PG0001

prime_field_polynomial_shift_exists

Construct a genuine trailing-zero prefix by the original beta-prefix extension theorem, including an empty source.

Alpha v34 checked-use · first admitted v34 · 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.

All products and sums use actual beta-coded coefficients; equality is formal coefficient equivalence, not equality of evaluations or raw codes. The zero gcd is included. Uniqueness is up to formal equivalence, not unique Bézout coefficients. This proves the polynomial gcd component, not G091 irreducible-polynomial existence or arbitrary prime-power-field construction.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ L. ∃ x. ∃ y. PolynomialShift(b,c,L,x,y)

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

Definition DAG

BetaPrefixEqual(b,c,d,e,l) · 1PolynomialShift(b,c,L,d,e) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

beta_prefix_extend · checked external prerequisite
Original expanded first-order statement 
forall b c L. (exists d e. ((forall mdr_i_pfp_shift_existsprefix mdr_a_pfp_shift_existsprefix. (exists mdr_gap_pfp_shift_existsprefixb. mdr_gap_pfp_shift_existsprefixb + S (mdr_i_pfp_shift_existsprefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_existsprefixo. ff_h_mdr_pfp_shift_existsprefixo + S (mdr_a_pfp_shift_existsprefix) = S ((S (mdr_i_pfp_shift_existsprefix)) * c)) /\ exists ff_q_mdr_pfp_shift_existsprefixo. b = ff_q_mdr_pfp_shift_existsprefixo * S ((S (mdr_i_pfp_shift_existsprefix)) * c) + (mdr_a_pfp_shift_existsprefix))) -> (((exists ff_h_mdr_pfp_shift_existsprefixn. ff_h_mdr_pfp_shift_existsprefixn + S (mdr_a_pfp_shift_existsprefix) = S ((S (mdr_i_pfp_shift_existsprefix)) * e)) /\ exists ff_q_mdr_pfp_shift_existsprefixn. d = ff_q_mdr_pfp_shift_existsprefixn * S ((S (mdr_i_pfp_shift_existsprefix)) * e) + (mdr_a_pfp_shift_existsprefix)))) /\ ((((exists ff_h_pfp_shift_existslast. ff_h_pfp_shift_existslast + S (0) = S ((S (L)) * e)) /\ exists ff_q_pfp_shift_existslast. d = ff_q_pfp_shift_existslast * S ((S (L)) * e) + (0))))))

Complete tactic proof in conservative notation

All 17 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

17 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro L
02Establish hL4–9

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

  1. L4
    have h : ∃ d. ∃ e. BetaAt(d,e,L,0) ∧ BetaPrefixEqual(b,c,d,e,L)Definitions: BetaAt(d,e,L,0)BetaPrefixEqual(b,c,d,e,L)Original native command in the exact edition
  2. L5
    specialize beta_prefix_extend (L)
  3. L6
    specialize beta_prefix_extend (b)
  4. L7
    specialize beta_prefix_extend (c)
  5. L8
    specialize beta_prefix_extend (0)
  6. L9
    apply beta_prefix_extend
03Separate the logical casesL10–12

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

  1. L10
    cases h
  2. L11
    cases h_witness
  3. L12
    cases h_witness_witness
04Construct an explicit witnessL13–14

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

  1. L13
    exists x
  2. L14
    exists x1
05Separate the logical casesL15–15

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

  1. L15
    split
06Use earlier factsL16–17

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

  1. L16
    exact h_witness_witness_right
  2. L17
    exact h_witness_witness_left

Library-wide reading audit

Original defined command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004have h : ∃ d. ∃ e. BetaAt(d,e,L,0) ∧ BetaPrefixEqual(b,c,d,e,L)
  5. 0005specialize beta_prefix_extend (L)
  6. 0006specialize beta_prefix_extend (b)
  7. 0007specialize beta_prefix_extend (c)
  8. 0008specialize beta_prefix_extend (0)
  9. 0009apply beta_prefix_extend
  10. 0010cases h
  11. 0011cases h_witness
  12. 0012cases h_witness_witness
  13. 0013exists x
  14. 0014exists x1
  15. 0015split
  16. 0016exact h_witness_witness_right
  17. 0017exact h_witness_witness_left