TH0001

hensel_predecessor_annihilates_residue

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

The predecessor of every positive modulus gives a witnessed subtraction-free negative residue.

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 k q. (exists hgcrt_mod_left_pth_negative hgcrt_mod_right_pth_negative. (q + k * q) + (S k) * hgcrt_mod_left_pth_negative = 0 + (S k) * hgcrt_mod_right_pth_negative)

Constructive proof overview

Generated structural guide

The predecessor of every positive modulus gives a witnessed subtraction-free negative residue.

The unchanged tactic script uses 2 declared prerequisites and contains 9 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mod_eq_predecessor_cancel Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

TH0009 hensel_correction_exists

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

9 script commands · 3 reading checkpoints · 1 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 assumptionsL1–2

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

  1. L1
    intro k
  2. L2
    intro q
02Use earlier factsL3–5

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

  1. L3
    specialize mod_eq_predecessor_cancel k
  2. L4
    specialize mod_eq_predecessor_cancel 0
  3. L5
    specialize mod_eq_predecessor_cancel q
03Establish hzL6–9

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply zero add.

  1. L6
    have hz : 0 + q = q
  2. L7
    apply zero_add
  3. L8
    rewrite hz at mod_eq_predecessor_cancel
  4. L9
    exact mod_eq_predecessor_cancel

Library-wide reading audit

Original exact command ledger · 9 lines
  1. 0001intro k
  2. 0002intro q
  3. 0003specialize mod_eq_predecessor_cancel k
  4. 0004specialize mod_eq_predecessor_cancel 0
  5. 0005specialize mod_eq_predecessor_cancel q
  6. 0006have hz : 0 + q = q
  7. 0007apply zero_add
  8. 0008rewrite hz at mod_eq_predecessor_cancel
  9. 0009exact mod_eq_predecessor_cancel

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