{
    "messages": [
    {
      "role": "system",
      "content": "As a mathematician familiar with Isabelle, your task is to translate the natural language problem into an Isabelle language version."
    },
    {
      "role": "user",
      "content": "Natural language version: \"How many seconds are in 7.8 minutes?\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes seconds minutes :: real\nassumes h0 : \"1 * minutes = 60 * seconds\"\nshows \"7.8 * minutes = 478 * seconds\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"Find the value of $x$ that satisfies $\\frac{\\sqrt{3x+5}}{\\sqrt{6x+5}}=\\frac{\\sqrt{5}}{3}$. Express your answer as a common fraction. The final Answer is $\\frac{20}{3}$.\"  \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes x :: real\nassumes h0 : \"3 * x + 5 \\<ge> 0\"\nand h1 : \"6 * x + 5 > 0\"\nand h2 : \"sqrt (3 * x + 5) / sqrt (6 * x + 5) = sqrt 5 / 3\"\nshows \"x = 20 / 3\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"A function $f$ is defined recursively by $f(1)=f(2)=1$ and $f(n)=f(n-1)-f(n-2)+n$ for all integers $n \\geq 3$. What is $f(2018)$? The final Answer is $2017$.\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes f :: \"int \\<Rightarrow> int\"\nassumes h0 : \"f(1) = 1\" \nand h1 : \"f(2) = 1\"\nand h2 : \"\\<forall>n. n \\<ge> 3 \\<longrightarrow> f n = f (n-1) - f (n-2) + n\"\nshows \"f 2018 = -2017\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"Simplify $(9 - 4i) - (-3 - 4i)$. The final answer is $12$.\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes a b :: complex\nassumes h0 : \"a = Complex (Re 9) (Im (-4))\"\nand h1 : \"b = Complex (Re (-3)) (Im (-4))\"\nshows \"a - b = 12\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"Simplify $(9 - 4i) - (-3 - 4i)$. The final answer is $12$.\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes a b :: complex\nassumes h0 : \"a = Complex (Re 9) (Im (-4))\"\nand h1 : \"b = Complex (Re (-3)) (Im (-4))\"\nshows \"a - b = 12\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"If $f(x) = 5x - 12$, find a value for $x$ so that $f^{-1}(x) = f(x + 1)$. The final answer is $\\frac{47}{24}$.\"\nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes x :: real and f :: \"real \\<Rightarrow> real\"\nassumes h0 : \"bij f\"\nand f : \"\\<forall> x. f x = 5 * x - 12\"\nand h1 : \"f (x + 1) = (inv f) x\"\nshows \"x = 47 / 24\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"Find the quotient when $x^5 + 7$ is divided by $x + 1.$ The final Answer is $x^4 - x^3 + x^2 - x + 1$.\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes x :: real and P Q R :: \"real \\<Rightarrow> real\"\nassumes P_def : \"\\<forall> x. P x = x powr 5 + 7\"\nand Q_def : \"\\<forall> x. Q x = x + 1\"\nand h0 : \"P x dvd Q x\"\nand h1 : \"\\<forall> x. P x = Q x * R x\"\nshows \"R x = x powr 4 - x powr 3 + x powr 2 - x + 1\""
    },
    {
      "role": "user",
      "content": "Natural language version: \"Given that $x^2 + y^2 = 1$, find the largest possible value of $x + y$. The final answer is $\\sqrt{2}$.\" \nTranslate the natural language version to an Isabelle version:"
    },
    {
      "role": "assistant",
      "content": "theorem\nfixes x y :: real\nassumes h0 : \"x powr 2 + y powr 2 = 1\"\nshows \"sqrt(2) = (GREATEST a. \\<exists> x y. x + y = a)\""
    }
  ]
}