TH000B

hensel_correction_exists_unique

Every coprime formal derivative at a nonzero modulus has exactly one canonical bounded correction digit.

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

∀ d. ∀ p. ∀ q. ¬p = 0 → Coprime(d,p) → ∃ x. HenselCorrection(d,p,q,x) ∧ (∀ y. HenselCorrection(d,p,q,y) → y = x)

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

Definition DAG

HenselCorrection(d,p,q,t) · 2Coprime(a,b) · 1

Actual proof prerequisites

hensel_correction_existshensel_correction_unique
Original expanded first-order statement 
forall d p q. ~(p = 0) -> (forall hmi_divisor_pth_correction. (exists hmi_left_factor_pth_correction. d = hmi_divisor_pth_correction * hmi_left_factor_pth_correction) -> (exists hmi_right_factor_pth_correction. p = hmi_divisor_pth_correction * hmi_right_factor_pth_correction) -> hmi_divisor_pth_correction = 1) -> exists t. ((((exists ff_lt_pth_correction_t_bound. ff_lt_pth_correction_t_bound + S t = p) /\ (exists hgcrt_mod_left_pth_correction_t_annihilation hgcrt_mod_right_pth_correction_t_annihilation. (q + d * t) + p * hgcrt_mod_left_pth_correction_t_annihilation = 0 + p * hgcrt_mod_right_pth_correction_t_annihilation))) /\ forall u. (((exists ff_lt_pth_correction_u_bound. ff_lt_pth_correction_u_bound + S u = p) /\ (exists hgcrt_mod_left_pth_correction_u_annihilation hgcrt_mod_right_pth_correction_u_annihilation. (q + d * u) + p * hgcrt_mod_left_pth_correction_u_annihilation = 0 + p * hgcrt_mod_right_pth_correction_u_annihilation))) -> u = t)

Complete unchanged native tactic proof

All 28 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

28 script commands · 8 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.

Named ingredients (2)
hensel correction existshensel correction unique
01Fix variables and assumptionsL1–5

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

  1. L1
    intro d
  2. L2
    intro p
  3. L3
    intro q
  4. L4
    intro hp
  5. L5
    intro hcop
02Establish hexistsL6–12

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

  1. L6
    have hexists : exists t. (((exists ff_lt_pth_unique_exists_bound. ff_lt_pth_unique_exists_bound + S t = p) /\ (exists hgcrt_mod_left_pth_unique_exists_annihilation hgcrt_mod_right_pth_unique_exists_annihilation. (q + d * t) + p * hgcrt_mod_left_pth_unique_exists_annihilation = 0 + p * hgcrt_mod_right_pth_unique_exists_annihilation)))
  2. L7
    specialize hensel_correction_exists d
  3. L8
    specialize hensel_correction_exists p
  4. L9
    specialize hensel_correction_exists q
  5. L10
    apply hensel_correction_exists
  6. L11
    exact hp
  7. L12
    exact hcop
03Separate the logical casesL13–13

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

  1. L13
    cases hexists
04Construct an explicit witnessL14–14

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

  1. L14
    exists x
05Separate the logical casesL15–15

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

  1. L15
    split
06Use earlier factsL16–16

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

  1. L16
    exact hexists_witness
07Fix variables and assumptionsL17–18

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

  1. L17
    intro u
  2. L18
    intro hu
08Use earlier factsL19–28

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

  1. L19
    specialize hensel_correction_unique d
  2. L20
    specialize hensel_correction_unique p
  3. L21
    specialize hensel_correction_unique q
  4. L22
    specialize hensel_correction_unique u
  5. L23
    specialize hensel_correction_unique x
  6. L24
    apply hensel_correction_unique
  7. L25
    exact hp
  8. L26
    exact hcop
  9. L27
    exact hu
  10. L28
    exact hexists_witness

Library-wide reading audit

Original defined command ledger · 28 lines
  1. 0001intro d
  2. 0002intro p
  3. 0003intro q
  4. 0004intro hp
  5. 0005intro hcop
  6. 0006have hexists : exists t. (((exists ff_lt_pth_unique_exists_bound. ff_lt_pth_unique_exists_bound + S t = p) /\ (exists hgcrt_mod_left_pth_unique_exists_annihilation hgcrt_mod_right_pth_unique_exists_annihilation. (q + d * t) + p * hgcrt_mod_left_pth_unique_exists_annihilation = 0 + p * hgcrt_mod_right_pth_unique_exists_annihilation)))
  7. 0007specialize hensel_correction_exists d
  8. 0008specialize hensel_correction_exists p
  9. 0009specialize hensel_correction_exists q
  10. 0010apply hensel_correction_exists
  11. 0011exact hp
  12. 0012exact hcop
  13. 0013cases hexists
  14. 0014exists x
  15. 0015split
  16. 0016exact hexists_witness
  17. 0017intro u
  18. 0018intro hu
  19. 0019specialize hensel_correction_unique d
  20. 0020specialize hensel_correction_unique p
  21. 0021specialize hensel_correction_unique q
  22. 0022specialize hensel_correction_unique u
  23. 0023specialize hensel_correction_unique x
  24. 0024apply hensel_correction_unique
  25. 0025exact hp
  26. 0026exact hcop
  27. 0027exact hu
  28. 0028exact hexists_witness