TH000F

hensel_correction_implies_multiple

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

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.

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 12 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mod_eq_zero_to_dvd_nonzero Alpha theorem; checked-use authorized

Direct dependents

TH0012 beta_horner_hensel_lift_divisibility

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

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 exact 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

Separate complete second-wave branches: Full G095 proof · Alpha v27.