{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
  "old_string": "  have hu2fac : (b ^ 2 * s + b) * eA \u2264 (0.2124 : \u211d) := by\n    -- u\u2082 antitone on [\u221a20, \u221e): derivative eA\u00b7(b\u00b2 \u2212 a(b\u00b2s+b)) \u2264 0\n    set u2 : \u211d \u2192 \u211d := fun t => (b ^ 2 * t + b) * Real.exp (-a * t) with hu2def\n    have hu2der : \u2200 x : \u211d, HasDerivAt u2\n        (Real.exp (-a * x) * (b ^ 2 - a * (b ^ 2 * x + b))) x := by\n      intro x\n      have hp : HasDerivAt (fun t => b ^ 2 * t + b) (b ^ 2 * 1) x :=\n        ((hasDerivAt_id' x).const_mul (b ^ 2)).add_const b\n      have he : HasDerivAt (fun t => Real.exp (-a * t))\n          (Real.exp (-a * x) * (-a * 1)) x :=\n        ((hasDerivAt_id' x).const_mul (-a)).exp\n      have h := hp.mul he\n      have e : b ^ 2 * 1 * Real.exp (-a * x) + (b ^ 2 * x + b) *\n          (Real.exp (-a * x) * (-a * 1)) =\n          Real.exp (-a * x) * (b ^ 2 - a * (b ^ 2 * x + b)) := by ring\n      rw [e] at h\n      exact h\n    have hb2lo : (1.9218 : \u211d) \u2264 b ^ 2 := by nlinarith [hblo]\n    have hb2up : b ^ 2 \u2264 (1.92182 : \u211d) := by nlinarith [hb, hbpos]\n    have hCD : ContDiff \u211d 2 u2 := by\n      simp only [hu2def]\n      exact ((contDiff_const.mul contDiff_id).add contDiff_const).mul\n        ((contDiff_const.mul contDiff_id).exp)\n    have hu2anti : AntitoneOn u2 (Set.Ici (Real.sqrt 20)) := by\n      apply antitoneOn_of_deriv_nonpos (convex_Ici _)\n      \u00b7 exact hCD.continuous.continuousOn\n      \u00b7 exact (hCD.differentiable (by norm_num)).differentiableOn.mono interior_subset\n      \u00b7 intro x hx\n        rw [interior_Ici] at hx\n        have hx4 : (4.4721 : \u211d) \u2264 x := by\n          have := Set.mem_Ioi.mp hx\n          linarith\n        have hbig : (8.8461 : \u211d) \u2264 a * (b ^ 2 * x + b) := by\n          have hb2x : (9.9809 : \u211d) \u2264 b ^ 2 * x + b := by\n            have h1 : (1.9218 : \u211d) * 4.4721 \u2264 b ^ 2 * x :=\n              mul_le_mul hb2lo hx4 (by norm_num) (sq_nonneg b)\n            nlinarith [hblo]\n          calc (8.8461 : \u211d) \u2264 0.8862943606 * 9.9809 := by norm_num\n            _ \u2264 a * (b ^ 2 * x + b) :=\n              mul_le_mul ha hb2x (by norm_num) hapos.le\n        rw [(hu2der x).deriv]\n        have h : b ^ 2 - a * (b ^ 2 * x + b) \u2264 0 := by linarith\n        exact mul_nonpos_of_nonneg_of_nonpos (Real.exp_pos _).le h\n    have hu2le : u2 s \u2264 u2 (Real.sqrt 20) :=\n      hu2anti (Set.mem_Ici.mpr hsqrthi.le) (Set.mem_Ici.mpr hs)\n    have hu2val : u2 (Real.sqrt 20) \u2264 (0.2124 : \u211d) := by\n      have hexp : Real.exp (-a * Real.sqrt 20) \u2264 (0.0212721 : \u211d) := by\n        have ht : (3.9636 : \u211d) \u2264 a * Real.sqrt 20 := by\n          calc (3.9636 : \u211d) \u2264 0.8862943606 * 4.4721 := by norm_num\n            _ \u2264 a * Real.sqrt 20 :=\n              mul_le_mul ha hsqrtlo (by norm_num) hapos.le\n        have hsum : (47.01 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) := by\n          have h1 := Real.sum_le_exp_of_nonneg (x := (3.9636 : \u211d)) (by norm_num) 7\n          have h2 : Real.exp (3.9636 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) :=\n            Real.exp_le_exp.mpr ht\n          have h3 : (47.01 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7,\n              (3.9636 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [Real.exp_neg]\n        calc (Real.exp (a * Real.sqrt 20))\u207b\u00b9 \u2264 1 / 47.01 :=\n              one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 47.01) hsum\n          _ \u2264 0.0212721 := by\n              rw [div_le_iff\u2080 (by norm_num : (0 : \u211d) < 47.01)]; norm_num\n      have hpoly : b ^ 2 * Real.sqrt 20 + b \u2264 (9.9811 : \u211d) := by\n        have h1 : b ^ 2 * Real.sqrt 20 \u2264 1.92182 * 4.4722 :=\n          mul_le_mul hb2up hsqrthi (by norm_num) (sq_nonneg b)\n        nlinarith [hb]\n      calc u2 (Real.sqrt 20) = (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) := rfl\n        _ \u2264 9.9811 * 0.0212721 :=\n          mul_le_mul hpoly hexp (Real.exp_pos _).le (by norm_num)\n        _ \u2264 0.2124 := by norm_num\n    calc (b ^ 2 * s + b) * eA = u2 s := rfl\n      _ \u2264 u2 (Real.sqrt 20) := hu2le\n      _ \u2264 0.2124 := hu2val",
  "new_string": "  have hu2fac : (b ^ 2 * s + b) * eA \u2264 (0.2124 : \u211d) := by\n    have hAid : iimA = a := by unfold iimA; rw [\u2190 hLdef, \u2190 hadef]\n    have hBid : iimB = b := by unfold iimB; rw [\u2190 hLdef, \u2190 hbdef]\n    have hanti := iim_u2_antitone\n    rw [hAid, hBid] at hanti\n    have hu2le : (b ^ 2 * s + b) * Real.exp (-a * s) \u2264\n        (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) :=\n      hanti (Set.mem_Ici.mpr (le_refl _)) (Set.mem_Ici.mpr hs)\n    have hu2val : (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) \u2264\n        (0.2124 : \u211d) := by\n      have hb2up : b ^ 2 \u2264 (1.92182 : \u211d) := by nlinarith [hb, hbpos]\n      have hexp : Real.exp (-a * Real.sqrt 20) \u2264 (0.0212721 : \u211d) := by\n        have ht : (3.9636 : \u211d) \u2264 a * Real.sqrt 20 := by\n          calc (3.9636 : \u211d) \u2264 0.8862943606 * 4.4721 := by norm_num\n            _ \u2264 a * Real.sqrt 20 :=\n              mul_le_mul ha hsqrtlo (by norm_num) hapos.le\n        have hsum : (47.01 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) := by\n          have h1 := Real.sum_le_exp_of_nonneg (x := (3.9636 : \u211d)) (by norm_num) 7\n          have h2 : Real.exp (3.9636 : \u211d) \u2264 Real.exp (a * Real.sqrt 20) :=\n            Real.exp_le_exp.mpr ht\n          have h3 : (47.01 : \u211d) \u2264 \u2211 i \u2208 Finset.range 7,\n              (3.9636 : \u211d) ^ i / (Nat.factorial i : \u211d) := by\n            norm_num [Finset.sum_range_succ]\n          linarith\n        rw [Real.exp_neg]\n        calc (Real.exp (a * Real.sqrt 20))\u207b\u00b9 \u2264 1 / 47.01 :=\n              one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 47.01) hsum\n          _ \u2264 0.0212721 := by\n              rw [div_le_iff\u2080 (by norm_num : (0 : \u211d) < 47.01)]; norm_num\n      have hpoly : b ^ 2 * Real.sqrt 20 + b \u2264 (9.9811 : \u211d) := by\n        have h1 : b ^ 2 * Real.sqrt 20 \u2264 1.92182 * 4.4722 :=\n          mul_le_mul hb2up hsqrthi (by norm_num) (sq_nonneg b)\n        nlinarith [hb]\n      exact le_trans (mul_le_mul hpoly hexp (Real.exp_pos _).le (by norm_num))\n        (by norm_num)\n    calc (b ^ 2 * s + b) * eA = (b ^ 2 * s + b) * Real.exp (-a * s) := by rw [heAdef]\n      _ \u2264 (b ^ 2 * Real.sqrt 20 + b) * Real.exp (-a * Real.sqrt 20) := hu2le\n      _ \u2264 0.2124 := hu2val"
}

{
  "filePath": "D:\\Python\\quantum-deploy\\docroot\\app.cgi",
  "oldString": "def _dbg(msg):\n    if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n        fh.write(msg + \"\\n\")",
  "newString": "def _dbg(msg):\n    if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n            fh.write(msg + \"\\n\")",
  "originalFile": "#!/opt/alt/python311/bin/python3.11\n\"\"\"WSGI->CGI adapter \u2014 runs the RH-WEBSITE Django app on Apache shared hosting.\n\nApache rewrites every non-file request to /app.cgi/<path> (see .htaccess).\nwsgiref's CGIHandler bridges the CGI environment onto Django's WSGI app.\n\"\"\"\nimport os\nimport sys\n\nDOCROOT = os.path.dirname(os.path.abspath(__file__))\nAPP = os.path.join(DOCROOT, \"app\")\n\n\ndef _dbg(msg):\n    if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n        fh.write(msg + \"\\n\")\n\n\n_dbg(\"TOP fsenc=\" + sys.getfilesystemencoding()\n     + \" PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))\n\n# Vendored pure-Python deps (Django 5.2, markdown) + the project package.\nsys.path.insert(0, os.path.join(APP, \"vendor\"))\nsys.path.insert(0, os.path.join(APP, \"web\"))\n\nos.environ.setdefault(\"DJANGO_SETTINGS_MODULE\", \"rhweb.settings\")\n# Apache mounts us at /app.cgi/<path>; present clean root URLs to Django.\nos.environ[\"SCRIPT_NAME\"] = \"\"\n\nfrom wsgiref.handlers import CGIHandler\n\nfrom rhweb.wsgi import application\n\n_dbg(\"AFTER-IMPORT PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))\n\nfrom urllib.parse import unquote_to_bytes\n\n\ndef _pep3333_paths(environ, start_response):\n    \"\"\"Normalize Apache's path vars to the PEP 3333 form Django needs.\n\n    Apache hands these vars either percent-encoded (mod_rewrite path)\n    or UTF-8-decoded (direct CGI path). Django (a) crashes with\n    UnicodeEncodeError on any non-latin-1 char (500) and (b) needs the\n    decoded path for URL resolution (else the resolver sees %XX and 404s).\n    So: bytes -> unquote -> re-mangle as latin-1. SCRIPT_URL/REDIRECT_URL\n    (set by mod_rewrite) are consulted by get_script_name BEFORE\n    PATH_INFO \u2014 all three must be normalized.\"\"\"\n    for key in (\"PATH_INFO\", \"SCRIPT_URL\", \"REDIRECT_URL\"):\n        v = environ.get(key)\n        if v:\n            raw = unquote_to_bytes(v)  # %XX-decodes; utf-8-encodes str first\n            environ[key] = raw.decode(\"latin-1\")\n    if \"resources\" in (environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n            fh.write(repr({k: environ.get(k) for k in\n                           (\"PATH_INFO\", \"SCRIPT_NAME\", \"SCRIPT_URL\",\n                            \"REDIRECT_URL\", \"REQUEST_METHOD\", \"HTTP_HOST\")})\n                     + \"\\n\")\n            try:\n                from django.core.handlers.wsgi import WSGIRequest\n                req = WSGIRequest(dict(environ))\n                fh.write(\"path_info=\" + repr(req.path_info) + \"\\n\")\n                from django.urls import resolve\n                m = resolve(req.path_info)\n                fh.write(\"match=\" + repr(m.kwargs) + \"\\n\")\n            except Exception as e:  # noqa: BLE001\n                fh.write(\"ERR \" + repr(e) + \"\\n\")\n            try:\n                raw = open(\"/proc/self/environ\", \"rb\").read()\n                for entry in raw.split(b\"\\x00\"):\n                    if entry.startswith((b\"PATH_INFO=\", b\"SCRIPT_URL=\")):\n                        fh.write(\"ENVBYTES \" + repr(entry) + \"\\n\")\n            except Exception as e:  # noqa: BLE001\n                fh.write(\"ENVBYTES ERR \" + repr(e) + \"\\n\")\n    return application(environ, start_response)\n\n\nCGIHandler().run(_pep3333_paths)\n",
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      "oldStart": 15,
      "oldLines": 7,
      "newStart": 15,
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      "lines": [
        "     if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):",
        "         with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",",
        "                   encoding=\"utf-8\") as fh:",
        "-        fh.write(msg + \"\\n\")",
        "+            fh.write(msg + \"\\n\")",
        " ",
        " ",
        " _dbg(\"TOP fsenc=\" + sys.getfilesystemencoding()"
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
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