PQ0042

prime_field_polynomial_monic_normalization_constant

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

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. FpMonicNormalization(p,k,ab,ac,bb,bc,1) → Repeat(bb,bc,1,1)

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

Definition DAG

FpMonicNormalization(p,k,ab,ac,bb,bc,L) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_polynomial_monic_constantprime_field_polynomial_monic_normalization_monic
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 20 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

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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