PP002D

prime_field_polynomial_horner_constant

A one-coefficient prefix evaluates to that actual constant, including zero and characteristic two.

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. ∀ a. Prime(p) → Lt(t,p) → Lt(a,p) → BetaAt(b,c,0,a) → FpHorner(p,b,c,t,1,a)

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

Definition DAG

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

Actual proof prerequisites

prime_field_polynomial_horner_successor_constructprime_field_polynomial_horner_empty_constructprime_field_multiply_zero_left · checked external prerequisiteprime_field_add_zero_left · checked external prerequisite
Original expanded first-order statement 
forall p b c t a. (~((p) = 1) /\ forall pfa_factor_left_constant_prime pfa_factor_right_constant_prime. (p) = pfa_factor_left_constant_prime * pfa_factor_right_constant_prime -> pfa_factor_left_constant_prime = 1 \/ pfa_factor_right_constant_prime = 1) -> (exists pfa_gap_constant_base. pfa_gap_constant_base + S (t) = (p)) -> (exists pfa_gap_constant_value. pfa_gap_constant_value + S (a) = (p)) -> (((exists ff_h_pfp_constant_coefficient. ff_h_pfp_constant_coefficient + S (a) = S ((S (0)) * c)) /\ exists ff_q_pfp_constant_coefficient. b = ff_q_pfp_constant_coefficient * S ((S (0)) * c) + (a))) -> (exists pfh_trace_code_constant_execution pfh_trace_scale_constant_execution. (((exists pfa_gap_constant_executiontracebase. pfa_gap_constant_executiontracebase + S (t) = (p)) /\ (((((exists ff_h_pfp_constant_executiontraceinitial. ff_h_pfp_constant_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontraceinitial. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontraceinitial * S ((S (0)) * pfh_trace_scale_constant_execution) + (0))) /\ (((((exists ff_h_pfp_constant_executiontraceterminal. ff_h_pfp_constant_executiontraceterminal + S (a) = S ((S (1)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontraceterminal. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontraceterminal * S ((S (1)) * pfh_trace_scale_constant_execution) + (a))) /\ ((forall pfh_index_constant_executiontracesteps. (exists pfa_gap_constant_executiontracestepsindex. pfa_gap_constant_executiontracestepsindex + S (pfh_index_constant_executiontracesteps) = (1)) -> (exists pfh_coefficient_constant_executiontracestepsstep pfh_before_constant_executiontracestepsstep pfh_after_constant_executiontracestepsstep pfh_product_constant_executiontracestepsstep. ((((exists ff_h_pfp_constant_executiontracestepsstepcoefficient. ff_h_pfp_constant_executiontracestepsstepcoefficient + S (pfh_coefficient_constant_executiontracestepsstep) = S ((S (pfh_index_constant_executiontracesteps)) * c)) /\ exists ff_q_pfp_constant_executiontracestepsstepcoefficient. b = ff_q_pfp_constant_executiontracestepsstepcoefficient * S ((S (pfh_index_constant_executiontracesteps)) * c) + (pfh_coefficient_constant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_executiontracestepsstepbefore. ff_h_pfp_constant_executiontracestepsstepbefore + S (pfh_before_constant_executiontracestepsstep) = S ((S (pfh_index_constant_executiontracesteps)) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontracestepsstepbefore. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontracestepsstepbefore * S ((S (pfh_index_constant_executiontracesteps)) * pfh_trace_scale_constant_execution) + (pfh_before_constant_executiontracestepsstep))) /\ (((((exists ff_h_pfp_constant_executiontracestepsstepafter. ff_h_pfp_constant_executiontracestepsstepafter + S (pfh_after_constant_executiontracestepsstep) = S ((S (S (pfh_index_constant_executiontracesteps))) * pfh_trace_scale_constant_execution)) /\ exists ff_q_pfp_constant_executiontracestepsstepafter. pfh_trace_code_constant_execution = ff_q_pfp_constant_executiontracestepsstepafter * S ((S (S (pfh_index_constant_executiontracesteps))) * pfh_trace_scale_constant_execution) + (pfh_after_constant_executiontracestepsstep))) /\ (((((exists pfa_gap_constant_executiontracestepsstepmultiplyleft. pfa_gap_constant_executiontracestepsstepmultiplyleft + S (pfh_before_constant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_executiontracestepsstepmultiplyright. pfa_gap_constant_executiontracestepsstepmultiplyright + S (t) = (p)) /\ ((((exists pfa_gap_constant_executiontracestepsstepmultiplyresultbound. pfa_gap_constant_executiontracestepsstepmultiplyresultbound + S (pfh_product_constant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_constant_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_constant_executiontracestepsstep) * (t)) + (p) * pfa_offset_left_constant_executiontracestepsstepmultiplyresultcongruence = (pfh_product_constant_executiontracestepsstep) + (p) * pfa_offset_right_constant_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_constant_executiontracestepsstepaddleft. pfa_gap_constant_executiontracestepsstepaddleft + S (pfh_product_constant_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_constant_executiontracestepsstepaddright. pfa_gap_constant_executiontracestepsstepaddright + S (pfh_coefficient_constant_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_constant_executiontracestepsstepaddresultbound. pfa_gap_constant_executiontracestepsstepaddresultbound + S (pfh_after_constant_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_constant_executiontracestepsstepaddresultcongruence pfa_offset_right_constant_executiontracestepsstepaddresultcongruence. ((pfh_product_constant_executiontracestepsstep) + (pfh_coefficient_constant_executiontracestepsstep)) + (p) * pfa_offset_left_constant_executiontracestepsstepaddresultcongruence = (pfh_after_constant_executiontracestepsstep) + (p) * pfa_offset_right_constant_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

Complete tactic proof in conservative notation

All 38 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

38 script commands · 4 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 horner successor constructprime field polynomial horner empty construct
01Fix variables and assumptionsL1–9

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 a
  6. L6
    intro hp
  7. L7
    intro ht
  8. L8
    intro ha
  9. L9
    intro hentry
02Use earlier factsL10–19

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

  1. L10
    specialize prime_field_polynomial_horner_successor_construct (p)
  2. L11
    specialize prime_field_polynomial_horner_successor_construct (b)
  3. L12
    specialize prime_field_polynomial_horner_successor_construct (c)
  4. L13
    specialize prime_field_polynomial_horner_successor_construct (t)
  5. L14
    specialize prime_field_polynomial_horner_successor_construct (0)
  6. L15
    specialize prime_field_polynomial_horner_successor_construct (a)
  7. L16
    specialize prime_field_polynomial_horner_successor_construct (0)
  8. L17
    specialize prime_field_polynomial_horner_successor_construct (0)
  9. L18
    specialize prime_field_polynomial_horner_successor_construct (a)
  10. L19
    apply prime_field_polynomial_horner_successor_construct
03Use earlier factsL20–29

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

  1. L20
    exact hp
  2. L21
    exact hentry
  3. L22
    specialize prime_field_polynomial_horner_empty_construct (p)
  4. L23
    specialize prime_field_polynomial_horner_empty_construct (b)
  5. L24
    specialize prime_field_polynomial_horner_empty_construct (c)
  6. L25
    specialize prime_field_polynomial_horner_empty_construct (t)
  7. L26
    apply prime_field_polynomial_horner_empty_construct
  8. L27
    exact hp
  9. L28
    exact ht
  10. L29
    specialize prime_field_multiply_zero_left (p)
04Use earlier factsL30–38

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

  1. L30
    specialize prime_field_multiply_zero_left (t)
  2. L31
    apply prime_field_multiply_zero_left
  3. L32
    exact hp
  4. L33
    exact ht
  5. L34
    specialize prime_field_add_zero_left (p)
  6. L35
    specialize prime_field_add_zero_left (a)
  7. L36
    apply prime_field_add_zero_left
  8. L37
    exact hp
  9. L38
    exact ha

Library-wide reading audit

Original defined command ledger · 38 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro t
  5. 0005intro a
  6. 0006intro hp
  7. 0007intro ht
  8. 0008intro ha
  9. 0009intro hentry
  10. 0010specialize prime_field_polynomial_horner_successor_construct (p)
  11. 0011specialize prime_field_polynomial_horner_successor_construct (b)
  12. 0012specialize prime_field_polynomial_horner_successor_construct (c)
  13. 0013specialize prime_field_polynomial_horner_successor_construct (t)
  14. 0014specialize prime_field_polynomial_horner_successor_construct (0)
  15. 0015specialize prime_field_polynomial_horner_successor_construct (a)
  16. 0016specialize prime_field_polynomial_horner_successor_construct (0)
  17. 0017specialize prime_field_polynomial_horner_successor_construct (0)
  18. 0018specialize prime_field_polynomial_horner_successor_construct (a)
  19. 0019apply prime_field_polynomial_horner_successor_construct
  20. 0020exact hp
  21. 0021exact hentry
  22. 0022specialize prime_field_polynomial_horner_empty_construct (p)
  23. 0023specialize prime_field_polynomial_horner_empty_construct (b)
  24. 0024specialize prime_field_polynomial_horner_empty_construct (c)
  25. 0025specialize prime_field_polynomial_horner_empty_construct (t)
  26. 0026apply prime_field_polynomial_horner_empty_construct
  27. 0027exact hp
  28. 0028exact ht
  29. 0029specialize prime_field_multiply_zero_left (p)
  30. 0030specialize prime_field_multiply_zero_left (t)
  31. 0031apply prime_field_multiply_zero_left
  32. 0032exact hp
  33. 0033exact ht
  34. 0034specialize prime_field_add_zero_left (p)
  35. 0035specialize prime_field_add_zero_left (a)
  36. 0036apply prime_field_add_zero_left
  37. 0037exact hp
  38. 0038exact ha