PQ0045

prime_field_polynomial_horner_trace_prefix

Every bounded prefix of a genuine Horner history is a genuine execution with its actually decoded terminal state.

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. ∀ l. ∀ r. ∀ u. ∀ v. ∀ n. ∀ h. FpHornerTrace(p,b,c,a,l,r,u,v) → Le(n,l) → BetaAt(u,v,n,h) → FpHorner(p,b,c,a,n,h)

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

Definition DAG

FpHornerTrace(p,b,c,x,l,r,u,v) · 1FpHorner(p,b,c,x,l,r) · 1Le(a,b) · 1BetaAt(b,c,i,x) · 1

Actual proof prerequisites

le_trans · checked external prerequisite
Original expanded first-order statement 
forall p b c a l r u v n h. (((exists pfa_gap_prefix_tracebase. pfa_gap_prefix_tracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_prefix_traceinitial. ff_h_pfp_prefix_traceinitial + S (0) = S ((S (0)) * v)) /\ exists ff_q_pfp_prefix_traceinitial. u = ff_q_pfp_prefix_traceinitial * S ((S (0)) * v) + (0))) /\ (((((exists ff_h_pfp_prefix_traceterminal. ff_h_pfp_prefix_traceterminal + S (r) = S ((S (l)) * v)) /\ exists ff_q_pfp_prefix_traceterminal. u = ff_q_pfp_prefix_traceterminal * S ((S (l)) * v) + (r))) /\ ((forall pfh_index_prefix_tracesteps. (exists pfa_gap_prefix_tracestepsindex. pfa_gap_prefix_tracestepsindex + S (pfh_index_prefix_tracesteps) = (l)) -> (exists pfh_coefficient_prefix_tracestepsstep pfh_before_prefix_tracestepsstep pfh_after_prefix_tracestepsstep pfh_product_prefix_tracestepsstep. ((((exists ff_h_pfp_prefix_tracestepsstepcoefficient. ff_h_pfp_prefix_tracestepsstepcoefficient + S (pfh_coefficient_prefix_tracestepsstep) = S ((S (pfh_index_prefix_tracesteps)) * c)) /\ exists ff_q_pfp_prefix_tracestepsstepcoefficient. b = ff_q_pfp_prefix_tracestepsstepcoefficient * S ((S (pfh_index_prefix_tracesteps)) * c) + (pfh_coefficient_prefix_tracestepsstep))) /\ (((((exists ff_h_pfp_prefix_tracestepsstepbefore. ff_h_pfp_prefix_tracestepsstepbefore + S (pfh_before_prefix_tracestepsstep) = S ((S (pfh_index_prefix_tracesteps)) * v)) /\ exists ff_q_pfp_prefix_tracestepsstepbefore. u = ff_q_pfp_prefix_tracestepsstepbefore * S ((S (pfh_index_prefix_tracesteps)) * v) + (pfh_before_prefix_tracestepsstep))) /\ (((((exists ff_h_pfp_prefix_tracestepsstepafter. ff_h_pfp_prefix_tracestepsstepafter + S (pfh_after_prefix_tracestepsstep) = S ((S (S (pfh_index_prefix_tracesteps))) * v)) /\ exists ff_q_pfp_prefix_tracestepsstepafter. u = ff_q_pfp_prefix_tracestepsstepafter * S ((S (S (pfh_index_prefix_tracesteps))) * v) + (pfh_after_prefix_tracestepsstep))) /\ (((((exists pfa_gap_prefix_tracestepsstepmultiplyleft. pfa_gap_prefix_tracestepsstepmultiplyleft + S (pfh_before_prefix_tracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_tracestepsstepmultiplyright. pfa_gap_prefix_tracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_prefix_tracestepsstepmultiplyresultbound. pfa_gap_prefix_tracestepsstepmultiplyresultbound + S (pfh_product_prefix_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_tracestepsstepmultiplyresultcongruence pfa_offset_right_prefix_tracestepsstepmultiplyresultcongruence. ((pfh_before_prefix_tracestepsstep) * (a)) + (p) * pfa_offset_left_prefix_tracestepsstepmultiplyresultcongruence = (pfh_product_prefix_tracestepsstep) + (p) * pfa_offset_right_prefix_tracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_prefix_tracestepsstepaddleft. pfa_gap_prefix_tracestepsstepaddleft + S (pfh_product_prefix_tracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_tracestepsstepaddright. pfa_gap_prefix_tracestepsstepaddright + S (pfh_coefficient_prefix_tracestepsstep) = (p)) /\ ((((exists pfa_gap_prefix_tracestepsstepaddresultbound. pfa_gap_prefix_tracestepsstepaddresultbound + S (pfh_after_prefix_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_tracestepsstepaddresultcongruence pfa_offset_right_prefix_tracestepsstepaddresultcongruence. ((pfh_product_prefix_tracestepsstep) + (pfh_coefficient_prefix_tracestepsstep)) + (p) * pfa_offset_left_prefix_tracestepsstepaddresultcongruence = (pfh_after_prefix_tracestepsstep) + (p) * pfa_offset_right_prefix_tracestepsstepaddresultcongruence)))))))))))))))))))))))))) -> (exists pfc_gap_prefix_length. pfc_gap_prefix_length+(n)=(l)) -> (((exists ff_h_pfp_prefix_state. ff_h_pfp_prefix_state + S (h) = S ((S (n)) * v)) /\ exists ff_q_pfp_prefix_state. u = ff_q_pfp_prefix_state * S ((S (n)) * v) + (h))) -> (exists pfh_trace_code_prefix_execution pfh_trace_scale_prefix_execution. (((exists pfa_gap_prefix_executiontracebase. pfa_gap_prefix_executiontracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_prefix_executiontraceinitial. ff_h_pfp_prefix_executiontraceinitial + S (0) = S ((S (0)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontraceinitial. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontraceinitial * S ((S (0)) * pfh_trace_scale_prefix_execution) + (0))) /\ (((((exists ff_h_pfp_prefix_executiontraceterminal. ff_h_pfp_prefix_executiontraceterminal + S (h) = S ((S (n)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontraceterminal. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontraceterminal * S ((S (n)) * pfh_trace_scale_prefix_execution) + (h))) /\ ((forall pfh_index_prefix_executiontracesteps. (exists pfa_gap_prefix_executiontracestepsindex. pfa_gap_prefix_executiontracestepsindex + S (pfh_index_prefix_executiontracesteps) = (n)) -> (exists pfh_coefficient_prefix_executiontracestepsstep pfh_before_prefix_executiontracestepsstep pfh_after_prefix_executiontracestepsstep pfh_product_prefix_executiontracestepsstep. ((((exists ff_h_pfp_prefix_executiontracestepsstepcoefficient. ff_h_pfp_prefix_executiontracestepsstepcoefficient + S (pfh_coefficient_prefix_executiontracestepsstep) = S ((S (pfh_index_prefix_executiontracesteps)) * c)) /\ exists ff_q_pfp_prefix_executiontracestepsstepcoefficient. b = ff_q_pfp_prefix_executiontracestepsstepcoefficient * S ((S (pfh_index_prefix_executiontracesteps)) * c) + (pfh_coefficient_prefix_executiontracestepsstep))) /\ (((((exists ff_h_pfp_prefix_executiontracestepsstepbefore. ff_h_pfp_prefix_executiontracestepsstepbefore + S (pfh_before_prefix_executiontracestepsstep) = S ((S (pfh_index_prefix_executiontracesteps)) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontracestepsstepbefore. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontracestepsstepbefore * S ((S (pfh_index_prefix_executiontracesteps)) * pfh_trace_scale_prefix_execution) + (pfh_before_prefix_executiontracestepsstep))) /\ (((((exists ff_h_pfp_prefix_executiontracestepsstepafter. ff_h_pfp_prefix_executiontracestepsstepafter + S (pfh_after_prefix_executiontracestepsstep) = S ((S (S (pfh_index_prefix_executiontracesteps))) * pfh_trace_scale_prefix_execution)) /\ exists ff_q_pfp_prefix_executiontracestepsstepafter. pfh_trace_code_prefix_execution = ff_q_pfp_prefix_executiontracestepsstepafter * S ((S (S (pfh_index_prefix_executiontracesteps))) * pfh_trace_scale_prefix_execution) + (pfh_after_prefix_executiontracestepsstep))) /\ (((((exists pfa_gap_prefix_executiontracestepsstepmultiplyleft. pfa_gap_prefix_executiontracestepsstepmultiplyleft + S (pfh_before_prefix_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_executiontracestepsstepmultiplyright. pfa_gap_prefix_executiontracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_prefix_executiontracestepsstepmultiplyresultbound. pfa_gap_prefix_executiontracestepsstepmultiplyresultbound + S (pfh_product_prefix_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_executiontracestepsstepmultiplyresultcongruence pfa_offset_right_prefix_executiontracestepsstepmultiplyresultcongruence. ((pfh_before_prefix_executiontracestepsstep) * (a)) + (p) * pfa_offset_left_prefix_executiontracestepsstepmultiplyresultcongruence = (pfh_product_prefix_executiontracestepsstep) + (p) * pfa_offset_right_prefix_executiontracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_prefix_executiontracestepsstepaddleft. pfa_gap_prefix_executiontracestepsstepaddleft + S (pfh_product_prefix_executiontracestepsstep) = (p)) /\ (((exists pfa_gap_prefix_executiontracestepsstepaddright. pfa_gap_prefix_executiontracestepsstepaddright + S (pfh_coefficient_prefix_executiontracestepsstep) = (p)) /\ ((((exists pfa_gap_prefix_executiontracestepsstepaddresultbound. pfa_gap_prefix_executiontracestepsstepaddresultbound + S (pfh_after_prefix_executiontracestepsstep) = (p)) /\ ((exists pfa_offset_left_prefix_executiontracestepsstepaddresultcongruence pfa_offset_right_prefix_executiontracestepsstepaddresultcongruence. ((pfh_product_prefix_executiontracestepsstep) + (pfh_coefficient_prefix_executiontracestepsstep)) + (p) * pfa_offset_left_prefix_executiontracestepsstepaddresultcongruence = (pfh_after_prefix_executiontracestepsstep) + (p) * pfa_offset_right_prefix_executiontracestepsstepaddresultcongruence)))))))))))))))))))))))))))

Complete tactic proof in conservative notation

All 34 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

34 script commands · 12 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.

01Fix variables and assumptionsL1–10

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 l
  6. L6
    intro r
  7. L7
    intro u
  8. L8
    intro v
  9. L9
    intro n
  10. L10
    intro h
02Fix variables and assumptionsL11–13

Work with arbitrary variables or the premises of the current implication.

  1. L11
    intro ht
  2. L12
    intro hn
  3. L13
    intro hh
03Separate the logical casesL14–16

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

  1. L14
    cases ht
  2. L15
    cases ht_right
  3. L16
    cases ht_right_right
04Construct an explicit witnessL17–18

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

  1. L17
    exists u
  2. L18
    exists v
05Separate the logical casesL19–19

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

  1. L19
    split
06Use earlier factsL20–20

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

  1. L20
    exact ht_left
07Separate the logical casesL21–21

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

  1. L21
    split
08Use earlier factsL22–22

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

  1. L22
    exact ht_right_left
09Separate the logical casesL23–23

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

  1. L23
    split
10Use earlier factsL24–24

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

  1. L24
    exact hh
11Fix variables and assumptionsL25–26

Work with arbitrary variables or the premises of the current implication.

  1. L25
    intro i
  2. L26
    intro hi
12Use earlier factsL27–34

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

  1. L27
    specialize ht_right_right_right (i)
  2. L28
    apply ht_right_right_right
  3. L29
    specialize le_trans (S i)
  4. L30
    specialize le_trans (n)
  5. L31
    specialize le_trans (l)
  6. L32
    apply le_trans
  7. L33
    exact hi
  8. L34
    exact hn

Library-wide reading audit

Original defined command ledger · 34 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro a
  5. 0005intro l
  6. 0006intro r
  7. 0007intro u
  8. 0008intro v
  9. 0009intro n
  10. 0010intro h
  11. 0011intro ht
  12. 0012intro hn
  13. 0013intro hh
  14. 0014cases ht
  15. 0015cases ht_right
  16. 0016cases ht_right_right
  17. 0017exists u
  18. 0018exists v
  19. 0019split
  20. 0020exact ht_left
  21. 0021split
  22. 0022exact ht_right_left
  23. 0023split
  24. 0024exact hh
  25. 0025intro i
  26. 0026intro hi
  27. 0027specialize ht_right_right_right (i)
  28. 0028apply ht_right_right_right
  29. 0029specialize le_trans (S i)
  30. 0030specialize le_trans (n)
  31. 0031specialize le_trans (l)
  32. 0032apply le_trans
  33. 0033exact hi
  34. 0034exact hn