SK0016

positive_power_nonzero_base

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

An actual positive-degree power with positive value has a nonzero natural base.

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 n k r. ~(n = 0) -> ~(k = 0) -> (exists pa_b_pvs_positive_output pa_c_pvs_positive_output. ((forall pa_i_pvs_positive_output_repeat. (exists pa_lt_pvs_positive_output_repeat_bound. pa_lt_pvs_positive_output_repeat_bound + S pa_i_pvs_positive_output_repeat = k) -> (((exists pa_h_pvs_positive_output_repeat_decoded. pa_h_pvs_positive_output_repeat_decoded + S (r) = S ((S (pa_i_pvs_positive_output_repeat)) * pa_c_pvs_positive_output)) /\ exists pa_q_pvs_positive_output_repeat_decoded. pa_b_pvs_positive_output = pa_q_pvs_positive_output_repeat_decoded * S ((S (pa_i_pvs_positive_output_repeat)) * pa_c_pvs_positive_output) + (r)))) /\ (exists pa_u_pvs_positive_output_product pa_v_pvs_positive_output_product. ((((exists pa_h_pvs_positive_output_product_start. pa_h_pvs_positive_output_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_output_product)) /\ exists pa_q_pvs_positive_output_product_start. pa_u_pvs_positive_output_product = pa_q_pvs_positive_output_product_start * S ((S (0)) * pa_v_pvs_positive_output_product) + (1))) /\ ((((exists pa_h_pvs_positive_output_product_terminal. pa_h_pvs_positive_output_product_terminal + S (n) = S ((S (k)) * pa_v_pvs_positive_output_product)) /\ exists pa_q_pvs_positive_output_product_terminal. pa_u_pvs_positive_output_product = pa_q_pvs_positive_output_product_terminal * S ((S (k)) * pa_v_pvs_positive_output_product) + (n))) /\ forall pa_i_pvs_positive_output_product. (exists pa_lt_pvs_positive_output_product_bound. pa_lt_pvs_positive_output_product_bound + S pa_i_pvs_positive_output_product = k) -> exists pa_p_pvs_positive_output_product pa_r_pvs_positive_output_product pa_s_pvs_positive_output_product. ((((exists pa_h_pvs_positive_output_product_factor. pa_h_pvs_positive_output_product_factor + S (pa_p_pvs_positive_output_product) = S ((S (pa_i_pvs_positive_output_product)) * pa_c_pvs_positive_output)) /\ exists pa_q_pvs_positive_output_product_factor. pa_b_pvs_positive_output = pa_q_pvs_positive_output_product_factor * S ((S (pa_i_pvs_positive_output_product)) * pa_c_pvs_positive_output) + (pa_p_pvs_positive_output_product))) /\ ((((exists pa_h_pvs_positive_output_product_partial. pa_h_pvs_positive_output_product_partial + S (pa_r_pvs_positive_output_product) = S ((S (pa_i_pvs_positive_output_product)) * pa_v_pvs_positive_output_product)) /\ exists pa_q_pvs_positive_output_product_partial. pa_u_pvs_positive_output_product = pa_q_pvs_positive_output_product_partial * S ((S (pa_i_pvs_positive_output_product)) * pa_v_pvs_positive_output_product) + (pa_r_pvs_positive_output_product))) /\ ((((exists pa_h_pvs_positive_output_product_successor. pa_h_pvs_positive_output_product_successor + S (pa_s_pvs_positive_output_product) = S ((S (S pa_i_pvs_positive_output_product)) * pa_v_pvs_positive_output_product)) /\ exists pa_q_pvs_positive_output_product_successor. pa_u_pvs_positive_output_product = pa_q_pvs_positive_output_product_successor * S ((S (S pa_i_pvs_positive_output_product)) * pa_v_pvs_positive_output_product) + (pa_s_pvs_positive_output_product))) /\ pa_s_pvs_positive_output_product = pa_r_pvs_positive_output_product * pa_p_pvs_positive_output_product)))))))) -> ~(r = 0)

Constructive proof overview

Generated structural guide

An actual positive-degree power with positive value has a nonzero natural base.

The unchanged tactic script uses 2 declared prerequisites and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

pow_positive_exponent_base_divides Alpha theorem; checked-use authorized mul_zero_left Stable theorem; checked-use authorized

Direct dependents

SK0017 positive_power_prime_valuations_divisible

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

20 script commands · 8 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.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro n
  2. L2
    intro k
  3. L3
    intro r
  4. L4
    intro hn
  5. L5
    intro hk
  6. L6
    intro hpow
  7. L7
    intro hr
02Establish hdivL8–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply pow positive exponent base divides.

  1. L8
    have hdiv : exists pvs_factor_positive_base_divisor. (n) = (r) * pvs_factor_positive_base_divisor
  2. L9
    specialize pow_positive_exponent_base_divides (r)
  3. L10
    specialize pow_positive_exponent_base_divides (k)
  4. L11
    specialize pow_positive_exponent_base_divides (n)
  5. L12
    apply pow_positive_exponent_base_divides
  6. L13
    exact hk
  7. L14
    exact hpow
03Separate the logical casesL15–15

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

  1. L15
    cases hdiv
04Use earlier factsL16–16

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

  1. L16
    apply hn
05Calculate and transport equalitiesL17–17

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

  1. L17
    trans r * x
06Use earlier factsL18–18

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

  1. L18
    exact hdiv_witness
07Calculate and transport equalitiesL19–19

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

  1. L19
    rewrite hr
08Use earlier factsL20–20

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

  1. L20
    apply mul_zero_left

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro n
  2. 0002intro k
  3. 0003intro r
  4. 0004intro hn
  5. 0005intro hk
  6. 0006intro hpow
  7. 0007intro hr
  8. 0008have hdiv : exists pvs_factor_positive_base_divisor. (n) = (r) * pvs_factor_positive_base_divisor
  9. 0009specialize pow_positive_exponent_base_divides (r)
  10. 0010specialize pow_positive_exponent_base_divides (k)
  11. 0011specialize pow_positive_exponent_base_divides (n)
  12. 0012apply pow_positive_exponent_base_divides
  13. 0013exact hk
  14. 0014exact hpow
  15. 0015cases hdiv
  16. 0016apply hn
  17. 0017trans r * x
  18. 0018exact hdiv_witness
  19. 0019rewrite hr
  20. 0020apply mul_zero_left