≠ 0 → ∃ r s : R, gcd a b = r*a + s*b)\n  (h2 : ∀ a : ℕ → R, (∀ i j : ℕ, i < j → a i ∣ a j) →\n  ∃ N : ℕ, ∀ n ≥ N, ∃ u : R, IsUnit u ∧ a n = u * a N) :\n  IsPrincipalIdealRing R :=", "goal": "R : Type u_1\ninst✝³ : Ring R\ninst✝² : NoZeroDivisors R\ninst✝¹ : CancelCommMonoidWithZero R\ninst✝ : GCDMonoid R\nh1 : ∀ (a b : R), a ≠ 0 → b ≠ 0 → ∃ r s, gcd a b = r * a + s * b\nh2 : ∀ (a : ℕ → R), (∀ (i j : ℕ), i < j → a i ∣ a j) → ∃ N, ∀ n ≥ N, ∃ u, IsUnit u ∧ a n = u * a N\n⊢ IsPrincipalIdealRing R", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_5a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_8_3_5a {n : ℤ} (hn0 : n > 3) (hn1 : Squarefree n) :\n  Irreducible (2 : Zsqrtd $ -n) ∧\n  Irreducible (⟨0, 1⟩ : Zsqrtd $ -n) ∧\n  Irreducible (1 + ⟨0, 1⟩ : Zsqrtd $ -n) :=", "goal": "n : ℤ\nhn0 : n > 3\nhn1 : Squarefree n\n⊢ Irreducible 2 ∧ Irreducible { re := 0, im := 1 } ∧ Irreducible (1 + { re := 0, im := 1 })", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_8_3_6b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_8_3_6b {q : ℕ} (hq0 : q.Prime)\n  (hq1 : q ≡ 3 [ZMOD 4]) {R : Type} [Ring R]\n  (hR : R = (GaussianInt ⧸ span ({↑q} : Set GaussianInt))) :\n  IsField R ∧ ∃ finR : Fintype R, @card R finR = q^2 :=", "goal": "q : ℕ\nhq0 : q.Prime\nhq1 : ↑q ≡ 3 [ZMOD 4]\nR : Type\ninst✝ : Ring R\nhR : R = (GaussianInt ⧸ span {↑q})\n⊢ IsField R ∧ ∃ finR, card R = q ^ 2", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_1_10", "split": "test", "informal_prefix": "/-- 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.-/\n", "formal_statement": "theorem exercise_9_1_10 {f : ℕ → MvPolynomial ℕ ℤ}\n  (hf : f = λ i => MvPolynomial.X i * MvPolynomial.X (i+1)):\n  Infinite (minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f))) :=", "goal": "f : ℕ → MvPolynomial ℕ ℤ\nhf : f = fun i => MvPolynomial.X i * MvPolynomial.X (i + 1)\n⊢ Infinite ↑(minimalPrimes (MvPolynomial ℕ ℤ ⧸ span (range f)))", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2a", "split": "test", "informal_prefix": "/-- Prove that $x^4-4x^3+6$ is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2a : Irreducible (X^4 - 4*X^3 + 6 : Polynomial ℤ) :=", "goal": "⊢ Irreducible (X ^ 4 - 4 * X ^ 3 + 6)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_2c", "split": "test", "informal_prefix": "/-- Prove that $x^4+4x^3+6x^2+2x+1$ is irreducible in $\\mathbb{Z}[x]$.-/\n", "formal_statement": "theorem exercise_9_4_2c : Irreducible\n  (X^4 + 4*X^3 + 6*X^2 + 2*X + 1 : Polynomial ℤ) :=", "goal": "⊢ Irreducible (X ^ 4 + 4 * X ^ 3 + 6 * X ^ 2 + 2 * X + 1)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_9_4_9", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_9_4_9 :\n  Irreducible (X^2 - C Zsqrtd.sqrtd : Polynomial (Zsqrtd 2)) :=", "goal": "⊢ Irreducible (X ^ 2 - C Zsqrtd.sqrtd)", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_11_1_13", "split": "test", "informal_prefix": "/-- Prove that as vector spaces over $\\mathbb{Q}, \\mathbb{R}^n \\cong \\mathbb{R}$, for all $n \\in \\mathbb{Z}^{+}$.-/\n", "formal_statement": "def exercise_11_1_13 {ι : Type*} [Fintype ι] :\n  (ι → ℝ) ≃ₗ[ℚ] ℝ :=", "goal": "ι : Type u_1\ninst✝ : Fintype ι\n⊢ (ι → ℝ) ≃ₗ[ℚ] ℝ", "header": "import Mathlib\n\nopen Fintype Subgroup Set Polynomial Ideal\nopen scoped BigOperators\n\n"}
{"name": "exercise_13_3b", "split": "test", "informal_prefix": "/-- Show that the collection $$\\mathcal{T}_\\infty = \\{U | X - U \\text{ is infinite or empty or all of X}\\}$$ does not need to be a topology on the set $X$.-/\n", "formal_statement": "theorem exercise_13_3b : ¬ ∀ X : Type, ∀s : Set (Set X),\n  (∀ t : Set X, t ∈ s → (Set.Infinite tᶜ ∨ t = ∅ ∨ t = ⊤)) →\n  (Set.Infinite (⋃₀ s)ᶜ ∨ (⋃₀ s) = ∅ ∨ (⋃₀ s) = ⊤) :=", "goal": "⊢ ¬∀ (X : Type) (s : Set (Set X)), (∀ t ∈ s, tᶜ.Infinite ∨ t = ∅ ∨ t = ⊤) → (⋃₀ s)ᶜ.Infinite ∨ ⋃₀ s = ∅ ∨ ⋃₀ s = ⊤", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_13_4a2", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_13_4a2 :\n  ∃ (X I : Type*) (T : I → Set (Set X)),\n  (∀ i, is_topology X (T i)) ∧ ¬  is_topology X (⋂ i : I, T i) :=", "goal": "⊢ ∃ X I T, (∀ (i : I), is_topology X (T i)) ∧ ¬is_topology X (⋂ i, T i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n  univ ∈ T ∧\n  (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n  (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_4b2", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_13_4b2 (X I : Type*) (T : I → Set (Set X)) (h : ∀ i, is_topology X (T i)) :\n  ∃! T', is_topology X T' ∧ (∀ i, T' ⊆ T i) ∧\n  ∀ T'', is_topology X T'' → (∀ i, T'' ⊆ T i) → T' ⊆ T'' :=", "goal": "X : Type u_1\nI : Type u_2\nT : I → Set (Set X)\nh : ∀ (i : I), is_topology X (T i)\n⊢ ∃! T',\n    is_topology X T' ∧\n      (∀ (i : I), T' ⊆ T i) ∧ ∀ (T'' : Set (Set X)), is_topology X T'' → (∀ (i : I), T'' ⊆ T i) → T' ⊆ T''", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n  univ ∈ T ∧\n  (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n  (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_5b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_13_5b {X : Type*}\n  [t : TopologicalSpace X] (A : Set (Set X)) (hA : t = generateFrom A) :\n  generateFrom A = generateFrom (sInter {T | is_topology X T ∧ A ⊆ T}) :=", "goal": "X : Type u_1\nt : TopologicalSpace X\nA : Set (Set X)\nhA : t = generateFrom A\n⊢ generateFrom A = generateFrom (⋂₀ {T | is_topology X T ∧ A ⊆ T})", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef is_topology (X : Type*) (T : Set (Set X)) :=\n  univ ∈ T ∧\n  (∀ s t, s ∈ T → t ∈ T → s ∩ t ∈ T) ∧\n  (∀s, (∀t ∈ s, t ∈ T) → sUnion s ∈ T)\n\n"}
{"name": "exercise_13_8a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_13_8a :\n  IsTopologicalBasis {S : Set ℝ | ∃ a b : ℚ, a < b ∧ S = Ioo ↑a ↑b} :=", "goal": "⊢ IsTopologicalBasis {S | ∃ a b, a < b ∧ S = Ioo ↑a ↑b}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_16_1", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_16_1 {X : Type*} [TopologicalSpace X]\n  (Y : Set X)\n  (A : Set Y) :\n  ∀ U : Set A, IsOpen U ↔ IsOpen (Subtype.val '' U) :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nY : Set X\nA : Set ↑Y\n⊢ ∀ (U : Set ↑A), IsOpen U ↔ IsOpen (Subtype.val '' U)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_16_6", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_16_6\n  (S : Set (Set (ℝ × ℝ)))\n  (hS : ∀ s, s ∈ S → ∃ a b c d, (rational a ∧ rational b ∧ rational c ∧ rational d\n  ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d})) :\n  IsTopologicalBasis S :=", "goal": "S : Set (Set (ℝ × ℝ))\nhS :\n  ∀ s ∈ S,\n    ∃ a b c d,\n      rational a ∧\n        rational b ∧ rational c ∧ rational d ∧ s = {x | ∃ x₁ x₂, x = (x₁, x₂) ∧ a < x₁ ∧ x₁ < b ∧ c < x₂ ∧ x₂ < d}\n⊢ IsTopologicalBasis S", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef rational (x : ℝ) := x ∈ range ((↑) : ℚ → ℝ)\n\n"}
{"name": "exercise_18_8a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_18_8a {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  [LinearOrder Y] [OrderTopology Y] {f g : X → Y}\n  (hf : Continuous f) (hg : Continuous g) :\n  IsClosed {x | f x ≤ g x} :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : LinearOrder Y\ninst✝ : OrderTopology Y\nf g : X → Y\nhf : Continuous f\nhg : Continuous g\n⊢ IsClosed {x | f x ≤ g x}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_18_13", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_18_13\n  {X : Type*} [TopologicalSpace X] {Y : Type*} [TopologicalSpace Y]\n  [T2Space Y] {A : Set X} {f : A → Y} (hf : Continuous f)\n  (g : closure A → Y)\n  (g_con : Continuous g) :\n  ∀ (g' : closure A → Y), Continuous g' →  (∀ (x : closure A), g x = g' x) :=", "goal": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nA : Set X\nf : ↑A → Y\nhf : Continuous f\ng : ↑(closure A) → Y\ng_con : Continuous g\n⊢ ∀ (g' : ↑(closure A) → Y), Continuous g' → ∀ (x : ↑(closure A)), g x = g' x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_20_2", "split": "test", "informal_prefix": "/-- Show that $\\mathbb{R} \\times \\mathbb{R}$ in the dictionary order topology is metrizable.-/\n", "formal_statement": "theorem exercise_20_2\n  [TopologicalSpace (ℝ ×ₗ ℝ)] [OrderTopology (ℝ ×ₗ ℝ)]\n  : MetrizableSpace (ℝ ×ₗ ℝ) :=", "goal": "inst✝¹ : TopologicalSpace (Lex (ℝ × ℝ))\ninst✝ : OrderTopology (Lex (ℝ × ℝ))\n⊢ MetrizableSpace (Lex (ℝ × ℝ))", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_21_6b", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_21_6b\n  (f : ℕ → I → ℝ )\n  (h : ∀ x n, f n x = x ^ n) :\n  ¬ ∃ f₀, TendstoUniformly f f₀ atTop :=", "goal": "f : ℕ → ↑I → ℝ\nh : ∀ (x : ↑I) (n : ℕ), f n x = ↑x ^ n\n⊢ ¬∃ f₀, TendstoUniformly f f₀ atTop", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_22_2a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_22_2a {X Y : Type*} [TopologicalSpace X]\n  [TopologicalSpace Y] (p : X → Y) (h : Continuous p) :\n  QuotientMap p ↔ ∃ (f : Y → X), Continuous f ∧ p ∘ f = id :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nh : Continuous p\n⊢ QuotientMap p ↔ ∃ f, Continuous f ∧ p ∘ f = id", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_22_5", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_22_5 {X Y : Type*} [TopologicalSpace X]\n  [TopologicalSpace Y] (p : X → Y) (hp : IsOpenMap p)\n  (A : Set X) (hA : IsOpen A) : IsOpenMap (p ∘ Subtype.val : A → Y) :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nhp : IsOpenMap p\nA : Set X\nhA : IsOpen A\n⊢ IsOpenMap (p ∘ Subtype.val)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_3", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_23_3 {X : Type*} [TopologicalSpace X]\n  [TopologicalSpace X] {A : ℕ → Set X}\n  (hAn : ∀ n, IsConnected (A n))\n  (A₀ : Set X)\n  (hA : IsConnected A₀)\n  (h : ∀ n, A₀ ∩ A n ≠ ∅) :\n  IsConnected (A₀ ∪ (⋃ n, A n)) :=", "goal": "X : Type u_1\ninst✝¹ inst✝ : TopologicalSpace X\nA : ℕ → Set X\nhAn : ∀ (n : ℕ), IsConnected (A n)\nA₀ : Set X\nhA : IsConnected A₀\nh : ∀ (n : ℕ), A₀ ∩ A n ≠ ∅\n⊢ IsConnected (A₀ ∪ ⋃ n, A n)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_6", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_23_6 {X : Type*}\n  [TopologicalSpace X] {A C : Set X} (hc : IsConnected C)\n  (hCA : C ∩ A ≠ ∅) (hCXA : C ∩ Aᶜ ≠ ∅) :\n  C ∩ (frontier A) ≠ ∅ :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nA C : Set X\nhc : IsConnected C\nhCA : C ∩ A ≠ ∅\nhCXA : C ∩ Aᶜ ≠ ∅\n⊢ C ∩ frontier A ≠ ∅", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_23_11", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_23_11 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  (p : X → Y) (hq : QuotientMap p)\n  (hY : ConnectedSpace Y) (hX : ∀ y : Y, IsConnected (p ⁻¹' {y})) :\n  ConnectedSpace X :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nhq : QuotientMap p\nhY : ConnectedSpace Y\nhX : ∀ (y : Y), IsConnected (p ⁻¹' {y})\n⊢ ConnectedSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_24_3a", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_24_3a [TopologicalSpace I] [CompactSpace I]\n  (f : I → I) (hf : Continuous f) :\n  ∃ (x : I), f x = x :=", "goal": "I : Type u_1\ninst✝¹ : TopologicalSpace I\ninst✝ : CompactSpace I\nf : I → I\nhf : Continuous f\n⊢ ∃ x, f x = x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_25_9", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_25_9 {G : Type*} [TopologicalSpace G] [Group G]\n  [TopologicalGroup G] (C : Set G) (h : C = connectedComponent 1) :\n  IsNormalSubgroup C :=", "goal": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : TopologicalGroup G\nC : Set G\nh : C = connectedComponent 1\n⊢ IsNormalSubgroup C", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_26_12", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_26_12 {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]\n  (p : X → Y) (h : Function.Surjective p) (hc : Continuous p) (hp : ∀ y, IsCompact (p ⁻¹' {y}))\n  (hY : CompactSpace Y) : CompactSpace X :=", "goal": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\np : X → Y\nh : Function.Surjective p\nhc : Continuous p\nhp : ∀ (y : Y), IsCompact (p ⁻¹' {y})\nhY : CompactSpace Y\n⊢ CompactSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_28_4", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_28_4 {X : Type*}\n  [TopologicalSpace X] (hT1 : T1Space X) :\n  countably_compact X ↔ limit_point_compact X :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhT1 : T1Space X\n⊢ countably_compact X ↔ limit_point_compact X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\ndef countably_compact (X : Type*) [TopologicalSpace X] :=\n  ∀ U : ℕ → Set X,\n  (∀ i, IsOpen (U i)) ∧ ((univ : Set X) ⊆ ⋃ i, U i) →\n  (∃ t : Finset ℕ, (univ : Set X) ⊆ ⋃ i ∈ t, U i)\n\ndef limit_point_compact (X : Type*) [TopologicalSpace X] :=\n  ∀ U : Set X, Infinite U → ∃ x ∈ U, ClusterPt x (𝓟 U)\n\n"}
{"name": "exercise_28_6", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_28_6 {X : Type*} [MetricSpace X]\n  [CompactSpace X] {f : X → X} (hf : Isometry f) :\n  Function.Bijective f :=", "goal": "X : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompactSpace X\nf : X → X\nhf : Isometry f\n⊢ Function.Bijective f", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_29_4", "split": "test", "informal_prefix": "/-- Show that $[0, 1]^\\omega$ is not locally compact in the uniform topology.-/\n", "formal_statement": "theorem exercise_29_4 [TopologicalSpace (ℕ → I)] :\n  ¬ LocallyCompactSpace (ℕ → I) :=", "goal": "inst✝ : TopologicalSpace (ℕ → ↑I)\n⊢ ¬LocallyCompactSpace (ℕ → ↑I)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_30_10", "split": "test", "informal_prefix": "/-- Show that if $X$ is a countable product of spaces having countable dense subsets, then $X$ has a countable dense subset.-/\n", "formal_statement": "theorem exercise_30_10\n  {X : ℕ → Type*} [∀ i, TopologicalSpace (X i)]\n  (h : ∀ i, ∃ (s : Set (X i)), Countable s ∧ Dense s) :\n  ∃ (s : Set (Π i, X i)), Countable s ∧ Dense s :=", "goal": "X : ℕ → Type u_1\ninst✝ : (i : ℕ) → TopologicalSpace (X i)\nh : ∀ (i : ℕ), ∃ s, Countable ↑s ∧ Dense s\n⊢ ∃ s, Countable ↑s ∧ Dense s", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_31_1", "split": "test", "informal_prefix": "/-- Show that if $X$ is regular, every pair of points of $X$ have neighborhoods whose closures are disjoint.-/\n", "formal_statement": "theorem exercise_31_1 {X : Type*} [TopologicalSpace X]\n  (hX : RegularSpace X) (x y : X) :\n  ∃ (U V : Set X), IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅ :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : RegularSpace X\nx y : X\n⊢ ∃ U V, IsOpen U ∧ IsOpen V ∧ x ∈ U ∧ y ∈ V ∧ closure U ∩ closure V = ∅", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_31_3", "split": "test", "informal_prefix": "/-- Show that every order topology is regular.-/\n", "formal_statement": "theorem exercise_31_3 {α : Type*} [PartialOrder α]\n  [TopologicalSpace α] (h : OrderTopology α) : RegularSpace α :=", "goal": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : TopologicalSpace α\nh : OrderTopology α\n⊢ RegularSpace α", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_2a", "split": "test", "informal_prefix": "/-- Show that if $\\prod X_\\alpha$ is Hausdorff, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.-/\n", "formal_statement": "theorem exercise_32_2a\n  {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]\n  (h : ∀ i, Nonempty (X i)) (h2 : T2Space (Π i, X i)) :\n  ∀ i, T2Space (X i) :=", "goal": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nh : ∀ (i : ι), Nonempty (X i)\nh2 : T2Space ((i : ι) → X i)\n⊢ ∀ (i : ι), T2Space (X i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_32_2c", "split": "test", "informal_prefix": "/-- Show that if $\\prod X_\\alpha$ is normal, then so is $X_\\alpha$. Assume that each $X_\\alpha$ is nonempty.-/\n", "formal_statement": "theorem exercise_32_2c\n  {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)]\n  (h : ∀ i, Nonempty (X i)) (h2 : NormalSpace (Π i, X i)) :\n  ∀ i, NormalSpace (X i) :=", "goal": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\nh : ∀ (i : ι), Nonempty (X i)\nh2 : NormalSpace ((i : ι) → X i)\n⊢ ∀ (i : ι), NormalSpace (X i)", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_33_7", "split": "test", "informal_prefix": "/-- Show that every locally compact Hausdorff space is completely regular.-/\n", "formal_statement": "theorem exercise_33_7 {X : Type*} [TopologicalSpace X]\n  (hX : LocallyCompactSpace X) (hX' : T2Space X) :\n  ∀ x A, IsClosed A ∧ ¬ x ∈ A →\n  ∃ (f : X → I), Continuous f ∧ f x = 1 ∧ f '' A = {0} :=", "goal": "X : Type u_1\ninst✝ : TopologicalSpace X\nhX : LocallyCompactSpace X\nhX' : T2Space X\n⊢ ∀ (x : X) (A : Set X), IsClosed A ∧ x ∉ A → ∃ f, Continuous f ∧ f x = 1 ∧ f '' A = {0}", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\nabbrev I : Set ℝ := Icc 0 1\n\n"}
{"name": "exercise_34_9", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_34_9\n  (X : Type*) [TopologicalSpace X] [CompactSpace X]\n  (X1 X2 : Set X) (hX1 : IsClosed X1) (hX2 : IsClosed X2)\n  (hX : X1 ∪ X2 = univ) (hX1m : MetrizableSpace X1)\n  (hX2m : MetrizableSpace X2) : MetrizableSpace X :=", "goal": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nX1 X2 : Set X\nhX1 : IsClosed X1\nhX2 : IsClosed X2\nhX : X1 ∪ X2 = univ\nhX1m : MetrizableSpace ↑X1\nhX2m : MetrizableSpace ↑X2\n⊢ MetrizableSpace X", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_43_2", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_43_2 {X : Type*} [MetricSpace X]\n  {Y : Type*} [MetricSpace Y] [CompleteSpace Y] (A : Set X)\n  (f : X → Y) (hf : UniformContinuousOn f A) :\n  ∃! (g : X → Y), ContinuousOn g (closure A) ∧\n  UniformContinuousOn g (closure A) ∧ ∀ (x : A), g x = f x :=", "goal": "X : Type u_1\ninst✝² : MetricSpace X\nY : Type u_2\ninst✝¹ : MetricSpace Y\ninst✝ : CompleteSpace Y\nA : Set X\nf : X → Y\nhf : UniformContinuousOn f A\n⊢ ∃! g, ContinuousOn g (closure A) ∧ UniformContinuousOn g (closure A) ∧ ∀ (x : ↑A), g ↑x = f ↑x", "header": "import Mathlib\n\nopen Filter Set TopologicalSpace\nopen scoped Topology\n\n"}
{"name": "exercise_1_30", "split": "test", "informal_prefix": "/-- Prove that $\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}$ is not an integer.-/\n", "formal_statement": "theorem exercise_1_30 {n : ℕ} :\n  ¬ ∃ a : ℤ, ∑ i : Fin n, (1 : ℚ) / (n+2) = a :=", "goal": "n : ℕ\n⊢ ¬∃ a, ∑ i : Fin n, 1 / (↑n + 2) = ↑a", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_4", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_2_4 {a : ℤ} (ha : a ≠ 0)\n  (f_a := λ n m : ℕ => Int.gcd (a^(2^n) + 1) (a^(2^m)+1)) {n m : ℕ}\n  (hnm : n > m) :\n  (Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2) :=", "goal": "a : ℤ\nha : a ≠ 0\nf_a : optParam (ℕ → ℕ → ℕ) fun n m => (a ^ 2 ^ n + 1).gcd (a ^ 2 ^ m + 1)\nn m : ℕ\nhnm : n > m\n⊢ (Odd a → f_a n m = 1) ∧ (Even a → f_a n m = 2)", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2_27a", "split": "test", "informal_prefix": "/-- Show that $\\sum^{\\prime} 1 / n$, the sum being over square free integers, diverges.-/\n", "formal_statement": "theorem exercise_2_27a :\n  ¬ Summable (λ i : {p : ℤ // Squarefree p} => (1 : ℚ) / i) :=", "goal": "⊢ ¬Summable fun i => 1 / ↑↑i", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_4", "split": "test", "informal_prefix": "/-- Show that the equation $3 x^{2}+2=y^{2}$ has no solution in integers.-/\n", "formal_statement": "theorem exercise_3_4 : ¬ ∃ x y : ℤ, 3*x^2 + 2 = y^2 :=", "goal": "⊢ ¬∃ x y, 3 * x ^ 2 + 2 = y ^ 2", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_3_10", "split": "test", "informal_prefix": "/-- If $n$ is not a prime, show that $(n-1) ! \\equiv 0(n)$, except when $n=4$.-/\n", "formal_statement": "theorem exercise_3_10 {n : ℕ} (hn0 : ¬ n.Prime) (hn1 : n ≠ 4) :\n  Nat.factorial (n-1) ≡ 0 [MOD n] :=", "goal": "n : ℕ\nhn0 : ¬n.Prime\nhn1 : n ≠ 4\n⊢ (n - 1).factorial ≡ 0 [MOD n]", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_4", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_4_4 {p t: ℕ} (hp0 : p.Prime) (hp1 : p = 4*t + 1)\n  (a : ZMod p) :\n  IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p :=", "goal": "p t : ℕ\nhp0 : p.Prime\nhp1 : p = 4 * t + 1\na : ZMod p\n⊢ IsPrimitiveRoot a p ↔ IsPrimitiveRoot (-a) p", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_6", "split": "test", "informal_prefix": "/-- If $p=2^{n}+1$ is a Fermat prime, show that 3 is a primitive root modulo $p$.-/\n", "formal_statement": "theorem exercise_4_6 {p n : ℕ} (hp : p.Prime) (hpn : p = 2^n + 1) :\n  IsPrimitiveRoot 3 p :=", "goal": "p n : ℕ\nhp : p.Prime\nhpn : p = 2 ^ n + 1\n⊢ IsPrimitiveRoot 3 p", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_4_11", "split": "test", "informal_prefix": "/-- Prove that $1^{k}+2^{k}+\\cdots+(p-1)^{k} \\equiv 0(p)$ if $p-1 \\nmid k$ and $-1(p)$ if $p-1 \\mid k$.-/\n", "formal_statement": "theorem exercise_4_11 {p : ℕ} (hp : p.Prime) (k s: ℕ)\n  (s := ∑ n : Fin p, (n : ℕ) ^ k) :\n  ((¬ p - 1 ∣ k) → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p]) :=", "goal": "p : ℕ\nhp : p.Prime\nk s✝ : ℕ\ns : optParam ℕ (∑ n : Fin p, ↑n ^ k)\n⊢ (¬p - 1 ∣ k → s ≡ 0 [MOD p]) ∧ (p - 1 ∣ k → s ≡ 0 [MOD p])", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_5_28", "split": "test", "informal_prefix": "/-- 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", "formal_statement": "theorem exercise_5_28 {p : ℕ} (hp : p.Prime) (hp1 : p ≡ 1 [MOD 4]):\n  ∃ x, x^4 ≡ 2 [MOD p] ↔ ∃ A B, p = A^2 + 64*B^2 :=", "goal": "p : ℕ\nhp : p.Prime\nhp1 : p ≡ 1 [MOD 4]\n⊢ ∃ x, x ^ 4 ≡ 2 [MOD p] ↔ ∃ A B, p = A ^ 2 + 64 * B ^ 2", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_12_12", "split": "test", "informal_prefix": "/-- Show that $\\sin (\\pi / 12)$ is an algebraic number.-/\n", "formal_statement": "theorem exercise_12_12 : IsAlgebraic ℚ (sin (Real.pi/12)) :=", "goal": "⊢ IsAlgebraic ℚ (π / 12).sin", "header": "import Mathlib\n\nopen Real\nopen scoped BigOperators\n\n"}
{"name": "exercise_2018_a5", "split": "test", "informal_prefix": "/-- Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be an infinitely differentiable function satisfying $f(0)=0, f(1)=1$, and $f(x) \\geq 0$ for all $x \\in$ $\\mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x)<0$.-/\n", "formal_statement": "theorem exercise_2018_a5 (f : ℝ → ℝ) (hf : ContDiff ℝ ⊤ f)\n  (hf0 : f 0 = 0) (hf1 : f 1 = 1) (hf2 : ∀ x, f x ≥ 0) :\n  ∃ (n : ℕ) (x : ℝ), iteratedDeriv n f x = 0 :=", "goal": "f : ℝ → ℝ\nhf : ContDiff ℝ ⊤ f\nhf0 : f 0 = 0\nhf1 : f 1 = 1\nhf2 : ∀ (x : ℝ), f x ≥ 0\n⊢ ∃ n x, iteratedDeriv n f x = 0", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2018_b4", "split": "test", "informal_prefix": "/-- Given a real number $a$, we define a sequence by $x_{0}=1$, $x_{1}=x_{2}=a$, and $x_{n+1}=2 x_{n} x_{n-1}-x_{n-2}$ for $n \\geq 2$. Prove that if $x_{n}=0$ for some $n$, then the sequence is periodic.-/\n", "formal_statement": "theorem exercise_2018_b4 (a : ℝ) (x : ℕ → ℝ) (hx0 : x 0 = a)\n  (hx1 : x 1 = a)\n  (hxn : ∀ n : ℕ, n ≥ 2 → x (n+1) = 2*(x n)*(x (n-1)) - x (n-2))\n  (h : ∃ n, x n = 0) :\n  ∃ c, Function.Periodic x c :=", "goal": "a : ℝ\nx : ℕ → ℝ\nhx0 : x 0 = a\nhx1 : x 1 = a\nhxn : ∀ n ≥ 2, x (n + 1) = 2 * x n * x (n - 1) - x (n - 2)\nh : ∃ n, x n = 0\n⊢ ∃ c, Function.Periodic x c", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2014_a5", "split": "test", "informal_prefix": "/-- Let-/\n", "formal_statement": "theorem exercise_2014_a5 (P : ℕ → Polynomial ℤ)\n  (hP : ∀ n, P n = ∑ i : Fin n, (n+1) * Polynomial.X ^ n) :\n  ∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k) :=", "goal": "P : ℕ → Polynomial ℤ\nhP : ∀ (n : ℕ), P n = ∑ i : Fin n, (↑n + 1) * Polynomial.X ^ n\n⊢ ∀ (j k : ℕ), j ≠ k → IsCoprime (P j) (P k)", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_2001_a5", "split": "test", "informal_prefix": "/-- Prove that there are unique positive integers $a, n$ such that $a^{n+1}-(a+1)^n=2001$.-/\n", "formal_statement": "theorem exercise_2001_a5 :\n  ∃! a : ℕ, ∃! n : ℕ, a > 0 ∧ n > 0 ∧ a^(n+1) - (a+1)^n = 2001 :=", "goal": "⊢ ∃! a, ∃! n, a > 0 ∧ n > 0 ∧ a ^ (n + 1) - (a + 1) ^ n = 2001", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1999_b4", "split": "test", "informal_prefix": "/-- Let $f$ be a real function with a continuous third derivative such that $f(x), f^{\\prime}(x), f^{\\prime \\prime}(x), f^{\\prime \\prime \\prime}(x)$ are positive for all $x$. Suppose that $f^{\\prime \\prime \\prime}(x) \\leq f(x)$ for all $x$. Show that $f^{\\prime}(x)<2 f(x)$ for all $x$.-/\n", "formal_statement": "theorem exercise_1999_b4 (f : ℝ → ℝ) (hf: ContDiff ℝ 3 f)\n  (hf1 : ∀ n ≤ 3, ∀ x : ℝ, iteratedDeriv n f x > 0)\n  (hf2 : ∀ x : ℝ, iteratedDeriv 3 f x ≤ f x) :\n  ∀ x : ℝ, deriv f x < 2 * f x :=", "goal": "f : ℝ → ℝ\nhf : ContDiff ℝ 3 f\nhf1 : ∀ n ≤ 3, ∀ (x : ℝ), iteratedDeriv n f x > 0\nhf2 : ∀ (x : ℝ), iteratedDeriv 3 f x ≤ f x\n⊢ ∀ (x : ℝ), deriv f x < 2 * f x", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}
{"name": "exercise_1998_b6", "split": "test", "informal_prefix": "/-- Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\\sqrt{n^3+a n^2+b n+c}$ is not an integer.-/\n", "formal_statement": "theorem exercise_1998_b6 (a b c : ℤ) :\n  ∃ n : ℤ, n > 0 ∧ ¬ ∃ m : ℤ, Real.sqrt (n^3 + a*n^2 + b*n + c) = m :=", "goal": "a b c : ℤ\n⊢ ∃ n > 0, ¬∃ m, √(↑n ^ 3 + ↑a * ↑n ^ 2 + ↑b * ↑n + ↑c) = ↑m", "header": "import Mathlib\n\nopen scoped BigOperators\n\n"}