PQ004D

prime_field_polynomial_horner_constant_value

An actual one-step execution returns the decoded constant coefficient; coefficient bounds follow from the execution itself.

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. ∀ r. ∀ v. Prime(p) → FpHorner(p,b,c,a,1,r) → BetaAt(b,c,0,v) → r = v

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

Definition DAG

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

Actual proof prerequisites

prime_field_polynomial_horner_input_bounds · checked external prerequisiteprime_field_polynomial_horner_functional · checked external prerequisiteprime_field_polynomial_horner_constant · checked external prerequisitematrix_rank_bounded_prefix_value · checked external prerequisite
Original expanded first-order statement 
forall p b c a r v. (~((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 pfh_trace_code_constant_execution pfh_trace_scale_constant_execution. (((exists pfa_gap_constant_executiontracebase. pfa_gap_constant_executiontracebase + S (a) = (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 (r) = 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) + (r))) /\ ((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 (a) = (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) * (a)) + (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))))))))))))))))))))))))))) -> (((exists ff_h_pfp_constant_actual_coefficient. ff_h_pfp_constant_actual_coefficient + S (v) = S ((S (0)) * c)) /\ exists ff_q_pfp_constant_actual_coefficient. b = ff_q_pfp_constant_actual_coefficient * S ((S (0)) * c) + (v))) -> r=v

Complete tactic proof in conservative notation

All 49 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

49 script commands · 9 reading checkpoints · 1 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.

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 a
  5. L5
    intro r
  6. L6
    intro v
  7. L7
    intro hp
  8. L8
    intro he
  9. L9
    intro hv
02Establish hbL10–18

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

  1. L10
    have hb : Lt(a,p) ∧ BetaPrefixInto(b,c,1,p)Definitions: Lt(a,p)BetaPrefixInto(b,c,1,p)Original native command in the exact edition
  2. L11
    specialize prime_field_polynomial_horner_input_bounds (p)
  3. L12
    specialize prime_field_polynomial_horner_input_bounds (b)
  4. L13
    specialize prime_field_polynomial_horner_input_bounds (c)
  5. L14
    specialize prime_field_polynomial_horner_input_bounds (a)
  6. L15
    specialize prime_field_polynomial_horner_input_bounds (1)
  7. L16
    specialize prime_field_polynomial_horner_input_bounds (r)
  8. L17
    apply prime_field_polynomial_horner_input_bounds
  9. L18
    exact he
03Separate the logical casesL19–19

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

  1. L19
    cases hb
04Use earlier factsL20–29

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

  1. L20
    specialize prime_field_polynomial_horner_functional (p)
  2. L21
    specialize prime_field_polynomial_horner_functional (b)
  3. L22
    specialize prime_field_polynomial_horner_functional (c)
  4. L23
    specialize prime_field_polynomial_horner_functional (a)
  5. L24
    specialize prime_field_polynomial_horner_functional (1)
  6. L25
    specialize prime_field_polynomial_horner_functional (r)
  7. L26
    specialize prime_field_polynomial_horner_functional (v)
  8. L27
    apply prime_field_polynomial_horner_functional
  9. L28
    exact hp
  10. L29
    exact he
05Use earlier factsL30–39

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

  1. L30
    specialize prime_field_polynomial_horner_constant (p)
  2. L31
    specialize prime_field_polynomial_horner_constant (b)
  3. L32
    specialize prime_field_polynomial_horner_constant (c)
  4. L33
    specialize prime_field_polynomial_horner_constant (a)
  5. L34
    specialize prime_field_polynomial_horner_constant (v)
  6. L35
    apply prime_field_polynomial_horner_constant
  7. L36
    exact hp
  8. L37
    exact hb_left
  9. L38
    specialize matrix_rank_bounded_prefix_value (b)
  10. L39
    specialize matrix_rank_bounded_prefix_value (c)
06Use earlier factsL40–45

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

  1. L40
    specialize matrix_rank_bounded_prefix_value (1)
  2. L41
    specialize matrix_rank_bounded_prefix_value (p)
  3. L42
    specialize matrix_rank_bounded_prefix_value (0)
  4. L43
    specialize matrix_rank_bounded_prefix_value (v)
  5. L44
    apply matrix_rank_bounded_prefix_value
  6. L45
    exact hb_right
07Construct an explicit witnessL46–46

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

  1. L46
    exists 0
08Calculate and transport equalitiesL47–47

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

  1. L47
    simp
09Use earlier factsL48–49

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

  1. L48
    exact hv
  2. L49
    exact hv

Library-wide reading audit

Original defined command ledger · 49 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro a
  5. 0005intro r
  6. 0006intro v
  7. 0007intro hp
  8. 0008intro he
  9. 0009intro hv
  10. 0010have hb : Lt(a,p) ∧ BetaPrefixInto(b,c,1,p)
  11. 0011specialize prime_field_polynomial_horner_input_bounds (p)
  12. 0012specialize prime_field_polynomial_horner_input_bounds (b)
  13. 0013specialize prime_field_polynomial_horner_input_bounds (c)
  14. 0014specialize prime_field_polynomial_horner_input_bounds (a)
  15. 0015specialize prime_field_polynomial_horner_input_bounds (1)
  16. 0016specialize prime_field_polynomial_horner_input_bounds (r)
  17. 0017apply prime_field_polynomial_horner_input_bounds
  18. 0018exact he
  19. 0019cases hb
  20. 0020specialize prime_field_polynomial_horner_functional (p)
  21. 0021specialize prime_field_polynomial_horner_functional (b)
  22. 0022specialize prime_field_polynomial_horner_functional (c)
  23. 0023specialize prime_field_polynomial_horner_functional (a)
  24. 0024specialize prime_field_polynomial_horner_functional (1)
  25. 0025specialize prime_field_polynomial_horner_functional (r)
  26. 0026specialize prime_field_polynomial_horner_functional (v)
  27. 0027apply prime_field_polynomial_horner_functional
  28. 0028exact hp
  29. 0029exact he
  30. 0030specialize prime_field_polynomial_horner_constant (p)
  31. 0031specialize prime_field_polynomial_horner_constant (b)
  32. 0032specialize prime_field_polynomial_horner_constant (c)
  33. 0033specialize prime_field_polynomial_horner_constant (a)
  34. 0034specialize prime_field_polynomial_horner_constant (v)
  35. 0035apply prime_field_polynomial_horner_constant
  36. 0036exact hp
  37. 0037exact hb_left
  38. 0038specialize matrix_rank_bounded_prefix_value (b)
  39. 0039specialize matrix_rank_bounded_prefix_value (c)
  40. 0040specialize matrix_rank_bounded_prefix_value (1)
  41. 0041specialize matrix_rank_bounded_prefix_value (p)
  42. 0042specialize matrix_rank_bounded_prefix_value (0)
  43. 0043specialize matrix_rank_bounded_prefix_value (v)
  44. 0044apply matrix_rank_bounded_prefix_value
  45. 0045exact hb_right
  46. 0046exists 0
  47. 0047simp
  48. 0048exact hv
  49. 0049exact hv