PQ001D

prime_field_polynomial_suffix_bounded

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

A genuine suffix ending at the annotated input length inherits every canonical coefficient bound.

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 b c L t d e M. L=t+M -> (forall fom_index_pfp_suffix_input_coefficients. (exists fom_gap_pfp_suffix_input_coefficients_index_bound. fom_gap_pfp_suffix_input_coefficients_index_bound + S (fom_index_pfp_suffix_input_coefficients) = L) -> exists fom_value_pfp_suffix_input_coefficients. ((((exists fom_beta_height_pfp_suffix_input_coefficients_entry. fom_beta_height_pfp_suffix_input_coefficients_entry + S (fom_value_pfp_suffix_input_coefficients) = S ((S (fom_index_pfp_suffix_input_coefficients)) * c)) /\ exists fom_beta_quotient_pfp_suffix_input_coefficients_entry. b = fom_beta_quotient_pfp_suffix_input_coefficients_entry * S ((S (fom_index_pfp_suffix_input_coefficients)) * c) + (fom_value_pfp_suffix_input_coefficients))) /\ (exists fom_gap_pfp_suffix_input_coefficients_value_bound. fom_gap_pfp_suffix_input_coefficients_value_bound + S (fom_value_pfp_suffix_input_coefficients) = p))) -> (forall pftrim_index_suffix_bounded pftrim_value_suffix_bounded. (exists pfa_gap_suffix_boundedbound. pfa_gap_suffix_boundedbound + S (pftrim_index_suffix_bounded) = (M)) -> (((exists ff_h_pfp_suffix_boundedsource. ff_h_pfp_suffix_boundedsource + S (pftrim_value_suffix_bounded) = S ((S ((t)+pftrim_index_suffix_bounded)) * c)) /\ exists ff_q_pfp_suffix_boundedsource. b = ff_q_pfp_suffix_boundedsource * S ((S ((t)+pftrim_index_suffix_bounded)) * c) + (pftrim_value_suffix_bounded))) -> (((exists ff_h_pfp_suffix_boundedoutput. ff_h_pfp_suffix_boundedoutput + S (pftrim_value_suffix_bounded) = S ((S (pftrim_index_suffix_bounded)) * e)) /\ exists ff_q_pfp_suffix_boundedoutput. d = ff_q_pfp_suffix_boundedoutput * S ((S (pftrim_index_suffix_bounded)) * e) + (pftrim_value_suffix_bounded)))) -> (forall fom_index_pfp_suffix_output_coefficients. (exists fom_gap_pfp_suffix_output_coefficients_index_bound. fom_gap_pfp_suffix_output_coefficients_index_bound + S (fom_index_pfp_suffix_output_coefficients) = M) -> exists fom_value_pfp_suffix_output_coefficients. ((((exists fom_beta_height_pfp_suffix_output_coefficients_entry. fom_beta_height_pfp_suffix_output_coefficients_entry + S (fom_value_pfp_suffix_output_coefficients) = S ((S (fom_index_pfp_suffix_output_coefficients)) * e)) /\ exists fom_beta_quotient_pfp_suffix_output_coefficients_entry. d = fom_beta_quotient_pfp_suffix_output_coefficients_entry * S ((S (fom_index_pfp_suffix_output_coefficients)) * e) + (fom_value_pfp_suffix_output_coefficients))) /\ (exists fom_gap_pfp_suffix_output_coefficients_value_bound. fom_gap_pfp_suffix_output_coefficients_value_bound + S (fom_value_pfp_suffix_output_coefficients) = p)))

Constructive proof overview

Generated structural guide

A genuine suffix ending at the annotated input length inherits every canonical coefficient bound.

The unchanged tactic script uses 1 declared prerequisite and contains 34 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

matrix_recursive_lt_add_left Alpha theorem; checked-use authorized

Direct dependents

PQ0023 prime_field_polynomial_trim_output_coefficients

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

34 script commands · 8 reading checkpoints · 2 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.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro L
  5. L5
    intro t
  6. L6
    intro d
  7. L7
    intro e
  8. L8
    intro M
  9. L9
    intro hlen
  10. L10
    intro hc
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hs
  2. L12
    intro i
  3. L13
    intro hi
03Establish hshiftL14–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply matrix recursive lt add left.

  1. L14
    have hshift : exists pfa_gap_suffix_shift_bound. pfa_gap_suffix_shift_bound + S (t+i) = (L)
  2. L15
    rewrite hlen
  3. L16
    specialize matrix_recursive_lt_add_left (i)
  4. L17
    specialize matrix_recursive_lt_add_left (M)
  5. L18
    specialize matrix_recursive_lt_add_left (t)
  6. L19
    apply matrix_recursive_lt_add_left
  7. L20
    exact hi
04Establish haL21–24

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

  1. L21
    have ha : exists a. ((((exists ff_h_pfp_suffix_bounded_source. ff_h_pfp_suffix_bounded_source + S (a) = S ((S (t+i)) * c)) /\ exists ff_q_pfp_suffix_bounded_source. b = ff_q_pfp_suffix_bounded_source * S ((S (t+i)) * c) + (a))) /\ ((exists pfa_gap_suffix_bounded_value. pfa_gap_suffix_bounded_value + S (a) = (p))))
  2. L22
    specialize hc (t+i)
  3. L23
    apply hc
  4. L24
    exact hshift
05Separate the logical casesL25–26

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

  1. L25
    cases ha
  2. L26
    cases ha_witness
06Construct an explicit witnessL27–27

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

  1. L27
    exists x
07Separate the logical casesL28–28

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

  1. L28
    split
08Use earlier factsL29–34

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

  1. L29
    specialize hs (i)
  2. L30
    specialize hs (x)
  3. L31
    apply hs
  4. L32
    exact hi
  5. L33
    exact ha_witness_left
  6. L34
    exact ha_witness_right

Library-wide reading audit

Original exact command ledger · 34 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro L
  5. 0005intro t
  6. 0006intro d
  7. 0007intro e
  8. 0008intro M
  9. 0009intro hlen
  10. 0010intro hc
  11. 0011intro hs
  12. 0012intro i
  13. 0013intro hi
  14. 0014have hshift : exists pfa_gap_suffix_shift_bound. pfa_gap_suffix_shift_bound + S (t+i) = (L)
  15. 0015rewrite hlen
  16. 0016specialize matrix_recursive_lt_add_left (i)
  17. 0017specialize matrix_recursive_lt_add_left (M)
  18. 0018specialize matrix_recursive_lt_add_left (t)
  19. 0019apply matrix_recursive_lt_add_left
  20. 0020exact hi
  21. 0021have ha : exists a. ((((exists ff_h_pfp_suffix_bounded_source. ff_h_pfp_suffix_bounded_source + S (a) = S ((S (t+i)) * c)) /\ exists ff_q_pfp_suffix_bounded_source. b = ff_q_pfp_suffix_bounded_source * S ((S (t+i)) * c) + (a))) /\ ((exists pfa_gap_suffix_bounded_value. pfa_gap_suffix_bounded_value + S (a) = (p))))
  22. 0022specialize hc (t+i)
  23. 0023apply hc
  24. 0024exact hshift
  25. 0025cases ha
  26. 0026cases ha_witness
  27. 0027exists x
  28. 0028split
  29. 0029specialize hs (i)
  30. 0030specialize hs (x)
  31. 0031apply hs
  32. 0032exact hi
  33. 0033exact ha_witness_left
  34. 0034exact ha_witness_right