HL001D

hensel_signed_derivative_unit_mod_transport

The same bounded inverse transports an integer derivative through congruent positive and negative components.

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

∀ p. ∀ dp. ∀ dn. ∀ ep. ∀ en. ModEq(p,dp,ep) → ModEq(p,dn,en) → SignedDerivativeUnit(p,dp,dn) → SignedDerivativeUnit(p,ep,en)

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

Definition DAG

SignedDerivativeUnit(p,dp,dn) · 2ModEq(m,a,b) · 3

Actual proof prerequisites

mod_eq_trans · checked external prerequisitemod_eq_mul_right · checked external prerequisitemod_eq_symm · checked external prerequisitemod_eq_add · checked external prerequisitemod_eq_refl · checked external prerequisite
Original expanded first-order statement 
forall p dp dn ep en. (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. dp + p * hgcrt_mod_left_hpl_mod = ep + p * hgcrt_mod_right_hpl_mod) -> (exists hgcrt_mod_left_hpl_mod hgcrt_mod_right_hpl_mod. dn + p * hgcrt_mod_left_hpl_mod = en + p * hgcrt_mod_right_hpl_mod) -> (exists sph_inverse_unit. ((exists hpl_gap_unit. hpl_gap_unit + S (sph_inverse_unit) = (p)) /\ (exists hgcrt_mod_left_hpl_unit hgcrt_mod_right_hpl_unit. (dp * sph_inverse_unit) + p * hgcrt_mod_left_hpl_unit = (1 + dn * sph_inverse_unit) + p * hgcrt_mod_right_hpl_unit))) -> (exists sph_inverse_unit. ((exists hpl_gap_unit. hpl_gap_unit + S (sph_inverse_unit) = (p)) /\ (exists hgcrt_mod_left_hpl_unit hgcrt_mod_right_hpl_unit. (ep * sph_inverse_unit) + p * hgcrt_mod_left_hpl_unit = (1 + en * sph_inverse_unit) + p * hgcrt_mod_right_hpl_unit)))

Complete tactic proof in conservative notation

All 51 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

51 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro dp
  3. L3
    intro dn
  4. L4
    intro ep
  5. L5
    intro en
  6. L6
    intro hpositive
  7. L7
    intro hnegative
  8. L8
    intro hunit
02Separate the logical casesL9–10

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

  1. L9
    cases hunit
  2. L10
    cases hunit_witness
03Establish hmiddleL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mod eq trans.

  1. L11
    have hmiddle : ModEq(p,dp · x,1 + en · x)Definitions: ModEq(p,dp · x,1 + en · x)Original native command in the exact edition
  2. L12
    specialize mod_eq_trans p
  3. L13
    specialize mod_eq_trans (dp * x)
  4. L14
    specialize mod_eq_trans (1 + dn * x)
  5. L15
    specialize mod_eq_trans (1 + en * x)
  6. L16
    apply mod_eq_trans
  7. L17
    exact hunit_witness_right
  8. L18
    specialize mod_eq_add p
  9. L19
    specialize mod_eq_add 1
  10. L20
    specialize mod_eq_add 1
04Use earlier factsL21–30

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

  1. L21
    specialize mod_eq_add (dn * x)
  2. L22
    specialize mod_eq_add (en * x)
  3. L23
    apply mod_eq_add
  4. L24
    specialize mod_eq_refl p
  5. L25
    specialize mod_eq_refl 1
  6. L26
    apply mod_eq_refl
  7. L27
    specialize mod_eq_mul_right p
  8. L28
    specialize mod_eq_mul_right dn
  9. L29
    specialize mod_eq_mul_right en
  10. L30
    specialize mod_eq_mul_right x
05Use earlier factsL31–32

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

  1. L31
    apply mod_eq_mul_right
  2. L32
    exact hnegative
06Construct an explicit witnessL33–33

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

  1. L33
    exists x
07Separate the logical casesL34–34

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

  1. L34
    split
08Use earlier factsL35–44

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

  1. L35
    exact hunit_witness_left
  2. L36
    specialize mod_eq_trans p
  3. L37
    specialize mod_eq_trans (ep * x)
  4. L38
    specialize mod_eq_trans (dp * x)
  5. L39
    specialize mod_eq_trans (1 + en * x)
  6. L40
    apply mod_eq_trans
  7. L41
    specialize mod_eq_mul_right p
  8. L42
    specialize mod_eq_mul_right ep
  9. L43
    specialize mod_eq_mul_right dp
  10. L44
    specialize mod_eq_mul_right x
09Use earlier factsL45–51

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

  1. L45
    apply mod_eq_mul_right
  2. L46
    specialize mod_eq_symm p
  3. L47
    specialize mod_eq_symm dp
  4. L48
    specialize mod_eq_symm ep
  5. L49
    apply mod_eq_symm
  6. L50
    exact hpositive
  7. L51
    exact hmiddle

Library-wide reading audit

Original defined command ledger · 51 lines
  1. 0001intro p
  2. 0002intro dp
  3. 0003intro dn
  4. 0004intro ep
  5. 0005intro en
  6. 0006intro hpositive
  7. 0007intro hnegative
  8. 0008intro hunit
  9. 0009cases hunit
  10. 0010cases hunit_witness
  11. 0011have hmiddle : ModEq(p,dp · x,1 + en · x)
  12. 0012specialize mod_eq_trans p
  13. 0013specialize mod_eq_trans (dp * x)
  14. 0014specialize mod_eq_trans (1 + dn * x)
  15. 0015specialize mod_eq_trans (1 + en * x)
  16. 0016apply mod_eq_trans
  17. 0017exact hunit_witness_right
  18. 0018specialize mod_eq_add p
  19. 0019specialize mod_eq_add 1
  20. 0020specialize mod_eq_add 1
  21. 0021specialize mod_eq_add (dn * x)
  22. 0022specialize mod_eq_add (en * x)
  23. 0023apply mod_eq_add
  24. 0024specialize mod_eq_refl p
  25. 0025specialize mod_eq_refl 1
  26. 0026apply mod_eq_refl
  27. 0027specialize mod_eq_mul_right p
  28. 0028specialize mod_eq_mul_right dn
  29. 0029specialize mod_eq_mul_right en
  30. 0030specialize mod_eq_mul_right x
  31. 0031apply mod_eq_mul_right
  32. 0032exact hnegative
  33. 0033exists x
  34. 0034split
  35. 0035exact hunit_witness_left
  36. 0036specialize mod_eq_trans p
  37. 0037specialize mod_eq_trans (ep * x)
  38. 0038specialize mod_eq_trans (dp * x)
  39. 0039specialize mod_eq_trans (1 + en * x)
  40. 0040apply mod_eq_trans
  41. 0041specialize mod_eq_mul_right p
  42. 0042specialize mod_eq_mul_right ep
  43. 0043specialize mod_eq_mul_right dp
  44. 0044specialize mod_eq_mul_right x
  45. 0045apply mod_eq_mul_right
  46. 0046specialize mod_eq_symm p
  47. 0047specialize mod_eq_symm dp
  48. 0048specialize mod_eq_symm ep
  49. 0049apply mod_eq_symm
  50. 0050exact hpositive
  51. 0051exact hmiddle