PP000B

prime_field_polynomial_zero_exists

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

Every prime admits an actual all-zero coefficient table of every finite representation length.

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 expanded first-order arithmetic statement

forall p l. (~((p) = 1) /\ forall pfa_factor_left_zero_domain pfa_factor_right_zero_domain. (p) = pfa_factor_left_zero_domain * pfa_factor_right_zero_domain -> pfa_factor_left_zero_domain = 1 \/ pfa_factor_right_zero_domain = 1) -> exists b c. ((forall fom_index_pfp_zero_table. (exists fom_gap_pfp_zero_table_index_bound. fom_gap_pfp_zero_table_index_bound + S (fom_index_pfp_zero_table) = l) -> exists fom_value_pfp_zero_table. ((((exists fom_beta_height_pfp_zero_table_entry. fom_beta_height_pfp_zero_table_entry + S (fom_value_pfp_zero_table) = S ((S (fom_index_pfp_zero_table)) * c)) /\ exists fom_beta_quotient_pfp_zero_table_entry. b = fom_beta_quotient_pfp_zero_table_entry * S ((S (fom_index_pfp_zero_table)) * c) + (fom_value_pfp_zero_table))) /\ (exists fom_gap_pfp_zero_table_value_bound. fom_gap_pfp_zero_table_value_bound + S (fom_value_pfp_zero_table) = p))) /\ ((forall pfp_repeat_index_zero_value. (exists pfa_gap_zero_valueindex. pfa_gap_zero_valueindex + S (pfp_repeat_index_zero_value) = (l)) -> (((exists ff_h_pfp_zero_valueentry. ff_h_pfp_zero_valueentry + S (0) = S ((S (pfp_repeat_index_zero_value)) * c)) /\ exists ff_q_pfp_zero_valueentry. b = ff_q_pfp_zero_valueentry * S ((S (pfp_repeat_index_zero_value)) * c) + (0))))))

Constructive proof overview

Generated structural guide

Every prime admits an actual all-zero coefficient table of every finite representation length.

The unchanged tactic script uses 2 declared prerequisites and contains 10 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

PP000A prime_field_polynomial_repeat_exists prime_field_zero_below_prime Alpha theorem; checked-use authorized

Direct dependents

none

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

10 script commands · 2 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.

Named ingredients (1)
prime field polynomial repeat exists
01Fix variables and assumptionsL1–3

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

  1. L1
    intro p
  2. L2
    intro l
  3. L3
    intro hp
02Use earlier factsL4–10

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

  1. L4
    specialize prime_field_polynomial_repeat_exists (p)
  2. L5
    specialize prime_field_polynomial_repeat_exists (0)
  3. L6
    specialize prime_field_polynomial_repeat_exists (l)
  4. L7
    apply prime_field_polynomial_repeat_exists
  5. L8
    specialize prime_field_zero_below_prime (p)
  6. L9
    apply prime_field_zero_below_prime
  7. L10
    exact hp

Library-wide reading audit

Original exact command ledger · 10 lines
  1. 0001intro p
  2. 0002intro l
  3. 0003intro hp
  4. 0004specialize prime_field_polynomial_repeat_exists (p)
  5. 0005specialize prime_field_polynomial_repeat_exists (0)
  6. 0006specialize prime_field_polynomial_repeat_exists (l)
  7. 0007apply prime_field_polynomial_repeat_exists
  8. 0008specialize prime_field_zero_below_prime (p)
  9. 0009apply prime_field_zero_below_prime
  10. 0010exact hp