TH0011

hensel_square_shift_multiple

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

Whenever the old modulus contains its lifting factor, every squared modulus shift is divisible by the next modulus.

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 m t p s. m = p * s -> exists w. (m * t) * (m * t) = (p * m) * w

Constructive proof overview

Generated structural guide

Whenever the old modulus contains its lifting factor, every squared modulus shift is divisible by the next modulus.

The unchanged tactic script uses 3 declared prerequisites and contains 24 exact native proof lines.

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

Proof neighborhood

Direct dependencies

mul_shuffle_four Alpha theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable 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

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.

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 exact 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

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