PP002B

prime_field_polynomial_horner_empty_construct

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Construct an actual zero-result execution of every empty coefficient prefix, retaining the canonical base guard.

Exact expanded first-order arithmetic statement

forall p b c t. (~((p) = 1) /\ forall pfa_factor_left_empty_construct_prime pfa_factor_right_empty_construct_prime. (p) = pfa_factor_left_empty_construct_prime * pfa_factor_right_empty_construct_prime -> pfa_factor_left_empty_construct_prime = 1 \/ pfa_factor_right_empty_construct_prime = 1) -> (exists pfa_gap_empty_construct_base. pfa_gap_empty_construct_base + S (t) = (p)) -> (exists pfh_trace_code_empty_construct_execution pfh_trace_scale_empty_construct_execution. (((exists pfa_gap_empty_construct_executiontracebase. pfa_gap_empty_construct_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_empty_construct_executiontraceinitial. ff_h_pfp_empty_construct_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontraceinitial. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontraceinitial * S ((S (0)) * pfh_trace_scale_empty_construct_execution) + (0))) /\ (((((exists ff_h_pfp_empty_construct_executiontraceterminal. ff_h_pfp_empty_construct_executiontraceterminal + S (0) = S ((S (0)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontraceterminal. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontraceterminal * S ((S (0)) * pfh_trace_scale_empty_construct_execution) + (0))) /\ ((forall pfh_index_empty_construct_executiontracesteps. (exists pfa_gap_empty_construct_executiontracestepsindex. pfa_gap_empty_construct_executiontracestepsindex + S (pfh_index_empty_construct_executiontracesteps) = (0)) -> (exists pfh_coefficient_empty_construct_executiontracestepsstep pfh_before_empty_construct_executiontracestepsstep pfh_after_empty_construct_executiontracestepsstep pfh_product_empty_construct_executiontracestepsstep. ((((exists ff_h_pfp_empty_construct_executiontracestepsstepcoefficient. ff_h_pfp_empty_construct_executiontracestepsstepcoefficient + S (pfh_coefficient_empty_construct_executiontracestepsstep) = S ((S (pfh_index_empty_construct_executiontracesteps)) * c)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepcoefficient. b = ff_q_pfp_empty_construct_executiontracestepsstepcoefficient * S ((S (pfh_index_empty_construct_executiontracesteps)) * c) + (pfh_coefficient_empty_construct_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_executiontracestepsstepbefore. ff_h_pfp_empty_construct_executiontracestepsstepbefore + S (pfh_before_empty_construct_executiontracestepsstep) = S ((S (pfh_index_empty_construct_executiontracesteps)) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepbefore. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontracestepsstepbefore * S ((S (pfh_index_empty_construct_executiontracesteps)) * pfh_trace_scale_empty_construct_execution) + (pfh_before_empty_construct_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_executiontracestepsstepafter. ff_h_pfp_empty_construct_executiontracestepsstepafter + S (pfh_after_empty_construct_executiontracestepsstep) = S ((S (S (pfh_index_empty_construct_executiontracesteps))) * pfh_trace_scale_empty_construct_execution)) /\ exists ff_q_pfp_empty_construct_executiontracestepsstepafter. pfh_trace_code_empty_construct_execution = ff_q_pfp_empty_construct_executiontracestepsstepafter * S ((S (S (pfh_index_empty_construct_executiontracesteps))) * pfh_trace_scale_empty_construct_execution) + (pfh_after_empty_construct_executiontracestepsstep))) /\ (((((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyleft. pfa_gap_empty_construct_executiontracestepsstepmultiplyleft + S (pfh_before_empty_construct_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyright. pfa_gap_empty_construct_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepmultiplyresultbound. pfa_gap_empty_construct_executiontracestepsstepmultiplyresultbound + S (pfh_product_empty_construct_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_empty_construct_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_empty_construct_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_empty_construct_executiontracestepsstepmultiplyresultcongruence = (pfh_product_empty_construct_executiontracestepsstep) + (p) * pfa_offset_right_empty_construct_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepaddleft. pfa_gap_empty_construct_executiontracestepsstepaddleft + S (pfh_product_empty_construct_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_executiontracestepsstepaddright. pfa_gap_empty_construct_executiontracestepsstepaddright + S (pfh_coefficient_empty_construct_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_empty_construct_executiontracestepsstepaddresultbound. pfa_gap_empty_construct_executiontracestepsstepaddresultbound + S (pfh_after_empty_construct_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_executiontracestepsstepaddresultcongruence pfa_offset_right_empty_construct_executiontracestepsstepaddresultcongruence. ((pfh_product_empty_construct_executiontracestepsstep) + (pfh_coefficient_empty_construct_executiontracestepsstep)) + (p) * pfa_offset_left_empty_construct_executiontracestepsstepaddresultcongruence = (pfh_after_empty_construct_executiontracestepsstep) + (p) * pfa_offset_right_empty_construct_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

Constructive proof overview

Generated structural guide

Construct an actual zero-result execution of every empty coefficient prefix, retaining the canonical base guard.

The unchanged tactic script uses 3 declared prerequisites and contains 31 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

PP0022 prime_field_polynomial_horner_exists matrix_rank_bounded_prefix_empty Alpha theorem; checked-use authorized PP0024 prime_field_polynomial_horner_empty

Direct dependents

PP002D prime_field_polynomial_horner_constant PP002E prime_field_polynomial_horner_zero

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

Read the argument

Proof checkpoints

31 script commands · 6 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.

Named ingredients (2)
prime field polynomial horner existsprime field polynomial horner empty

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–6

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 t
  5. L5
    intro hp
  6. L6
    intro ht
02Establish heL7–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial horner exists.

  1. L7
    have he : ∃ r. FpHorner(p,b,c,t,0,r)Definitions: FpHorner
  2. L8
    specialize prime_field_polynomial_horner_exists (p)
  3. L9
    specialize prime_field_polynomial_horner_exists (b)
  4. L10
    specialize prime_field_polynomial_horner_exists (c)
  5. L11
    specialize prime_field_polynomial_horner_exists (t)
  6. L12
    specialize prime_field_polynomial_horner_exists (0)
  7. L13
    apply prime_field_polynomial_horner_exists
  8. L14
    exact hp
  9. L15
    specialize matrix_rank_bounded_prefix_empty (b)
  10. L16
    specialize matrix_rank_bounded_prefix_empty (c)
03Use earlier factsL17–19

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

  1. L17
    specialize matrix_rank_bounded_prefix_empty (p)
  2. L18
    apply matrix_rank_bounded_prefix_empty
  3. L19
    exact ht
04Separate the logical casesL20–20

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

  1. L20
    cases he
05Establish hzL21–30

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial horner empty.

  1. L21
    have hz : x=0
  2. L22
    specialize prime_field_polynomial_horner_empty (p)
  3. L23
    specialize prime_field_polynomial_horner_empty (b)
  4. L24
    specialize prime_field_polynomial_horner_empty (c)
  5. L25
    specialize prime_field_polynomial_horner_empty (t)
  6. L26
    specialize prime_field_polynomial_horner_empty (x)
  7. L27
    apply prime_field_polynomial_horner_empty
  8. L28
    exact he_witness
  9. L29
    rewrite hz at he_witness
  10. L30
    rewrite hz at he_witness
06Use earlier factsL31–31

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

  1. L31
    exact he_witness

Library-wide reading audit

Original exact command ledger · 31 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro t
  5. 0005intro hp
  6. 0006intro ht
  7. 0007have he : exists r. (exists pfh_trace_code_empty_construct_chosen pfh_trace_scale_empty_construct_chosen. (((exists pfa_gap_empty_construct_chosentracebase. pfa_gap_empty_construct_chosentracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_empty_construct_chosentraceinitial. ff_h_pfp_empty_construct_chosentraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_empty_construct_chosen)) /\ exists ff_q_pfp_empty_construct_chosentraceinitial. pfh_trace_code_empty_construct_chosen = ff_q_pfp_empty_construct_chosentraceinitial * S ((S (0)) * pfh_trace_scale_empty_construct_chosen) + (0))) /\ (((((exists ff_h_pfp_empty_construct_chosentraceterminal. ff_h_pfp_empty_construct_chosentraceterminal + S (r) = S ((S (0)) * pfh_trace_scale_empty_construct_chosen)) /\ exists ff_q_pfp_empty_construct_chosentraceterminal. pfh_trace_code_empty_construct_chosen = ff_q_pfp_empty_construct_chosentraceterminal * S ((S (0)) * pfh_trace_scale_empty_construct_chosen) + (r))) /\ ((forall pfh_index_empty_construct_chosentracesteps. (exists pfa_gap_empty_construct_chosentracestepsindex. pfa_gap_empty_construct_chosentracestepsindex + S (pfh_index_empty_construct_chosentracesteps) = (0)) -> (exists pfh_coefficient_empty_construct_chosentracestepsstep pfh_before_empty_construct_chosentracestepsstep pfh_after_empty_construct_chosentracestepsstep pfh_product_empty_construct_chosentracestepsstep. ((((exists ff_h_pfp_empty_construct_chosentracestepsstepcoefficient. ff_h_pfp_empty_construct_chosentracestepsstepcoefficient + S (pfh_coefficient_empty_construct_chosentracestepsstep) = S ((S (pfh_index_empty_construct_chosentracesteps)) * c)) /\ exists ff_q_pfp_empty_construct_chosentracestepsstepcoefficient. b = ff_q_pfp_empty_construct_chosentracestepsstepcoefficient * S ((S (pfh_index_empty_construct_chosentracesteps)) * c) + (pfh_coefficient_empty_construct_chosentracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_chosentracestepsstepbefore. ff_h_pfp_empty_construct_chosentracestepsstepbefore + S (pfh_before_empty_construct_chosentracestepsstep) = S ((S (pfh_index_empty_construct_chosentracesteps)) * pfh_trace_scale_empty_construct_chosen)) /\ exists ff_q_pfp_empty_construct_chosentracestepsstepbefore. pfh_trace_code_empty_construct_chosen = ff_q_pfp_empty_construct_chosentracestepsstepbefore * S ((S (pfh_index_empty_construct_chosentracesteps)) * pfh_trace_scale_empty_construct_chosen) + (pfh_before_empty_construct_chosentracestepsstep))) /\ (((((exists ff_h_pfp_empty_construct_chosentracestepsstepafter. ff_h_pfp_empty_construct_chosentracestepsstepafter + S (pfh_after_empty_construct_chosentracestepsstep) = S ((S (S (pfh_index_empty_construct_chosentracesteps))) * pfh_trace_scale_empty_construct_chosen)) /\ exists ff_q_pfp_empty_construct_chosentracestepsstepafter. pfh_trace_code_empty_construct_chosen = ff_q_pfp_empty_construct_chosentracestepsstepafter * S ((S (S (pfh_index_empty_construct_chosentracesteps))) * pfh_trace_scale_empty_construct_chosen) + (pfh_after_empty_construct_chosentracestepsstep))) /\ (((((exists pfa_gap_empty_construct_chosentracestepsstepmultiplyleft. pfa_gap_empty_construct_chosentracestepsstepmultiplyleft + S (pfh_before_empty_construct_chosentracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_chosentracestepsstepmultiplyright. pfa_gap_empty_construct_chosentracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_empty_construct_chosentracestepsstepmultiplyresultbound. pfa_gap_empty_construct_chosentracestepsstepmultiplyresultbound + S (pfh_product_empty_construct_chosentracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_chosentracestepsstepmultiplyresultcongruence pfa_offset_right_empty_construct_chosentracestepsstepmultiplyresultcongruence. ((pfh_before_empty_construct_chosentracestepsstep) * (t)) + (p) * pfa_offset_left_empty_construct_chosentracestepsstepmultiplyresultcongruence = (pfh_product_empty_construct_chosentracestepsstep) + (p) * pfa_offset_right_empty_construct_chosentracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_empty_construct_chosentracestepsstepaddleft. pfa_gap_empty_construct_chosentracestepsstepaddleft + S (pfh_product_empty_construct_chosentracestepsstep) = (p)) /\ (((exists pfa_gap_empty_construct_chosentracestepsstepaddright. pfa_gap_empty_construct_chosentracestepsstepaddright + S (pfh_coefficient_empty_construct_chosentracestepsstep) = (p)) /\ ((((exists pfa_gap_empty_construct_chosentracestepsstepaddresultbound. pfa_gap_empty_construct_chosentracestepsstepaddresultbound + S (pfh_after_empty_construct_chosentracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_construct_chosentracestepsstepaddresultcongruence pfa_offset_right_empty_construct_chosentracestepsstepaddresultcongruence. ((pfh_product_empty_construct_chosentracestepsstep) + (pfh_coefficient_empty_construct_chosentracestepsstep)) + (p) * pfa_offset_left_empty_construct_chosentracestepsstepaddresultcongruence = (pfh_after_empty_construct_chosentracestepsstep) + (p) * pfa_offset_right_empty_construct_chosentracestepsstepaddresultcongruence)))))))))))))))))))))))))))
  8. 0008specialize prime_field_polynomial_horner_exists (p)
  9. 0009specialize prime_field_polynomial_horner_exists (b)
  10. 0010specialize prime_field_polynomial_horner_exists (c)
  11. 0011specialize prime_field_polynomial_horner_exists (t)
  12. 0012specialize prime_field_polynomial_horner_exists (0)
  13. 0013apply prime_field_polynomial_horner_exists
  14. 0014exact hp
  15. 0015specialize matrix_rank_bounded_prefix_empty (b)
  16. 0016specialize matrix_rank_bounded_prefix_empty (c)
  17. 0017specialize matrix_rank_bounded_prefix_empty (p)
  18. 0018apply matrix_rank_bounded_prefix_empty
  19. 0019exact ht
  20. 0020cases he
  21. 0021have hz : x=0
  22. 0022specialize prime_field_polynomial_horner_empty (p)
  23. 0023specialize prime_field_polynomial_horner_empty (b)
  24. 0024specialize prime_field_polynomial_horner_empty (c)
  25. 0025specialize prime_field_polynomial_horner_empty (t)
  26. 0026specialize prime_field_polynomial_horner_empty (x)
  27. 0027apply prime_field_polynomial_horner_empty
  28. 0028exact he_witness
  29. 0029rewrite hz at he_witness
  30. 0030rewrite hz at he_witness
  31. 0031exact he_witness