PQ0022

prime_field_polynomial_trim_empty_input

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

Every pair of output beta codes is a valid empty trim of every empty input, including modulus zero; raw encodings are deliberately unconstrained.

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 d e. ((((0)=(0)+(0)) /\ (((forall fom_index_pfp_empty_triminput. (exists fom_gap_pfp_empty_triminput_index_bound. fom_gap_pfp_empty_triminput_index_bound + S (fom_index_pfp_empty_triminput) = 0) -> exists fom_value_pfp_empty_triminput. ((((exists fom_beta_height_pfp_empty_triminput_entry. fom_beta_height_pfp_empty_triminput_entry + S (fom_value_pfp_empty_triminput) = S ((S (fom_index_pfp_empty_triminput)) * c)) /\ exists fom_beta_quotient_pfp_empty_triminput_entry. b = fom_beta_quotient_pfp_empty_triminput_entry * S ((S (fom_index_pfp_empty_triminput)) * c) + (fom_value_pfp_empty_triminput))) /\ (exists fom_gap_pfp_empty_triminput_value_bound. fom_gap_pfp_empty_triminput_value_bound + S (fom_value_pfp_empty_triminput) = p))) /\ (((forall pfp_repeat_index_empty_trimremoved. (exists pfa_gap_empty_trimremovedindex. pfa_gap_empty_trimremovedindex + S (pfp_repeat_index_empty_trimremoved) = (0)) -> (((exists ff_h_pfp_empty_trimremovedentry. ff_h_pfp_empty_trimremovedentry + S (0) = S ((S (pfp_repeat_index_empty_trimremoved)) * c)) /\ exists ff_q_pfp_empty_trimremovedentry. b = ff_q_pfp_empty_trimremovedentry * S ((S (pfp_repeat_index_empty_trimremoved)) * c) + (0)))) /\ (((forall pftrim_index_empty_trimsuffix pftrim_value_empty_trimsuffix. (exists pfa_gap_empty_trimsuffixbound. pfa_gap_empty_trimsuffixbound + S (pftrim_index_empty_trimsuffix) = (0)) -> (((exists ff_h_pfp_empty_trimsuffixsource. ff_h_pfp_empty_trimsuffixsource + S (pftrim_value_empty_trimsuffix) = S ((S ((0)+pftrim_index_empty_trimsuffix)) * c)) /\ exists ff_q_pfp_empty_trimsuffixsource. b = ff_q_pfp_empty_trimsuffixsource * S ((S ((0)+pftrim_index_empty_trimsuffix)) * c) + (pftrim_value_empty_trimsuffix))) -> (((exists ff_h_pfp_empty_trimsuffixoutput. ff_h_pfp_empty_trimsuffixoutput + S (pftrim_value_empty_trimsuffix) = S ((S (pftrim_index_empty_trimsuffix)) * e)) /\ exists ff_q_pfp_empty_trimsuffixoutput. d = ff_q_pfp_empty_trimsuffixoutput * S ((S (pftrim_index_empty_trimsuffix)) * e) + (pftrim_value_empty_trimsuffix)))) /\ (((0)=0 \/ (exists pftrim_leading_empty_trimnormal. ((((exists ff_h_pfp_empty_trimnormalentry. ff_h_pfp_empty_trimnormalentry + S (pftrim_leading_empty_trimnormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_empty_trimnormalentry. d = ff_q_pfp_empty_trimnormalentry * S ((S (0)) * e) + (pftrim_leading_empty_trimnormal))) /\ ((~(pftrim_leading_empty_trimnormal=0)))))))))))))))

Constructive proof overview

Generated structural guide

Every pair of output beta codes is a valid empty trim of every empty input, including modulus zero; raw encodings are deliberately unconstrained.

The unchanged tactic script uses 3 declared prerequisites and contains 30 exact native proof lines.

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

Proof neighborhood

Direct dependencies

matrix_rank_bounded_prefix_empty Alpha theorem; checked-use authorized beta_repeat_empty Stable theorem; checked-use authorized matrix_rank_no_index_below_zero Alpha theorem; checked-use authorized

Direct dependents

none

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

30 script commands · 14 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.

01Fix variables and assumptionsL1–5

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 d
  5. L5
    intro e
02Separate the logical casesL6–6

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

  1. L6
    split
03Calculate and transport equalitiesL7–7

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    simp
04Separate the logical casesL8–8

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

  1. L8
    split
05Use earlier factsL9–12

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

  1. L9
    specialize matrix_rank_bounded_prefix_empty (b)
  2. L10
    specialize matrix_rank_bounded_prefix_empty (c)
  3. L11
    specialize matrix_rank_bounded_prefix_empty (p)
  4. L12
    apply matrix_rank_bounded_prefix_empty
06Separate the logical casesL13–13

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

  1. L13
    split
07Use earlier factsL14–18

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

  1. L14
    specialize beta_repeat_empty (b)
  2. L15
    specialize beta_repeat_empty (c)
  3. L16
    specialize beta_repeat_empty (0)
  4. L17
    specialize beta_repeat_empty (0)
  5. L18
    apply beta_repeat_empty
08Calculate and transport equalitiesL19–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    refl
09Separate the logical casesL20–20

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

  1. L20
    split
10Fix variables and assumptionsL21–24

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

  1. L21
    intro i
  2. L22
    intro a
  3. L23
    intro hi
  4. L24
    intro ha
11Separate the logical casesL25–25

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

  1. L25
    exfalso
12Use earlier factsL26–28

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

  1. L26
    specialize matrix_rank_no_index_below_zero (i)
  2. L27
    apply matrix_rank_no_index_below_zero
  3. L28
    exact hi
13Separate the logical casesL29–29

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

  1. L29
    left
14Calculate and transport equalitiesL30–30

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L30
    refl

Library-wide reading audit

Original exact command ledger · 30 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006split
  7. 0007simp
  8. 0008split
  9. 0009specialize matrix_rank_bounded_prefix_empty (b)
  10. 0010specialize matrix_rank_bounded_prefix_empty (c)
  11. 0011specialize matrix_rank_bounded_prefix_empty (p)
  12. 0012apply matrix_rank_bounded_prefix_empty
  13. 0013split
  14. 0014specialize beta_repeat_empty (b)
  15. 0015specialize beta_repeat_empty (c)
  16. 0016specialize beta_repeat_empty (0)
  17. 0017specialize beta_repeat_empty (0)
  18. 0018apply beta_repeat_empty
  19. 0019refl
  20. 0020split
  21. 0021intro i
  22. 0022intro a
  23. 0023intro hi
  24. 0024intro ha
  25. 0025exfalso
  26. 0026specialize matrix_rank_no_index_below_zero (i)
  27. 0027apply matrix_rank_no_index_below_zero
  28. 0028exact hi
  29. 0029left
  30. 0030refl