SK0015

power_divisible_exponent_root

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

A witnessed exponent quotient constructs the corresponding natural root of an actual prime power; this algebraic lemma needs no prime assumption.

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 p e k t P. e = k * t -> (exists pa_b_pvs_exponent_source pa_c_pvs_exponent_source. ((forall pa_i_pvs_exponent_source_repeat. (exists pa_lt_pvs_exponent_source_repeat_bound. pa_lt_pvs_exponent_source_repeat_bound + S pa_i_pvs_exponent_source_repeat = e) -> (((exists pa_h_pvs_exponent_source_repeat_decoded. pa_h_pvs_exponent_source_repeat_decoded + S (p) = S ((S (pa_i_pvs_exponent_source_repeat)) * pa_c_pvs_exponent_source)) /\ exists pa_q_pvs_exponent_source_repeat_decoded. pa_b_pvs_exponent_source = pa_q_pvs_exponent_source_repeat_decoded * S ((S (pa_i_pvs_exponent_source_repeat)) * pa_c_pvs_exponent_source) + (p)))) /\ (exists pa_u_pvs_exponent_source_product pa_v_pvs_exponent_source_product. ((((exists pa_h_pvs_exponent_source_product_start. pa_h_pvs_exponent_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_exponent_source_product)) /\ exists pa_q_pvs_exponent_source_product_start. pa_u_pvs_exponent_source_product = pa_q_pvs_exponent_source_product_start * S ((S (0)) * pa_v_pvs_exponent_source_product) + (1))) /\ ((((exists pa_h_pvs_exponent_source_product_terminal. pa_h_pvs_exponent_source_product_terminal + S (P) = S ((S (e)) * pa_v_pvs_exponent_source_product)) /\ exists pa_q_pvs_exponent_source_product_terminal. pa_u_pvs_exponent_source_product = pa_q_pvs_exponent_source_product_terminal * S ((S (e)) * pa_v_pvs_exponent_source_product) + (P))) /\ forall pa_i_pvs_exponent_source_product. (exists pa_lt_pvs_exponent_source_product_bound. pa_lt_pvs_exponent_source_product_bound + S pa_i_pvs_exponent_source_product = e) -> exists pa_p_pvs_exponent_source_product pa_r_pvs_exponent_source_product pa_s_pvs_exponent_source_product. ((((exists pa_h_pvs_exponent_source_product_factor. pa_h_pvs_exponent_source_product_factor + S (pa_p_pvs_exponent_source_product) = S ((S (pa_i_pvs_exponent_source_product)) * pa_c_pvs_exponent_source)) /\ exists pa_q_pvs_exponent_source_product_factor. pa_b_pvs_exponent_source = pa_q_pvs_exponent_source_product_factor * S ((S (pa_i_pvs_exponent_source_product)) * pa_c_pvs_exponent_source) + (pa_p_pvs_exponent_source_product))) /\ ((((exists pa_h_pvs_exponent_source_product_partial. pa_h_pvs_exponent_source_product_partial + S (pa_r_pvs_exponent_source_product) = S ((S (pa_i_pvs_exponent_source_product)) * pa_v_pvs_exponent_source_product)) /\ exists pa_q_pvs_exponent_source_product_partial. pa_u_pvs_exponent_source_product = pa_q_pvs_exponent_source_product_partial * S ((S (pa_i_pvs_exponent_source_product)) * pa_v_pvs_exponent_source_product) + (pa_r_pvs_exponent_source_product))) /\ ((((exists pa_h_pvs_exponent_source_product_successor. pa_h_pvs_exponent_source_product_successor + S (pa_s_pvs_exponent_source_product) = S ((S (S pa_i_pvs_exponent_source_product)) * pa_v_pvs_exponent_source_product)) /\ exists pa_q_pvs_exponent_source_product_successor. pa_u_pvs_exponent_source_product = pa_q_pvs_exponent_source_product_successor * S ((S (S pa_i_pvs_exponent_source_product)) * pa_v_pvs_exponent_source_product) + (pa_s_pvs_exponent_source_product))) /\ pa_s_pvs_exponent_source_product = pa_r_pvs_exponent_source_product * pa_p_pvs_exponent_source_product)))))))) -> exists r. (exists pa_b_pvs_exponent_root pa_c_pvs_exponent_root. ((forall pa_i_pvs_exponent_root_repeat. (exists pa_lt_pvs_exponent_root_repeat_bound. pa_lt_pvs_exponent_root_repeat_bound + S pa_i_pvs_exponent_root_repeat = k) -> (((exists pa_h_pvs_exponent_root_repeat_decoded. pa_h_pvs_exponent_root_repeat_decoded + S (r) = S ((S (pa_i_pvs_exponent_root_repeat)) * pa_c_pvs_exponent_root)) /\ exists pa_q_pvs_exponent_root_repeat_decoded. pa_b_pvs_exponent_root = pa_q_pvs_exponent_root_repeat_decoded * S ((S (pa_i_pvs_exponent_root_repeat)) * pa_c_pvs_exponent_root) + (r)))) /\ (exists pa_u_pvs_exponent_root_product pa_v_pvs_exponent_root_product. ((((exists pa_h_pvs_exponent_root_product_start. pa_h_pvs_exponent_root_product_start + S (1) = S ((S (0)) * pa_v_pvs_exponent_root_product)) /\ exists pa_q_pvs_exponent_root_product_start. pa_u_pvs_exponent_root_product = pa_q_pvs_exponent_root_product_start * S ((S (0)) * pa_v_pvs_exponent_root_product) + (1))) /\ ((((exists pa_h_pvs_exponent_root_product_terminal. pa_h_pvs_exponent_root_product_terminal + S (P) = S ((S (k)) * pa_v_pvs_exponent_root_product)) /\ exists pa_q_pvs_exponent_root_product_terminal. pa_u_pvs_exponent_root_product = pa_q_pvs_exponent_root_product_terminal * S ((S (k)) * pa_v_pvs_exponent_root_product) + (P))) /\ forall pa_i_pvs_exponent_root_product. (exists pa_lt_pvs_exponent_root_product_bound. pa_lt_pvs_exponent_root_product_bound + S pa_i_pvs_exponent_root_product = k) -> exists pa_p_pvs_exponent_root_product pa_r_pvs_exponent_root_product pa_s_pvs_exponent_root_product. ((((exists pa_h_pvs_exponent_root_product_factor. pa_h_pvs_exponent_root_product_factor + S (pa_p_pvs_exponent_root_product) = S ((S (pa_i_pvs_exponent_root_product)) * pa_c_pvs_exponent_root)) /\ exists pa_q_pvs_exponent_root_product_factor. pa_b_pvs_exponent_root = pa_q_pvs_exponent_root_product_factor * S ((S (pa_i_pvs_exponent_root_product)) * pa_c_pvs_exponent_root) + (pa_p_pvs_exponent_root_product))) /\ ((((exists pa_h_pvs_exponent_root_product_partial. pa_h_pvs_exponent_root_product_partial + S (pa_r_pvs_exponent_root_product) = S ((S (pa_i_pvs_exponent_root_product)) * pa_v_pvs_exponent_root_product)) /\ exists pa_q_pvs_exponent_root_product_partial. pa_u_pvs_exponent_root_product = pa_q_pvs_exponent_root_product_partial * S ((S (pa_i_pvs_exponent_root_product)) * pa_v_pvs_exponent_root_product) + (pa_r_pvs_exponent_root_product))) /\ ((((exists pa_h_pvs_exponent_root_product_successor. pa_h_pvs_exponent_root_product_successor + S (pa_s_pvs_exponent_root_product) = S ((S (S pa_i_pvs_exponent_root_product)) * pa_v_pvs_exponent_root_product)) /\ exists pa_q_pvs_exponent_root_product_successor. pa_u_pvs_exponent_root_product = pa_q_pvs_exponent_root_product_successor * S ((S (S pa_i_pvs_exponent_root_product)) * pa_v_pvs_exponent_root_product) + (pa_s_pvs_exponent_root_product))) /\ pa_s_pvs_exponent_root_product = pa_r_pvs_exponent_root_product * pa_p_pvs_exponent_root_product))))))))

Constructive proof overview

Generated structural guide

A witnessed exponent quotient constructs the corresponding natural root of an actual prime power; this algebraic lemma needs no prime assumption.

The unchanged tactic script uses 4 declared prerequisites and contains 38 exact native proof lines.

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

Proof neighborhood

Direct dependencies

pow_exists Stable theorem; checked-use authorized pow_mul_exp Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized SK0011 power_value_eq_transport

Direct dependents

SK0019 prime_valuation_divisible_power_root_bounded

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

38 script commands · 10 reading checkpoints · 2 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.

Named ingredients (1)
power value eq transport

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–7

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

  1. L1
    intro p
  2. L2
    intro e
  3. L3
    intro k
  4. L4
    intro t
  5. L5
    intro P
  6. L6
    intro heq
  7. L7
    intro hpow
02Establish hrootL8–11

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

  1. L8
    have hroot : ∃ r. Pow(p,t,r)Definitions: Pow
  2. L9
    specialize pow_exists (p)
  3. L10
    specialize pow_exists (t)
  4. L11
    apply pow_exists
03Separate the logical casesL12–12

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

  1. L12
    cases hroot
04Establish houterL13–16

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

  1. L13
    have houter : ∃ z. Pow(x,k,z)Definitions: Pow
  2. L14
    specialize pow_exists (x)
  3. L15
    specialize pow_exists (k)
  4. L16
    apply pow_exists
05Separate the logical casesL17–17

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

  1. L17
    cases houter
06Construct an explicit witnessL18–18

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists x
07Use earlier factsL19–28

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

  1. L19
    specialize power_value_eq_transport (x)
  2. L20
    specialize power_value_eq_transport (k)
  3. L21
    specialize power_value_eq_transport (x1)
  4. L22
    specialize power_value_eq_transport (P)
  5. L23
    apply power_value_eq_transport
  6. L24
    specialize pow_mul_exp (p)
  7. L25
    specialize pow_mul_exp (t)
  8. L26
    specialize pow_mul_exp (k)
  9. L27
    specialize pow_mul_exp (e)
  10. L28
    specialize pow_mul_exp (x)
08Use earlier factsL29–31

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

  1. L29
    specialize pow_mul_exp (x1)
  2. L30
    specialize pow_mul_exp (P)
  3. L31
    apply pow_mul_exp
09Calculate and transport equalitiesL32–32

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

  1. L32
    trans k * t
10Use earlier factsL33–38

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

  1. L33
    exact heq
  2. L34
    apply mul_comm
  3. L35
    exact hroot_witness
  4. L36
    exact houter_witness
  5. L37
    exact hpow
  6. L38
    exact houter_witness

Library-wide reading audit

Original exact command ledger · 38 lines
  1. 0001intro p
  2. 0002intro e
  3. 0003intro k
  4. 0004intro t
  5. 0005intro P
  6. 0006intro heq
  7. 0007intro hpow
  8. 0008have hroot : exists r. (exists pa_b_pvs_exponent_inner pa_c_pvs_exponent_inner. ((forall pa_i_pvs_exponent_inner_repeat. (exists pa_lt_pvs_exponent_inner_repeat_bound. pa_lt_pvs_exponent_inner_repeat_bound + S pa_i_pvs_exponent_inner_repeat = t) -> (((exists pa_h_pvs_exponent_inner_repeat_decoded. pa_h_pvs_exponent_inner_repeat_decoded + S (p) = S ((S (pa_i_pvs_exponent_inner_repeat)) * pa_c_pvs_exponent_inner)) /\ exists pa_q_pvs_exponent_inner_repeat_decoded. pa_b_pvs_exponent_inner = pa_q_pvs_exponent_inner_repeat_decoded * S ((S (pa_i_pvs_exponent_inner_repeat)) * pa_c_pvs_exponent_inner) + (p)))) /\ (exists pa_u_pvs_exponent_inner_product pa_v_pvs_exponent_inner_product. ((((exists pa_h_pvs_exponent_inner_product_start. pa_h_pvs_exponent_inner_product_start + S (1) = S ((S (0)) * pa_v_pvs_exponent_inner_product)) /\ exists pa_q_pvs_exponent_inner_product_start. pa_u_pvs_exponent_inner_product = pa_q_pvs_exponent_inner_product_start * S ((S (0)) * pa_v_pvs_exponent_inner_product) + (1))) /\ ((((exists pa_h_pvs_exponent_inner_product_terminal. pa_h_pvs_exponent_inner_product_terminal + S (r) = S ((S (t)) * pa_v_pvs_exponent_inner_product)) /\ exists pa_q_pvs_exponent_inner_product_terminal. pa_u_pvs_exponent_inner_product = pa_q_pvs_exponent_inner_product_terminal * S ((S (t)) * pa_v_pvs_exponent_inner_product) + (r))) /\ forall pa_i_pvs_exponent_inner_product. (exists pa_lt_pvs_exponent_inner_product_bound. pa_lt_pvs_exponent_inner_product_bound + S pa_i_pvs_exponent_inner_product = t) -> exists pa_p_pvs_exponent_inner_product pa_r_pvs_exponent_inner_product pa_s_pvs_exponent_inner_product. ((((exists pa_h_pvs_exponent_inner_product_factor. pa_h_pvs_exponent_inner_product_factor + S (pa_p_pvs_exponent_inner_product) = S ((S (pa_i_pvs_exponent_inner_product)) * pa_c_pvs_exponent_inner)) /\ exists pa_q_pvs_exponent_inner_product_factor. pa_b_pvs_exponent_inner = pa_q_pvs_exponent_inner_product_factor * S ((S (pa_i_pvs_exponent_inner_product)) * pa_c_pvs_exponent_inner) + (pa_p_pvs_exponent_inner_product))) /\ ((((exists pa_h_pvs_exponent_inner_product_partial. pa_h_pvs_exponent_inner_product_partial + S (pa_r_pvs_exponent_inner_product) = S ((S (pa_i_pvs_exponent_inner_product)) * pa_v_pvs_exponent_inner_product)) /\ exists pa_q_pvs_exponent_inner_product_partial. pa_u_pvs_exponent_inner_product = pa_q_pvs_exponent_inner_product_partial * S ((S (pa_i_pvs_exponent_inner_product)) * pa_v_pvs_exponent_inner_product) + (pa_r_pvs_exponent_inner_product))) /\ ((((exists pa_h_pvs_exponent_inner_product_successor. pa_h_pvs_exponent_inner_product_successor + S (pa_s_pvs_exponent_inner_product) = S ((S (S pa_i_pvs_exponent_inner_product)) * pa_v_pvs_exponent_inner_product)) /\ exists pa_q_pvs_exponent_inner_product_successor. pa_u_pvs_exponent_inner_product = pa_q_pvs_exponent_inner_product_successor * S ((S (S pa_i_pvs_exponent_inner_product)) * pa_v_pvs_exponent_inner_product) + (pa_s_pvs_exponent_inner_product))) /\ pa_s_pvs_exponent_inner_product = pa_r_pvs_exponent_inner_product * pa_p_pvs_exponent_inner_product))))))))
  9. 0009specialize pow_exists (p)
  10. 0010specialize pow_exists (t)
  11. 0011apply pow_exists
  12. 0012cases hroot
  13. 0013have houter : exists z. (exists pa_b_pvs_exponent_outer pa_c_pvs_exponent_outer. ((forall pa_i_pvs_exponent_outer_repeat. (exists pa_lt_pvs_exponent_outer_repeat_bound. pa_lt_pvs_exponent_outer_repeat_bound + S pa_i_pvs_exponent_outer_repeat = k) -> (((exists pa_h_pvs_exponent_outer_repeat_decoded. pa_h_pvs_exponent_outer_repeat_decoded + S (x) = S ((S (pa_i_pvs_exponent_outer_repeat)) * pa_c_pvs_exponent_outer)) /\ exists pa_q_pvs_exponent_outer_repeat_decoded. pa_b_pvs_exponent_outer = pa_q_pvs_exponent_outer_repeat_decoded * S ((S (pa_i_pvs_exponent_outer_repeat)) * pa_c_pvs_exponent_outer) + (x)))) /\ (exists pa_u_pvs_exponent_outer_product pa_v_pvs_exponent_outer_product. ((((exists pa_h_pvs_exponent_outer_product_start. pa_h_pvs_exponent_outer_product_start + S (1) = S ((S (0)) * pa_v_pvs_exponent_outer_product)) /\ exists pa_q_pvs_exponent_outer_product_start. pa_u_pvs_exponent_outer_product = pa_q_pvs_exponent_outer_product_start * S ((S (0)) * pa_v_pvs_exponent_outer_product) + (1))) /\ ((((exists pa_h_pvs_exponent_outer_product_terminal. pa_h_pvs_exponent_outer_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_exponent_outer_product)) /\ exists pa_q_pvs_exponent_outer_product_terminal. pa_u_pvs_exponent_outer_product = pa_q_pvs_exponent_outer_product_terminal * S ((S (k)) * pa_v_pvs_exponent_outer_product) + (z))) /\ forall pa_i_pvs_exponent_outer_product. (exists pa_lt_pvs_exponent_outer_product_bound. pa_lt_pvs_exponent_outer_product_bound + S pa_i_pvs_exponent_outer_product = k) -> exists pa_p_pvs_exponent_outer_product pa_r_pvs_exponent_outer_product pa_s_pvs_exponent_outer_product. ((((exists pa_h_pvs_exponent_outer_product_factor. pa_h_pvs_exponent_outer_product_factor + S (pa_p_pvs_exponent_outer_product) = S ((S (pa_i_pvs_exponent_outer_product)) * pa_c_pvs_exponent_outer)) /\ exists pa_q_pvs_exponent_outer_product_factor. pa_b_pvs_exponent_outer = pa_q_pvs_exponent_outer_product_factor * S ((S (pa_i_pvs_exponent_outer_product)) * pa_c_pvs_exponent_outer) + (pa_p_pvs_exponent_outer_product))) /\ ((((exists pa_h_pvs_exponent_outer_product_partial. pa_h_pvs_exponent_outer_product_partial + S (pa_r_pvs_exponent_outer_product) = S ((S (pa_i_pvs_exponent_outer_product)) * pa_v_pvs_exponent_outer_product)) /\ exists pa_q_pvs_exponent_outer_product_partial. pa_u_pvs_exponent_outer_product = pa_q_pvs_exponent_outer_product_partial * S ((S (pa_i_pvs_exponent_outer_product)) * pa_v_pvs_exponent_outer_product) + (pa_r_pvs_exponent_outer_product))) /\ ((((exists pa_h_pvs_exponent_outer_product_successor. pa_h_pvs_exponent_outer_product_successor + S (pa_s_pvs_exponent_outer_product) = S ((S (S pa_i_pvs_exponent_outer_product)) * pa_v_pvs_exponent_outer_product)) /\ exists pa_q_pvs_exponent_outer_product_successor. pa_u_pvs_exponent_outer_product = pa_q_pvs_exponent_outer_product_successor * S ((S (S pa_i_pvs_exponent_outer_product)) * pa_v_pvs_exponent_outer_product) + (pa_s_pvs_exponent_outer_product))) /\ pa_s_pvs_exponent_outer_product = pa_r_pvs_exponent_outer_product * pa_p_pvs_exponent_outer_product))))))))
  14. 0014specialize pow_exists (x)
  15. 0015specialize pow_exists (k)
  16. 0016apply pow_exists
  17. 0017cases houter
  18. 0018exists x
  19. 0019specialize power_value_eq_transport (x)
  20. 0020specialize power_value_eq_transport (k)
  21. 0021specialize power_value_eq_transport (x1)
  22. 0022specialize power_value_eq_transport (P)
  23. 0023apply power_value_eq_transport
  24. 0024specialize pow_mul_exp (p)
  25. 0025specialize pow_mul_exp (t)
  26. 0026specialize pow_mul_exp (k)
  27. 0027specialize pow_mul_exp (e)
  28. 0028specialize pow_mul_exp (x)
  29. 0029specialize pow_mul_exp (x1)
  30. 0030specialize pow_mul_exp (P)
  31. 0031apply pow_mul_exp
  32. 0032trans k * t
  33. 0033exact heq
  34. 0034apply mul_comm
  35. 0035exact hroot_witness
  36. 0036exact houter_witness
  37. 0037exact hpow
  38. 0038exact houter_witness