PQ0038

prime_field_polynomial_monic_normalization_bounded

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

Exact expanded first-order arithmetic 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))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 1 declared prerequisite and contains 24 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

prime_field_polynomial_scale_bounded Alpha theorem; checked-use authorized

Direct dependents

PQ003A prime_field_polynomial_monic_normalization_monic

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

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.

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
  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 exact 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 : ((exists pfa_gap_normalization_bound_scalescalar. pfa_gap_normalization_bound_scalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_bound_scale. (exists pfa_gap_normalization_bound_scaleindex. pfa_gap_normalization_bound_scaleindex + S (pfp_index_normalization_bound_scale) = (L)) -> exists pfp_source_normalization_bound_scale pfp_value_normalization_bound_scale. ((((exists ff_h_pfp_normalization_bound_scalesource. ff_h_pfp_normalization_bound_scalesource + S (pfp_source_normalization_bound_scale) = S ((S (pfp_index_normalization_bound_scale)) * ac)) /\ exists ff_q_pfp_normalization_bound_scalesource. ab = ff_q_pfp_normalization_bound_scalesource * S ((S (pfp_index_normalization_bound_scale)) * ac) + (pfp_source_normalization_bound_scale))) /\ (((((exists ff_h_pfp_normalization_bound_scaletarget. ff_h_pfp_normalization_bound_scaletarget + S (pfp_value_normalization_bound_scale) = S ((S (pfp_index_normalization_bound_scale)) * bc)) /\ exists ff_q_pfp_normalization_bound_scaletarget. bb = ff_q_pfp_normalization_bound_scaletarget * S ((S (pfp_index_normalization_bound_scale)) * bc) + (pfp_value_normalization_bound_scale))) /\ ((((exists pfa_gap_normalization_bound_scaleoperationleft. pfa_gap_normalization_bound_scaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_bound_scaleoperationright. pfa_gap_normalization_bound_scaleoperationright + S (pfp_source_normalization_bound_scale) = (p)) /\ ((((exists pfa_gap_normalization_bound_scaleoperationresultbound. pfa_gap_normalization_bound_scaleoperationresultbound + S (pfp_value_normalization_bound_scale) = (p)) /\ ((exists pfa_offset_left_normalization_bound_scaleoperationresultcongruence pfa_offset_right_normalization_bound_scaleoperationresultcongruence. ((k) * (pfp_source_normalization_bound_scale)) + (p) * pfa_offset_left_normalization_bound_scaleoperationresultcongruence = (pfp_value_normalization_bound_scale) + (p) * pfa_offset_right_normalization_bound_scaleoperationresultcongruence))))))))))))))))
  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