PP000A

prime_field_polynomial_repeat_exists

Construct a finite coefficient table containing exactly the chosen canonical coefficient at every position.

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. ∀ a. ∀ l. Lt(a,p) → ∃ x. ∃ y. BetaPrefixInto(x,y,l,p) ∧ Repeat(x,y,a,l)

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 1Lt(a,b) · 1Repeat(b,c,a,l) · 2

Actual proof prerequisites

beta_repeat_exists · checked external prerequisiteprime_field_polynomial_repeat_coefficients
Original expanded first-order statement 
forall p a l. (exists pfa_gap_repeat_domain. pfa_gap_repeat_domain + S (a) = (p)) -> exists b c. ((forall fom_index_pfp_repeat_table. (exists fom_gap_pfp_repeat_table_index_bound. fom_gap_pfp_repeat_table_index_bound + S (fom_index_pfp_repeat_table) = l) -> exists fom_value_pfp_repeat_table. ((((exists fom_beta_height_pfp_repeat_table_entry. fom_beta_height_pfp_repeat_table_entry + S (fom_value_pfp_repeat_table) = S ((S (fom_index_pfp_repeat_table)) * c)) /\ exists fom_beta_quotient_pfp_repeat_table_entry. b = fom_beta_quotient_pfp_repeat_table_entry * S ((S (fom_index_pfp_repeat_table)) * c) + (fom_value_pfp_repeat_table))) /\ (exists fom_gap_pfp_repeat_table_value_bound. fom_gap_pfp_repeat_table_value_bound + S (fom_value_pfp_repeat_table) = p))) /\ ((forall pfp_repeat_index_repeat_value. (exists pfa_gap_repeat_valueindex. pfa_gap_repeat_valueindex + S (pfp_repeat_index_repeat_value) = (l)) -> (((exists ff_h_pfp_repeat_valueentry. ff_h_pfp_repeat_valueentry + S (a) = S ((S (pfp_repeat_index_repeat_value)) * c)) /\ exists ff_q_pfp_repeat_valueentry. b = ff_q_pfp_repeat_valueentry * S ((S (pfp_repeat_index_repeat_value)) * c) + (a))))))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 6 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)
prime field polynomial repeat coefficients
01Fix variables and assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro l
  4. L4
    intro ha
02Establish hrL5–8

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

  1. L5
    have hr : ∃ b. ∃ c. Repeat(b,c,a,l)Definitions: Repeat(b,c,a,l)Original native command in the exact edition
  2. L6
    specialize beta_repeat_exists (a)
  3. L7
    specialize beta_repeat_exists (l)
  4. L8
    apply beta_repeat_exists
03Separate the logical casesL9–10

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

  1. L9
    cases hr
  2. L10
    cases hr_witness
04Construct an explicit witnessL11–12

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

  1. L11
    exists x
  2. L12
    exists x1
05Separate the logical casesL13–13

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

  1. L13
    split
06Use earlier factsL14–22

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

  1. L14
    specialize prime_field_polynomial_repeat_coefficients (p)
  2. L15
    specialize prime_field_polynomial_repeat_coefficients (x)
  3. L16
    specialize prime_field_polynomial_repeat_coefficients (x1)
  4. L17
    specialize prime_field_polynomial_repeat_coefficients (a)
  5. L18
    specialize prime_field_polynomial_repeat_coefficients (l)
  6. L19
    apply prime_field_polynomial_repeat_coefficients
  7. L20
    exact ha
  8. L21
    exact hr_witness_witness
  9. L22
    exact hr_witness_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro l
  4. 0004intro ha
  5. 0005have hr : ∃ b. ∃ c. Repeat(b,c,a,l)
  6. 0006specialize beta_repeat_exists (a)
  7. 0007specialize beta_repeat_exists (l)
  8. 0008apply beta_repeat_exists
  9. 0009cases hr
  10. 0010cases hr_witness
  11. 0011exists x
  12. 0012exists x1
  13. 0013split
  14. 0014specialize prime_field_polynomial_repeat_coefficients (p)
  15. 0015specialize prime_field_polynomial_repeat_coefficients (x)
  16. 0016specialize prime_field_polynomial_repeat_coefficients (x1)
  17. 0017specialize prime_field_polynomial_repeat_coefficients (a)
  18. 0018specialize prime_field_polynomial_repeat_coefficients (l)
  19. 0019apply prime_field_polynomial_repeat_coefficients
  20. 0020exact ha
  21. 0021exact hr_witness_witness
  22. 0022exact hr_witness_witness