PQ0026

prime_field_polynomial_trim_zero_of_empty

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

Alpha v34 checked-use · first admitted v32 · 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 coefficients retain the established highest-degree-first order. Trimming handles empty and all-zero prefixes; monic normalization requires a nonzero leading coefficient. Synthetic division has a nonempty input of length S n and a quotient of length n, unique in decoded values. Its coefficient recurrence, actual evaluation remainder and positive-degree drop are checked. General polynomial Euclidean division, gcd/Bezout, an arbitrary-convolution factor theorem, irreducible-polynomial existence and the full G091 prime-power-field endpoint remain open. These exact theorems are first admitted to Alpha v32; Stable remains unchanged.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ L. ∀ t. ∀ d. ∀ e. ∀ M. FpPolynomialTrim(p,b,c,L,t,d,e,M) → M = 0 → Repeat(b,c,0,L)

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

Definition DAG

FpPolynomialTrim(p,b,c,L,t,d,e,M) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

none
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 21 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

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.

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 assumptionsL1–10

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 defined 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