PQ0025

prime_field_polynomial_trim_leading_source_nonzero

When the retained length is positive, every actual input decoding at the cut position is nonzero.

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

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

Definition DAG

FpPolynomialTrim(p,b,c,L,t,d,e,M) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

one_le_of_ne_zero · checked external prerequisitebeta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall p b c L t d e M a. ((((L)=(t)+(M)) /\ (((forall fom_index_pfp_trim_nonzero_sourceinput. (exists fom_gap_pfp_trim_nonzero_sourceinput_index_bound. fom_gap_pfp_trim_nonzero_sourceinput_index_bound + S (fom_index_pfp_trim_nonzero_sourceinput) = L) -> exists fom_value_pfp_trim_nonzero_sourceinput. ((((exists fom_beta_height_pfp_trim_nonzero_sourceinput_entry. fom_beta_height_pfp_trim_nonzero_sourceinput_entry + S (fom_value_pfp_trim_nonzero_sourceinput) = S ((S (fom_index_pfp_trim_nonzero_sourceinput)) * c)) /\ exists fom_beta_quotient_pfp_trim_nonzero_sourceinput_entry. b = fom_beta_quotient_pfp_trim_nonzero_sourceinput_entry * S ((S (fom_index_pfp_trim_nonzero_sourceinput)) * c) + (fom_value_pfp_trim_nonzero_sourceinput))) /\ (exists fom_gap_pfp_trim_nonzero_sourceinput_value_bound. fom_gap_pfp_trim_nonzero_sourceinput_value_bound + S (fom_value_pfp_trim_nonzero_sourceinput) = p))) /\ (((forall pfp_repeat_index_trim_nonzero_sourceremoved. (exists pfa_gap_trim_nonzero_sourceremovedindex. pfa_gap_trim_nonzero_sourceremovedindex + S (pfp_repeat_index_trim_nonzero_sourceremoved) = (t)) -> (((exists ff_h_pfp_trim_nonzero_sourceremovedentry. ff_h_pfp_trim_nonzero_sourceremovedentry + S (0) = S ((S (pfp_repeat_index_trim_nonzero_sourceremoved)) * c)) /\ exists ff_q_pfp_trim_nonzero_sourceremovedentry. b = ff_q_pfp_trim_nonzero_sourceremovedentry * S ((S (pfp_repeat_index_trim_nonzero_sourceremoved)) * c) + (0)))) /\ (((forall pftrim_index_trim_nonzero_sourcesuffix pftrim_value_trim_nonzero_sourcesuffix. (exists pfa_gap_trim_nonzero_sourcesuffixbound. pfa_gap_trim_nonzero_sourcesuffixbound + S (pftrim_index_trim_nonzero_sourcesuffix) = (M)) -> (((exists ff_h_pfp_trim_nonzero_sourcesuffixsource. ff_h_pfp_trim_nonzero_sourcesuffixsource + S (pftrim_value_trim_nonzero_sourcesuffix) = S ((S ((t)+pftrim_index_trim_nonzero_sourcesuffix)) * c)) /\ exists ff_q_pfp_trim_nonzero_sourcesuffixsource. b = ff_q_pfp_trim_nonzero_sourcesuffixsource * S ((S ((t)+pftrim_index_trim_nonzero_sourcesuffix)) * c) + (pftrim_value_trim_nonzero_sourcesuffix))) -> (((exists ff_h_pfp_trim_nonzero_sourcesuffixoutput. ff_h_pfp_trim_nonzero_sourcesuffixoutput + S (pftrim_value_trim_nonzero_sourcesuffix) = S ((S (pftrim_index_trim_nonzero_sourcesuffix)) * e)) /\ exists ff_q_pfp_trim_nonzero_sourcesuffixoutput. d = ff_q_pfp_trim_nonzero_sourcesuffixoutput * S ((S (pftrim_index_trim_nonzero_sourcesuffix)) * e) + (pftrim_value_trim_nonzero_sourcesuffix)))) /\ (((M)=0 \/ (exists pftrim_leading_trim_nonzero_sourcenormal. ((((exists ff_h_pfp_trim_nonzero_sourcenormalentry. ff_h_pfp_trim_nonzero_sourcenormalentry + S (pftrim_leading_trim_nonzero_sourcenormal) = S ((S (0)) * e)) /\ exists ff_q_pfp_trim_nonzero_sourcenormalentry. d = ff_q_pfp_trim_nonzero_sourcenormalentry * S ((S (0)) * e) + (pftrim_leading_trim_nonzero_sourcenormal))) /\ ((~(pftrim_leading_trim_nonzero_sourcenormal=0))))))))))))))) -> ~(M=0) -> (((exists ff_h_pfp_trim_leading_input. ff_h_pfp_trim_leading_input + S (a) = S ((S (t)) * c)) /\ exists ff_q_pfp_trim_leading_input. b = ff_q_pfp_trim_leading_input * S ((S (t)) * c) + (a))) -> ~(a=0)

Complete tactic proof in conservative notation

All 49 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

49 script commands · 11 reading checkpoints · 3 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 a
  10. L10
    intro h
02Fix variables and assumptionsL11–12

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

  1. L11
    intro hM
  2. L12
    intro ha
03Separate the logical casesL13–18

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

  1. L13
    cases h
  2. L14
    cases h_right
  3. L15
    cases h_right_right
  4. L16
    cases h_right_right_right
  5. L17
    cases h_right_right_right_right
  6. L18
    exfalso
04Use earlier factsL19–20

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

  1. L19
    apply hM
  2. L20
    exact h_right_right_right_right_left
05Separate the logical casesL21–22

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

  1. L21
    cases h_right_right_right_right_right
  2. L22
    cases h_right_right_right_right_right_witness
06Establish houtL23–29

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

  1. L23
    have hout : BetaAt(d,e,0,a)Definitions: BetaAt(d,e,0,a)Original native command in the exact edition
  2. L24
    specialize h_right_right_right_left (0)
  3. L25
    specialize h_right_right_right_left (a)
  4. L26
    apply h_right_right_right_left
  5. L27
    specialize one_le_of_ne_zero (M)
  6. L28
    apply one_le_of_ne_zero
  7. L29
    exact hM
07Establish hindexL30–34

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

  1. L30
    have hindex : t+0=t
  2. L31
    simp
  3. L32
    rewrite hindex
  4. L33
    rewrite hindex
  5. L34
    exact ha
08Establish heqL35–44

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

  1. L35
    have heq : a=x
  2. L36
    specialize beta_at_unique (d)
  3. L37
    specialize beta_at_unique (e)
  4. L38
    specialize beta_at_unique (0)
  5. L39
    specialize beta_at_unique (a)
  6. L40
    specialize beta_at_unique (x)
  7. L41
    apply beta_at_unique
  8. L42
    exact hout
  9. L43
    exact h_right_right_right_right_right_witness_left
  10. L44
    intro hz
09Use earlier factsL45–45

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

  1. L45
    apply h_right_right_right_right_right_witness_right
10Calculate and transport equalitiesL46–47

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

  1. L46
    trans a
  2. L47
    symm
11Use earlier factsL48–49

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

  1. L48
    exact heq
  2. L49
    exact hz

Library-wide reading audit

Original defined command ledger · 49 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 a
  10. 0010intro h
  11. 0011intro hM
  12. 0012intro ha
  13. 0013cases h
  14. 0014cases h_right
  15. 0015cases h_right_right
  16. 0016cases h_right_right_right
  17. 0017cases h_right_right_right_right
  18. 0018exfalso
  19. 0019apply hM
  20. 0020exact h_right_right_right_right_left
  21. 0021cases h_right_right_right_right_right
  22. 0022cases h_right_right_right_right_right_witness
  23. 0023have hout : BetaAt(d,e,0,a)
  24. 0024specialize h_right_right_right_left (0)
  25. 0025specialize h_right_right_right_left (a)
  26. 0026apply h_right_right_right_left
  27. 0027specialize one_le_of_ne_zero (M)
  28. 0028apply one_le_of_ne_zero
  29. 0029exact hM
  30. 0030have hindex : t+0=t
  31. 0031simp
  32. 0032rewrite hindex
  33. 0033rewrite hindex
  34. 0034exact ha
  35. 0035have heq : a=x
  36. 0036specialize beta_at_unique (d)
  37. 0037specialize beta_at_unique (e)
  38. 0038specialize beta_at_unique (0)
  39. 0039specialize beta_at_unique (a)
  40. 0040specialize beta_at_unique (x)
  41. 0041apply beta_at_unique
  42. 0042exact hout
  43. 0043exact h_right_right_right_right_right_witness_left
  44. 0044intro hz
  45. 0045apply h_right_right_right_right_right_witness_right
  46. 0046trans a
  47. 0047symm
  48. 0048exact heq
  49. 0049exact hz