HL000E

hensel_canonical_horner_root_exists_unique

At iteration zero every unrestricted root has exactly one representative in its own canonical residue interval.

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

∀ b. ∀ c. ∀ a. ∀ l. ∀ m. ¬m = 0 → HornerRootModulo(b,c,a,l,m) → ∃ x. CanonicalHornerLift(b,c,l,m,a,m,x) ∧ (∀ y. CanonicalHornerLift(b,c,l,m,a,m,y) → y = x)

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

Definition DAG

HornerRootModulo(b,c,a,l,m) · 1CanonicalHornerLift(b,c,l,m,a,M,r) · 2Lt(a,b) · 1ModEq(m,a,b) · 1

Actual proof prerequisites

hensel_canonical_residue_existsbeta_horner_root_mod_transportmod_eq_symm · checked external prerequisitemod_eq_trans · checked external prerequisitemod_eq_bounded_unique · checked external prerequisite
Original expanded first-order statement 
forall b c a l m. ~(m = 0) -> (exists hpl_value_root. ((exists ff_u_ph_hpl_root ff_v_ph_hpl_root. ((((exists fs_h_ph_hpl_root_body_start. fs_h_ph_hpl_root_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_start. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_start * S ((S (0)) * ff_v_ph_hpl_root) + (0))) /\ ((((exists fs_h_ph_hpl_root_body_terminal. fs_h_ph_hpl_root_body_terminal + S (hpl_value_root) = S ((S (l)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_terminal. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_terminal * S ((S (l)) * ff_v_ph_hpl_root) + (hpl_value_root))) /\ forall ff_i_ph_hpl_root_body_steps. (exists ph_bound_hpl_root_body_steps. ph_bound_hpl_root_body_steps + S ff_i_ph_hpl_root_body_steps = l) -> exists ff_coefficient_ph_hpl_root_body_steps ff_previous_ph_hpl_root_body_steps ff_current_ph_hpl_root_body_steps. ((((exists fs_h_ph_hpl_root_body_steps_coefficient. fs_h_ph_hpl_root_body_steps_coefficient + S (ff_coefficient_ph_hpl_root_body_steps) = S ((S (ff_i_ph_hpl_root_body_steps)) * c)) /\ exists fs_q_ph_hpl_root_body_steps_coefficient. b = fs_q_ph_hpl_root_body_steps_coefficient * S ((S (ff_i_ph_hpl_root_body_steps)) * c) + (ff_coefficient_ph_hpl_root_body_steps))) /\ ((((exists fs_h_ph_hpl_root_body_steps_before. fs_h_ph_hpl_root_body_steps_before + S (ff_previous_ph_hpl_root_body_steps) = S ((S (ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_steps_before. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_steps_before * S ((S (ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root) + (ff_previous_ph_hpl_root_body_steps))) /\ ((((exists fs_h_ph_hpl_root_body_steps_after. fs_h_ph_hpl_root_body_steps_after + S (ff_current_ph_hpl_root_body_steps) = S ((S (S ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root)) /\ exists fs_q_ph_hpl_root_body_steps_after. ff_u_ph_hpl_root = fs_q_ph_hpl_root_body_steps_after * S ((S (S ff_i_ph_hpl_root_body_steps)) * ff_v_ph_hpl_root) + (ff_current_ph_hpl_root_body_steps))) /\ ff_current_ph_hpl_root_body_steps = ff_previous_ph_hpl_root_body_steps * a + ff_coefficient_ph_hpl_root_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_root hgcrt_mod_right_hpl_root. hpl_value_root + m * hgcrt_mod_left_hpl_root = 0 + m * hgcrt_mod_right_hpl_root))) -> exists r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. r + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * r + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + m * hgcrt_mod_left_hpl_lift = 0 + m * hgcrt_mod_right_hpl_lift)))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (m)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. z + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists hpl_value_lift. ((exists ff_u_ph_hpl_lift ff_v_ph_hpl_lift. ((((exists fs_h_ph_hpl_lift_body_start. fs_h_ph_hpl_lift_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_start. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_start * S ((S (0)) * ff_v_ph_hpl_lift) + (0))) /\ ((((exists fs_h_ph_hpl_lift_body_terminal. fs_h_ph_hpl_lift_body_terminal + S (hpl_value_lift) = S ((S (l)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_terminal. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_terminal * S ((S (l)) * ff_v_ph_hpl_lift) + (hpl_value_lift))) /\ forall ff_i_ph_hpl_lift_body_steps. (exists ph_bound_hpl_lift_body_steps. ph_bound_hpl_lift_body_steps + S ff_i_ph_hpl_lift_body_steps = l) -> exists ff_coefficient_ph_hpl_lift_body_steps ff_previous_ph_hpl_lift_body_steps ff_current_ph_hpl_lift_body_steps. ((((exists fs_h_ph_hpl_lift_body_steps_coefficient. fs_h_ph_hpl_lift_body_steps_coefficient + S (ff_coefficient_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * c)) /\ exists fs_q_ph_hpl_lift_body_steps_coefficient. b = fs_q_ph_hpl_lift_body_steps_coefficient * S ((S (ff_i_ph_hpl_lift_body_steps)) * c) + (ff_coefficient_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_before. fs_h_ph_hpl_lift_body_steps_before + S (ff_previous_ph_hpl_lift_body_steps) = S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_before. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_before * S ((S (ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_previous_ph_hpl_lift_body_steps))) /\ ((((exists fs_h_ph_hpl_lift_body_steps_after. fs_h_ph_hpl_lift_body_steps_after + S (ff_current_ph_hpl_lift_body_steps) = S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift)) /\ exists fs_q_ph_hpl_lift_body_steps_after. ff_u_ph_hpl_lift = fs_q_ph_hpl_lift_body_steps_after * S ((S (S ff_i_ph_hpl_lift_body_steps)) * ff_v_ph_hpl_lift) + (ff_current_ph_hpl_lift_body_steps))) /\ ff_current_ph_hpl_lift_body_steps = ff_previous_ph_hpl_lift_body_steps * z + ff_coefficient_ph_hpl_lift_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. hpl_value_lift + m * hgcrt_mod_left_hpl_lift = 0 + m * hgcrt_mod_right_hpl_lift)))))) -> z = r)

Complete tactic proof in conservative notation

All 50 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

50 script commands · 13 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 canonical residue existsbeta horner root mod transport
01Fix variables and assumptionsL1–7

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro a
  4. L4
    intro l
  5. L5
    intro m
  6. L6
    intro hm
  7. L7
    intro hroot
02Establish hresidueL8–12

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

  1. L8
    have hresidue : ∃ r. Lt(r,m) ∧ ModEq(m,a,r)Definitions: Lt(r,m)ModEq(m,a,r)Original native command in the exact edition
  2. L9
    specialize hensel_canonical_residue_exists m
  3. L10
    specialize hensel_canonical_residue_exists a
  4. L11
    apply hensel_canonical_residue_exists
  5. L12
    exact hm
03Separate the logical casesL13–14

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

  1. L13
    cases hresidue
  2. L14
    cases hresidue_witness
04Construct an explicit witnessL15–15

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

  1. L15
    exists x
05Separate the logical casesL16–17

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

  1. L16
    split
  2. L17
    split
06Use earlier factsL18–18

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

  1. L18
    exact hresidue_witness_left
07Separate the logical casesL19–19

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

  1. L19
    split
08Use earlier factsL20–29

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

  1. L20
    specialize mod_eq_symm m
  2. L21
    specialize mod_eq_symm a
  3. L22
    specialize mod_eq_symm x
  4. L23
    apply mod_eq_symm
  5. L24
    exact hresidue_witness_right
  6. L25
    specialize beta_horner_root_mod_transport b
  7. L26
    specialize beta_horner_root_mod_transport c
  8. L27
    specialize beta_horner_root_mod_transport a
  9. L28
    specialize beta_horner_root_mod_transport x
  10. L29
    specialize beta_horner_root_mod_transport l
09Use earlier factsL30–33

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

  1. L30
    specialize beta_horner_root_mod_transport m
  2. L31
    apply beta_horner_root_mod_transport
  3. L32
    exact hresidue_witness_right
  4. L33
    exact hroot
10Fix variables and assumptionsL34–35

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

  1. L34
    intro z
  2. L35
    intro hz
11Separate the logical casesL36–37

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

  1. L36
    cases hz
  2. L37
    cases hz_right
12Use earlier factsL38–47

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

  1. L38
    specialize mod_eq_bounded_unique m
  2. L39
    specialize mod_eq_bounded_unique z
  3. L40
    specialize mod_eq_bounded_unique x
  4. L41
    apply mod_eq_bounded_unique
  5. L42
    exact hz_left
  6. L43
    exact hresidue_witness_left
  7. L44
    specialize mod_eq_trans m
  8. L45
    specialize mod_eq_trans z
  9. L46
    specialize mod_eq_trans a
  10. L47
    specialize mod_eq_trans x
13Use earlier factsL48–50

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

  1. L48
    apply mod_eq_trans
  2. L49
    exact hz_right_left
  3. L50
    exact hresidue_witness_right

Library-wide reading audit

Original defined command ledger · 50 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro a
  4. 0004intro l
  5. 0005intro m
  6. 0006intro hm
  7. 0007intro hroot
  8. 0008have hresidue : ∃ r. Lt(r,m) ∧ ModEq(m,a,r)
  9. 0009specialize hensel_canonical_residue_exists m
  10. 0010specialize hensel_canonical_residue_exists a
  11. 0011apply hensel_canonical_residue_exists
  12. 0012exact hm
  13. 0013cases hresidue
  14. 0014cases hresidue_witness
  15. 0015exists x
  16. 0016split
  17. 0017split
  18. 0018exact hresidue_witness_left
  19. 0019split
  20. 0020specialize mod_eq_symm m
  21. 0021specialize mod_eq_symm a
  22. 0022specialize mod_eq_symm x
  23. 0023apply mod_eq_symm
  24. 0024exact hresidue_witness_right
  25. 0025specialize beta_horner_root_mod_transport b
  26. 0026specialize beta_horner_root_mod_transport c
  27. 0027specialize beta_horner_root_mod_transport a
  28. 0028specialize beta_horner_root_mod_transport x
  29. 0029specialize beta_horner_root_mod_transport l
  30. 0030specialize beta_horner_root_mod_transport m
  31. 0031apply beta_horner_root_mod_transport
  32. 0032exact hresidue_witness_right
  33. 0033exact hroot
  34. 0034intro z
  35. 0035intro hz
  36. 0036cases hz
  37. 0037cases hz_right
  38. 0038specialize mod_eq_bounded_unique m
  39. 0039specialize mod_eq_bounded_unique z
  40. 0040specialize mod_eq_bounded_unique x
  41. 0041apply mod_eq_bounded_unique
  42. 0042exact hz_left
  43. 0043exact hresidue_witness_left
  44. 0044specialize mod_eq_trans m
  45. 0045specialize mod_eq_trans z
  46. 0046specialize mod_eq_trans a
  47. 0047specialize mod_eq_trans x
  48. 0048apply mod_eq_trans
  49. 0049exact hz_right_left
  50. 0050exact hresidue_witness_right