PQ0033

prime_field_polynomial_monic_transport

Actual prefix reencoding preserves monicity, without constraining any outside entry.

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. ∀ B. ∀ C. ∀ L. (∀ x. ∀ y. Lt(x,L) → BetaAt(b,c,x,y) → BetaAt(B,C,x,y)) → FpMonic(p,b,c,L) → FpMonic(p,B,C,L)

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

Definition DAG

FpMonic(p,b,c,L) · 2Lt(a,b) · 1BetaAt(b,c,i,x) · 2

Actual proof prerequisites

matrix_rank_bounded_prefix_transport · checked external prerequisiteone_le_of_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall p b c B C L. (forall mdr_i_pfp_monic_recode mdr_a_pfp_monic_recode. (exists mdr_gap_pfp_monic_recodeb. mdr_gap_pfp_monic_recodeb + S (mdr_i_pfp_monic_recode) = (L)) -> (((exists ff_h_mdr_pfp_monic_recodeo. ff_h_mdr_pfp_monic_recodeo + S (mdr_a_pfp_monic_recode) = S ((S (mdr_i_pfp_monic_recode)) * c)) /\ exists ff_q_mdr_pfp_monic_recodeo. b = ff_q_mdr_pfp_monic_recodeo * S ((S (mdr_i_pfp_monic_recode)) * c) + (mdr_a_pfp_monic_recode))) -> (((exists ff_h_mdr_pfp_monic_recoden. ff_h_mdr_pfp_monic_recoden + S (mdr_a_pfp_monic_recode) = S ((S (mdr_i_pfp_monic_recode)) * C)) /\ exists ff_q_mdr_pfp_monic_recoden. B = ff_q_mdr_pfp_monic_recoden * S ((S (mdr_i_pfp_monic_recode)) * C) + (mdr_a_pfp_monic_recode)))) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_oldcoefficients. (exists fom_gap_pfp_monic_oldcoefficients_index_bound. fom_gap_pfp_monic_oldcoefficients_index_bound + S (fom_index_pfp_monic_oldcoefficients) = L) -> exists fom_value_pfp_monic_oldcoefficients. ((((exists fom_beta_height_pfp_monic_oldcoefficients_entry. fom_beta_height_pfp_monic_oldcoefficients_entry + S (fom_value_pfp_monic_oldcoefficients) = S ((S (fom_index_pfp_monic_oldcoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_oldcoefficients_entry. b = fom_beta_quotient_pfp_monic_oldcoefficients_entry * S ((S (fom_index_pfp_monic_oldcoefficients)) * c) + (fom_value_pfp_monic_oldcoefficients))) /\ (exists fom_gap_pfp_monic_oldcoefficients_value_bound. fom_gap_pfp_monic_oldcoefficients_value_bound + S (fom_value_pfp_monic_oldcoefficients) = p))) /\ ((((exists ff_h_pfp_monic_oldleading. ff_h_pfp_monic_oldleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_oldleading. b = ff_q_pfp_monic_oldleading * S ((S (0)) * c) + (1)))))))) -> (((~((L) = 0)) /\ (((forall fom_index_pfp_monic_newcoefficients. (exists fom_gap_pfp_monic_newcoefficients_index_bound. fom_gap_pfp_monic_newcoefficients_index_bound + S (fom_index_pfp_monic_newcoefficients) = L) -> exists fom_value_pfp_monic_newcoefficients. ((((exists fom_beta_height_pfp_monic_newcoefficients_entry. fom_beta_height_pfp_monic_newcoefficients_entry + S (fom_value_pfp_monic_newcoefficients) = S ((S (fom_index_pfp_monic_newcoefficients)) * C)) /\ exists fom_beta_quotient_pfp_monic_newcoefficients_entry. B = fom_beta_quotient_pfp_monic_newcoefficients_entry * S ((S (fom_index_pfp_monic_newcoefficients)) * C) + (fom_value_pfp_monic_newcoefficients))) /\ (exists fom_gap_pfp_monic_newcoefficients_value_bound. fom_gap_pfp_monic_newcoefficients_value_bound + S (fom_value_pfp_monic_newcoefficients) = p))) /\ ((((exists ff_h_pfp_monic_newleading. ff_h_pfp_monic_newleading + S (1) = S ((S (0)) * C)) /\ exists ff_q_pfp_monic_newleading. B = ff_q_pfp_monic_newleading * S ((S (0)) * C) + (1))))))))

Complete tactic proof in conservative notation

All 29 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

29 script commands · 6 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 assumptionsL1–8

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 B
  5. L5
    intro C
  6. L6
    intro L
  7. L7
    intro he
  8. L8
    intro h
02Separate the logical casesL9–11

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

  1. L9
    cases h
  2. L10
    cases h_right
  3. L11
    split
03Use earlier factsL12–12

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

  1. L12
    exact h_left
04Separate the logical casesL13–13

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

  1. L13
    split
05Use earlier factsL14–23

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

  1. L14
    specialize matrix_rank_bounded_prefix_transport (b)
  2. L15
    specialize matrix_rank_bounded_prefix_transport (c)
  3. L16
    specialize matrix_rank_bounded_prefix_transport (B)
  4. L17
    specialize matrix_rank_bounded_prefix_transport (C)
  5. L18
    specialize matrix_rank_bounded_prefix_transport (L)
  6. L19
    specialize matrix_rank_bounded_prefix_transport (p)
  7. L20
    apply matrix_rank_bounded_prefix_transport
  8. L21
    exact he
  9. L22
    exact h_right_left
  10. L23
    specialize he (0)
06Use earlier factsL24–29

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

  1. L24
    specialize he (1)
  2. L25
    apply he
  3. L26
    specialize one_le_of_ne_zero (L)
  4. L27
    apply one_le_of_ne_zero
  5. L28
    exact h_left
  6. L29
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro B
  5. 0005intro C
  6. 0006intro L
  7. 0007intro he
  8. 0008intro h
  9. 0009cases h
  10. 0010cases h_right
  11. 0011split
  12. 0012exact h_left
  13. 0013split
  14. 0014specialize matrix_rank_bounded_prefix_transport (b)
  15. 0015specialize matrix_rank_bounded_prefix_transport (c)
  16. 0016specialize matrix_rank_bounded_prefix_transport (B)
  17. 0017specialize matrix_rank_bounded_prefix_transport (C)
  18. 0018specialize matrix_rank_bounded_prefix_transport (L)
  19. 0019specialize matrix_rank_bounded_prefix_transport (p)
  20. 0020apply matrix_rank_bounded_prefix_transport
  21. 0021exact he
  22. 0022exact h_right_left
  23. 0023specialize he (0)
  24. 0024specialize he (1)
  25. 0025apply he
  26. 0026specialize one_le_of_ne_zero (L)
  27. 0027apply one_le_of_ne_zero
  28. 0028exact h_left
  29. 0029exact h_right_right