PV0006

pow_positive_exponent_base_divides

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

Every positive power has its base as an actual divisor; no prime or positivity oracle is needed.

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 a k z. ~(k = 0) -> (exists pa_b_pvs_positive_power pa_c_pvs_positive_power. ((forall pa_i_pvs_positive_power_repeat. (exists pa_lt_pvs_positive_power_repeat_bound. pa_lt_pvs_positive_power_repeat_bound + S pa_i_pvs_positive_power_repeat = k) -> (((exists pa_h_pvs_positive_power_repeat_decoded. pa_h_pvs_positive_power_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power)) /\ exists pa_q_pvs_positive_power_repeat_decoded. pa_b_pvs_positive_power = pa_q_pvs_positive_power_repeat_decoded * S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power) + (a)))) /\ (exists pa_u_pvs_positive_power_product pa_v_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_start. pa_h_pvs_positive_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_power_product)) /\ exists pa_q_pvs_positive_power_product_start. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_start * S ((S (0)) * pa_v_pvs_positive_power_product) + (1))) /\ ((((exists pa_h_pvs_positive_power_product_terminal. pa_h_pvs_positive_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_positive_power_product)) /\ exists pa_q_pvs_positive_power_product_terminal. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_terminal * S ((S (k)) * pa_v_pvs_positive_power_product) + (z))) /\ forall pa_i_pvs_positive_power_product. (exists pa_lt_pvs_positive_power_product_bound. pa_lt_pvs_positive_power_product_bound + S pa_i_pvs_positive_power_product = k) -> exists pa_p_pvs_positive_power_product pa_r_pvs_positive_power_product pa_s_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_factor. pa_h_pvs_positive_power_product_factor + S (pa_p_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power)) /\ exists pa_q_pvs_positive_power_product_factor. pa_b_pvs_positive_power = pa_q_pvs_positive_power_product_factor * S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power) + (pa_p_pvs_positive_power_product))) /\ ((((exists pa_h_pvs_positive_power_product_partial. pa_h_pvs_positive_power_product_partial + S (pa_r_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\ exists pa_q_pvs_positive_power_product_partial. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_partial * S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_r_pvs_positive_power_product))) /\ ((((exists pa_h_pvs_positive_power_product_successor. pa_h_pvs_positive_power_product_successor + S (pa_s_pvs_positive_power_product) = S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\ exists pa_q_pvs_positive_power_product_successor. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_successor * S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_s_pvs_positive_power_product))) /\ pa_s_pvs_positive_power_product = pa_r_pvs_positive_power_product * pa_p_pvs_positive_power_product)))))))) -> (exists pvs_factor_positive_divides. (z) = (a) * pvs_factor_positive_divides)

Constructive proof overview

Generated structural guide

Every positive power has its base as an actual divisor; no prime or positivity oracle is needed.

The unchanged tactic script uses 3 declared prerequisites and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v29 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

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

PV0011 prime_valuation_strict_cofactor_exists

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

24 script commands · 8 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 assumptionsL1–5

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

  1. L1
    intro a
  2. L2
    intro k
  3. L3
    intro z
  4. L4
    intro hk
  5. L5
    intro hpow
02Establish hsL6–9

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

  1. L6
    have hs : exists j. k = S j
  2. L7
    specialize nonzero_is_succ (k)
  3. L8
    apply nonzero_is_succ
  4. L9
    exact hk
03Separate the logical casesL10–10

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

  1. L10
    cases hs
04Establish hprevL11–18

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

  1. L11
    have hprev : ∃ r. Pow(a,x,r) ∧ z = r · aDefinitions: Pow
  2. L12
    specialize pow_successor_decompose (a)
  3. L13
    specialize pow_successor_decompose (x)
  4. L14
    specialize pow_successor_decompose (k)
  5. L15
    specialize pow_successor_decompose (z)
  6. L16
    apply pow_successor_decompose
  7. L17
    exact hs_witness
  8. L18
    exact hpow
05Separate the logical casesL19–20

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

  1. L19
    cases hprev
  2. L20
    cases hprev_witness
06Construct an explicit witnessL21–21

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

  1. L21
    exists x1
07Calculate and transport equalitiesL22–22

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

  1. L22
    trans x1 * a
08Use earlier factsL23–24

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

  1. L23
    exact hprev_witness_right
  2. L24
    apply mul_comm

Library-wide reading audit

Original exact command ledger · 24 lines
  1. 0001intro a
  2. 0002intro k
  3. 0003intro z
  4. 0004intro hk
  5. 0005intro hpow
  6. 0006have hs : exists j. k = S j
  7. 0007specialize nonzero_is_succ (k)
  8. 0008apply nonzero_is_succ
  9. 0009exact hk
  10. 0010cases hs
  11. 0011have hprev : exists r. (exists pa_b_pvs_positive_previous pa_c_pvs_positive_previous. ((forall pa_i_pvs_positive_previous_repeat. (exists pa_lt_pvs_positive_previous_repeat_bound. pa_lt_pvs_positive_previous_repeat_bound + S pa_i_pvs_positive_previous_repeat = x) -> (((exists pa_h_pvs_positive_previous_repeat_decoded. pa_h_pvs_positive_previous_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous)) /\ exists pa_q_pvs_positive_previous_repeat_decoded. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_repeat_decoded * S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous) + (a)))) /\ (exists pa_u_pvs_positive_previous_product pa_v_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_start. pa_h_pvs_positive_previous_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_previous_product)) /\ exists pa_q_pvs_positive_previous_product_start. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_start * S ((S (0)) * pa_v_pvs_positive_previous_product) + (1))) /\ ((((exists pa_h_pvs_positive_previous_product_terminal. pa_h_pvs_positive_previous_product_terminal + S (r) = S ((S (x)) * pa_v_pvs_positive_previous_product)) /\ exists pa_q_pvs_positive_previous_product_terminal. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_terminal * S ((S (x)) * pa_v_pvs_positive_previous_product) + (r))) /\ forall pa_i_pvs_positive_previous_product. (exists pa_lt_pvs_positive_previous_product_bound. pa_lt_pvs_positive_previous_product_bound + S pa_i_pvs_positive_previous_product = x) -> exists pa_p_pvs_positive_previous_product pa_r_pvs_positive_previous_product pa_s_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_factor. pa_h_pvs_positive_previous_product_factor + S (pa_p_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous)) /\ exists pa_q_pvs_positive_previous_product_factor. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_product_factor * S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous) + (pa_p_pvs_positive_previous_product))) /\ ((((exists pa_h_pvs_positive_previous_product_partial. pa_h_pvs_positive_previous_product_partial + S (pa_r_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\ exists pa_q_pvs_positive_previous_product_partial. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_partial * S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_r_pvs_positive_previous_product))) /\ ((((exists pa_h_pvs_positive_previous_product_successor. pa_h_pvs_positive_previous_product_successor + S (pa_s_pvs_positive_previous_product) = S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\ exists pa_q_pvs_positive_previous_product_successor. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_successor * S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_s_pvs_positive_previous_product))) /\ pa_s_pvs_positive_previous_product = pa_r_pvs_positive_previous_product * pa_p_pvs_positive_previous_product)))))))) /\ z = r * a
  12. 0012specialize pow_successor_decompose (a)
  13. 0013specialize pow_successor_decompose (x)
  14. 0014specialize pow_successor_decompose (k)
  15. 0015specialize pow_successor_decompose (z)
  16. 0016apply pow_successor_decompose
  17. 0017exact hs_witness
  18. 0018exact hpow
  19. 0019cases hprev
  20. 0020cases hprev_witness
  21. 0021exists x1
  22. 0022trans x1 * a
  23. 0023exact hprev_witness_right
  24. 0024apply mul_comm