SK0011

power_value_eq_transport

Transport the actual terminal value of a power trace along equality.

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

∀ a. ∀ k. ∀ u. ∀ v. u = v → Pow(a,k,u) → Pow(a,k,v)

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

Definition DAG

Pow(a,e,z) · 2

Actual proof prerequisites

none
Original expanded first-order statement 
forall a k u v. u = v -> (exists pa_b_pvs_transport_source pa_c_pvs_transport_source. ((forall pa_i_pvs_transport_source_repeat. (exists pa_lt_pvs_transport_source_repeat_bound. pa_lt_pvs_transport_source_repeat_bound + S pa_i_pvs_transport_source_repeat = k) -> (((exists pa_h_pvs_transport_source_repeat_decoded. pa_h_pvs_transport_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_transport_source_repeat)) * pa_c_pvs_transport_source)) /\ exists pa_q_pvs_transport_source_repeat_decoded. pa_b_pvs_transport_source = pa_q_pvs_transport_source_repeat_decoded * S ((S (pa_i_pvs_transport_source_repeat)) * pa_c_pvs_transport_source) + (a)))) /\ (exists pa_u_pvs_transport_source_product pa_v_pvs_transport_source_product. ((((exists pa_h_pvs_transport_source_product_start. pa_h_pvs_transport_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_transport_source_product)) /\ exists pa_q_pvs_transport_source_product_start. pa_u_pvs_transport_source_product = pa_q_pvs_transport_source_product_start * S ((S (0)) * pa_v_pvs_transport_source_product) + (1))) /\ ((((exists pa_h_pvs_transport_source_product_terminal. pa_h_pvs_transport_source_product_terminal + S (u) = S ((S (k)) * pa_v_pvs_transport_source_product)) /\ exists pa_q_pvs_transport_source_product_terminal. pa_u_pvs_transport_source_product = pa_q_pvs_transport_source_product_terminal * S ((S (k)) * pa_v_pvs_transport_source_product) + (u))) /\ forall pa_i_pvs_transport_source_product. (exists pa_lt_pvs_transport_source_product_bound. pa_lt_pvs_transport_source_product_bound + S pa_i_pvs_transport_source_product = k) -> exists pa_p_pvs_transport_source_product pa_r_pvs_transport_source_product pa_s_pvs_transport_source_product. ((((exists pa_h_pvs_transport_source_product_factor. pa_h_pvs_transport_source_product_factor + S (pa_p_pvs_transport_source_product) = S ((S (pa_i_pvs_transport_source_product)) * pa_c_pvs_transport_source)) /\ exists pa_q_pvs_transport_source_product_factor. pa_b_pvs_transport_source = pa_q_pvs_transport_source_product_factor * S ((S (pa_i_pvs_transport_source_product)) * pa_c_pvs_transport_source) + (pa_p_pvs_transport_source_product))) /\ ((((exists pa_h_pvs_transport_source_product_partial. pa_h_pvs_transport_source_product_partial + S (pa_r_pvs_transport_source_product) = S ((S (pa_i_pvs_transport_source_product)) * pa_v_pvs_transport_source_product)) /\ exists pa_q_pvs_transport_source_product_partial. pa_u_pvs_transport_source_product = pa_q_pvs_transport_source_product_partial * S ((S (pa_i_pvs_transport_source_product)) * pa_v_pvs_transport_source_product) + (pa_r_pvs_transport_source_product))) /\ ((((exists pa_h_pvs_transport_source_product_successor. pa_h_pvs_transport_source_product_successor + S (pa_s_pvs_transport_source_product) = S ((S (S pa_i_pvs_transport_source_product)) * pa_v_pvs_transport_source_product)) /\ exists pa_q_pvs_transport_source_product_successor. pa_u_pvs_transport_source_product = pa_q_pvs_transport_source_product_successor * S ((S (S pa_i_pvs_transport_source_product)) * pa_v_pvs_transport_source_product) + (pa_s_pvs_transport_source_product))) /\ pa_s_pvs_transport_source_product = pa_r_pvs_transport_source_product * pa_p_pvs_transport_source_product)))))))) -> (exists pa_b_pvs_transport_target pa_c_pvs_transport_target. ((forall pa_i_pvs_transport_target_repeat. (exists pa_lt_pvs_transport_target_repeat_bound. pa_lt_pvs_transport_target_repeat_bound + S pa_i_pvs_transport_target_repeat = k) -> (((exists pa_h_pvs_transport_target_repeat_decoded. pa_h_pvs_transport_target_repeat_decoded + S (a) = S ((S (pa_i_pvs_transport_target_repeat)) * pa_c_pvs_transport_target)) /\ exists pa_q_pvs_transport_target_repeat_decoded. pa_b_pvs_transport_target = pa_q_pvs_transport_target_repeat_decoded * S ((S (pa_i_pvs_transport_target_repeat)) * pa_c_pvs_transport_target) + (a)))) /\ (exists pa_u_pvs_transport_target_product pa_v_pvs_transport_target_product. ((((exists pa_h_pvs_transport_target_product_start. pa_h_pvs_transport_target_product_start + S (1) = S ((S (0)) * pa_v_pvs_transport_target_product)) /\ exists pa_q_pvs_transport_target_product_start. pa_u_pvs_transport_target_product = pa_q_pvs_transport_target_product_start * S ((S (0)) * pa_v_pvs_transport_target_product) + (1))) /\ ((((exists pa_h_pvs_transport_target_product_terminal. pa_h_pvs_transport_target_product_terminal + S (v) = S ((S (k)) * pa_v_pvs_transport_target_product)) /\ exists pa_q_pvs_transport_target_product_terminal. pa_u_pvs_transport_target_product = pa_q_pvs_transport_target_product_terminal * S ((S (k)) * pa_v_pvs_transport_target_product) + (v))) /\ forall pa_i_pvs_transport_target_product. (exists pa_lt_pvs_transport_target_product_bound. pa_lt_pvs_transport_target_product_bound + S pa_i_pvs_transport_target_product = k) -> exists pa_p_pvs_transport_target_product pa_r_pvs_transport_target_product pa_s_pvs_transport_target_product. ((((exists pa_h_pvs_transport_target_product_factor. pa_h_pvs_transport_target_product_factor + S (pa_p_pvs_transport_target_product) = S ((S (pa_i_pvs_transport_target_product)) * pa_c_pvs_transport_target)) /\ exists pa_q_pvs_transport_target_product_factor. pa_b_pvs_transport_target = pa_q_pvs_transport_target_product_factor * S ((S (pa_i_pvs_transport_target_product)) * pa_c_pvs_transport_target) + (pa_p_pvs_transport_target_product))) /\ ((((exists pa_h_pvs_transport_target_product_partial. pa_h_pvs_transport_target_product_partial + S (pa_r_pvs_transport_target_product) = S ((S (pa_i_pvs_transport_target_product)) * pa_v_pvs_transport_target_product)) /\ exists pa_q_pvs_transport_target_product_partial. pa_u_pvs_transport_target_product = pa_q_pvs_transport_target_product_partial * S ((S (pa_i_pvs_transport_target_product)) * pa_v_pvs_transport_target_product) + (pa_r_pvs_transport_target_product))) /\ ((((exists pa_h_pvs_transport_target_product_successor. pa_h_pvs_transport_target_product_successor + S (pa_s_pvs_transport_target_product) = S ((S (S pa_i_pvs_transport_target_product)) * pa_v_pvs_transport_target_product)) /\ exists pa_q_pvs_transport_target_product_successor. pa_u_pvs_transport_target_product = pa_q_pvs_transport_target_product_successor * S ((S (S pa_i_pvs_transport_target_product)) * pa_v_pvs_transport_target_product) + (pa_s_pvs_transport_target_product))) /\ pa_s_pvs_transport_target_product = pa_r_pvs_transport_target_product * pa_p_pvs_transport_target_product))))))))

Complete tactic proof in conservative notation

All 9 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

9 script commands · 3 reading checkpoints · 0 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 assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro k
  3. L3
    intro u
  4. L4
    intro v
  5. L5
    intro heq
  6. L6
    intro hpow
02Calculate and transport equalitiesL7–8

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

  1. L7
    rewrite heq at hpow
  2. L8
    rewrite heq at hpow
03Use earlier factsL9–9

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

  1. L9
    exact hpow

Library-wide reading audit

Original defined command ledger · 9 lines
  1. 0001intro a
  2. 0002intro k
  3. 0003intro u
  4. 0004intro v
  5. 0005intro heq
  6. 0006intro hpow
  7. 0007rewrite heq at hpow
  8. 0008rewrite heq at hpow
  9. 0009exact hpow