EL0013

lte_power_exponent_eq_transport

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

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.

All displayed hypotheses are required. Powers and positive differences are actual existential outputs. The proof constructs second-order correction identities and iterates the prime step; no binomial expansion or LTE oracle is assumed. The 2-adic variants remain separate open targets.

Exact theorem in conservative defined notation

∀ a. ∀ n. ∀ m. ∀ A. n = m → Pow(a,n,A) → Pow(a,m,A)

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 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))))))))

Complete tactic proof in conservative notation

All 11 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

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.

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 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 defined 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