PQ0052

prime_field_polynomial_synthetic_represented_degree

Synthetic division of a nonzero-leading polynomial of positive represented degree S n produces a quotient of represented degree exactly n.

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. ∀ a. ∀ n. ∀ qb. ∀ qc. ∀ r. Prime(p) → FpRepresentedDegree(p,b,c,S S n,S n) → FpSyntheticDivision(p,b,c,a,S n,qb,qc,r) → FpRepresentedDegree(p,qb,qc,S n,n)

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

Definition DAG

FpRepresentedDegree(p,b,c,L,d) · 2FpSyntheticDivision(p,b,c,a,n,qb,qc,r) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_polynomial_synthetic_quotient_boundedprime_field_polynomial_synthetic_leading_coefficient
Original expanded first-order statement 
forall p b c a n qb qc r. (~((p) = 1) /\ forall pfa_factor_left_degree_prime pfa_factor_right_degree_prime. (p) = pfa_factor_left_degree_prime * pfa_factor_right_degree_prime -> pfa_factor_left_degree_prime = 1 \/ pfa_factor_right_degree_prime = 1) -> ((((S (S n))=S (S n)) /\ (((forall fom_index_pfp_degree_inputcoefficients. (exists fom_gap_pfp_degree_inputcoefficients_index_bound. fom_gap_pfp_degree_inputcoefficients_index_bound + S (fom_index_pfp_degree_inputcoefficients) = S (S n)) -> exists fom_value_pfp_degree_inputcoefficients. ((((exists fom_beta_height_pfp_degree_inputcoefficients_entry. fom_beta_height_pfp_degree_inputcoefficients_entry + S (fom_value_pfp_degree_inputcoefficients) = S ((S (fom_index_pfp_degree_inputcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_degree_inputcoefficients_entry. b = fom_beta_quotient_pfp_degree_inputcoefficients_entry * S ((S (fom_index_pfp_degree_inputcoefficients)) * c) + (fom_value_pfp_degree_inputcoefficients))) /\ (exists fom_gap_pfp_degree_inputcoefficients_value_bound. fom_gap_pfp_degree_inputcoefficients_value_bound + S (fom_value_pfp_degree_inputcoefficients) = p))) /\ ((exists pfd_leading_degree_input. ((((exists ff_h_pfp_degree_inputentry. ff_h_pfp_degree_inputentry + S (pfd_leading_degree_input) = S ((S (0)) * c)) /\ exists ff_q_pfp_degree_inputentry. b = ff_q_pfp_degree_inputentry * S ((S (0)) * c) + (pfd_leading_degree_input))) /\ ((~(pfd_leading_degree_input=0)))))))))) -> (exists pfs_history_code_degree_division pfs_history_scale_degree_division. ((((exists pfa_gap_degree_divisiontracebase. pfa_gap_degree_divisiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_degree_divisiontraceinitial. ff_h_pfp_degree_divisiontraceinitial + S (0) = S ((S (0)) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontraceinitial. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontraceinitial * S ((S (0)) * pfs_history_scale_degree_division) + (0))) /\ (((((exists ff_h_pfp_degree_divisiontraceterminal. ff_h_pfp_degree_divisiontraceterminal + S (r) = S ((S (S (S n))) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontraceterminal. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontraceterminal * S ((S (S (S n))) * pfs_history_scale_degree_division) + (r))) /\ ((forall pfh_index_degree_divisiontracesteps. (exists pfa_gap_degree_divisiontracestepsindex. pfa_gap_degree_divisiontracestepsindex + S (pfh_index_degree_divisiontracesteps) = (S (S n))) -> (exists pfh_coefficient_degree_divisiontracestepsstep pfh_before_degree_divisiontracestepsstep pfh_after_degree_divisiontracestepsstep pfh_product_degree_divisiontracestepsstep. ((((exists ff_h_pfp_degree_divisiontracestepsstepcoefficient. ff_h_pfp_degree_divisiontracestepsstepcoefficient + S (pfh_coefficient_degree_divisiontracestepsstep) = S ((S (pfh_index_degree_divisiontracesteps)) * c)) /\ exists ff_q_pfp_degree_divisiontracestepsstepcoefficient. b = ff_q_pfp_degree_divisiontracestepsstepcoefficient * S ((S (pfh_index_degree_divisiontracesteps)) * c) + (pfh_coefficient_degree_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_degree_divisiontracestepsstepbefore. ff_h_pfp_degree_divisiontracestepsstepbefore + S (pfh_before_degree_divisiontracestepsstep) = S ((S (pfh_index_degree_divisiontracesteps)) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontracestepsstepbefore. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontracestepsstepbefore * S ((S (pfh_index_degree_divisiontracesteps)) * pfs_history_scale_degree_division) + (pfh_before_degree_divisiontracestepsstep))) /\ (((((exists ff_h_pfp_degree_divisiontracestepsstepafter. ff_h_pfp_degree_divisiontracestepsstepafter + S (pfh_after_degree_divisiontracestepsstep) = S ((S (S (pfh_index_degree_divisiontracesteps))) * pfs_history_scale_degree_division)) /\ exists ff_q_pfp_degree_divisiontracestepsstepafter. pfs_history_code_degree_division = ff_q_pfp_degree_divisiontracestepsstepafter * S ((S (S (pfh_index_degree_divisiontracesteps))) * pfs_history_scale_degree_division) + (pfh_after_degree_divisiontracestepsstep))) /\ (((((exists pfa_gap_degree_divisiontracestepsstepmultiplyleft. pfa_gap_degree_divisiontracestepsstepmultiplyleft + S (pfh_before_degree_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_degree_divisiontracestepsstepmultiplyright. pfa_gap_degree_divisiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_degree_divisiontracestepsstepmultiplyresultbound. pfa_gap_degree_divisiontracestepsstepmultiplyresultbound + S (pfh_product_degree_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_degree_divisiontracestepsstepmultiplyresultcongruence pfa_offset_right_degree_divisiontracestepsstepmultiplyresultcongruence. ((pfh_before_degree_divisiontracestepsstep) * (a)) + (p) * pfa_offset_left_degree_divisiontracestepsstepmultiplyresultcongruence = (pfh_product_degree_divisiontracestepsstep) + (p) * pfa_offset_right_degree_divisiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_degree_divisiontracestepsstepaddleft. pfa_gap_degree_divisiontracestepsstepaddleft + S (pfh_product_degree_divisiontracestepsstep) = (p)) /\ (((exists pfa_gap_degree_divisiontracestepsstepaddright. pfa_gap_degree_divisiontracestepsstepaddright + S (pfh_coefficient_degree_divisiontracestepsstep) = (p)) /\ ((((exists pfa_gap_degree_divisiontracestepsstepaddresultbound. pfa_gap_degree_divisiontracestepsstepaddresultbound + S (pfh_after_degree_divisiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_degree_divisiontracestepsstepaddresultcongruence pfa_offset_right_degree_divisiontracestepsstepaddresultcongruence. ((pfh_product_degree_divisiontracestepsstep) + (pfh_coefficient_degree_divisiontracestepsstep)) + (p) * pfa_offset_left_degree_divisiontracestepsstepaddresultcongruence = (pfh_after_degree_divisiontracestepsstep) + (p) * pfa_offset_right_degree_divisiontracestepsstepaddresultcongruence)))))))))))))))))))))))))) /\ ((forall ff_index_mcp_pfs_degree_divisionquotient ff_source_mcp_pfs_degree_divisionquotient ff_target_mcp_pfs_degree_divisionquotient. (exists mcp_gap_pfs_degree_divisionquotient_bound. mcp_gap_pfs_degree_divisionquotient_bound + S (ff_index_mcp_pfs_degree_divisionquotient) = (S n)) -> (((exists fs_h_mcp_pfs_degree_divisionquotient_source. fs_h_mcp_pfs_degree_divisionquotient_source + S (ff_source_mcp_pfs_degree_divisionquotient) = S ((S ((1) + (1) * ff_index_mcp_pfs_degree_divisionquotient)) * pfs_history_scale_degree_division)) /\ exists fs_q_mcp_pfs_degree_divisionquotient_source. pfs_history_code_degree_division = fs_q_mcp_pfs_degree_divisionquotient_source * S ((S ((1) + (1) * ff_index_mcp_pfs_degree_divisionquotient)) * pfs_history_scale_degree_division) + (ff_source_mcp_pfs_degree_divisionquotient))) -> (((exists fs_h_mcp_pfs_degree_divisionquotient_target. fs_h_mcp_pfs_degree_divisionquotient_target + S (ff_target_mcp_pfs_degree_divisionquotient) = S ((S (ff_index_mcp_pfs_degree_divisionquotient)) * qc)) /\ exists fs_q_mcp_pfs_degree_divisionquotient_target. qb = fs_q_mcp_pfs_degree_divisionquotient_target * S ((S (ff_index_mcp_pfs_degree_divisionquotient)) * qc) + (ff_target_mcp_pfs_degree_divisionquotient))) -> ff_target_mcp_pfs_degree_divisionquotient = ff_source_mcp_pfs_degree_divisionquotient)))) -> ((((S n)=S (n)) /\ (((forall fom_index_pfp_degree_quotientcoefficients. (exists fom_gap_pfp_degree_quotientcoefficients_index_bound. fom_gap_pfp_degree_quotientcoefficients_index_bound + S (fom_index_pfp_degree_quotientcoefficients) = S n) -> exists fom_value_pfp_degree_quotientcoefficients. ((((exists fom_beta_height_pfp_degree_quotientcoefficients_entry. fom_beta_height_pfp_degree_quotientcoefficients_entry + S (fom_value_pfp_degree_quotientcoefficients) = S ((S (fom_index_pfp_degree_quotientcoefficients)) * qc)) /\ exists fom_beta_quotient_pfp_degree_quotientcoefficients_entry. qb = fom_beta_quotient_pfp_degree_quotientcoefficients_entry * S ((S (fom_index_pfp_degree_quotientcoefficients)) * qc) + (fom_value_pfp_degree_quotientcoefficients))) /\ (exists fom_gap_pfp_degree_quotientcoefficients_value_bound. fom_gap_pfp_degree_quotientcoefficients_value_bound + S (fom_value_pfp_degree_quotientcoefficients) = p))) /\ ((exists pfd_leading_degree_quotient. ((((exists ff_h_pfp_degree_quotiententry. ff_h_pfp_degree_quotiententry + S (pfd_leading_degree_quotient) = S ((S (0)) * qc)) /\ exists ff_q_pfp_degree_quotiententry. qb = ff_q_pfp_degree_quotiententry * S ((S (0)) * qc) + (pfd_leading_degree_quotient))) /\ ((~(pfd_leading_degree_quotient=0))))))))))

Complete tactic proof in conservative notation

All 45 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

45 script commands · 12 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (2)
prime field polynomial synthetic quotient boundedprime field polynomial synthetic leading coefficient
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 a
  5. L5
    intro n
  6. L6
    intro qb
  7. L7
    intro qc
  8. L8
    intro r
  9. L9
    intro hp
  10. L10
    intro hd
02Fix variables and assumptionsL11–11

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

  1. L11
    intro hs
03Separate the logical casesL12–14

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

  1. L12
    cases hd
  2. L13
    cases hd_right
  3. L14
    split
04Calculate and transport equalitiesL15–15

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

  1. L15
    refl
05Separate the logical casesL16–16

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

  1. L16
    split
06Use earlier factsL17–26

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

  1. L17
    specialize prime_field_polynomial_synthetic_quotient_bounded (p)
  2. L18
    specialize prime_field_polynomial_synthetic_quotient_bounded (b)
  3. L19
    specialize prime_field_polynomial_synthetic_quotient_bounded (c)
  4. L20
    specialize prime_field_polynomial_synthetic_quotient_bounded (a)
  5. L21
    specialize prime_field_polynomial_synthetic_quotient_bounded (S n)
  6. L22
    specialize prime_field_polynomial_synthetic_quotient_bounded (qb)
  7. L23
    specialize prime_field_polynomial_synthetic_quotient_bounded (qc)
  8. L24
    specialize prime_field_polynomial_synthetic_quotient_bounded (r)
  9. L25
    apply prime_field_polynomial_synthetic_quotient_bounded
  10. L26
    exact hp
07Use earlier factsL27–27

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

  1. L27
    exact hs
08Separate the logical casesL28–29

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

  1. L28
    cases hd_right_right
  2. L29
    cases hd_right_right_witness
09Construct an explicit witnessL30–30

Supply the displayed value, then prove that it has the required property.

  1. L30
    exists x
10Separate the logical casesL31–31

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

  1. L31
    split
11Use earlier factsL32–41

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

  1. L32
    specialize prime_field_polynomial_synthetic_leading_coefficient (p)
  2. L33
    specialize prime_field_polynomial_synthetic_leading_coefficient (b)
  3. L34
    specialize prime_field_polynomial_synthetic_leading_coefficient (c)
  4. L35
    specialize prime_field_polynomial_synthetic_leading_coefficient (a)
  5. L36
    specialize prime_field_polynomial_synthetic_leading_coefficient (n)
  6. L37
    specialize prime_field_polynomial_synthetic_leading_coefficient (qb)
  7. L38
    specialize prime_field_polynomial_synthetic_leading_coefficient (qc)
  8. L39
    specialize prime_field_polynomial_synthetic_leading_coefficient (r)
  9. L40
    specialize prime_field_polynomial_synthetic_leading_coefficient (x)
  10. L41
    apply prime_field_polynomial_synthetic_leading_coefficient
12Use earlier factsL42–45

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

  1. L42
    exact hp
  2. L43
    exact hs
  3. L44
    exact hd_right_right_witness_left
  4. L45
    exact hd_right_right_witness_right

Library-wide reading audit

Original defined command ledger · 45 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro a
  5. 0005intro n
  6. 0006intro qb
  7. 0007intro qc
  8. 0008intro r
  9. 0009intro hp
  10. 0010intro hd
  11. 0011intro hs
  12. 0012cases hd
  13. 0013cases hd_right
  14. 0014split
  15. 0015refl
  16. 0016split
  17. 0017specialize prime_field_polynomial_synthetic_quotient_bounded (p)
  18. 0018specialize prime_field_polynomial_synthetic_quotient_bounded (b)
  19. 0019specialize prime_field_polynomial_synthetic_quotient_bounded (c)
  20. 0020specialize prime_field_polynomial_synthetic_quotient_bounded (a)
  21. 0021specialize prime_field_polynomial_synthetic_quotient_bounded (S n)
  22. 0022specialize prime_field_polynomial_synthetic_quotient_bounded (qb)
  23. 0023specialize prime_field_polynomial_synthetic_quotient_bounded (qc)
  24. 0024specialize prime_field_polynomial_synthetic_quotient_bounded (r)
  25. 0025apply prime_field_polynomial_synthetic_quotient_bounded
  26. 0026exact hp
  27. 0027exact hs
  28. 0028cases hd_right_right
  29. 0029cases hd_right_right_witness
  30. 0030exists x
  31. 0031split
  32. 0032specialize prime_field_polynomial_synthetic_leading_coefficient (p)
  33. 0033specialize prime_field_polynomial_synthetic_leading_coefficient (b)
  34. 0034specialize prime_field_polynomial_synthetic_leading_coefficient (c)
  35. 0035specialize prime_field_polynomial_synthetic_leading_coefficient (a)
  36. 0036specialize prime_field_polynomial_synthetic_leading_coefficient (n)
  37. 0037specialize prime_field_polynomial_synthetic_leading_coefficient (qb)
  38. 0038specialize prime_field_polynomial_synthetic_leading_coefficient (qc)
  39. 0039specialize prime_field_polynomial_synthetic_leading_coefficient (r)
  40. 0040specialize prime_field_polynomial_synthetic_leading_coefficient (x)
  41. 0041apply prime_field_polynomial_synthetic_leading_coefficient
  42. 0042exact hp
  43. 0043exact hs
  44. 0044exact hd_right_right_witness_left
  45. 0045exact hd_right_right_witness_right