PQ0046

prime_field_polynomial_horner_trace_state_bounded

All actually decoded states of a canonical Horner history are canonical field elements, including its initial and terminal states.

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

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

Definition DAG

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

Actual proof prerequisites

prime_field_polynomial_horner_result_bounded · checked external prerequisiteprime_field_polynomial_horner_trace_prefix
Original expanded first-order statement 
forall p b c a l r u v n h. (~((p) = 1) /\ forall pfa_factor_left_state_prime pfa_factor_right_state_prime. (p) = pfa_factor_left_state_prime * pfa_factor_right_state_prime -> pfa_factor_left_state_prime = 1 \/ pfa_factor_right_state_prime = 1) -> (((exists pfa_gap_state_tracebase. pfa_gap_state_tracebase + S (a) = (p)) /\ (((((exists ff_h_pfp_state_traceinitial. ff_h_pfp_state_traceinitial + S (0) = S ((S (0)) * v)) /\ exists ff_q_pfp_state_traceinitial. u = ff_q_pfp_state_traceinitial * S ((S (0)) * v) + (0))) /\ (((((exists ff_h_pfp_state_traceterminal. ff_h_pfp_state_traceterminal + S (r) = S ((S (l)) * v)) /\ exists ff_q_pfp_state_traceterminal. u = ff_q_pfp_state_traceterminal * S ((S (l)) * v) + (r))) /\ ((forall pfh_index_state_tracesteps. (exists pfa_gap_state_tracestepsindex. pfa_gap_state_tracestepsindex + S (pfh_index_state_tracesteps) = (l)) -> (exists pfh_coefficient_state_tracestepsstep pfh_before_state_tracestepsstep pfh_after_state_tracestepsstep pfh_product_state_tracestepsstep. ((((exists ff_h_pfp_state_tracestepsstepcoefficient. ff_h_pfp_state_tracestepsstepcoefficient + S (pfh_coefficient_state_tracestepsstep) = S ((S (pfh_index_state_tracesteps)) * c)) /\ exists ff_q_pfp_state_tracestepsstepcoefficient. b = ff_q_pfp_state_tracestepsstepcoefficient * S ((S (pfh_index_state_tracesteps)) * c) + (pfh_coefficient_state_tracestepsstep))) /\ (((((exists ff_h_pfp_state_tracestepsstepbefore. ff_h_pfp_state_tracestepsstepbefore + S (pfh_before_state_tracestepsstep) = S ((S (pfh_index_state_tracesteps)) * v)) /\ exists ff_q_pfp_state_tracestepsstepbefore. u = ff_q_pfp_state_tracestepsstepbefore * S ((S (pfh_index_state_tracesteps)) * v) + (pfh_before_state_tracestepsstep))) /\ (((((exists ff_h_pfp_state_tracestepsstepafter. ff_h_pfp_state_tracestepsstepafter + S (pfh_after_state_tracestepsstep) = S ((S (S (pfh_index_state_tracesteps))) * v)) /\ exists ff_q_pfp_state_tracestepsstepafter. u = ff_q_pfp_state_tracestepsstepafter * S ((S (S (pfh_index_state_tracesteps))) * v) + (pfh_after_state_tracestepsstep))) /\ (((((exists pfa_gap_state_tracestepsstepmultiplyleft. pfa_gap_state_tracestepsstepmultiplyleft + S (pfh_before_state_tracestepsstep) = (p)) /\ (((exists pfa_gap_state_tracestepsstepmultiplyright. pfa_gap_state_tracestepsstepmultiplyright + S (a) = (p)) /\ ((((exists pfa_gap_state_tracestepsstepmultiplyresultbound. pfa_gap_state_tracestepsstepmultiplyresultbound + S (pfh_product_state_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_state_tracestepsstepmultiplyresultcongruence pfa_offset_right_state_tracestepsstepmultiplyresultcongruence. ((pfh_before_state_tracestepsstep) * (a)) + (p) * pfa_offset_left_state_tracestepsstepmultiplyresultcongruence = (pfh_product_state_tracestepsstep) + (p) * pfa_offset_right_state_tracestepsstepmultiplyresultcongruence))))))))) /\ ((((exists pfa_gap_state_tracestepsstepaddleft. pfa_gap_state_tracestepsstepaddleft + S (pfh_product_state_tracestepsstep) = (p)) /\ (((exists pfa_gap_state_tracestepsstepaddright. pfa_gap_state_tracestepsstepaddright + S (pfh_coefficient_state_tracestepsstep) = (p)) /\ ((((exists pfa_gap_state_tracestepsstepaddresultbound. pfa_gap_state_tracestepsstepaddresultbound + S (pfh_after_state_tracestepsstep) = (p)) /\ ((exists pfa_offset_left_state_tracestepsstepaddresultcongruence pfa_offset_right_state_tracestepsstepaddresultcongruence. ((pfh_product_state_tracestepsstep) + (pfh_coefficient_state_tracestepsstep)) + (p) * pfa_offset_left_state_tracestepsstepaddresultcongruence = (pfh_after_state_tracestepsstep) + (p) * pfa_offset_right_state_tracestepsstepaddresultcongruence)))))))))))))))))))))))))) -> (exists pfc_gap_state_length. pfc_gap_state_length+(n)=(l)) -> (((exists ff_h_pfp_state_entry. ff_h_pfp_state_entry + S (h) = S ((S (n)) * v)) /\ exists ff_q_pfp_state_entry. u = ff_q_pfp_state_entry * S ((S (n)) * v) + (h))) -> (exists pfa_gap_state_bound. pfa_gap_state_bound + S (h) = (p))

Complete tactic proof in conservative notation

All 36 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

36 script commands · 5 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 (1)
prime field polynomial horner trace prefix
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 assumptionsL11–14

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

  1. L11
    intro hp
  2. L12
    intro ht
  3. L13
    intro hn
  4. L14
    intro hh
03Use earlier factsL15–24

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

  1. L15
    specialize prime_field_polynomial_horner_result_bounded (p)
  2. L16
    specialize prime_field_polynomial_horner_result_bounded (b)
  3. L17
    specialize prime_field_polynomial_horner_result_bounded (c)
  4. L18
    specialize prime_field_polynomial_horner_result_bounded (a)
  5. L19
    specialize prime_field_polynomial_horner_result_bounded (n)
  6. L20
    specialize prime_field_polynomial_horner_result_bounded (h)
  7. L21
    apply prime_field_polynomial_horner_result_bounded
  8. L22
    exact hp
  9. L23
    specialize prime_field_polynomial_horner_trace_prefix (p)
  10. L24
    specialize prime_field_polynomial_horner_trace_prefix (b)
04Use earlier factsL25–34

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

  1. L25
    specialize prime_field_polynomial_horner_trace_prefix (c)
  2. L26
    specialize prime_field_polynomial_horner_trace_prefix (a)
  3. L27
    specialize prime_field_polynomial_horner_trace_prefix (l)
  4. L28
    specialize prime_field_polynomial_horner_trace_prefix (r)
  5. L29
    specialize prime_field_polynomial_horner_trace_prefix (u)
  6. L30
    specialize prime_field_polynomial_horner_trace_prefix (v)
  7. L31
    specialize prime_field_polynomial_horner_trace_prefix (n)
  8. L32
    specialize prime_field_polynomial_horner_trace_prefix (h)
  9. L33
    apply prime_field_polynomial_horner_trace_prefix
  10. L34
    exact ht
05Use earlier factsL35–36

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

  1. L35
    exact hn
  2. L36
    exact hh

Library-wide reading audit

Original defined command ledger · 36 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 hp
  12. 0012intro ht
  13. 0013intro hn
  14. 0014intro hh
  15. 0015specialize prime_field_polynomial_horner_result_bounded (p)
  16. 0016specialize prime_field_polynomial_horner_result_bounded (b)
  17. 0017specialize prime_field_polynomial_horner_result_bounded (c)
  18. 0018specialize prime_field_polynomial_horner_result_bounded (a)
  19. 0019specialize prime_field_polynomial_horner_result_bounded (n)
  20. 0020specialize prime_field_polynomial_horner_result_bounded (h)
  21. 0021apply prime_field_polynomial_horner_result_bounded
  22. 0022exact hp
  23. 0023specialize prime_field_polynomial_horner_trace_prefix (p)
  24. 0024specialize prime_field_polynomial_horner_trace_prefix (b)
  25. 0025specialize prime_field_polynomial_horner_trace_prefix (c)
  26. 0026specialize prime_field_polynomial_horner_trace_prefix (a)
  27. 0027specialize prime_field_polynomial_horner_trace_prefix (l)
  28. 0028specialize prime_field_polynomial_horner_trace_prefix (r)
  29. 0029specialize prime_field_polynomial_horner_trace_prefix (u)
  30. 0030specialize prime_field_polynomial_horner_trace_prefix (v)
  31. 0031specialize prime_field_polynomial_horner_trace_prefix (n)
  32. 0032specialize prime_field_polynomial_horner_trace_prefix (h)
  33. 0033apply prime_field_polynomial_horner_trace_prefix
  34. 0034exact ht
  35. 0035exact hn
  36. 0036exact hh