TH0011

hensel_square_shift_multiple

Whenever the old modulus contains its lifting factor, every squared modulus shift is divisible by 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. ∀ t. ∀ p. ∀ s. m = p · s → Dvd(p · m,m · t · (m · t))

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

Definition DAG

Dvd(d,n) · 1

Actual proof prerequisites

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

Complete unchanged native tactic proof

All 24 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

24 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 assumptionsL1–5

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

  1. L1
    intro m
  2. L2
    intro t
  3. L3
    intro p
  4. L4
    intro s
  5. L5
    intro hfactor
02Construct an explicit witnessL6–6

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

  1. L6
    exists s * (t * t)
03Calculate and transport equalitiesL7–7

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

  1. L7
    trans (m * m) * (t * t)
04Use earlier factsL8–8

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

  1. L8
    apply mul_shuffle_four
05Calculate and transport equalitiesL9–12

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

  1. L9
    trans ((p * m) * s) * (t * t)
  2. L10
    congr
  3. L11
    trans (p * s) * m
  4. L12
    congr
06Use earlier factsL13–13

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

  1. L13
    exact hfactor
07Calculate and transport equalitiesL14–15

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

  1. L14
    refl
  2. L15
    trans p * (s * m)
08Use earlier factsL16–16

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

  1. L16
    apply mul_assoc
09Calculate and transport equalitiesL17–19

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

  1. L17
    trans p * (m * s)
  2. L18
    congr
  3. L19
    refl
10Use earlier factsL20–20

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

  1. L20
    apply mul_comm
11Calculate and transport equalitiesL21–21

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

  1. L21
    symm
12Use earlier factsL22–22

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

  1. L22
    apply mul_assoc
13Calculate and transport equalitiesL23–23

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

  1. L23
    refl
14Use earlier factsL24–24

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

  1. L24
    apply mul_assoc

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro m
  2. 0002intro t
  3. 0003intro p
  4. 0004intro s
  5. 0005intro hfactor
  6. 0006exists s * (t * t)
  7. 0007trans (m * m) * (t * t)
  8. 0008apply mul_shuffle_four
  9. 0009trans ((p * m) * s) * (t * t)
  10. 0010congr
  11. 0011trans (p * s) * m
  12. 0012congr
  13. 0013exact hfactor
  14. 0014refl
  15. 0015trans p * (s * m)
  16. 0016apply mul_assoc
  17. 0017trans p * (m * s)
  18. 0018congr
  19. 0019refl
  20. 0020apply mul_comm
  21. 0021symm
  22. 0022apply mul_assoc
  23. 0023refl
  24. 0024apply mul_assoc