HL000F

hensel_positive_power_factor

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

Every actual positive power of a nonzero base supplies both nonzeroness and an explicit base factor.

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 p k m. ~(p = 0) -> ~(k = 0) -> (exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = k) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (m) = S ((S (k)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (k)) * pa_v_hpl_power_product) + (m))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = k) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))) -> ~(m = 0) /\ exists s. m = p * s

Constructive proof overview

Generated structural guide

Every actual positive power of a nonzero base supplies both nonzeroness and an explicit base factor.

The unchanged tactic script uses 5 declared prerequisites and contains 36 exact native proof lines.

Alpha v34 checked-use · first admitted v27 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

pow_nonzero_of_one_le Alpha theorem; checked-use authorized one_le_of_ne_zero Stable theorem; checked-use authorized nonzero_is_succ Stable theorem; checked-use authorized pow_successor_decompose Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

HL0010 beta_horner_prime_power_hensel_lift_exists_unique HL0013 beta_horner_prime_power_iterated_lifts_exists_unique HL0021 beta_signed_horner_prime_power_hensel_lift_exists_unique HL0022 beta_signed_horner_prime_power_iterated_lifts_exists_unique

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

36 script commands · 11 reading checkpoints · 2 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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro m
  4. L4
    intro hp
  5. L5
    intro hk
  6. L6
    intro hpower
02Separate the logical casesL7–7

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

  1. L7
    split
03Fix variables and assumptionsL8–8

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

  1. L8
    intro hzero
04Use earlier factsL9–17

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

  1. L9
    specialize pow_nonzero_of_one_le p
  2. L10
    specialize pow_nonzero_of_one_le k
  3. L11
    specialize pow_nonzero_of_one_le m
  4. L12
    apply pow_nonzero_of_one_le
  5. L13
    specialize one_le_of_ne_zero p
  6. L14
    apply one_le_of_ne_zero
  7. L15
    exact hp
  8. L16
    exact hpower
  9. L17
    exact hzero
05Establish hkpositiveL18–21

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply nonzero is succ.

  1. L18
    have hkpositive : exists e. k = S e
  2. L19
    specialize nonzero_is_succ k
  3. L20
    apply nonzero_is_succ
  4. L21
    exact hk
06Separate the logical casesL22–22

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

  1. L22
    cases hkpositive
07Establish hpreviousL23–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow successor decompose.

  1. L23
    have hprevious : ∃ r. Pow(p,x,r) ∧ m = r · pDefinitions: Pow
  2. L24
    specialize pow_successor_decompose p
  3. L25
    specialize pow_successor_decompose x
  4. L26
    specialize pow_successor_decompose k
  5. L27
    specialize pow_successor_decompose m
  6. L28
    apply pow_successor_decompose
  7. L29
    exact hkpositive_witness
  8. L30
    exact hpower
08Separate the logical casesL31–32

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

  1. L31
    cases hprevious
  2. L32
    cases hprevious_witness
09Construct an explicit witnessL33–33

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

  1. L33
    exists x1
10Calculate and transport equalitiesL34–34

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

  1. L34
    trans x1 * p
11Use earlier factsL35–36

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

  1. L35
    exact hprevious_witness_right
  2. L36
    apply mul_comm

Library-wide reading audit

Original exact command ledger · 36 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro m
  4. 0004intro hp
  5. 0005intro hk
  6. 0006intro hpower
  7. 0007split
  8. 0008intro hzero
  9. 0009specialize pow_nonzero_of_one_le p
  10. 0010specialize pow_nonzero_of_one_le k
  11. 0011specialize pow_nonzero_of_one_le m
  12. 0012apply pow_nonzero_of_one_le
  13. 0013specialize one_le_of_ne_zero p
  14. 0014apply one_le_of_ne_zero
  15. 0015exact hp
  16. 0016exact hpower
  17. 0017exact hzero
  18. 0018have hkpositive : exists e. k = S e
  19. 0019specialize nonzero_is_succ k
  20. 0020apply nonzero_is_succ
  21. 0021exact hk
  22. 0022cases hkpositive
  23. 0023have hprevious : exists r. (exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = x) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (r) = S ((S (x)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (x)) * pa_v_hpl_power_product) + (r))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = x) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))) /\ m = r * p
  24. 0024specialize pow_successor_decompose p
  25. 0025specialize pow_successor_decompose x
  26. 0026specialize pow_successor_decompose k
  27. 0027specialize pow_successor_decompose m
  28. 0028apply pow_successor_decompose
  29. 0029exact hkpositive_witness
  30. 0030exact hpower
  31. 0031cases hprevious
  32. 0032cases hprevious_witness
  33. 0033exists x1
  34. 0034trans x1 * p
  35. 0035exact hprevious_witness_right
  36. 0036apply mul_comm