PX003E

prime_field_convolution_prefix_empty_left_zero

An actual ambient convolution prefix of an empty quotient consists entirely of zero coefficients, regardless of its requested length.

Alpha v34 checked-use · first admitted v33 · 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.

Coefficients are highest-degree-first. The divisor has a nonzero decoded head; primality supplies its actual inverse. Empty quotients and remainders are included. Functionality compares the constructed execution lengths and decoded coefficients, never arbitrary beta codes. Formal polynomial equivalence compares every coefficient, not evaluations on a finite field. The formal identity and remainder-degree bound are proved separately, not assumed by the execution graph. Arbitrary quotient/remainder-pair uniqueness from a formal identity, multiplication associativity, gcd/Bezout, irreducible-polynomial existence, and the full G091 prime-power-field goal remain open. The seven displayed new names are conservative first-order notation, not new kernel primitives.

Exact theorem in conservative defined notation

∀ p. ∀ qb. ∀ qc. ∀ bb. ∀ bc. ∀ M. ∀ pb. ∀ pc. ∀ L. ¬p = 0 → FpConvolutionPrefix(p,qb,qc,0,bb,bc,M,pb,pc,L) → Repeat(pb,pc,0,L)

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

Definition DAG

FpConvolutionCoefficient(p,ab,ac,L,bb,bc,M,i,r) · 1FpConvolutionPrefix(p,ab,ac,L,bb,bc,M,cb,cc,l) · 1BetaAt(b,c,i,x) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_convolution_coefficient_zero_left · checked external prerequisitelt_not_le · checked external prerequisitezero_le · checked external prerequisite
Original expanded first-order statement 
forall p qb qc bb bc M pb pc L. ~(p=0) -> (forall pfc_index_division_empty_product_source. (exists pfa_gap_division_empty_product_sourcebound. pfa_gap_division_empty_product_sourcebound + S (pfc_index_division_empty_product_source) = (L)) -> exists pfc_value_division_empty_product_source. ((((exists ff_h_pfp_division_empty_product_sourceentry. ff_h_pfp_division_empty_product_sourceentry + S (pfc_value_division_empty_product_source) = S ((S (pfc_index_division_empty_product_source)) * pc)) /\ exists ff_q_pfp_division_empty_product_sourceentry. pb = ff_q_pfp_division_empty_product_sourceentry * S ((S (pfc_index_division_empty_product_source)) * pc) + (pfc_value_division_empty_product_source))) /\ ((exists pfc_terms_code_division_empty_product_sourcecoefficient pfc_terms_scale_division_empty_product_sourcecoefficient pfc_natural_sum_division_empty_product_sourcecoefficient. ((forall pfc_index_division_empty_product_sourcecoefficientdiagonal. (exists pfa_gap_division_empty_product_sourcecoefficientdiagonalbound. pfa_gap_division_empty_product_sourcecoefficientdiagonalbound + S (pfc_index_division_empty_product_sourcecoefficientdiagonal) = (S (pfc_index_division_empty_product_source))) -> exists pfc_value_division_empty_product_sourcecoefficientdiagonal. ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonalentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonalentry + S (pfc_value_division_empty_product_sourcecoefficientdiagonal) = S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * pfc_terms_scale_division_empty_product_sourcecoefficient)) /\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonalentry. pfc_terms_code_division_empty_product_sourcecoefficient = ff_q_pfp_division_empty_product_sourcecoefficientdiagonalentry * S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * pfc_terms_scale_division_empty_product_sourcecoefficient) + (pfc_value_division_empty_product_sourcecoefficientdiagonal))) /\ ((exists pfc_complement_division_empty_product_sourcecoefficientdiagonalterm pfc_left_division_empty_product_sourcecoefficientdiagonalterm pfc_right_division_empty_product_sourcecoefficientdiagonalterm. (((pfc_index_division_empty_product_sourcecoefficientdiagonal)+pfc_complement_division_empty_product_sourcecoefficientdiagonalterm=(pfc_index_division_empty_product_source)) /\ ((((((exists pfa_gap_division_empty_product_sourcecoefficientdiagonaltermleftinside. pfa_gap_division_empty_product_sourcecoefficientdiagonaltermleftinside + S (pfc_index_division_empty_product_sourcecoefficientdiagonal) = (0)) /\ ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry + S (pfc_left_division_empty_product_sourcecoefficientdiagonalterm) = S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * qc)) /\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry. qb = ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermleftentry * S ((S (pfc_index_division_empty_product_sourcecoefficientdiagonal)) * qc) + (pfc_left_division_empty_product_sourcecoefficientdiagonalterm)))))) \/ (((exists pfc_gap_division_empty_product_sourcecoefficientdiagonaltermleftoutside. pfc_gap_division_empty_product_sourcecoefficientdiagonaltermleftoutside+(0)=(pfc_index_division_empty_product_sourcecoefficientdiagonal)) /\ (((pfc_left_division_empty_product_sourcecoefficientdiagonalterm)=0))))) /\ ((((((exists pfa_gap_division_empty_product_sourcecoefficientdiagonaltermrightinside. pfa_gap_division_empty_product_sourcecoefficientdiagonaltermrightinside + S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm) = (M)) /\ ((((exists ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry. ff_h_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry + S (pfc_right_division_empty_product_sourcecoefficientdiagonalterm) = S ((S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) * bc)) /\ exists ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry. bb = ff_q_pfp_division_empty_product_sourcecoefficientdiagonaltermrightentry * S ((S (pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) * bc) + (pfc_right_division_empty_product_sourcecoefficientdiagonalterm)))))) \/ (((exists pfc_gap_division_empty_product_sourcecoefficientdiagonaltermrightoutside. pfc_gap_division_empty_product_sourcecoefficientdiagonaltermrightoutside+(M)=(pfc_complement_division_empty_product_sourcecoefficientdiagonalterm)) /\ (((pfc_right_division_empty_product_sourcecoefficientdiagonalterm)=0))))) /\ (((pfc_value_division_empty_product_sourcecoefficientdiagonal)=pfc_left_division_empty_product_sourcecoefficientdiagonalterm*pfc_right_division_empty_product_sourcecoefficientdiagonalterm))))))))))) /\ (((exists fs_u_pfc_division_empty_product_sourcecoefficientsum fs_v_pfc_division_empty_product_sourcecoefficientsum. ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_start. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_start + S (0) = S ((S (0)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_start. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_start * S ((S (0)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (0))) /\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_terminal. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_terminal + S (pfc_natural_sum_division_empty_product_sourcecoefficient) = S ((S (S (pfc_index_division_empty_product_source))) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_terminal. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_terminal * S ((S (S (pfc_index_division_empty_product_source))) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (pfc_natural_sum_division_empty_product_sourcecoefficient))) /\ forall fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps. (exists fs_lt_pfc_division_empty_product_sourcecoefficientsum_body_steps_bound. fs_lt_pfc_division_empty_product_sourcecoefficientsum_body_steps_bound + S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps = S (pfc_index_division_empty_product_source)) -> exists fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps. ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand + S (fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_sourcecoefficient)) /\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand. pfc_terms_code_division_empty_product_sourcecoefficient = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_summand * S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * pfc_terms_scale_division_empty_product_sourcecoefficient) + (fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial + S (fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_partial * S ((S (fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\ ((((exists fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor. fs_h_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor + S (fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps) = S ((S (S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum)) /\ exists fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor. fs_u_pfc_division_empty_product_sourcecoefficientsum = fs_q_pfc_division_empty_product_sourcecoefficientsum_body_steps_successor * S ((S (S fs_i_pfc_division_empty_product_sourcecoefficientsum_body_steps)) * fs_v_pfc_division_empty_product_sourcecoefficientsum) + (fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps))) /\ fs_s_pfc_division_empty_product_sourcecoefficientsum_body_steps = fs_r_pfc_division_empty_product_sourcecoefficientsum_body_steps + fs_a_pfc_division_empty_product_sourcecoefficientsum_body_steps)))))) /\ ((((exists pfa_gap_division_empty_product_sourcecoefficientresiduebound. pfa_gap_division_empty_product_sourcecoefficientresiduebound + S (pfc_value_division_empty_product_source) = (p)) /\ ((exists pfa_offset_left_division_empty_product_sourcecoefficientresiduecongruence pfa_offset_right_division_empty_product_sourcecoefficientresiduecongruence. (pfc_natural_sum_division_empty_product_sourcecoefficient) + (p) * pfa_offset_left_division_empty_product_sourcecoefficientresiduecongruence = (pfc_value_division_empty_product_source) + (p) * pfa_offset_right_division_empty_product_sourcecoefficientresiduecongruence)))))))))))) -> (forall pfp_repeat_index_division_empty_product_result. (exists pfa_gap_division_empty_product_resultindex. pfa_gap_division_empty_product_resultindex + S (pfp_repeat_index_division_empty_product_result) = (L)) -> (((exists ff_h_pfp_division_empty_product_resultentry. ff_h_pfp_division_empty_product_resultentry + S (0) = S ((S (pfp_repeat_index_division_empty_product_result)) * pc)) /\ exists ff_q_pfp_division_empty_product_resultentry. pb = ff_q_pfp_division_empty_product_resultentry * S ((S (pfp_repeat_index_division_empty_product_result)) * pc) + (0))))

Complete tactic proof in conservative notation

All 44 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

44 script commands · 11 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.

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 qb
  3. L3
    intro qc
  4. L4
    intro bb
  5. L5
    intro bc
  6. L6
    intro M
  7. L7
    intro pb
  8. L8
    intro pc
  9. L9
    intro L
  10. L10
    intro hp
02Fix variables and assumptionsL11–13

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

  1. L11
    intro h
  2. L12
    intro i
  3. L13
    intro hi
03Establish hvL14–17

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

  1. L14
    have hv : ∃ r. BetaAt(pb,pc,i,r) ∧ FpConvolutionCoefficient(p,qb,qc,0,bb,bc,M,i,r)Definitions: BetaAt(pb,pc,i,r)FpConvolutionCoefficient(p,qb,qc,0,bb,bc,M,i,r)Original native command in the exact edition
  2. L15
    specialize h (i)
  3. L16
    apply h
  4. L17
    exact hi
04Separate the logical casesL18–19

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

  1. L18
    cases hv
  2. L19
    cases hv_witness
05Establish hzL20–29

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

  1. L20
    have hz : x=0
  2. L21
    specialize prime_field_convolution_coefficient_zero_left (p)
  3. L22
    specialize prime_field_convolution_coefficient_zero_left (qb)
  4. L23
    specialize prime_field_convolution_coefficient_zero_left (qc)
  5. L24
    specialize prime_field_convolution_coefficient_zero_left (0)
  6. L25
    specialize prime_field_convolution_coefficient_zero_left (bb)
  7. L26
    specialize prime_field_convolution_coefficient_zero_left (bc)
  8. L27
    specialize prime_field_convolution_coefficient_zero_left (M)
  9. L28
    specialize prime_field_convolution_coefficient_zero_left (i)
  10. L29
    specialize prime_field_convolution_coefficient_zero_left (x)
06Use earlier factsL30–31

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

  1. L30
    apply prime_field_convolution_coefficient_zero_left
  2. L31
    exact hp
07Fix variables and assumptionsL32–33

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

  1. L32
    intro j
  2. L33
    intro hj
08Separate the logical casesL34–34

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

  1. L34
    exfalso
09Use earlier factsL35–41

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

  1. L35
    specialize lt_not_le (j)
  2. L36
    specialize lt_not_le (0)
  3. L37
    apply lt_not_le
  4. L38
    exact hj
  5. L39
    specialize zero_le (j)
  6. L40
    apply zero_le
  7. L41
    exact hv_witness_right
10Calculate and transport equalitiesL42–43

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

  1. L42
    rewrite hz at hv_witness_left
  2. L43
    rewrite hz at hv_witness_left
11Use earlier factsL44–44

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

  1. L44
    exact hv_witness_left

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro p
  2. 0002intro qb
  3. 0003intro qc
  4. 0004intro bb
  5. 0005intro bc
  6. 0006intro M
  7. 0007intro pb
  8. 0008intro pc
  9. 0009intro L
  10. 0010intro hp
  11. 0011intro h
  12. 0012intro i
  13. 0013intro hi
  14. 0014have hv : ∃ r. BetaAt(pb,pc,i,r) ∧ FpConvolutionCoefficient(p,qb,qc,0,bb,bc,M,i,r)
  15. 0015specialize h (i)
  16. 0016apply h
  17. 0017exact hi
  18. 0018cases hv
  19. 0019cases hv_witness
  20. 0020have hz : x=0
  21. 0021specialize prime_field_convolution_coefficient_zero_left (p)
  22. 0022specialize prime_field_convolution_coefficient_zero_left (qb)
  23. 0023specialize prime_field_convolution_coefficient_zero_left (qc)
  24. 0024specialize prime_field_convolution_coefficient_zero_left (0)
  25. 0025specialize prime_field_convolution_coefficient_zero_left (bb)
  26. 0026specialize prime_field_convolution_coefficient_zero_left (bc)
  27. 0027specialize prime_field_convolution_coefficient_zero_left (M)
  28. 0028specialize prime_field_convolution_coefficient_zero_left (i)
  29. 0029specialize prime_field_convolution_coefficient_zero_left (x)
  30. 0030apply prime_field_convolution_coefficient_zero_left
  31. 0031exact hp
  32. 0032intro j
  33. 0033intro hj
  34. 0034exfalso
  35. 0035specialize lt_not_le (j)
  36. 0036specialize lt_not_le (0)
  37. 0037apply lt_not_le
  38. 0038exact hj
  39. 0039specialize zero_le (j)
  40. 0040apply zero_le
  41. 0041exact hv_witness_right
  42. 0042rewrite hz at hv_witness_left
  43. 0043rewrite hz at hv_witness_left
  44. 0044exact hv_witness_left