PQ0038

prime_field_polynomial_monic_normalization_bounded

The recorded scalar and every source and target coefficient are genuinely below the modulus.

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. ∀ k. ∀ ab. ∀ ac. ∀ bb. ∀ bc. ∀ L. FpMonicNormalization(p,k,ab,ac,bb,bc,L) → Lt(k,p) ∧ (BetaPrefixInto(ab,ac,L,p) ∧ BetaPrefixInto(bb,bc,L,p))

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

Definition DAG

BetaPrefixInto(b,c,l,B) · 2FpPolyScale(p,k,ab,ac,bb,bc,l) · 1FpMonicNormalization(p,k,ab,ac,bb,bc,L) · 1Lt(a,b) · 1

Actual proof prerequisites

prime_field_polynomial_scale_bounded · checked external prerequisite
Original expanded first-order statement 
forall p k ab ac bb bc L. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_bound_source. ((((exists ff_h_pfp_normalization_bound_sourcesource. ff_h_pfp_normalization_bound_sourcesource + S (pfm_leading_normalization_bound_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_bound_sourcesource. ab = ff_q_pfp_normalization_bound_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_bound_source))) /\ ((((~((pfm_leading_normalization_bound_source) = 0)) /\ ((((exists pfa_gap_normalization_bound_sourceinversemultiplicationleft. pfa_gap_normalization_bound_sourceinversemultiplicationleft + S (pfm_leading_normalization_bound_source) = (p)) /\ (((exists pfa_gap_normalization_bound_sourceinversemultiplicationright. pfa_gap_normalization_bound_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_bound_sourceinversemultiplicationresultbound. pfa_gap_normalization_bound_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_bound_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_bound_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_bound_source) * (k)) + (p) * pfa_offset_left_normalization_bound_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_bound_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_bound_sourcescalescalar. pfa_gap_normalization_bound_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_bound_sourcescale. (exists pfa_gap_normalization_bound_sourcescaleindex. pfa_gap_normalization_bound_sourcescaleindex + S (pfp_index_normalization_bound_sourcescale) = (L)) -> exists pfp_source_normalization_bound_sourcescale pfp_value_normalization_bound_sourcescale. ((((exists ff_h_pfp_normalization_bound_sourcescalesource. ff_h_pfp_normalization_bound_sourcescalesource + S (pfp_source_normalization_bound_sourcescale) = S ((S (pfp_index_normalization_bound_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_bound_sourcescalesource. ab = ff_q_pfp_normalization_bound_sourcescalesource * S ((S (pfp_index_normalization_bound_sourcescale)) * ac) + (pfp_source_normalization_bound_sourcescale))) /\ (((((exists ff_h_pfp_normalization_bound_sourcescaletarget. ff_h_pfp_normalization_bound_sourcescaletarget + S (pfp_value_normalization_bound_sourcescale) = S ((S (pfp_index_normalization_bound_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_bound_sourcescaletarget. bb = ff_q_pfp_normalization_bound_sourcescaletarget * S ((S (pfp_index_normalization_bound_sourcescale)) * bc) + (pfp_value_normalization_bound_sourcescale))) /\ ((((exists pfa_gap_normalization_bound_sourcescaleoperationleft. pfa_gap_normalization_bound_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_bound_sourcescaleoperationright. pfa_gap_normalization_bound_sourcescaleoperationright + S (pfp_source_normalization_bound_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_bound_sourcescaleoperationresultbound. pfa_gap_normalization_bound_sourcescaleoperationresultbound + S (pfp_value_normalization_bound_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_bound_sourcescaleoperationresultcongruence pfa_offset_right_normalization_bound_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_bound_sourcescale)) + (p) * pfa_offset_left_normalization_bound_sourcescaleoperationresultcongruence = (pfp_value_normalization_bound_sourcescale) + (p) * pfa_offset_right_normalization_bound_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists pfa_gap_normalization_bound_scalar. pfa_gap_normalization_bound_scalar + S (k) = (p)) /\ (((forall fom_index_pfp_normalization_bound_input. (exists fom_gap_pfp_normalization_bound_input_index_bound. fom_gap_pfp_normalization_bound_input_index_bound + S (fom_index_pfp_normalization_bound_input) = L) -> exists fom_value_pfp_normalization_bound_input. ((((exists fom_beta_height_pfp_normalization_bound_input_entry. fom_beta_height_pfp_normalization_bound_input_entry + S (fom_value_pfp_normalization_bound_input) = S ((S (fom_index_pfp_normalization_bound_input)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_bound_input_entry. ab = fom_beta_quotient_pfp_normalization_bound_input_entry * S ((S (fom_index_pfp_normalization_bound_input)) * ac) + (fom_value_pfp_normalization_bound_input))) /\ (exists fom_gap_pfp_normalization_bound_input_value_bound. fom_gap_pfp_normalization_bound_input_value_bound + S (fom_value_pfp_normalization_bound_input) = p))) /\ ((forall fom_index_pfp_normalization_bound_output. (exists fom_gap_pfp_normalization_bound_output_index_bound. fom_gap_pfp_normalization_bound_output_index_bound + S (fom_index_pfp_normalization_bound_output) = L) -> exists fom_value_pfp_normalization_bound_output. ((((exists fom_beta_height_pfp_normalization_bound_output_entry. fom_beta_height_pfp_normalization_bound_output_entry + S (fom_value_pfp_normalization_bound_output) = S ((S (fom_index_pfp_normalization_bound_output)) * bc)) /\ exists fom_beta_quotient_pfp_normalization_bound_output_entry. bb = fom_beta_quotient_pfp_normalization_bound_output_entry * S ((S (fom_index_pfp_normalization_bound_output)) * bc) + (fom_value_pfp_normalization_bound_output))) /\ (exists fom_gap_pfp_normalization_bound_output_value_bound. fom_gap_pfp_normalization_bound_output_value_bound + S (fom_value_pfp_normalization_bound_output) = p))))))))

Complete tactic proof in conservative notation

All 24 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

24 script commands · 5 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro k
  3. L3
    intro ab
  4. L4
    intro ac
  5. L5
    intro bb
  6. L6
    intro bc
  7. L7
    intro L
  8. L8
    intro h
02Separate the logical casesL9–10

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

  1. L9
    cases h
  2. L10
    cases h_right
03Establish hsL11–12

Establish this local claim before using it. It is not an additional assumption.

  1. L11
    have hs : FpPolyScale(p,k,ab,ac,bb,bc,L)Definitions: FpPolyScale(p,k,ab,ac,bb,bc,L)Original native command in the exact edition
  2. L12
    exact h_right_right
04Separate the logical casesL13–14

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

  1. L13
    cases hs
  2. L14
    split
05Use earlier factsL15–24

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

  1. L15
    exact hs_left
  2. L16
    specialize prime_field_polynomial_scale_bounded (p)
  3. L17
    specialize prime_field_polynomial_scale_bounded (k)
  4. L18
    specialize prime_field_polynomial_scale_bounded (ab)
  5. L19
    specialize prime_field_polynomial_scale_bounded (ac)
  6. L20
    specialize prime_field_polynomial_scale_bounded (bb)
  7. L21
    specialize prime_field_polynomial_scale_bounded (bc)
  8. L22
    specialize prime_field_polynomial_scale_bounded (L)
  9. L23
    apply prime_field_polynomial_scale_bounded
  10. L24
    exact h_right_right

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro L
  8. 0008intro h
  9. 0009cases h
  10. 0010cases h_right
  11. 0011have hs : FpPolyScale(p,k,ab,ac,bb,bc,L)
  12. 0012exact h_right_right
  13. 0013cases hs
  14. 0014split
  15. 0015exact hs_left
  16. 0016specialize prime_field_polynomial_scale_bounded (p)
  17. 0017specialize prime_field_polynomial_scale_bounded (k)
  18. 0018specialize prime_field_polynomial_scale_bounded (ab)
  19. 0019specialize prime_field_polynomial_scale_bounded (ac)
  20. 0020specialize prime_field_polynomial_scale_bounded (bb)
  21. 0021specialize prime_field_polynomial_scale_bounded (bc)
  22. 0022specialize prime_field_polynomial_scale_bounded (L)
  23. 0023apply prime_field_polynomial_scale_bounded
  24. 0024exact h_right_right