EL0015

lte_power_iteration_construct

Construct a composed power graph from two actual powers and their exact multiplied exponent.

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

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

Definition DAG

Pow(a,e,z) · 4

Actual proof prerequisites

pow_exists · checked external prerequisitepow_mul_exp · checked external prerequisitelte_power_value_eq_transport
Original expanded first-order statement 
forall a n k m A C. m = n * k -> (exists pa_b_olte_iteration_base pa_c_olte_iteration_base. ((forall pa_i_olte_iteration_base_repeat. (exists pa_lt_olte_iteration_base_repeat_bound. pa_lt_olte_iteration_base_repeat_bound + S pa_i_olte_iteration_base_repeat = n) -> (((exists pa_h_olte_iteration_base_repeat_decoded. pa_h_olte_iteration_base_repeat_decoded + S (a) = S ((S (pa_i_olte_iteration_base_repeat)) * pa_c_olte_iteration_base)) /\ exists pa_q_olte_iteration_base_repeat_decoded. pa_b_olte_iteration_base = pa_q_olte_iteration_base_repeat_decoded * S ((S (pa_i_olte_iteration_base_repeat)) * pa_c_olte_iteration_base) + (a)))) /\ (exists pa_u_olte_iteration_base_product pa_v_olte_iteration_base_product. ((((exists pa_h_olte_iteration_base_product_start. pa_h_olte_iteration_base_product_start + S (1) = S ((S (0)) * pa_v_olte_iteration_base_product)) /\ exists pa_q_olte_iteration_base_product_start. pa_u_olte_iteration_base_product = pa_q_olte_iteration_base_product_start * S ((S (0)) * pa_v_olte_iteration_base_product) + (1))) /\ ((((exists pa_h_olte_iteration_base_product_terminal. pa_h_olte_iteration_base_product_terminal + S (A) = S ((S (n)) * pa_v_olte_iteration_base_product)) /\ exists pa_q_olte_iteration_base_product_terminal. pa_u_olte_iteration_base_product = pa_q_olte_iteration_base_product_terminal * S ((S (n)) * pa_v_olte_iteration_base_product) + (A))) /\ forall pa_i_olte_iteration_base_product. (exists pa_lt_olte_iteration_base_product_bound. pa_lt_olte_iteration_base_product_bound + S pa_i_olte_iteration_base_product = n) -> exists pa_p_olte_iteration_base_product pa_r_olte_iteration_base_product pa_s_olte_iteration_base_product. ((((exists pa_h_olte_iteration_base_product_factor. pa_h_olte_iteration_base_product_factor + S (pa_p_olte_iteration_base_product) = S ((S (pa_i_olte_iteration_base_product)) * pa_c_olte_iteration_base)) /\ exists pa_q_olte_iteration_base_product_factor. pa_b_olte_iteration_base = pa_q_olte_iteration_base_product_factor * S ((S (pa_i_olte_iteration_base_product)) * pa_c_olte_iteration_base) + (pa_p_olte_iteration_base_product))) /\ ((((exists pa_h_olte_iteration_base_product_partial. pa_h_olte_iteration_base_product_partial + S (pa_r_olte_iteration_base_product) = S ((S (pa_i_olte_iteration_base_product)) * pa_v_olte_iteration_base_product)) /\ exists pa_q_olte_iteration_base_product_partial. pa_u_olte_iteration_base_product = pa_q_olte_iteration_base_product_partial * S ((S (pa_i_olte_iteration_base_product)) * pa_v_olte_iteration_base_product) + (pa_r_olte_iteration_base_product))) /\ ((((exists pa_h_olte_iteration_base_product_successor. pa_h_olte_iteration_base_product_successor + S (pa_s_olte_iteration_base_product) = S ((S (S pa_i_olte_iteration_base_product)) * pa_v_olte_iteration_base_product)) /\ exists pa_q_olte_iteration_base_product_successor. pa_u_olte_iteration_base_product = pa_q_olte_iteration_base_product_successor * S ((S (S pa_i_olte_iteration_base_product)) * pa_v_olte_iteration_base_product) + (pa_s_olte_iteration_base_product))) /\ pa_s_olte_iteration_base_product = pa_r_olte_iteration_base_product * pa_p_olte_iteration_base_product)))))))) -> (exists pa_b_olte_iteration_outer pa_c_olte_iteration_outer. ((forall pa_i_olte_iteration_outer_repeat. (exists pa_lt_olte_iteration_outer_repeat_bound. pa_lt_olte_iteration_outer_repeat_bound + S pa_i_olte_iteration_outer_repeat = k) -> (((exists pa_h_olte_iteration_outer_repeat_decoded. pa_h_olte_iteration_outer_repeat_decoded + S (A) = S ((S (pa_i_olte_iteration_outer_repeat)) * pa_c_olte_iteration_outer)) /\ exists pa_q_olte_iteration_outer_repeat_decoded. pa_b_olte_iteration_outer = pa_q_olte_iteration_outer_repeat_decoded * S ((S (pa_i_olte_iteration_outer_repeat)) * pa_c_olte_iteration_outer) + (A)))) /\ (exists pa_u_olte_iteration_outer_product pa_v_olte_iteration_outer_product. ((((exists pa_h_olte_iteration_outer_product_start. pa_h_olte_iteration_outer_product_start + S (1) = S ((S (0)) * pa_v_olte_iteration_outer_product)) /\ exists pa_q_olte_iteration_outer_product_start. pa_u_olte_iteration_outer_product = pa_q_olte_iteration_outer_product_start * S ((S (0)) * pa_v_olte_iteration_outer_product) + (1))) /\ ((((exists pa_h_olte_iteration_outer_product_terminal. pa_h_olte_iteration_outer_product_terminal + S (C) = S ((S (k)) * pa_v_olte_iteration_outer_product)) /\ exists pa_q_olte_iteration_outer_product_terminal. pa_u_olte_iteration_outer_product = pa_q_olte_iteration_outer_product_terminal * S ((S (k)) * pa_v_olte_iteration_outer_product) + (C))) /\ forall pa_i_olte_iteration_outer_product. (exists pa_lt_olte_iteration_outer_product_bound. pa_lt_olte_iteration_outer_product_bound + S pa_i_olte_iteration_outer_product = k) -> exists pa_p_olte_iteration_outer_product pa_r_olte_iteration_outer_product pa_s_olte_iteration_outer_product. ((((exists pa_h_olte_iteration_outer_product_factor. pa_h_olte_iteration_outer_product_factor + S (pa_p_olte_iteration_outer_product) = S ((S (pa_i_olte_iteration_outer_product)) * pa_c_olte_iteration_outer)) /\ exists pa_q_olte_iteration_outer_product_factor. pa_b_olte_iteration_outer = pa_q_olte_iteration_outer_product_factor * S ((S (pa_i_olte_iteration_outer_product)) * pa_c_olte_iteration_outer) + (pa_p_olte_iteration_outer_product))) /\ ((((exists pa_h_olte_iteration_outer_product_partial. pa_h_olte_iteration_outer_product_partial + S (pa_r_olte_iteration_outer_product) = S ((S (pa_i_olte_iteration_outer_product)) * pa_v_olte_iteration_outer_product)) /\ exists pa_q_olte_iteration_outer_product_partial. pa_u_olte_iteration_outer_product = pa_q_olte_iteration_outer_product_partial * S ((S (pa_i_olte_iteration_outer_product)) * pa_v_olte_iteration_outer_product) + (pa_r_olte_iteration_outer_product))) /\ ((((exists pa_h_olte_iteration_outer_product_successor. pa_h_olte_iteration_outer_product_successor + S (pa_s_olte_iteration_outer_product) = S ((S (S pa_i_olte_iteration_outer_product)) * pa_v_olte_iteration_outer_product)) /\ exists pa_q_olte_iteration_outer_product_successor. pa_u_olte_iteration_outer_product = pa_q_olte_iteration_outer_product_successor * S ((S (S pa_i_olte_iteration_outer_product)) * pa_v_olte_iteration_outer_product) + (pa_s_olte_iteration_outer_product))) /\ pa_s_olte_iteration_outer_product = pa_r_olte_iteration_outer_product * pa_p_olte_iteration_outer_product)))))))) -> (exists pa_b_olte_iteration_result pa_c_olte_iteration_result. ((forall pa_i_olte_iteration_result_repeat. (exists pa_lt_olte_iteration_result_repeat_bound. pa_lt_olte_iteration_result_repeat_bound + S pa_i_olte_iteration_result_repeat = m) -> (((exists pa_h_olte_iteration_result_repeat_decoded. pa_h_olte_iteration_result_repeat_decoded + S (a) = S ((S (pa_i_olte_iteration_result_repeat)) * pa_c_olte_iteration_result)) /\ exists pa_q_olte_iteration_result_repeat_decoded. pa_b_olte_iteration_result = pa_q_olte_iteration_result_repeat_decoded * S ((S (pa_i_olte_iteration_result_repeat)) * pa_c_olte_iteration_result) + (a)))) /\ (exists pa_u_olte_iteration_result_product pa_v_olte_iteration_result_product. ((((exists pa_h_olte_iteration_result_product_start. pa_h_olte_iteration_result_product_start + S (1) = S ((S (0)) * pa_v_olte_iteration_result_product)) /\ exists pa_q_olte_iteration_result_product_start. pa_u_olte_iteration_result_product = pa_q_olte_iteration_result_product_start * S ((S (0)) * pa_v_olte_iteration_result_product) + (1))) /\ ((((exists pa_h_olte_iteration_result_product_terminal. pa_h_olte_iteration_result_product_terminal + S (C) = S ((S (m)) * pa_v_olte_iteration_result_product)) /\ exists pa_q_olte_iteration_result_product_terminal. pa_u_olte_iteration_result_product = pa_q_olte_iteration_result_product_terminal * S ((S (m)) * pa_v_olte_iteration_result_product) + (C))) /\ forall pa_i_olte_iteration_result_product. (exists pa_lt_olte_iteration_result_product_bound. pa_lt_olte_iteration_result_product_bound + S pa_i_olte_iteration_result_product = m) -> exists pa_p_olte_iteration_result_product pa_r_olte_iteration_result_product pa_s_olte_iteration_result_product. ((((exists pa_h_olte_iteration_result_product_factor. pa_h_olte_iteration_result_product_factor + S (pa_p_olte_iteration_result_product) = S ((S (pa_i_olte_iteration_result_product)) * pa_c_olte_iteration_result)) /\ exists pa_q_olte_iteration_result_product_factor. pa_b_olte_iteration_result = pa_q_olte_iteration_result_product_factor * S ((S (pa_i_olte_iteration_result_product)) * pa_c_olte_iteration_result) + (pa_p_olte_iteration_result_product))) /\ ((((exists pa_h_olte_iteration_result_product_partial. pa_h_olte_iteration_result_product_partial + S (pa_r_olte_iteration_result_product) = S ((S (pa_i_olte_iteration_result_product)) * pa_v_olte_iteration_result_product)) /\ exists pa_q_olte_iteration_result_product_partial. pa_u_olte_iteration_result_product = pa_q_olte_iteration_result_product_partial * S ((S (pa_i_olte_iteration_result_product)) * pa_v_olte_iteration_result_product) + (pa_r_olte_iteration_result_product))) /\ ((((exists pa_h_olte_iteration_result_product_successor. pa_h_olte_iteration_result_product_successor + S (pa_s_olte_iteration_result_product) = S ((S (S pa_i_olte_iteration_result_product)) * pa_v_olte_iteration_result_product)) /\ exists pa_q_olte_iteration_result_product_successor. pa_u_olte_iteration_result_product = pa_q_olte_iteration_result_product_successor * S ((S (S pa_i_olte_iteration_result_product)) * pa_v_olte_iteration_result_product) + (pa_s_olte_iteration_result_product))) /\ pa_s_olte_iteration_result_product = pa_r_olte_iteration_result_product * pa_p_olte_iteration_result_product))))))))

Complete tactic proof in conservative notation

All 33 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

33 script commands · 7 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 (1)
lte power value eq transport
01Fix variables and assumptionsL1–9

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

  1. L1
    intro a
  2. L2
    intro n
  3. L3
    intro k
  4. L4
    intro m
  5. L5
    intro A
  6. L6
    intro C
  7. L7
    intro hm
  8. L8
    intro hA
  9. L9
    intro hC
02Establish hexL10–13

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

  1. L10
    have hex : ∃ z. Pow(a,m,z)Definitions: Pow(a,m,z)Original native command in the exact edition
  2. L11
    specialize pow_exists (a)
  3. L12
    specialize pow_exists (m)
  4. L13
    apply pow_exists
03Separate the logical casesL14–14

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

  1. L14
    cases hex
04Use earlier factsL15–19

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

  1. L15
    specialize lte_power_value_eq_transport (a)
  2. L16
    specialize lte_power_value_eq_transport (m)
  3. L17
    specialize lte_power_value_eq_transport (x)
  4. L18
    specialize lte_power_value_eq_transport (C)
  5. L19
    apply lte_power_value_eq_transport
05Calculate and transport equalitiesL20–20

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

  1. L20
    symm
06Use earlier factsL21–30

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

  1. L21
    specialize pow_mul_exp (a)
  2. L22
    specialize pow_mul_exp (n)
  3. L23
    specialize pow_mul_exp (k)
  4. L24
    specialize pow_mul_exp (m)
  5. L25
    specialize pow_mul_exp (A)
  6. L26
    specialize pow_mul_exp (C)
  7. L27
    specialize pow_mul_exp (x)
  8. L28
    apply pow_mul_exp
  9. L29
    exact hm
  10. L30
    exact hA
07Use earlier factsL31–33

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

  1. L31
    exact hC
  2. L32
    exact hex_witness
  3. L33
    exact hex_witness

Library-wide reading audit

Original defined command ledger · 33 lines
  1. 0001intro a
  2. 0002intro n
  3. 0003intro k
  4. 0004intro m
  5. 0005intro A
  6. 0006intro C
  7. 0007intro hm
  8. 0008intro hA
  9. 0009intro hC
  10. 0010have hex : ∃ z. Pow(a,m,z)
  11. 0011specialize pow_exists (a)
  12. 0012specialize pow_exists (m)
  13. 0013apply pow_exists
  14. 0014cases hex
  15. 0015specialize lte_power_value_eq_transport (a)
  16. 0016specialize lte_power_value_eq_transport (m)
  17. 0017specialize lte_power_value_eq_transport (x)
  18. 0018specialize lte_power_value_eq_transport (C)
  19. 0019apply lte_power_value_eq_transport
  20. 0020symm
  21. 0021specialize pow_mul_exp (a)
  22. 0022specialize pow_mul_exp (n)
  23. 0023specialize pow_mul_exp (k)
  24. 0024specialize pow_mul_exp (m)
  25. 0025specialize pow_mul_exp (A)
  26. 0026specialize pow_mul_exp (C)
  27. 0027specialize pow_mul_exp (x)
  28. 0028apply pow_mul_exp
  29. 0029exact hm
  30. 0030exact hA
  31. 0031exact hC
  32. 0032exact hex_witness
  33. 0033exact hex_witness