PP0024

prime_field_polynomial_horner_empty

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

The empty polynomial execution returns zero by its actual initial and terminal beta entries.

Exact expanded first-order arithmetic statement

forall p b c t r. (exists pfh_trace_code_empty_execution pfh_trace_scale_empty_execution. (((exists pfa_gap_empty_executiontracebase. pfa_gap_empty_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_empty_executiontraceinitial. ff_h_pfp_empty_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontraceinitial. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontraceinitial * S ((S (0)) * pfh_trace_scale_empty_execution) + (0))) /\ (((((exists ff_h_pfp_empty_executiontraceterminal. ff_h_pfp_empty_executiontraceterminal + S (r) = S ((S (0)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontraceterminal. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontraceterminal * S ((S (0)) * pfh_trace_scale_empty_execution) + (r))) /\ ((forall pfh_index_empty_executiontracesteps. (exists pfa_gap_empty_executiontracestepsindex. pfa_gap_empty_executiontracestepsindex + S (pfh_index_empty_executiontracesteps) = (0)) -> (exists pfh_coefficient_empty_executiontracestepsstep pfh_before_empty_executiontracestepsstep pfh_after_empty_executiontracestepsstep pfh_product_empty_executiontracestepsstep. ((((exists ff_h_pfp_empty_executiontracestepsstepcoefficient. ff_h_pfp_empty_executiontracestepsstepcoefficient + S (pfh_coefficient_empty_executiontracestepsstep) = S ((S (pfh_index_empty_executiontracesteps)) * c)) /\ exists ff_q_pfp_empty_executiontracestepsstepcoefficient. b = ff_q_pfp_empty_executiontracestepsstepcoefficient * S ((S (pfh_index_empty_executiontracesteps)) * c) + (pfh_coefficient_empty_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_executiontracestepsstepbefore. ff_h_pfp_empty_executiontracestepsstepbefore + S (pfh_before_empty_executiontracestepsstep) = S ((S (pfh_index_empty_executiontracesteps)) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontracestepsstepbefore. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontracestepsstepbefore * S ((S (pfh_index_empty_executiontracesteps)) * pfh_trace_scale_empty_execution) + (pfh_before_empty_executiontracestepsstep))) /\ (((((exists ff_h_pfp_empty_executiontracestepsstepafter. ff_h_pfp_empty_executiontracestepsstepafter + S (pfh_after_empty_executiontracestepsstep) = S ((S (S (pfh_index_empty_executiontracesteps))) * pfh_trace_scale_empty_execution)) /\ exists ff_q_pfp_empty_executiontracestepsstepafter. pfh_trace_code_empty_execution = ff_q_pfp_empty_executiontracestepsstepafter * S ((S (S (pfh_index_empty_executiontracesteps))) * pfh_trace_scale_empty_execution) + (pfh_after_empty_executiontracestepsstep))) /\ (((((exists pfa_gap_empty_executiontracestepsstepmultiplyleft. pfa_gap_empty_executiontracestepsstepmultiplyleft + S (pfh_before_empty_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_executiontracestepsstepmultiplyright. pfa_gap_empty_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_empty_executiontracestepsstepmultiplyresultbound. pfa_gap_empty_executiontracestepsstepmultiplyresultbound + S (pfh_product_empty_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_empty_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_empty_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_empty_executiontracestepsstepmultiplyresultcongruence = (pfh_product_empty_executiontracestepsstep) + (p) * pfa_offset_right_empty_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_empty_executiontracestepsstepaddleft. pfa_gap_empty_executiontracestepsstepaddleft + S (pfh_product_empty_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_empty_executiontracestepsstepaddright. pfa_gap_empty_executiontracestepsstepaddright + S (pfh_coefficient_empty_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_empty_executiontracestepsstepaddresultbound. pfa_gap_empty_executiontracestepsstepaddresultbound + S (pfh_after_empty_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_empty_executiontracestepsstepaddresultcongruence pfa_offset_right_empty_executiontracestepsstepaddresultcongruence. ((pfh_product_empty_executiontracestepsstep) + (pfh_coefficient_empty_executiontracestepsstep)) + (p) * pfa_offset_left_empty_executiontracestepsstepaddresultcongruence = (pfh_after_empty_executiontracestepsstep) + (p) * pfa_offset_right_empty_executiontracestepsstepaddresultcongruence))))))))))))))))))))))))))) -> r=0

Constructive proof overview

Generated structural guide

The empty polynomial execution returns zero by its actual initial and terminal beta entries.

The unchanged tactic script uses 1 declared prerequisite and contains 19 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

beta_at_unique Alpha theorem; checked-use authorized

Direct dependents

PP0027 prime_field_polynomial_horner_normalization_residue PP002B prime_field_polynomial_horner_empty_construct

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

19 script commands · 3 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.

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 r
  6. L6
    intro h
02Separate the logical casesL7–11

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

  1. L7
    cases h
  2. L8
    cases h_witness
  3. L9
    cases h_witness_witness
  4. L10
    cases h_witness_witness_right
  5. L11
    cases h_witness_witness_right_right
03Use earlier factsL12–19

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

  1. L12
    specialize beta_at_unique (x)
  2. L13
    specialize beta_at_unique (x1)
  3. L14
    specialize beta_at_unique (0)
  4. L15
    specialize beta_at_unique (r)
  5. L16
    specialize beta_at_unique (0)
  6. L17
    apply beta_at_unique
  7. L18
    exact h_witness_witness_right_right_left
  8. L19
    exact h_witness_witness_right_left

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro t
  5. 0005intro r
  6. 0006intro h
  7. 0007cases h
  8. 0008cases h_witness
  9. 0009cases h_witness_witness
  10. 0010cases h_witness_witness_right
  11. 0011cases h_witness_witness_right_right
  12. 0012specialize beta_at_unique (x)
  13. 0013specialize beta_at_unique (x1)
  14. 0014specialize beta_at_unique (0)
  15. 0015specialize beta_at_unique (r)
  16. 0016specialize beta_at_unique (0)
  17. 0017apply beta_at_unique
  18. 0018exact h_witness_witness_right_right_left
  19. 0019exact h_witness_witness_right_left