SK0013

power_one_base_exists

Construct the actual identity 1=1^k uniformly for all nonnegative exponents, in particular every positive exponent required by the n=1 profile.

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

∀ k. Pow(1,k,1)

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

Definition DAG

Pow(a,e,z) · 2

Actual proof prerequisites

pow_exists · checked external prerequisitepower_one_base_valuepower_value_eq_transport
Original expanded first-order statement 
forall k. (exists pa_b_pvs_uniform_unit pa_c_pvs_uniform_unit. ((forall pa_i_pvs_uniform_unit_repeat. (exists pa_lt_pvs_uniform_unit_repeat_bound. pa_lt_pvs_uniform_unit_repeat_bound + S pa_i_pvs_uniform_unit_repeat = k) -> (((exists pa_h_pvs_uniform_unit_repeat_decoded. pa_h_pvs_uniform_unit_repeat_decoded + S (1) = S ((S (pa_i_pvs_uniform_unit_repeat)) * pa_c_pvs_uniform_unit)) /\ exists pa_q_pvs_uniform_unit_repeat_decoded. pa_b_pvs_uniform_unit = pa_q_pvs_uniform_unit_repeat_decoded * S ((S (pa_i_pvs_uniform_unit_repeat)) * pa_c_pvs_uniform_unit) + (1)))) /\ (exists pa_u_pvs_uniform_unit_product pa_v_pvs_uniform_unit_product. ((((exists pa_h_pvs_uniform_unit_product_start. pa_h_pvs_uniform_unit_product_start + S (1) = S ((S (0)) * pa_v_pvs_uniform_unit_product)) /\ exists pa_q_pvs_uniform_unit_product_start. pa_u_pvs_uniform_unit_product = pa_q_pvs_uniform_unit_product_start * S ((S (0)) * pa_v_pvs_uniform_unit_product) + (1))) /\ ((((exists pa_h_pvs_uniform_unit_product_terminal. pa_h_pvs_uniform_unit_product_terminal + S (1) = S ((S (k)) * pa_v_pvs_uniform_unit_product)) /\ exists pa_q_pvs_uniform_unit_product_terminal. pa_u_pvs_uniform_unit_product = pa_q_pvs_uniform_unit_product_terminal * S ((S (k)) * pa_v_pvs_uniform_unit_product) + (1))) /\ forall pa_i_pvs_uniform_unit_product. (exists pa_lt_pvs_uniform_unit_product_bound. pa_lt_pvs_uniform_unit_product_bound + S pa_i_pvs_uniform_unit_product = k) -> exists pa_p_pvs_uniform_unit_product pa_r_pvs_uniform_unit_product pa_s_pvs_uniform_unit_product. ((((exists pa_h_pvs_uniform_unit_product_factor. pa_h_pvs_uniform_unit_product_factor + S (pa_p_pvs_uniform_unit_product) = S ((S (pa_i_pvs_uniform_unit_product)) * pa_c_pvs_uniform_unit)) /\ exists pa_q_pvs_uniform_unit_product_factor. pa_b_pvs_uniform_unit = pa_q_pvs_uniform_unit_product_factor * S ((S (pa_i_pvs_uniform_unit_product)) * pa_c_pvs_uniform_unit) + (pa_p_pvs_uniform_unit_product))) /\ ((((exists pa_h_pvs_uniform_unit_product_partial. pa_h_pvs_uniform_unit_product_partial + S (pa_r_pvs_uniform_unit_product) = S ((S (pa_i_pvs_uniform_unit_product)) * pa_v_pvs_uniform_unit_product)) /\ exists pa_q_pvs_uniform_unit_product_partial. pa_u_pvs_uniform_unit_product = pa_q_pvs_uniform_unit_product_partial * S ((S (pa_i_pvs_uniform_unit_product)) * pa_v_pvs_uniform_unit_product) + (pa_r_pvs_uniform_unit_product))) /\ ((((exists pa_h_pvs_uniform_unit_product_successor. pa_h_pvs_uniform_unit_product_successor + S (pa_s_pvs_uniform_unit_product) = S ((S (S pa_i_pvs_uniform_unit_product)) * pa_v_pvs_uniform_unit_product)) /\ exists pa_q_pvs_uniform_unit_product_successor. pa_u_pvs_uniform_unit_product = pa_q_pvs_uniform_unit_product_successor * S ((S (S pa_i_pvs_uniform_unit_product)) * pa_v_pvs_uniform_unit_product) + (pa_s_pvs_uniform_unit_product))) /\ pa_s_pvs_uniform_unit_product = pa_r_pvs_uniform_unit_product * pa_p_pvs_uniform_unit_product))))))))

Complete tactic proof in conservative notation

All 16 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

16 script commands · 4 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.

Named ingredients (2)
power value eq transportpower one base value
01Fix variables and assumptionsL1–1

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

  1. L1
    intro k
02Establish hexL2–5

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

  1. L2
    have hex : ∃ z. Pow(1,k,z)Definitions: Pow(1,k,z)Original native command in the exact edition
  2. L3
    specialize pow_exists (1)
  3. L4
    specialize pow_exists (k)
  4. L5
    apply pow_exists
03Separate the logical casesL6–6

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

  1. L6
    cases hex
04Use earlier factsL7–16

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

  1. L7
    specialize power_value_eq_transport (1)
  2. L8
    specialize power_value_eq_transport (k)
  3. L9
    specialize power_value_eq_transport (x)
  4. L10
    specialize power_value_eq_transport (1)
  5. L11
    apply power_value_eq_transport
  6. L12
    specialize power_one_base_value (k)
  7. L13
    specialize power_one_base_value (x)
  8. L14
    apply power_one_base_value
  9. L15
    exact hex_witness
  10. L16
    exact hex_witness

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro k
  2. 0002have hex : ∃ z. Pow(1,k,z)
  3. 0003specialize pow_exists (1)
  4. 0004specialize pow_exists (k)
  5. 0005apply pow_exists
  6. 0006cases hex
  7. 0007specialize power_value_eq_transport (1)
  8. 0008specialize power_value_eq_transport (k)
  9. 0009specialize power_value_eq_transport (x)
  10. 0010specialize power_value_eq_transport (1)
  11. 0011apply power_value_eq_transport
  12. 0012specialize power_one_base_value (k)
  13. 0013specialize power_one_base_value (x)
  14. 0014apply power_one_base_value
  15. 0015exact hex_witness
  16. 0016exact hex_witness