PV0003

prime_valuation_nondivisor_of_zero

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

Valuation zero excludes divisibility, with both intended domain guards explicit.

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 n. (~((p) = 1) /\ forall pvs_left_nondivisor_domain pvs_right_nondivisor_domain. (p) = pvs_left_nondivisor_domain * pvs_right_nondivisor_domain -> pvs_left_nondivisor_domain = 1 \/ pvs_right_nondivisor_domain = 1) -> ~(n = 0) -> (((exists bpd_gap_pvs_nondivisor_value_selected_bound. bpd_gap_pvs_nondivisor_value_selected_bound + (0) = (n)) /\ (exists bpvi_result_pvs_nondivisor_value_selected. ((exists bpvi_b_pvs_nondivisor_value_selected_power bpvi_c_pvs_nondivisor_value_selected_power. ((forall bpvi_i_pvs_nondivisor_value_selected_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_selected_power. bpvi_repeat_gap_pvs_nondivisor_value_selected_power + S bpvi_i_pvs_nondivisor_value_selected_power = 0) -> (((exists bpvi_h_pvs_nondivisor_value_selected_power_repeat. bpvi_h_pvs_nondivisor_value_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_repeat. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (p)))) /\ (exists bpvi_u_pvs_nondivisor_value_selected_power bpvi_v_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_start. bpvi_h_pvs_nondivisor_value_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_start. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_terminal. bpvi_h_pvs_nondivisor_value_selected_power_terminal + S (bpvi_result_pvs_nondivisor_value_selected) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_terminal. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_result_pvs_nondivisor_value_selected))) /\ forall bpvi_j_pvs_nondivisor_value_selected_power. (exists bpvi_product_gap_pvs_nondivisor_value_selected_power. bpvi_product_gap_pvs_nondivisor_value_selected_power + S bpvi_j_pvs_nondivisor_value_selected_power = 0) -> exists bpvi_factor_pvs_nondivisor_value_selected_power bpvi_partial_pvs_nondivisor_value_selected_power bpvi_successor_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_factor. bpvi_h_pvs_nondivisor_value_selected_power_factor + S (bpvi_factor_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_factor. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (bpvi_factor_pvs_nondivisor_value_selected_power))) /\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_partial. bpvi_h_pvs_nondivisor_value_selected_power_partial + S (bpvi_partial_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_partial. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_partial_pvs_nondivisor_value_selected_power))) /\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_successor. bpvi_h_pvs_nondivisor_value_selected_power_successor + S (bpvi_successor_pvs_nondivisor_value_selected_power) = S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\ exists bpvi_q_pvs_nondivisor_value_selected_power_successor. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_successor_pvs_nondivisor_value_selected_power))) /\ bpvi_successor_pvs_nondivisor_value_selected_power = bpvi_partial_pvs_nondivisor_value_selected_power * bpvi_factor_pvs_nondivisor_value_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_nondivisor_value_selected. n = bpvi_result_pvs_nondivisor_value_selected * bpvi_divisor_factor_pvs_nondivisor_value_selected))) /\ forall bpd_candidate_pvs_nondivisor_value. (exists bpd_gap_pvs_nondivisor_value_candidate_bound. bpd_gap_pvs_nondivisor_value_candidate_bound + (bpd_candidate_pvs_nondivisor_value) = (n)) -> (exists bpvi_result_pvs_nondivisor_value_candidate. ((exists bpvi_b_pvs_nondivisor_value_candidate_power bpvi_c_pvs_nondivisor_value_candidate_power. ((forall bpvi_i_pvs_nondivisor_value_candidate_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_candidate_power. bpvi_repeat_gap_pvs_nondivisor_value_candidate_power + S bpvi_i_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> (((exists bpvi_h_pvs_nondivisor_value_candidate_power_repeat. bpvi_h_pvs_nondivisor_value_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_repeat. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (p)))) /\ (exists bpvi_u_pvs_nondivisor_value_candidate_power bpvi_v_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_start. bpvi_h_pvs_nondivisor_value_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_start. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_terminal. bpvi_h_pvs_nondivisor_value_candidate_power_terminal + S (bpvi_result_pvs_nondivisor_value_candidate) = S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_terminal. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_terminal * S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_result_pvs_nondivisor_value_candidate))) /\ forall bpvi_j_pvs_nondivisor_value_candidate_power. (exists bpvi_product_gap_pvs_nondivisor_value_candidate_power. bpvi_product_gap_pvs_nondivisor_value_candidate_power + S bpvi_j_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> exists bpvi_factor_pvs_nondivisor_value_candidate_power bpvi_partial_pvs_nondivisor_value_candidate_power bpvi_successor_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_factor. bpvi_h_pvs_nondivisor_value_candidate_power_factor + S (bpvi_factor_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_factor. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (bpvi_factor_pvs_nondivisor_value_candidate_power))) /\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_partial. bpvi_h_pvs_nondivisor_value_candidate_power_partial + S (bpvi_partial_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_partial. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_partial_pvs_nondivisor_value_candidate_power))) /\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_successor. bpvi_h_pvs_nondivisor_value_candidate_power_successor + S (bpvi_successor_pvs_nondivisor_value_candidate_power) = S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\ exists bpvi_q_pvs_nondivisor_value_candidate_power_successor. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_successor_pvs_nondivisor_value_candidate_power))) /\ bpvi_successor_pvs_nondivisor_value_candidate_power = bpvi_partial_pvs_nondivisor_value_candidate_power * bpvi_factor_pvs_nondivisor_value_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_nondivisor_value_candidate. n = bpvi_result_pvs_nondivisor_value_candidate * bpvi_divisor_factor_pvs_nondivisor_value_candidate)) -> (exists bpd_gap_pvs_nondivisor_value_maximal. bpd_gap_pvs_nondivisor_value_maximal + (bpd_candidate_pvs_nondivisor_value) = (0))) -> ~(exists pvs_factor_nondivisor_result. (n) = (p) * pvs_factor_nondivisor_result)

Constructive proof overview

Generated structural guide

Valuation zero excludes divisibility, with both intended domain guards explicit.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

prime_power_valuation_zero_iff_not_divides Alpha theorem; checked-use authorized

Direct dependents

PV0008 prime_divisor_of_prime_power

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

18 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.

01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro hp
  4. L4
    intro hn
  5. L5
    intro hval
02Establish hiffL6–13

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime power valuation zero iff not divides.

  1. L6
    have hiff : (0 = 0 -> ~(exists pvs_factor_nondivisor_forward. (n) = (p) * pvs_factor_nondivisor_forward)) /\ (~(exists pvs_factor_nondivisor_reverse. (n) = (p) * pvs_factor_nondivisor_reverse) -> 0 = 0)
  2. L7
    specialize prime_power_valuation_zero_iff_not_divides (p)
  3. L8
    specialize prime_power_valuation_zero_iff_not_divides (n)
  4. L9
    specialize prime_power_valuation_zero_iff_not_divides (0)
  5. L10
    apply prime_power_valuation_zero_iff_not_divides
  6. L11
    exact hp
  7. L12
    exact hn
  8. L13
    exact hval
03Separate the logical casesL14–14

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

  1. L14
    cases hiff
04Fix variables and assumptionsL15–15

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

  1. L15
    intro hdiv
05Use earlier factsL16–16

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

  1. L16
    apply hiff_left
06Calculate and transport equalitiesL17–17

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

  1. L17
    refl
07Use earlier factsL18–18

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

  1. L18
    exact hdiv

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro hp
  4. 0004intro hn
  5. 0005intro hval
  6. 0006have hiff : (0 = 0 -> ~(exists pvs_factor_nondivisor_forward. (n) = (p) * pvs_factor_nondivisor_forward)) /\ (~(exists pvs_factor_nondivisor_reverse. (n) = (p) * pvs_factor_nondivisor_reverse) -> 0 = 0)
  7. 0007specialize prime_power_valuation_zero_iff_not_divides (p)
  8. 0008specialize prime_power_valuation_zero_iff_not_divides (n)
  9. 0009specialize prime_power_valuation_zero_iff_not_divides (0)
  10. 0010apply prime_power_valuation_zero_iff_not_divides
  11. 0011exact hp
  12. 0012exact hn
  13. 0013exact hval
  14. 0014cases hiff
  15. 0015intro hdiv
  16. 0016apply hiff_left
  17. 0017refl
  18. 0018exact hdiv