TH000F

hensel_correction_implies_multiple

Every verified bounded Hensel correction supplies an actual natural divisibility witness for its annihilated linear residual.

Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Historical partial components only: this chapter proves exact natural polynomial Taylor remainders, bounded corrections, and one-step divisibility lifts. G095 is now closed in the separate Alpha-v27 hensel-lifting branch for integer polynomials, unrestricted input roots, unique canonical representatives, and every positive prime power. Full G095 proof · Alpha v27

Exact theorem in conservative defined notation

∀ d. ∀ p. ∀ q. ∀ t. ¬p = 0 → HenselCorrection(d,p,q,t) → Dvd(p,q + d · t)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

HenselCorrection(d,p,q,t) · 1Dvd(d,n) · 1

Actual proof prerequisites

mod_eq_zero_to_dvd_nonzero · checked external prerequisite
Original expanded first-order statement 
forall d p q t. ~(p = 0) -> (((exists ff_lt_pth_correction_exists_bound. ff_lt_pth_correction_exists_bound + S t = p) /\ (exists hgcrt_mod_left_pth_correction_exists_annihilation hgcrt_mod_right_pth_correction_exists_annihilation. (q + d * t) + p * hgcrt_mod_left_pth_correction_exists_annihilation = 0 + p * hgcrt_mod_right_pth_correction_exists_annihilation))) -> exists w. q + d * t = p * w

Complete unchanged native tactic proof

All 12 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

12 script commands · 3 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–6

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro d
  2. L2
    intro p
  3. L3
    intro q
  4. L4
    intro t
  5. L5
    intro hp
  6. L6
    intro ht
02Separate the logical casesL7–7

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L7
    cases ht
03Use earlier factsL8–12

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L8
    specialize mod_eq_zero_to_dvd_nonzero p
  2. L9
    specialize mod_eq_zero_to_dvd_nonzero (q + d * t)
  3. L10
    apply mod_eq_zero_to_dvd_nonzero
  4. L11
    exact hp
  5. L12
    exact ht_right

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro d
  2. 0002intro p
  3. 0003intro q
  4. 0004intro t
  5. 0005intro hp
  6. 0006intro ht
  7. 0007cases ht
  8. 0008specialize mod_eq_zero_to_dvd_nonzero p
  9. 0009specialize mod_eq_zero_to_dvd_nonzero (q + d * t)
  10. 0010apply mod_eq_zero_to_dvd_nonzero
  11. 0011exact hp
  12. 0012exact ht_right