TH0010

hensel_linear_correction_multiple

A root-correction divisibility witness makes the complete first-order lifted value a multiple of the next modulus.

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

∀ m. ∀ d. ∀ q. ∀ t. ∀ p. Dvd(p,q + d · t) → Dvd(p · m,m · q + m · t · d)

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

Definition DAG

Dvd(d,n) · 2

Actual proof prerequisites

mul_add · checked external prerequisitemul_assoc · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall m d q t p. (exists w. q + d * t = p * w) -> exists w. m * q + (m * t) * d = (p * m) * w

Complete unchanged native tactic proof

All 27 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

27 script commands · 14 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 m
  2. L2
    intro d
  3. L3
    intro q
  4. L4
    intro t
  5. L5
    intro p
  6. L6
    intro hmultiple
02Separate the logical casesL7–7

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

  1. L7
    cases hmultiple
03Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists x
04Calculate and transport equalitiesL9–11

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    trans m * (q + d * t)
  2. L10
    symm
  3. L11
    trans m * q + m * (d * t)
05Use earlier factsL12–12

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

  1. L12
    apply mul_add
06Calculate and transport equalitiesL13–17

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L13
    congr
  2. L14
    refl
  3. L15
    trans m * (t * d)
  4. L16
    congr
  5. L17
    refl
07Use earlier factsL18–18

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

  1. L18
    apply mul_comm
08Calculate and transport equalitiesL19–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    symm
09Use earlier factsL20–20

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

  1. L20
    apply mul_assoc
10Calculate and transport equalitiesL21–23

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L21
    rewrite hmultiple_witness
  2. L22
    trans (m * p) * x
  3. L23
    symm
11Use earlier factsL24–24

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

  1. L24
    apply mul_assoc
12Calculate and transport equalitiesL25–25

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L25
    congr
13Use earlier factsL26–26

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

  1. L26
    apply mul_comm
14Calculate and transport equalitiesL27–27

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L27
    refl

Library-wide reading audit

Original defined command ledger · 27 lines
  1. 0001intro m
  2. 0002intro d
  3. 0003intro q
  4. 0004intro t
  5. 0005intro p
  6. 0006intro hmultiple
  7. 0007cases hmultiple
  8. 0008exists x
  9. 0009trans m * (q + d * t)
  10. 0010symm
  11. 0011trans m * q + m * (d * t)
  12. 0012apply mul_add
  13. 0013congr
  14. 0014refl
  15. 0015trans m * (t * d)
  16. 0016congr
  17. 0017refl
  18. 0018apply mul_comm
  19. 0019symm
  20. 0020apply mul_assoc
  21. 0021rewrite hmultiple_witness
  22. 0022trans (m * p) * x
  23. 0023symm
  24. 0024apply mul_assoc
  25. 0025congr
  26. 0026apply mul_comm
  27. 0027refl