HL0003

hensel_canonical_residue_exists

Every unrestricted natural input has a constructed canonical residue at every nonzero modulus.

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. ∀ a. ¬m = 0 → ∃ x. Lt(x,m) ∧ ModEq(m,a,x)

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

Definition DAG

Lt(a,b) · 2ModEq(m,a,b) · 1

Actual proof prerequisites

division_remainder_exists · checked external prerequisiteadd_comm · checked external prerequisite
Original expanded first-order statement 
forall m a. ~(m = 0) -> exists r. ((exists hpl_gap_bound. hpl_gap_bound + S (r) = (m)) /\ (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. a + m * hgcrt_mod_left_hpl_mod = r + m * hgcrt_mod_right_hpl_mod))

Complete tactic proof in conservative notation

All 21 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

21 script commands · 9 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.

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 assumptionsL1–3

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

  1. L1
    intro m
  2. L2
    intro a
  3. L3
    intro hm
02Establish hdivL4–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder exists.

  1. L4
    have hdiv : ∃ q. ∃ r. a = m · q + r ∧ Lt(r,m)Definitions: Lt(r,m)Original native command in the exact edition
  2. L5
    specialize division_remainder_exists m
  3. L6
    specialize division_remainder_exists a
  4. L7
    apply division_remainder_exists
  5. L8
    exact hm
03Separate the logical casesL9–11

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

  1. L9
    cases hdiv
  2. L10
    cases hdiv_witness
  3. L11
    cases hdiv_witness_witness
04Construct an explicit witnessL12–12

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

  1. L12
    exists x1
05Separate the logical casesL13–13

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

  1. L13
    split
06Use earlier factsL14–14

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

  1. L14
    exact hdiv_witness_witness_right
07Construct an explicit witnessL15–16

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

  1. L15
    exists 0
  2. L16
    exists x
08Calculate and transport equalitiesL17–19

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

  1. L17
    trans a
  2. L18
    simp
  3. L19
    trans m * x + x1
09Use earlier factsL20–21

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

  1. L20
    exact hdiv_witness_witness_left
  2. L21
    apply add_comm

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro m
  2. 0002intro a
  3. 0003intro hm
  4. 0004have hdiv : ∃ q. ∃ r. a = m · q + r ∧ Lt(r,m)
  5. 0005specialize division_remainder_exists m
  6. 0006specialize division_remainder_exists a
  7. 0007apply division_remainder_exists
  8. 0008exact hm
  9. 0009cases hdiv
  10. 0010cases hdiv_witness
  11. 0011cases hdiv_witness_witness
  12. 0012exists x1
  13. 0013split
  14. 0014exact hdiv_witness_witness_right
  15. 0015exists 0
  16. 0016exists x
  17. 0017trans a
  18. 0018simp
  19. 0019trans m * x + x1
  20. 0020exact hdiv_witness_witness_left
  21. 0021apply add_comm