{
  "statement": "Prove H4: Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
  "starting_statement": null,
  "target_statement": "Let \\(d,r:[0,\\infty)\\to\\mathbb R\\) be continuous, with \\(d>0\\) and \\(r\\ge0\\). Let \\(v:[0,\\infty)\\times\\mathbb R\\to\\mathbb R\\) be continuous and locally Lipschitz in space, and satisfy \\(|v(\\tau,h)|\\le C_T(1+|h|)\\) for \\(0\\le\\tau\\le T\\), for each finite \\(T\\).\nSuppose \\(F\\in C([0,\\infty)\\times\\mathbb R)\\cap C^{1,2}((0,\\infty)\\times\\mathbb R)\\) is bounded on each strip \\([0,T]\\times\\mathbb R\\) and satisfies\n\\[\nF_\\tau=d(\\tau)F_{hh}+v(\\tau,h)F_h+r(\\tau)F(1-F),\\qquad\n0\\le F(0,h)\\le1\\quad(h\\in\\mathbb R).\n\\]\n\nThen \\(0\\le F(\\tau,h)\\le1\\) for every \\(\\tau\\ge0\\) and \\(h\\in\\mathbb R\\).",
  "case_id": "SM-1-H0-to-H4",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}