HL0004

hensel_lift_digit_bound

A bounded old representative and bounded correction digit give the exact next-modulus bound.

Alpha v34 checked-use · first admitted v27 · 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.

The derivative-nonzero criterion supplies no inverse or power witness: both are constructed. Roots may be arbitrary natural representatives of signed integer polynomials. Singular-root classification and p-adic completion are separate milestones.

Exact theorem in conservative defined notation

∀ m. ∀ p. ∀ a. ∀ t. Lt(a,m) → Lt(t,p) → Lt(a + m · t,p · m)

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

Definition DAG

Lt(a,b) · 3

Actual proof prerequisites

division_block_upper · checked external prerequisitemul_le_mul_left · checked external prerequisitelt_of_lt_of_le · checked external prerequisiteadd_comm · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order statement 
forall m p a t. (exists hpl_gap_bound. hpl_gap_bound + S (a) = (m)) -> (exists hpl_gap_bound. hpl_gap_bound + S (t) = (p)) -> (exists hpl_gap_bound. hpl_gap_bound + S (a + m * t) = (p * m))

Complete tactic proof in conservative notation

All 26 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

26 script commands · 4 reading checkpoints · 2 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 p
  3. L3
    intro a
  4. L4
    intro t
  5. L5
    intro ha
  6. L6
    intro ht
02Establish hsumL7–9

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

  1. L7
    have hsum : a + m * t = m * t + a
  2. L8
    apply add_comm
  3. L9
    rewrite hsum
03Establish hproductL10–19

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

  1. L10
    have hproduct : p * m = m * p
  2. L11
    apply mul_comm
  3. L12
    rewrite hproduct
  4. L13
    specialize lt_of_lt_of_le (m * t + a)
  5. L14
    specialize lt_of_lt_of_le (m * S t)
  6. L15
    specialize lt_of_lt_of_le (m * p)
  7. L16
    apply lt_of_lt_of_le
  8. L17
    specialize division_block_upper m
  9. L18
    specialize division_block_upper t
  10. L19
    specialize division_block_upper a
04Use earlier factsL20–26

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

  1. L20
    apply division_block_upper
  2. L21
    exact ha
  3. L22
    specialize mul_le_mul_left (S t)
  4. L23
    specialize mul_le_mul_left p
  5. L24
    specialize mul_le_mul_left m
  6. L25
    apply mul_le_mul_left
  7. L26
    exact ht

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro m
  2. 0002intro p
  3. 0003intro a
  4. 0004intro t
  5. 0005intro ha
  6. 0006intro ht
  7. 0007have hsum : a + m * t = m * t + a
  8. 0008apply add_comm
  9. 0009rewrite hsum
  10. 0010have hproduct : p * m = m * p
  11. 0011apply mul_comm
  12. 0012rewrite hproduct
  13. 0013specialize lt_of_lt_of_le (m * t + a)
  14. 0014specialize lt_of_lt_of_le (m * S t)
  15. 0015specialize lt_of_lt_of_le (m * p)
  16. 0016apply lt_of_lt_of_le
  17. 0017specialize division_block_upper m
  18. 0018specialize division_block_upper t
  19. 0019specialize division_block_upper a
  20. 0020apply division_block_upper
  21. 0021exact ha
  22. 0022specialize mul_le_mul_left (S t)
  23. 0023specialize mul_le_mul_left p
  24. 0024specialize mul_le_mul_left m
  25. 0025apply mul_le_mul_left
  26. 0026exact ht