PQ0026

prime_field_polynomial_trim_zero_of_empty

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

An empty actual trim certifies that every coefficient of the entire input prefix is zero.

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_trim_empty_sourceinput. (exists fom_gap_pfp_trim_empty_sourceinput_index_bound. fom_gap_pfp_trim_empty_sourceinput_index_bound + S (fom_index_pfp_trim_empty_sourceinput) = L) -> exists fom_value_pfp_trim_empty_sourceinput. ((((exists fom_beta_height_pfp_trim_empty_sourceinput_entry. fom_beta_height_pfp_trim_empty_sourceinput_entry + S (fom_value_pfp_trim_empty_sourceinput) = S ((S (fom_index_pfp_trim_empty_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_empty_sourceinput_entry. b = fom_beta_quotient_pfp_trim_empty_sourceinput_entry * S ((S (fom_index_pfp_trim_empty_sourceinput)) * c) + (fom_value_pfp_trim_empty_sourceinput))) /\ (exists fom_gap_pfp_trim_empty_sourceinput_value_bound. fom_gap_pfp_trim_empty_sourceinput_value_bound + S (fom_value_pfp_trim_empty_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_empty_sourceremoved. (exists pfa_gap_trim_empty_sourceremovedindex. pfa_gap_trim_empty_sourceremovedindex + S (pfp_repeat_index_trim_empty_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_empty_sourceremovedentry. ff_h_pfp_trim_empty_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_empty_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_empty_sourceremovedentry. b = ff_q_pfp_trim_empty_sourceremovedentry * S ((S (pfp_repeat_index_trim_empty_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_empty_sourcesuffix pftrim_value_trim_empty_sourcesuffix. (exists pfa_gap_trim_empty_sourcesuffixbound. pfa_gap_trim_empty_sourcesuffixbound + S (pftrim_index_trim_empty_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_empty_sourcesuffixsource. ff_h_pfp_trim_empty_sourcesuffixsource + S (pftrim_value_trim_empty_sourcesuffix) = S ((S ((t)+pftrim_index_trim_empty_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_empty_sourcesuffixsource. b = ff_q_pfp_trim_empty_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_empty_sourcesuffix)) * c) + (pftrim_value_trim_empty_sourcesuffix))) -> (((exists ff_h_pfp_trim_empty_sourcesuffixoutput. ff_h_pfp_trim_empty_sourcesuffixoutput + S (pftrim_value_trim_empty_sourcesuffix) = S ((S (pftrim_index_trim_empty_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_empty_sourcesuffixoutput. d = ff_q_pfp_trim_empty_sourcesuffixoutput * S ((S (pftrim_index_trim_empty_sourcesuffix)) * e) + (pftrim_value_trim_empty_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_empty_sourcenormal. ((((exists ff_h_pfp_trim_empty_sourcenormalentry. ff_h_pfp_trim_empty_sourcenormalentry + S (pftrim_leading_trim_empty_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_empty_sourcenormalentry. d = ff_q_pfp_trim_empty_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_empty_sourcenormal))) /\ ((~(pftrim_leading_trim_empty_sourcenormal=0))))))))))))))) -> M=0 -> (forall pfp_repeat_index_trim_zero_input. (exists pfa_gap_trim_zero_inputindex. pfa_gap_trim_zero_inputindex + S (pfp_repeat_index_trim_zero_input) = (L)) -> (((exists ff_h_pfp_trim_zero_inputentry. ff_h_pfp_trim_zero_inputentry + S (0) = S ((S (pfp_repeat_index_trim_zero_input)) * c)) /\ exists ff_q_pfp_trim_zero_inputentry. b = ff_q_pfp_trim_zero_inputentry * S ((S (pfp_repeat_index_trim_zero_input)) * c) + (0))))

Constructive proof overview

Generated structural guide

An empty actual trim certifies that every coefficient of the entire input prefix is zero.

The unchanged tactic script uses 0 declared prerequisites and contains 21 exact native proof lines.

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

Proof neighborhood

Direct dependencies

none

Direct dependents

PQ0028 prime_field_polynomial_trim_zero_iff

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

21 script commands · 3 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.

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 h
  10. L10
    intro hM
02Separate the logical casesL11–14

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

  1. L11
    cases h
  2. L12
    cases h_right
  3. L13
    cases h_right_right
  4. L14
    cases h_right_right_right
03Establish hlenL15–21

Establish this local claim before using it. It is not an additional assumption.

  1. L15
    have hlen : L=t
  2. L16
    trans t+M
  3. L17
    exact h_left
  4. L18
    rewrite hM
  5. L19
    simp
  6. L20
    rewrite hlen
  7. L21
    exact h_right_right_left

Library-wide reading audit

Original exact command ledger · 21 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 h
  10. 0010intro hM
  11. 0011cases h
  12. 0012cases h_right
  13. 0013cases h_right_right
  14. 0014cases h_right_right_right
  15. 0015have hlen : L=t
  16. 0016trans t+M
  17. 0017exact h_left
  18. 0018rewrite hM
  19. 0019simp
  20. 0020rewrite hlen
  21. 0021exact h_right_right_left