PQ0035

prime_field_polynomial_monic_normalization_inverse

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

The recorded scalar is an actual inverse of every decoding of the source leading coefficient.

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 a. (((~((L) = 0)) /\ (((exists pfm_leading_normalization_inverse_source. ((((exists ff_h_pfp_normalization_inverse_sourcesource. ff_h_pfp_normalization_inverse_sourcesource + S (pfm_leading_normalization_inverse_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcesource. ab = ff_q_pfp_normalization_inverse_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_inverse_source))) /\ ((((~((pfm_leading_normalization_inverse_source) = 0)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationleft. pfa_gap_normalization_inverse_sourceinversemultiplicationleft + S (pfm_leading_normalization_inverse_source) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourceinversemultiplicationright. pfa_gap_normalization_inverse_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound. pfa_gap_normalization_inverse_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_inverse_source) * (k)) + (p) * pfa_offset_left_normalization_inverse_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_inverse_sourcescalescalar. pfa_gap_normalization_inverse_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_inverse_sourcescale. (exists pfa_gap_normalization_inverse_sourcescaleindex. pfa_gap_normalization_inverse_sourcescaleindex + S (pfp_index_normalization_inverse_sourcescale) = (L)) -> exists pfp_source_normalization_inverse_sourcescale pfp_value_normalization_inverse_sourcescale. ((((exists ff_h_pfp_normalization_inverse_sourcescalesource. ff_h_pfp_normalization_inverse_sourcescalesource + S (pfp_source_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_inverse_sourcescalesource. ab = ff_q_pfp_normalization_inverse_sourcescalesource * S ((S (pfp_index_normalization_inverse_sourcescale)) * ac) + (pfp_source_normalization_inverse_sourcescale))) /\ (((((exists ff_h_pfp_normalization_inverse_sourcescaletarget. ff_h_pfp_normalization_inverse_sourcescaletarget + S (pfp_value_normalization_inverse_sourcescale) = S ((S (pfp_index_normalization_inverse_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_inverse_sourcescaletarget. bb = ff_q_pfp_normalization_inverse_sourcescaletarget * S ((S (pfp_index_normalization_inverse_sourcescale)) * bc) + (pfp_value_normalization_inverse_sourcescale))) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationleft. pfa_gap_normalization_inverse_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_inverse_sourcescaleoperationright. pfa_gap_normalization_inverse_sourcescaleoperationright + S (pfp_source_normalization_inverse_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_inverse_sourcescaleoperationresultbound. pfa_gap_normalization_inverse_sourcescaleoperationresultbound + S (pfp_value_normalization_inverse_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_inverse_sourcescale)) + (p) * pfa_offset_left_normalization_inverse_sourcescaleoperationresultcongruence = (pfp_value_normalization_inverse_sourcescale) + (p) * pfa_offset_right_normalization_inverse_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (((exists ff_h_pfp_normalization_inverse_entry. ff_h_pfp_normalization_inverse_entry + S (a) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_inverse_entry. ab = ff_q_pfp_normalization_inverse_entry * S ((S (0)) * ac) + (a))) -> (((~((a) = 0)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationleft. pfa_gap_normalization_inverse_resultmultiplicationleft + S (a) = (p)) /\ (((exists pfa_gap_normalization_inverse_resultmultiplicationright. pfa_gap_normalization_inverse_resultmultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_inverse_resultmultiplicationresultbound. pfa_gap_normalization_inverse_resultmultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence. ((a) * (k)) + (p) * pfa_offset_left_normalization_inverse_resultmultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_inverse_resultmultiplicationresultcongruence))))))))))))

Constructive proof overview

Generated structural guide

The recorded scalar is an actual inverse of every decoding of the source leading coefficient.

The unchanged tactic script uses 1 declared prerequisite and contains 27 exact native proof lines.

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

Proof neighborhood

Direct dependencies

beta_at_unique Stable theorem; checked-use authorized

Direct dependents

none

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

27 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.

01Fix variables and assumptionsL1–10

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 a
  9. L9
    intro h
  10. L10
    intro ha
02Separate the logical casesL11–14

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

  1. L11
    cases h
  2. L12
    cases h_right
  3. L13
    cases h_right_left
  4. L14
    cases h_right_left_witness
03Establish heL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L15
    have he : a=x
  2. L16
    specialize beta_at_unique (ab)
  3. L17
    specialize beta_at_unique (ac)
  4. L18
    specialize beta_at_unique (0)
  5. L19
    specialize beta_at_unique (a)
  6. L20
    specialize beta_at_unique (x)
  7. L21
    apply beta_at_unique
  8. L22
    exact ha
  9. L23
    exact h_right_left_witness_left
  10. L24
    rewrite he
04Calculate and transport equalitiesL25–26

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L25
    rewrite he
  2. L26
    rewrite he
05Use earlier factsL27–27

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

  1. L27
    exact h_right_left_witness_right

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro L
  8. 0008intro a
  9. 0009intro h
  10. 0010intro ha
  11. 0011cases h
  12. 0012cases h_right
  13. 0013cases h_right_left
  14. 0014cases h_right_left_witness
  15. 0015have he : a=x
  16. 0016specialize beta_at_unique (ab)
  17. 0017specialize beta_at_unique (ac)
  18. 0018specialize beta_at_unique (0)
  19. 0019specialize beta_at_unique (a)
  20. 0020specialize beta_at_unique (x)
  21. 0021apply beta_at_unique
  22. 0022exact ha
  23. 0023exact h_right_left_witness_left
  24. 0024rewrite he
  25. 0025rewrite he
  26. 0026rewrite he
  27. 0027exact h_right_left_witness_right