EL0013

lte_power_exponent_eq_transport

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

Equality transports one actual relational-power argument without changing the graph or adding a choice principle.

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 a n m A. n = m -> (exists pa_b_olte_lte_power_exponent_eq_transportsource pa_c_olte_lte_power_exponent_eq_transportsource. ((forall pa_i_olte_lte_power_exponent_eq_transportsource_repeat. (exists pa_lt_olte_lte_power_exponent_eq_transportsource_repeat_bound. pa_lt_olte_lte_power_exponent_eq_transportsource_repeat_bound + S pa_i_olte_lte_power_exponent_eq_transportsource_repeat = n) -> (((exists pa_h_olte_lte_power_exponent_eq_transportsource_repeat_decoded. pa_h_olte_lte_power_exponent_eq_transportsource_repeat_decoded + S (a) = S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_repeat)) * pa_c_olte_lte_power_exponent_eq_transportsource)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_repeat_decoded. pa_b_olte_lte_power_exponent_eq_transportsource = pa_q_olte_lte_power_exponent_eq_transportsource_repeat_decoded * S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_repeat)) * pa_c_olte_lte_power_exponent_eq_transportsource) + (a)))) /\ (exists pa_u_olte_lte_power_exponent_eq_transportsource_product pa_v_olte_lte_power_exponent_eq_transportsource_product. ((((exists pa_h_olte_lte_power_exponent_eq_transportsource_product_start. pa_h_olte_lte_power_exponent_eq_transportsource_product_start + S (1) = S ((S (0)) * pa_v_olte_lte_power_exponent_eq_transportsource_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_product_start. pa_u_olte_lte_power_exponent_eq_transportsource_product = pa_q_olte_lte_power_exponent_eq_transportsource_product_start * S ((S (0)) * pa_v_olte_lte_power_exponent_eq_transportsource_product) + (1))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportsource_product_terminal. pa_h_olte_lte_power_exponent_eq_transportsource_product_terminal + S (A) = S ((S (n)) * pa_v_olte_lte_power_exponent_eq_transportsource_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_product_terminal. pa_u_olte_lte_power_exponent_eq_transportsource_product = pa_q_olte_lte_power_exponent_eq_transportsource_product_terminal * S ((S (n)) * pa_v_olte_lte_power_exponent_eq_transportsource_product) + (A))) /\ forall pa_i_olte_lte_power_exponent_eq_transportsource_product. (exists pa_lt_olte_lte_power_exponent_eq_transportsource_product_bound. pa_lt_olte_lte_power_exponent_eq_transportsource_product_bound + S pa_i_olte_lte_power_exponent_eq_transportsource_product = n) -> exists pa_p_olte_lte_power_exponent_eq_transportsource_product pa_r_olte_lte_power_exponent_eq_transportsource_product pa_s_olte_lte_power_exponent_eq_transportsource_product. ((((exists pa_h_olte_lte_power_exponent_eq_transportsource_product_factor. pa_h_olte_lte_power_exponent_eq_transportsource_product_factor + S (pa_p_olte_lte_power_exponent_eq_transportsource_product) = S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_c_olte_lte_power_exponent_eq_transportsource)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_product_factor. pa_b_olte_lte_power_exponent_eq_transportsource = pa_q_olte_lte_power_exponent_eq_transportsource_product_factor * S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_c_olte_lte_power_exponent_eq_transportsource) + (pa_p_olte_lte_power_exponent_eq_transportsource_product))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportsource_product_partial. pa_h_olte_lte_power_exponent_eq_transportsource_product_partial + S (pa_r_olte_lte_power_exponent_eq_transportsource_product) = S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_v_olte_lte_power_exponent_eq_transportsource_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_product_partial. pa_u_olte_lte_power_exponent_eq_transportsource_product = pa_q_olte_lte_power_exponent_eq_transportsource_product_partial * S ((S (pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_v_olte_lte_power_exponent_eq_transportsource_product) + (pa_r_olte_lte_power_exponent_eq_transportsource_product))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportsource_product_successor. pa_h_olte_lte_power_exponent_eq_transportsource_product_successor + S (pa_s_olte_lte_power_exponent_eq_transportsource_product) = S ((S (S pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_v_olte_lte_power_exponent_eq_transportsource_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportsource_product_successor. pa_u_olte_lte_power_exponent_eq_transportsource_product = pa_q_olte_lte_power_exponent_eq_transportsource_product_successor * S ((S (S pa_i_olte_lte_power_exponent_eq_transportsource_product)) * pa_v_olte_lte_power_exponent_eq_transportsource_product) + (pa_s_olte_lte_power_exponent_eq_transportsource_product))) /\ pa_s_olte_lte_power_exponent_eq_transportsource_product = pa_r_olte_lte_power_exponent_eq_transportsource_product * pa_p_olte_lte_power_exponent_eq_transportsource_product)))))))) -> (exists pa_b_olte_lte_power_exponent_eq_transportresult pa_c_olte_lte_power_exponent_eq_transportresult. ((forall pa_i_olte_lte_power_exponent_eq_transportresult_repeat. (exists pa_lt_olte_lte_power_exponent_eq_transportresult_repeat_bound. pa_lt_olte_lte_power_exponent_eq_transportresult_repeat_bound + S pa_i_olte_lte_power_exponent_eq_transportresult_repeat = m) -> (((exists pa_h_olte_lte_power_exponent_eq_transportresult_repeat_decoded. pa_h_olte_lte_power_exponent_eq_transportresult_repeat_decoded + S (a) = S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_repeat)) * pa_c_olte_lte_power_exponent_eq_transportresult)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_repeat_decoded. pa_b_olte_lte_power_exponent_eq_transportresult = pa_q_olte_lte_power_exponent_eq_transportresult_repeat_decoded * S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_repeat)) * pa_c_olte_lte_power_exponent_eq_transportresult) + (a)))) /\ (exists pa_u_olte_lte_power_exponent_eq_transportresult_product pa_v_olte_lte_power_exponent_eq_transportresult_product. ((((exists pa_h_olte_lte_power_exponent_eq_transportresult_product_start. pa_h_olte_lte_power_exponent_eq_transportresult_product_start + S (1) = S ((S (0)) * pa_v_olte_lte_power_exponent_eq_transportresult_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_product_start. pa_u_olte_lte_power_exponent_eq_transportresult_product = pa_q_olte_lte_power_exponent_eq_transportresult_product_start * S ((S (0)) * pa_v_olte_lte_power_exponent_eq_transportresult_product) + (1))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportresult_product_terminal. pa_h_olte_lte_power_exponent_eq_transportresult_product_terminal + S (A) = S ((S (m)) * pa_v_olte_lte_power_exponent_eq_transportresult_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_product_terminal. pa_u_olte_lte_power_exponent_eq_transportresult_product = pa_q_olte_lte_power_exponent_eq_transportresult_product_terminal * S ((S (m)) * pa_v_olte_lte_power_exponent_eq_transportresult_product) + (A))) /\ forall pa_i_olte_lte_power_exponent_eq_transportresult_product. (exists pa_lt_olte_lte_power_exponent_eq_transportresult_product_bound. pa_lt_olte_lte_power_exponent_eq_transportresult_product_bound + S pa_i_olte_lte_power_exponent_eq_transportresult_product = m) -> exists pa_p_olte_lte_power_exponent_eq_transportresult_product pa_r_olte_lte_power_exponent_eq_transportresult_product pa_s_olte_lte_power_exponent_eq_transportresult_product. ((((exists pa_h_olte_lte_power_exponent_eq_transportresult_product_factor. pa_h_olte_lte_power_exponent_eq_transportresult_product_factor + S (pa_p_olte_lte_power_exponent_eq_transportresult_product) = S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_c_olte_lte_power_exponent_eq_transportresult)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_product_factor. pa_b_olte_lte_power_exponent_eq_transportresult = pa_q_olte_lte_power_exponent_eq_transportresult_product_factor * S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_c_olte_lte_power_exponent_eq_transportresult) + (pa_p_olte_lte_power_exponent_eq_transportresult_product))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportresult_product_partial. pa_h_olte_lte_power_exponent_eq_transportresult_product_partial + S (pa_r_olte_lte_power_exponent_eq_transportresult_product) = S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_v_olte_lte_power_exponent_eq_transportresult_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_product_partial. pa_u_olte_lte_power_exponent_eq_transportresult_product = pa_q_olte_lte_power_exponent_eq_transportresult_product_partial * S ((S (pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_v_olte_lte_power_exponent_eq_transportresult_product) + (pa_r_olte_lte_power_exponent_eq_transportresult_product))) /\ ((((exists pa_h_olte_lte_power_exponent_eq_transportresult_product_successor. pa_h_olte_lte_power_exponent_eq_transportresult_product_successor + S (pa_s_olte_lte_power_exponent_eq_transportresult_product) = S ((S (S pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_v_olte_lte_power_exponent_eq_transportresult_product)) /\ exists pa_q_olte_lte_power_exponent_eq_transportresult_product_successor. pa_u_olte_lte_power_exponent_eq_transportresult_product = pa_q_olte_lte_power_exponent_eq_transportresult_product_successor * S ((S (S pa_i_olte_lte_power_exponent_eq_transportresult_product)) * pa_v_olte_lte_power_exponent_eq_transportresult_product) + (pa_s_olte_lte_power_exponent_eq_transportresult_product))) /\ pa_s_olte_lte_power_exponent_eq_transportresult_product = pa_r_olte_lte_power_exponent_eq_transportresult_product * pa_p_olte_lte_power_exponent_eq_transportresult_product))))))))

Constructive proof overview

Generated structural guide

Equality transports one actual relational-power argument without changing the graph or adding a choice principle.

The unchanged tactic script uses 0 declared prerequisites and contains 11 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

EL001B lte_odd_prime_power_difference_quotient EL001C lte_coprime_power_difference_quotient

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

11 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.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro a
  2. L2
    intro n
  3. L3
    intro m
  4. L4
    intro A
  5. L5
    intro heq
  6. L6
    intro hpow
02Calculate and transport equalitiesL7–10

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
  3. L9
    rewrite heq at hpow
  4. L10
    rewrite heq at hpow
03Use earlier factsL11–11

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

  1. L11
    exact hpow

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro a
  2. 0002intro n
  3. 0003intro m
  4. 0004intro A
  5. 0005intro heq
  6. 0006intro hpow
  7. 0007rewrite heq at hpow
  8. 0008rewrite heq at hpow
  9. 0009rewrite heq at hpow
  10. 0010rewrite heq at hpow
  11. 0011exact hpow