PP002B

prime_field_polynomial_horner_empty_construct

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

Alpha v34 checked-use · first admitted v31 · 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.

Length is representation length, not polynomial degree. Leading zeros and the empty zero polynomial are allowed; the canonical argument guard x<p also applies to the empty case. Evaluation is defined by actual field-operation steps, not an assumed residue invariant. Polynomial division, gcd, irreducibles and general prime-power extension fields remain open; this does not close G091.

Exact theorem in conservative defined notation

∀ p. ∀ b. ∀ c. ∀ t. Prime(p) → Lt(t,p) → FpHorner(p,b,c,t,0,0)

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

Definition DAG

FpHorner(p,b,c,x,l,r) · 2Lt(a,b) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_polynomial_horner_existsmatrix_rank_bounded_prefix_empty · checked external prerequisiteprime_field_polynomial_horner_empty
Original expanded first-order 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)))))))))))))))))))))))))))

Complete tactic proof in conservative notation

All 31 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

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.

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 horner existsprime field polynomial horner empty
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(p,b,c,t,0,r)Original native command in the exact edition
  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 defined 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 : ∃ r. FpHorner(p,b,c,t,0,r)
  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