{
  "statement": "H1: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.\n\nProve H2: Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
  "starting_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit.",
  "target_statement": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nLet \\(a\\in\\mathbb R\\). Suppose \\(p\\in C^1(\\mathbb R)\\) is strictly positive, is \\(C^2\\) on each open half-line, and satisfies\n\\[\n\\frac c2p''=(a-cH_-)p+(b-cH_-)tp'-\\frac{c^2}{2}(p'J'+pJ'')\\qquad(t\\ne0).\n\\]\nAssume \\(p,p'\\) grow at most polynomially.\n\nThe following improper integral, with zero omitted from its integrand, is absolutely convergent and equals zero:\n\\[\n\\int_{\\mathbb R}p\\left\\{c\\left[(f')^2-2f^2(1-f)\\right]-(b-a-c)f(1-f)\\right\\}\\,dt=0.\n\\]",
  "case_id": "SM-1-H1-to-H2",
  "given_context": {
    "premise": "Let \\(0<c<1\\), \\(b=(1+c)/2\\), and \\(H_-(t)=\\mathbf1_{t<0}\\) for \\(t\\ne0\\).\nLet \\(J\\in C^1(\\mathbb R)\\) be \\(C^4\\) on each open half-line, with finite one-sided derivatives through order four at zero, and satisfy\n\\[\n\\frac c2J''=(-b+cH_-)tJ'+J-H_-+\\frac{c^2}{4}(J')^2\\qquad(t\\ne0).\n\\]\nFor some \\(C_0,q,\\eta>0\\), assume\n\\[\n|J(t)-H_-(t)|+\\sum_{j=1}^4|J^{(j)}(t)|\n\\le C_0(1+|t|)^qe^{-\\eta t^2}\n\\]\nfor all sufficiently large \\(|t|\\). For \\(t\\ne0\\), put \\(f(t)=H_-(t)+cJ''(t)/2\\).\n\nThe function \\(f\\) extends to \\(C^2(\\mathbb R)\\) and satisfies\n\\[\n\\frac c2f''+\\left((b-cH_-)t-\\frac{c^2}{2}J'\\right)f'+cf(1-f)=0\n\\]\non the entire real line, where the coefficient at \\(t=0\\) is defined by its continuous limit."
  },
  "protocol": "single"
}