PV0006

pow_positive_exponent_base_divides

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

Alpha v34 checked-use · first admitted v29 · 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.

This is a shared constructive tool, not an additional major blueprint goal. The list covers every prime divisor, has no repeated primes, and contains actual prime-power values. One uses the empty support; zero is excluded.

Exact theorem in conservative defined notation

∀ a. ∀ k. ∀ z. ¬k = 0 → Pow(a,k,z) → Dvd(a,z)

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

Definition DAG

Dvd(d,n) · 1Pow(a,e,z) · 2

Actual proof prerequisites

nonzero_is_succ · checked external prerequisitepow_successor_decompose · checked external prerequisitemul_comm · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 24 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

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.

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 assumptionsL1–5

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(a,x,r)Original native command in the exact edition
  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 defined 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 : ∃ r. Pow(a,x,r) ∧ 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