TH000B

hensel_correction_exists_unique

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 28 exact native proof lines.

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

Proof neighborhood

Direct dependencies

TH0009 hensel_correction_exists TH000A hensel_correction_unique

Direct dependents

none

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

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.

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 exact 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

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