SK0016

positive_power_nonzero_base

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

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.

For n>1 the exponent gcd and a real beta table classify and witness all positive root degrees. The unit n=1 has a separate uniform certificate for every positive degree. Zero is excluded. NaturalSquarefreeDecomposition is deliberately distinct from the unrelated polynomial definition.

Exact theorem in conservative defined notation

∀ n. ∀ k. ∀ r. ¬n = 0 → ¬k = 0 → Pow(r,k,n) → ¬r = 0

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

Definition DAG

Dvd(d,n) · 1Pow(a,e,z) · 1

Actual proof prerequisites

pow_positive_exponent_base_divides · checked external prerequisitemul_zero_left · checked external prerequisite
Original expanded first-order 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)

Complete tactic proof in conservative notation

All 20 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

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.

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 assumptionsL1–7

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
  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 defined 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 : Dvd(r,n)
  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