{
  "statement": "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\\]\n\nProve H3: 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 \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,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\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\\]",
  "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 \\(p:\\mathbb R\\to(0,\\infty)\\) be continuous and grow at most polynomially.\n\nSuppose there are \\(\\gamma_\\infty>0\\), positive scalars \\(\\gamma(y),A_y\\), and real scalars \\(I_y\\ne0,Q_y\\), for \\(y\\ge1\\), such that\n\\[\n\\gamma(y)y^c\\longrightarrow\\gamma_\\infty,\\qquad\nyI_y=\\frac{\\gamma(y)}2Q_y,\n\\]\n\\[\n\\frac{I_y}{A_y}\\longrightarrow\\int_{\\mathbb R}pf^2(1-f)\\,dt,\\qquad\n\\frac{\\gamma_\\infty Q_y}{A_y y^{2b}}\\longrightarrow\\int_{\\mathbb R}p(f')^2\\,dt.\n\\]\nThe integrals are improper integrals over \\(\\mathbb R\\setminus\\{0\\}\\), and both limits are finite real numbers.\n\nThen\n\\[\n\\int_{\\mathbb R}p(f')^2\\,dt=2\\int_{\\mathbb R}pf^2(1-f)\\,dt>0.\n\\]",
  "case_id": "SM-1-H2-to-H3",
  "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\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\\]"
  },
  "protocol": "single"
}