{
  "analysis": "c1-c3-training-dynamics-and-c0-mode-suppression-v1",
  "c0_mode_suppression_and_c3_recovery": {
    "by_c0_count_floor": {
      "1": {
        "c0_modes": 73635,
        "c0_modes_absent_from_c1": 54330,
        "minimum_c0_observations": 1,
        "recovery_fraction": 0.23064605190502485,
        "suppressed_modes_present_in_c3": 12531,
        "theorems_with_c3_recovery": 5631,
        "theorems_with_suppressed_mode": 7633
      },
      "2": {
        "c0_modes": 30069,
        "c0_modes_absent_from_c1": 16515,
        "minimum_c0_observations": 2,
        "recovery_fraction": 0.4332425068119891,
        "suppressed_modes_present_in_c3": 7155,
        "theorems_with_c3_recovery": 4255,
        "theorems_with_suppressed_mode": 5992
      },
      "4": {
        "c0_modes": 12028,
        "c0_modes_absent_from_c1": 4931,
        "minimum_c0_observations": 4,
        "recovery_fraction": 0.5881160008111945,
        "suppressed_modes_present_in_c3": 2900,
        "theorems_with_c3_recovery": 2321,
        "theorems_with_suppressed_mode": 3383
      }
    },
    "top_recovered_examples": [
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 32,
        "c3_example_proof": "  rw [sin_three_mul]\n```",
        "mode_id": "6727b129361f964fc8a2040a9b2bc5b69b94619b7acbe03bf2fe19e5f2dea9d0",
        "tactic_heads": [
          "rw"
        ],
        "theorem": "lean_workbook_plus_34186"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 32,
        "c3_example_proof": "  ring_nf\n  nlinarith [ha, hb, hc, hab]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_55178"
      },
      {
        "c0_count": 5,
        "c1_count": 0,
        "c3_count": 30,
        "c3_example_proof": "  ring_nf\n  nlinarith [hx, hy, hz]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_54730"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 30,
        "c3_example_proof": "  ring_nf\n  nlinarith [ha, hb, hc]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_117"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 29,
        "c3_example_proof": "  ring_nf at h\u2081\n  linarith only [h\u2080, h\u2081]\n```",
        "mode_id": "dcc5b8fffe5aeee64658f3c384b303afd1a48d87776097efe517969a1517283a",
        "tactic_heads": [
          "ring_nf",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_25242"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 28,
        "c3_example_proof": "  ring_nf at h1 h2 \u22a2\n  nlinarith [sq_nonneg (a + b + c + d - 4), sq_nonneg (a * b * c * d - 1)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_55988"
      },
      {
        "c0_count": 12,
        "c1_count": 0,
        "c3_count": 28,
        "c3_example_proof": "  congr 1\n```",
        "mode_id": "f57692659b8340f7746fca35d4d27557bc11621a8f443ea8c0e500144b3bc546",
        "tactic_heads": [
          "congr"
        ],
        "theorem": "lean_workbook_plus_29910"
      },
      {
        "c0_count": 19,
        "c1_count": 0,
        "c3_count": 27,
        "c3_example_proof": "  simp only [sub_eq_add_neg, mul_neg, mul_assoc]\n  positivity\n```",
        "mode_id": "e6d8a1c844c6993d9ca6ce4904b7e6fe1d43c167f8541c9c0701040ff2de7d67",
        "tactic_heads": [
          "simp",
          "positivity"
        ],
        "theorem": "lean_workbook_plus_31055"
      },
      {
        "c0_count": 19,
        "c1_count": 0,
        "c3_count": 26,
        "c3_example_proof": "  simp only [add_comm, add_left_comm, add_assoc, zero_add, add_zero]\n  nlinarith [sq_nonneg (x^2 - x)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_8714"
      },
      {
        "c0_count": 29,
        "c1_count": 0,
        "c3_count": 26,
        "c3_example_proof": "  simp [mul_add, add_mul, mul_comm, mul_left_comm]\n  nlinarith [sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (b - c), sq_nonneg (a + b), sq_nonneg (a + c), sq_nonneg (b + c)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_73337"
      },
      {
        "c0_count": 15,
        "c1_count": 0,
        "c3_count": 25,
        "c3_example_proof": "  have := sq_nonneg ((x - y) * (y - z) * (z - x))\n  linarith [sq_nonneg (x * y + y * z + z * x), sq_nonneg (x * y * z)]\n```",
        "mode_id": "11c2883840300770292bef981e24cabafaaf7b831344e370541584c86206d7e1",
        "tactic_heads": [
          "have",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_53745"
      },
      {
        "c0_count": 16,
        "c1_count": 0,
        "c3_count": 25,
        "c3_example_proof": "  simp [sq, sub_eq_add_neg, add_assoc]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_64633"
      },
      {
        "c0_count": 17,
        "c1_count": 0,
        "c3_count": 25,
        "c3_example_proof": "  simp [mul_add, mul_comm, mul_left_comm]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_9036"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 24,
        "c3_example_proof": "  norm_num [h\u2081, h\u2082]\n```",
        "mode_id": "1527b27f2e1bc951dba0fe6ba8a371d859c076a7f6659b13a63dabb14b40558e",
        "tactic_heads": [
          "norm_num"
        ],
        "theorem": "lean_workbook_plus_40315"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 24,
        "c3_example_proof": "  rw [add_comm]\n  ring_nf\n```",
        "mode_id": "546b453cfe13b54953cc919f1d179eccd00877e039933fa4198c4befde4159db",
        "tactic_heads": [
          "rw",
          "ring_nf"
        ],
        "theorem": "lean_workbook_plus_47713"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 24,
        "c3_example_proof": "  exact hf\n```",
        "mode_id": "c9680df45a04a31d289589dee59ba91ccf7927d589105ff7a089fdd9a8f9e341",
        "tactic_heads": [
          "exact"
        ],
        "theorem": "lean_workbook_plus_67755"
      },
      {
        "c0_count": 12,
        "c1_count": 0,
        "c3_count": 24,
        "c3_example_proof": "  have := sq_nonneg (a - b)\n  have := sq_nonneg (b - c)\n  have := sq_nonneg (c - a)\n  nlinarith\n```",
        "mode_id": "986c1e7e9b6ac7f68095ae93d6922d880ba2da90cc6e8c7d11026852e10fea2c",
        "tactic_heads": [
          "have",
          "have",
          "have",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_42088"
      },
      {
        "c0_count": 21,
        "c1_count": 0,
        "c3_count": 24,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_comm, add_left_comm, mul_add, add_mul]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_78829"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  rw [sub_sq, add_comm]\n  ring\n```",
        "mode_id": "b877a1abd88518eaa88f177646389a5ded26bdca716c3c38ba5bdc6a7bdee784",
        "tactic_heads": [
          "rw",
          "ring"
        ],
        "theorem": "lean_workbook_plus_33531"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  ring_nf\n  nlinarith [ha, hb, hc, hab, h]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_40161"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  ring_nf at h \u22a2\n  nlinarith [sq_nonneg (a^3 - 3*a), h]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_16044"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  simp [sq, add_assoc]\n  nlinarith [sq_nonneg (x + y - 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_63712"
      },
      {
        "c0_count": 18,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_42668"
      },
      {
        "c0_count": 21,
        "c1_count": 0,
        "c3_count": 23,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_comm, add_left_comm]\n  nlinarith [sq_nonneg (x^2 - 1), sq_nonneg x]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_9565"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  nlinarith [sq_nonneg (y * z - x^2),sq_nonneg (z * x - y^2),sq_nonneg (x * y - z^2)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_43770"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  rw [Real.cos_three_mul]\n```",
        "mode_id": "6727b129361f964fc8a2040a9b2bc5b69b94619b7acbe03bf2fe19e5f2dea9d0",
        "tactic_heads": [
          "rw"
        ],
        "theorem": "lean_workbook_plus_4399"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  ring_nf\n  nlinarith [sq_nonneg (x - y), sq_nonneg (y - z), sq_nonneg (z - x)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_13349"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  contrapose! h\n  nlinarith [h.1, h.2.1, h.2.2]\n```",
        "mode_id": "45381077fe10a2fa3602472a75a50f2652daeb8b5d68ebd10495c46cc1a95af6",
        "tactic_heads": [
          "contrapose",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_27320"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  ring_nf at h \u22a2\n  nlinarith [sq_nonneg (x + y + z)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_55251"
      },
      {
        "c0_count": 9,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  subst hf\n  simp [hx]\n```",
        "mode_id": "1a9868bb662f25d8aea5b674903ac7cc9f4dfdb32bdf527e021b8521d0a95087",
        "tactic_heads": [
          "subst",
          "simp"
        ],
        "theorem": "lean_workbook_plus_75048"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  simp [sq, sub_eq_add_neg, add_assoc]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_25308"
      },
      {
        "c0_count": 11,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  simp [sq]\n  nlinarith [sq_nonneg (x - y)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_62199"
      },
      {
        "c0_count": 14,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  simp [sq, add_comm, add_left_comm, add_assoc]\n  nlinarith [sq_nonneg (c + d - 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_74871"
      },
      {
        "c0_count": 16,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  simp only [sq]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_32628"
      },
      {
        "c0_count": 17,
        "c1_count": 0,
        "c3_count": 22,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_comm, add_left_comm, mul_comm, mul_left_comm]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_49026"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 n\n```",
        "mode_id": "341efbf0b85bb18d7d4c81de63f908f1773aa22b2419756e43ae77e05d9a9e28",
        "tactic_heads": [
          "simpa"
        ],
        "theorem": "lean_workbook_plus_56812"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  ring_nf at h \u22a2\n  nlinarith [sq_nonneg (x - y)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_72091"
      },
      {
        "c0_count": 5,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  ring_nf at *\n```",
        "mode_id": "ddae671e6fa9c5bb5a710d748819a3550644d1bfb0e2d877af6f7b61aac2bee1",
        "tactic_heads": [
          "ring_nf"
        ],
        "theorem": "lean_workbook_plus_9174"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  simp [add_assoc, add_comm, add_left_comm]\n  nlinarith [sq_nonneg (a - b), sq_nonneg (a + b), sq_nonneg (a - 1), sq_nonneg (b - 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_65634"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  simp [sq, add_assoc, add_comm, add_left_comm]\n  nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_3379"
      },
      {
        "c0_count": 17,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  simp [add_comm]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_34089"
      },
      {
        "c0_count": 20,
        "c1_count": 0,
        "c3_count": 21,
        "c3_example_proof": "  have h\u2083 := sq_nonneg (a - b)\n  nlinarith [h\u2081, h\u2082, h\u2083]\n```",
        "mode_id": "7f203f60d53c01904fc7c7e40a21a362bc6f122f0c87549d371c494399b0e71e",
        "tactic_heads": [
          "have",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_45659"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  congr 1\n```",
        "mode_id": "f57692659b8340f7746fca35d4d27557bc11621a8f443ea8c0e500144b3bc546",
        "tactic_heads": [
          "congr"
        ],
        "theorem": "lean_workbook_plus_44283"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  nlinarith [hx.1, hx.2.1, hx.2.2, hab, hbc, hca]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_53042"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  congr 1\n```",
        "mode_id": "f57692659b8340f7746fca35d4d27557bc11621a8f443ea8c0e500144b3bc546",
        "tactic_heads": [
          "congr"
        ],
        "theorem": "lean_workbook_plus_82449"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  nlinarith [sq_nonneg (a - d), sq_nonneg (b - c), sq_nonneg (c - b), sq_nonneg (d - e), sq_nonneg (e - a)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_51285"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  have := sq_nonneg (a - b)\n  have := sq_nonneg (a - c)\n  have := sq_nonneg (b - c)\n  nlinarith [sq_nonneg (a + b - 2 * c), sq_nonneg (a + c - 2 * b), sq_nonneg (b + c - 2 * a)]\n```",
        "mode_id": "986c1e7e9b6ac7f68095ae93d6922d880ba2da90cc6e8c7d11026852e10fea2c",
        "tactic_heads": [
          "have",
          "have",
          "have",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_31508"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  simp [\u2190 pow_add]\n```",
        "mode_id": "3db44c329926f443b09e3ca6f154e7791d1ebb924454ba535b37564daafe689d",
        "tactic_heads": [
          "simp"
        ],
        "theorem": "lean_workbook_plus_20172"
      },
      {
        "c0_count": 15,
        "c1_count": 0,
        "c3_count": 20,
        "c3_example_proof": "  simp [sq, sub_eq_add_neg, add_assoc, add_comm, add_left_comm]\n  nlinarith [sq_nonneg (x + (-y + (-z)))]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_27601"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  refine' \u27e8_, _, _, _, _\u27e9\n  all_goals linarith [h1.1, h1.2.1, h1.2.2.1, h1.2.2.2.1, h2.1, h2.2.1, h2.2.2.1, h2.2.2.2]\n```",
        "mode_id": "77d03d87f325a00a1664e88e9613d813f82ad8a5400b46e7f5b46474ae6df56a",
        "tactic_heads": [
          "refine'",
          "all_goals"
        ],
        "theorem": "lean_workbook_plus_28637"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  nlinarith [sq_nonneg (a * b - b * c), sq_nonneg (b * c - c * a), sq_nonneg (a * b - c * a)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_44927"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  use (a^2 + b^2 + c^2)\n  have h1 : c = -a - b := by linarith\n  rw [h1]\n  ring\n```",
        "mode_id": "8ce89596f844765b9fd73ee5a81eda51d3441c223bf49e041ceb82c575a16dfd",
        "tactic_heads": [
          "use",
          "have",
          "rw",
          "ring"
        ],
        "theorem": "lean_workbook_plus_52507"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  ring_nf\n  nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (a - c)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_80927"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  have := sq_nonneg (a - b)\n  have := sq_nonneg (a - c)\n  have := sq_nonneg (b - c)\n  nlinarith [sq_nonneg (a + b - 2 * c), sq_nonneg (a + c - 2 * b), sq_nonneg (b + c - 2 * a)]\n```",
        "mode_id": "986c1e7e9b6ac7f68095ae93d6922d880ba2da90cc6e8c7d11026852e10fea2c",
        "tactic_heads": [
          "have",
          "have",
          "have",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_45111"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_1379"
      },
      {
        "c0_count": 12,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  simp [mul_add, add_mul, mul_comm, mul_left_comm]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_27005"
      },
      {
        "c0_count": 16,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  simp [pow_add, mul_add, add_mul, mul_comm, mul_left_comm]\n  nlinarith [hx, hn]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_30809"
      },
      {
        "c0_count": 16,
        "c1_count": 0,
        "c3_count": 19,
        "c3_example_proof": "  simp [mul_add, mul_comm, mul_left_comm, pow_two, pow_three]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_63764"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  subst h\u2081 h\u2082\n  ring\n```",
        "mode_id": "df36e84afec6f410e844fd2174415bc6d9610f5d5d428fc004e945461ae640c9",
        "tactic_heads": [
          "subst",
          "ring"
        ],
        "theorem": "lean_workbook_plus_40842"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  cases' h\u2081 with h\u2081_left h\u2081_right\n  nlinarith [sq_nonneg (a (5 * k - 4)), sq_nonneg (a (5 * k - 3)), sq_nonneg (a (5 * k - 2)), sq_nonneg (a (5 * k - 1)), sq_nonneg (a (5 * k))]\n```",
        "mode_id": "d6a61111c06a48a41e3859bc73def6111eda7252ce2529c9d2ef179b92f4a1d9",
        "tactic_heads": [
          "cases'",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_5909"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  refine' \u27e8_, rfl\u27e9\n```",
        "mode_id": "c052c7897414bb50b3b9f03bd14567b65f2409c93df75dcfe1d963a0a225e140",
        "tactic_heads": [
          "refine'"
        ],
        "theorem": "lean_workbook_plus_30741"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  constructor\n  intro h1\n  omega\n  intro h2\n  omega\n```",
        "mode_id": "c01aac464cf3a5517d84655476f71b8e188199fa23ca2760cecb04685258d7fb",
        "tactic_heads": [
          "constructor",
          "intro",
          "omega",
          "intro",
          "omega"
        ],
        "theorem": "lean_workbook_plus_37865"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  use fun _ => 0\n  constructor\n  exact continuous_const\n  intro a ha b\n  norm_num\n```",
        "mode_id": "f9d3c64914af0deeaeb2c7f9e35680f9b4b5906325b7abdcf7cd4ceb3da5c3ea",
        "tactic_heads": [
          "use",
          "constructor",
          "exact",
          "intro",
          "norm_num"
        ],
        "theorem": "lean_workbook_plus_49092"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  refine' \u27e8_, _\u27e9\n  linarith\n  linarith\n```",
        "mode_id": "15643b585ce4e6bda9854c47d915f617dc79ad72e9098bb0c3f804c3a3f66735",
        "tactic_heads": [
          "refine'",
          "linarith",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_66893"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  rw [sub_eq_iff_eq_add]\n  ring\n```",
        "mode_id": "b877a1abd88518eaa88f177646389a5ded26bdca716c3c38ba5bdc6a7bdee784",
        "tactic_heads": [
          "rw",
          "ring"
        ],
        "theorem": "lean_workbook_plus_32564"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  field_simp [ha, hb]\n  ring_nf\n```",
        "mode_id": "f39c1b3c6d96667d68c15490d2684f055bd42192dd2f3014fc58d2bebc26f345",
        "tactic_heads": [
          "field_simp",
          "ring_nf"
        ],
        "theorem": "lean_workbook_plus_66099"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  ring_nf\n  nlinarith [h]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_19040"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  have := log_le_sub_one_of_pos hx\n  linarith [Real.log_le_sub_one_of_pos hx]\n```",
        "mode_id": "11c2883840300770292bef981e24cabafaaf7b831344e370541584c86206d7e1",
        "tactic_heads": [
          "have",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_29002"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  rintro a b c \u27e8ha, hb, hc\u27e9\n  simp [Real.logb, ha.ne', hb.ne', hc.ne', log_mul]\n  ring\n```",
        "mode_id": "0f2e8ce5a00b86b26b985bb0ff27e9b0fbba578461576a647211956968972456",
        "tactic_heads": [
          "rintro",
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_34341"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  simp [norm_mul, h]\n```",
        "mode_id": "3db44c329926f443b09e3ca6f154e7791d1ebb924454ba535b37564daafe689d",
        "tactic_heads": [
          "simp"
        ],
        "theorem": "lean_workbook_plus_37727"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_comm, add_left_comm]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_43792"
      },
      {
        "c0_count": 10,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  simp [add_comm, add_left_comm]\n  nlinarith [sq_nonneg (x + z)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_49846"
      },
      {
        "c0_count": 13,
        "c1_count": 0,
        "c3_count": 18,
        "c3_example_proof": "  simp [sub_eq_add_neg, mul_add, mul_comm, mul_left_comm]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_22085"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  rw [add_assoc]\n  ring_nf\n```",
        "mode_id": "546b453cfe13b54953cc919f1d179eccd00877e039933fa4198c4befde4159db",
        "tactic_heads": [
          "rw",
          "ring_nf"
        ],
        "theorem": "lean_workbook_plus_11942"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  exact (by norm_num : (3^33 + 77) % 100 = 0)\n```",
        "mode_id": "c9680df45a04a31d289589dee59ba91ccf7927d589105ff7a089fdd9a8f9e341",
        "tactic_heads": [
          "exact"
        ],
        "theorem": "lean_workbook_plus_3463"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  refine' (log_le_sub_one_of_pos _).trans _\n  linarith\n  linarith\n```",
        "mode_id": "15643b585ce4e6bda9854c47d915f617dc79ad72e9098bb0c3f804c3a3f66735",
        "tactic_heads": [
          "refine'",
          "linarith",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_55466"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  rw [sq]\n  nlinarith [sq_nonneg (x + 100 - x)]\n```",
        "mode_id": "643ae3604974a3819953094fe5218fe1d4a02c5b2a6e2be15bc3eaf17ffd767a",
        "tactic_heads": [
          "rw",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_76123"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  intro x\n  constructor\n  intro hx\n  nlinarith\n  intro hx\n  nlinarith [hx, sq_nonneg (x - 1)]\n```",
        "mode_id": "6082d31b497c787e251d07fdbd163f723cbe133a1d861494d1e52983eb020b36",
        "tactic_heads": [
          "intro",
          "constructor",
          "intro",
          "nlinarith",
          "intro",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_7907"
      },
      {
        "c0_count": 3,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  specialize h x\n  assumption\n```",
        "mode_id": "d931c2c2c80991129424af43921b49044eeb1b3ef5e5e4f4c2661be92273ad8a",
        "tactic_heads": [
          "specialize",
          "assumption"
        ],
        "theorem": "lean_workbook_plus_29855"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  intro a b c d\n  simp [mul_add, add_mul, mul_comm, mul_left_comm]\n  linarith [sq_nonneg (a - b), sq_nonneg (a - c), sq_nonneg (a - d), sq_nonneg (b - c), sq_nonneg (b - d), sq_nonneg (c - d)]\n```",
        "mode_id": "f5aa2735530bdfd4a50dcf0b4ab6ce055966baf9128184fea56330c632f02bbe",
        "tactic_heads": [
          "intro",
          "simp",
          "linarith"
        ],
        "theorem": "lean_workbook_plus_12893"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  ring_nf\n  nlinarith [h\u2081, h\u2082]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_14309"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  ring_nf\n  nlinarith [sq_nonneg (a - b), sq_nonneg (a + b), sq_nonneg (a - 1/2), sq_nonneg (b - 1/2)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_74750"
      },
      {
        "c0_count": 4,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  ring_nf\n  nlinarith [sq_nonneg (a + b), sq_nonneg (a - b)]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_79425"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp only [sq]\n  nlinarith\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_13023"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  intro n\n  simp [add_assoc, add_comm, add_left_comm]\n  omega\n```",
        "mode_id": "ff3334c38e17e31a28401bbb69ceb863d776b23e588377d3597b898b142e37ea",
        "tactic_heads": [
          "intro",
          "simp",
          "omega"
        ],
        "theorem": "lean_workbook_plus_18708"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  nlinarith [mul_pos (sub_pos.mpr hy) (sub_pos.mpr hz)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_29043"
      },
      {
        "c0_count": 6,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  norm_num [hx]\n```",
        "mode_id": "1527b27f2e1bc951dba0fe6ba8a371d859c076a7f6659b13a63dabb14b40558e",
        "tactic_heads": [
          "norm_num"
        ],
        "theorem": "lean_workbook_plus_63992"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  ring_nf\n  nlinarith [sq_nonneg (x - y), sq_nonneg (x - z), sq_nonneg (y - z), h\u2081, h\u2082]\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_13465"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  ring_nf at *\n  nlinarith\n```",
        "mode_id": "a5a40efc2a9247817ec7260c8e47ebffb12f04f19cf26b21d8f826aa01eccc2c",
        "tactic_heads": [
          "ring_nf",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_60085"
      },
      {
        "c0_count": 7,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  nlinarith [sq_nonneg (x + y + z), sq_nonneg (x - y), sq_nonneg (x - z), sq_nonneg (y - z)]\n```",
        "mode_id": "686c8788912d81560c26b3f4eb23766d99eb0152dc5f0d63eb620c88632f2e58",
        "tactic_heads": [
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_69024"
      },
      {
        "c0_count": 8,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  refine' Set.ext fun x => _\n  constructor <;> simp (config := { contextual := true })\n```",
        "mode_id": "7f168235786d1c4879d7f0bfc187d153b8f9e8b4cc60aa54b55c133d480cc80c",
        "tactic_heads": [
          "refine'",
          "constructor"
        ],
        "theorem": "lean_workbook_plus_62290"
      },
      {
        "c0_count": 9,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  use 0\n  simp only [h, zero_mul, pow_two]\n  nlinarith\n```",
        "mode_id": "831b532301553cea954291506a5da98aa1ed2a6cf68d00b9c565b6ebd7096c77",
        "tactic_heads": [
          "use",
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_3459"
      },
      {
        "c0_count": 9,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp [sub_eq_add_neg, add_comm, add_left_comm, mul_comm, mul_left_comm]\n  nlinarith [sq_nonneg (u - v), sq_nonneg (u + v), sq_nonneg (v - u)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_8773"
      },
      {
        "c0_count": 11,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  constructor\n  intro h1\n  omega\n  intro h2\n  omega\n```",
        "mode_id": "c01aac464cf3a5517d84655476f71b8e188199fa23ca2760cecb04685258d7fb",
        "tactic_heads": [
          "constructor",
          "intro",
          "omega",
          "intro",
          "omega"
        ],
        "theorem": "lean_workbook_plus_41584"
      },
      {
        "c0_count": 12,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp [sq]\n  nlinarith [sq_nonneg (a - 1), sq_nonneg (a + 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_48060"
      },
      {
        "c0_count": 14,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp [add_assoc]\n  nlinarith [sq_nonneg (p * q - (p + q))]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_78157"
      },
      {
        "c0_count": 17,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp [mul_add, mul_comm, mul_left_comm]\n  ring\n```",
        "mode_id": "fa785e66ce14a18a8f82a85504c717a60c9777a067d937dc264fa6b3a4115fb9",
        "tactic_heads": [
          "simp",
          "ring"
        ],
        "theorem": "lean_workbook_plus_79951"
      },
      {
        "c0_count": 22,
        "c1_count": 0,
        "c3_count": 17,
        "c3_example_proof": "  simp [add_mul, mul_add]\n  nlinarith [sq_nonneg (x - y), sq_nonneg (x - 1), sq_nonneg (y - 1)]\n```",
        "mode_id": "f16ad6b82d29ebdc24e37dc75dae87d57447b38441c795de93f53f58d7d2bd56",
        "tactic_heads": [
          "simp",
          "nlinarith"
        ],
        "theorem": "lean_workbook_plus_2944"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 16,
        "c3_example_proof": "  rintro m \u27e8h1, h2\u27e9\n  use Real.arcsin m\n  constructor\n  exact \u27e8by linarith [neg_pi_div_two_le_arcsin m], by linarith [arcsin_le_pi_div_two m]\u27e9\n  rw [Real.sin_arcsin h1 h2]\n```",
        "mode_id": "cbb1709f891ff663878eddd6759e3f6cb9141f4b0a8c4c0f15b18b9061711f23",
        "tactic_heads": [
          "rintro",
          "use",
          "constructor",
          "exact",
          "rw"
        ],
        "theorem": "lean_workbook_plus_43176"
      },
      {
        "c0_count": 2,
        "c1_count": 0,
        "c3_count": 16,
        "c3_example_proof": "  rw [add_assoc]\n  ring\n```",
        "mode_id": "b877a1abd88518eaa88f177646389a5ded26bdca716c3c38ba5bdc6a7bdee784",
        "tactic_heads": [
          "rw",
          "ring"
        ],
        "theorem": "lean_workbook_plus_47860"
      }
    ]
  },
  "claim_boundary": "Mode absence or recovery is relative to finite 32-proposal samples and deterministic tactic signatures; it is not proof of semantic strategy loss or creation.",
  "correct_rollout_rarefaction": {
    "c1": {
      "1": {
        "correct_draws": 1,
        "eligible_theorems": 7932,
        "mean_expected_modes": 1.0
      },
      "16": {
        "correct_draws": 16,
        "eligible_theorems": 6901,
        "mean_expected_modes": 3.638497925129634
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      "2": {
        "correct_draws": 2,
        "eligible_theorems": 7735,
        "mean_expected_modes": 1.4594932798319882
      },
      "4": {
        "correct_draws": 4,
        "eligible_theorems": 7554,
        "mean_expected_modes": 2.044726319896371
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      "8": {
        "correct_draws": 8,
        "eligible_theorems": 7271,
        "mean_expected_modes": 2.7712636972705305
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    },
    "c3": {
      "1": {
        "correct_draws": 1,
        "eligible_theorems": 8078,
        "mean_expected_modes": 1.0
      },
      "16": {
        "correct_draws": 16,
        "eligible_theorems": 6864,
        "mean_expected_modes": 5.471650216672157
      },
      "2": {
        "correct_draws": 2,
        "eligible_theorems": 7884,
        "mean_expected_modes": 1.6565950180959415
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      "4": {
        "correct_draws": 4,
        "eligible_theorems": 7653,
        "mean_expected_modes": 2.595138830074835
      },
      "8": {
        "correct_draws": 8,
        "eligible_theorems": 7346,
        "mean_expected_modes": 3.861833247090533
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    },
    "paired_common_theorems": {
      "1": {
        "c1_mean_expected_modes": 1.0,
        "c3_mean_expected_modes": 1.0,
        "common_eligible_theorems": 7811,
        "correct_draws": 1,
        "mean_delta_c3_minus_c1": 0.0,
        "theorems_c1_higher": 0,
        "theorems_c3_higher": 0,
        "theorems_equal": 7811
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      "16": {
        "c1_mean_expected_modes": 3.638618786133925,
        "c3_mean_expected_modes": 5.462243982177484,
        "common_eligible_theorems": 6644,
        "correct_draws": 16,
        "mean_delta_c3_minus_c1": 1.8236251960435577,
        "theorems_c1_higher": 1094,
        "theorems_c3_higher": 5413,
        "theorems_equal": 137
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      "2": {
        "c1_mean_expected_modes": 1.4587119620501299,
        "c3_mean_expected_modes": 1.655427287926616,
        "common_eligible_theorems": 7629,
        "correct_draws": 2,
        "mean_delta_c3_minus_c1": 0.1967153258764905,
        "theorems_c1_higher": 1617,
        "theorems_c3_higher": 5809,
        "theorems_equal": 203
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      "4": {
        "c1_mean_expected_modes": 2.0437348201634706,
        "c3_mean_expected_modes": 2.589989882666942,
        "common_eligible_theorems": 7415,
        "correct_draws": 4,
        "mean_delta_c3_minus_c1": 0.5462550625034702,
        "theorems_c1_higher": 1486,
        "theorems_c3_higher": 5772,
        "theorems_equal": 157
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      "8": {
        "c1_mean_expected_modes": 2.770240177275343,
        "c3_mean_expected_modes": 3.855179862644322,
        "common_eligible_theorems": 7112,
        "correct_draws": 8,
        "mean_delta_c3_minus_c1": 1.0849396853689868,
        "theorems_c1_higher": 1273,
        "theorems_c3_higher": 5693,
        "theorems_equal": 146
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  "overall": {
    "c1": {
      "blocked_correct": 0,
      "correct": 215482,
      "correct_mode_coverage": 34336,
      "correct_rate": 0.6974430346970482,
      "exact_proof_coverage": 66515,
      "exact_proofs_per_correct_rollout": 0.30868007536592384,
      "mean_effective_modes": 2.4918579252831035,
      "mean_modes_per_solved_theorem": 4.328794755421079,
      "mean_top_mode_share": 0.6745715465727168,
      "modes_per_correct_rollout": 0.15934509611011594,
      "proposals": 308960,
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      "theorems": 9655
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    "c3": {
      "blocked_correct": 13906,
      "correct": 211635,
      "correct_mode_coverage": 53825,
      "correct_rate": 0.6849915846711548,
      "exact_proof_coverage": 111570,
      "exact_proofs_per_correct_rollout": 0.527181231837834,
      "mean_effective_modes": 3.6728016545235254,
      "mean_modes_per_solved_theorem": 6.663159197821243,
      "mean_top_mode_share": 0.5191676632722008,
      "modes_per_correct_rollout": 0.2543293878611761,
      "proposals": 308960,
      "solved_theorems": 8078,
      "theorems": 9655
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  },
  "sources": {
    "c0_archive": {
      "path": "/scratch/memoozd/rl/restriction/runs/c0-base-20260804-seed42-complete/mode_archive.json",
      "sha256": "fcffb4a3dc9baf837d4780ec30855308ebc7f0c5086d607162889938b5e426d3"
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      "physical_proposals": 308992,
      "proof_log": "/scratch/memoozd/rl/restriction/runs/c1-grpo-default-full-20260814-seed42-5ced4c3/artifacts/proofs/global_step_604.jsonl",
      "proof_log_sha256": "1db715569b8d1d8d7abf558bfd0c0c9b59779fd2ae7e959af0a31f0bb622d9f0",
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      "proof_log_sha256": "6e3d8b0df13b877f7749728d101cfa127345cf6175859f91f2bf84902108ad95",
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