PG0001

prime_field_polynomial_shift_exists

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

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

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 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))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 17 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_prefix_extend Alpha theorem; checked-use authorized

Direct dependents

PG000D prime_field_polynomial_convolution_shift_right_exists PG001B prime_field_polynomial_append_shift_constant_decomposition_exists PG001C prime_field_convolution_coefficient_right_append_add PG001D prime_field_polynomial_shift_scale_aligned_sum_exists

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

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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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: BetaPrefixEqualBetaAt
  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 exact command ledger · 17 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro L
  4. 0004have h : exists d e. ((((exists ff_h_pfp_shift_exists_last. ff_h_pfp_shift_exists_last + S (0) = S ((S (L)) * e)) /\ exists ff_q_pfp_shift_exists_last. d = ff_q_pfp_shift_exists_last * S ((S (L)) * e) + (0))) /\ ((forall mdr_i_pfp_shift_exists_prefix mdr_a_pfp_shift_exists_prefix. (exists mdr_gap_pfp_shift_exists_prefixb. mdr_gap_pfp_shift_exists_prefixb + S (mdr_i_pfp_shift_exists_prefix) = (L)) -> (((exists ff_h_mdr_pfp_shift_exists_prefixo. ff_h_mdr_pfp_shift_exists_prefixo + S (mdr_a_pfp_shift_exists_prefix) = S ((S (mdr_i_pfp_shift_exists_prefix)) * c)) /\ exists ff_q_mdr_pfp_shift_exists_prefixo. b = ff_q_mdr_pfp_shift_exists_prefixo * S ((S (mdr_i_pfp_shift_exists_prefix)) * c) + (mdr_a_pfp_shift_exists_prefix))) -> (((exists ff_h_mdr_pfp_shift_exists_prefixn. ff_h_mdr_pfp_shift_exists_prefixn + S (mdr_a_pfp_shift_exists_prefix) = S ((S (mdr_i_pfp_shift_exists_prefix)) * e)) /\ exists ff_q_mdr_pfp_shift_exists_prefixn. d = ff_q_mdr_pfp_shift_exists_prefixn * S ((S (mdr_i_pfp_shift_exists_prefix)) * e) + (mdr_a_pfp_shift_exists_prefix))))))
  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