PP000B

prime_field_polynomial_zero_exists

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

Alpha v34 checked-use · first admitted v31 · 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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ l. Prime(p) → ∃ x. ∃ y. BetaPrefixInto(x,y,l,p) ∧ Repeat(x,y,0,l)

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 1Prime(p) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_polynomial_repeat_existsprime_field_zero_below_prime · checked external prerequisite
Original expanded first-order 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))))))

Complete tactic proof in conservative notation

All 10 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

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.

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)
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 defined 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