PQ0042

prime_field_polynomial_monic_normalization_constant

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

Every actual normalization of a nonzero constant representation is the constant one.

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. (((~((1) = 0)) /\ (((exists pfm_leading_normalization_constant_source. ((((exists ff_h_pfp_normalization_constant_sourcesource. ff_h_pfp_normalization_constant_sourcesource + S (pfm_leading_normalization_constant_source) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_constant_sourcesource. ab = ff_q_pfp_normalization_constant_sourcesource * S ((S (0)) * ac) + (pfm_leading_normalization_constant_source))) /\ ((((~((pfm_leading_normalization_constant_source) = 0)) /\ ((((exists pfa_gap_normalization_constant_sourceinversemultiplicationleft. pfa_gap_normalization_constant_sourceinversemultiplicationleft + S (pfm_leading_normalization_constant_source) = (p)) /\ (((exists pfa_gap_normalization_constant_sourceinversemultiplicationright. pfa_gap_normalization_constant_sourceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_constant_sourceinversemultiplicationresultbound. pfa_gap_normalization_constant_sourceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_constant_sourceinversemultiplicationresultcongruence pfa_offset_right_normalization_constant_sourceinversemultiplicationresultcongruence. ((pfm_leading_normalization_constant_source) * (k)) + (p) * pfa_offset_left_normalization_constant_sourceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_constant_sourceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_constant_sourcescalescalar. pfa_gap_normalization_constant_sourcescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_constant_sourcescale. (exists pfa_gap_normalization_constant_sourcescaleindex. pfa_gap_normalization_constant_sourcescaleindex + S (pfp_index_normalization_constant_sourcescale) = (1)) -> exists pfp_source_normalization_constant_sourcescale pfp_value_normalization_constant_sourcescale. ((((exists ff_h_pfp_normalization_constant_sourcescalesource. ff_h_pfp_normalization_constant_sourcescalesource + S (pfp_source_normalization_constant_sourcescale) = S ((S (pfp_index_normalization_constant_sourcescale)) * ac)) /\ exists ff_q_pfp_normalization_constant_sourcescalesource. ab = ff_q_pfp_normalization_constant_sourcescalesource * S ((S (pfp_index_normalization_constant_sourcescale)) * ac) + (pfp_source_normalization_constant_sourcescale))) /\ (((((exists ff_h_pfp_normalization_constant_sourcescaletarget. ff_h_pfp_normalization_constant_sourcescaletarget + S (pfp_value_normalization_constant_sourcescale) = S ((S (pfp_index_normalization_constant_sourcescale)) * bc)) /\ exists ff_q_pfp_normalization_constant_sourcescaletarget. bb = ff_q_pfp_normalization_constant_sourcescaletarget * S ((S (pfp_index_normalization_constant_sourcescale)) * bc) + (pfp_value_normalization_constant_sourcescale))) /\ ((((exists pfa_gap_normalization_constant_sourcescaleoperationleft. pfa_gap_normalization_constant_sourcescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_constant_sourcescaleoperationright. pfa_gap_normalization_constant_sourcescaleoperationright + S (pfp_source_normalization_constant_sourcescale) = (p)) /\ ((((exists pfa_gap_normalization_constant_sourcescaleoperationresultbound. pfa_gap_normalization_constant_sourcescaleoperationresultbound + S (pfp_value_normalization_constant_sourcescale) = (p)) /\ ((exists pfa_offset_left_normalization_constant_sourcescaleoperationresultcongruence pfa_offset_right_normalization_constant_sourcescaleoperationresultcongruence. ((k) * (pfp_source_normalization_constant_sourcescale)) + (p) * pfa_offset_left_normalization_constant_sourcescaleoperationresultcongruence = (pfp_value_normalization_constant_sourcescale) + (p) * pfa_offset_right_normalization_constant_sourcescaleoperationresultcongruence)))))))))))))))))))))) -> (forall pfp_repeat_index_normalization_constant_result. (exists pfa_gap_normalization_constant_resultindex. pfa_gap_normalization_constant_resultindex + S (pfp_repeat_index_normalization_constant_result) = (1)) -> (((exists ff_h_pfp_normalization_constant_resultentry. ff_h_pfp_normalization_constant_resultentry + S (1) = S ((S (pfp_repeat_index_normalization_constant_result)) * bc)) /\ exists ff_q_pfp_normalization_constant_resultentry. bb = ff_q_pfp_normalization_constant_resultentry * S ((S (pfp_repeat_index_normalization_constant_result)) * bc) + (1))))

Constructive proof overview

Generated structural guide

Every actual normalization of a nonzero constant representation is the constant one.

The unchanged tactic script uses 2 declared prerequisites and contains 20 exact native proof lines.

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

Proof neighborhood

Direct dependencies

PQ0034 prime_field_polynomial_monic_constant PQ003A prime_field_polynomial_monic_normalization_monic

Direct dependents

PQ0044 prime_field_polynomial_monic_normalization_degree_zero_exists

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

20 script commands · 3 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.

Named ingredients (2)
prime field polynomial monic constantprime field polynomial monic normalization monic
01Fix variables and assumptionsL1–7

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 h
02Use earlier factsL8–17

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

  1. L8
    specialize prime_field_polynomial_monic_constant (p)
  2. L9
    specialize prime_field_polynomial_monic_constant (bb)
  3. L10
    specialize prime_field_polynomial_monic_constant (bc)
  4. L11
    apply prime_field_polynomial_monic_constant
  5. L12
    specialize prime_field_polynomial_monic_normalization_monic (p)
  6. L13
    specialize prime_field_polynomial_monic_normalization_monic (k)
  7. L14
    specialize prime_field_polynomial_monic_normalization_monic (ab)
  8. L15
    specialize prime_field_polynomial_monic_normalization_monic (ac)
  9. L16
    specialize prime_field_polynomial_monic_normalization_monic (bb)
  10. L17
    specialize prime_field_polynomial_monic_normalization_monic (bc)
03Use earlier factsL18–20

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

  1. L18
    specialize prime_field_polynomial_monic_normalization_monic (1)
  2. L19
    apply prime_field_polynomial_monic_normalization_monic
  3. L20
    exact h

Library-wide reading audit

Original exact command ledger · 20 lines
  1. 0001intro p
  2. 0002intro k
  3. 0003intro ab
  4. 0004intro ac
  5. 0005intro bb
  6. 0006intro bc
  7. 0007intro h
  8. 0008specialize prime_field_polynomial_monic_constant (p)
  9. 0009specialize prime_field_polynomial_monic_constant (bb)
  10. 0010specialize prime_field_polynomial_monic_constant (bc)
  11. 0011apply prime_field_polynomial_monic_constant
  12. 0012specialize prime_field_polynomial_monic_normalization_monic (p)
  13. 0013specialize prime_field_polynomial_monic_normalization_monic (k)
  14. 0014specialize prime_field_polynomial_monic_normalization_monic (ab)
  15. 0015specialize prime_field_polynomial_monic_normalization_monic (ac)
  16. 0016specialize prime_field_polynomial_monic_normalization_monic (bb)
  17. 0017specialize prime_field_polynomial_monic_normalization_monic (bc)
  18. 0018specialize prime_field_polynomial_monic_normalization_monic (1)
  19. 0019apply prime_field_polynomial_monic_normalization_monic
  20. 0020exact h