PQ0044

prime_field_polynomial_monic_normalization_degree_zero_exists

Construct the normalized constant-one prefix from every actual nonzero represented constant over a prime field.

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. ∀ ab. ∀ ac. Prime(p) → FpRepresentedDegree(p,ab,ac,1,0) → ∃ x. ∃ y. ∃ z. FpMonicNormalization(p,x,ab,ac,y,z,1) ∧ Repeat(y,z,1,1)

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

Definition DAG

FpRepresentedDegree(p,b,c,L,d) · 1FpMonicNormalization(p,k,ab,ac,bb,bc,L) · 2Prime(p) · 1Repeat(b,c,a,l) · 1

Actual proof prerequisites

prime_field_polynomial_monic_normalization_existsprime_field_polynomial_monic_normalization_constant
Original expanded first-order statement 
forall p ab ac. (~((p) = 1) /\ forall pfa_factor_left_normalization_zero_prime pfa_factor_right_normalization_zero_prime. (p) = pfa_factor_left_normalization_zero_prime * pfa_factor_right_normalization_zero_prime -> pfa_factor_left_normalization_zero_prime = 1 \/ pfa_factor_right_normalization_zero_prime = 1) -> ((((1)=S (0)) /\ (((forall fom_index_pfp_normalization_zero_inputcoefficients. (exists fom_gap_pfp_normalization_zero_inputcoefficients_index_bound. fom_gap_pfp_normalization_zero_inputcoefficients_index_bound + S (fom_index_pfp_normalization_zero_inputcoefficients) = 1) -> exists fom_value_pfp_normalization_zero_inputcoefficients. ((((exists fom_beta_height_pfp_normalization_zero_inputcoefficients_entry. fom_beta_height_pfp_normalization_zero_inputcoefficients_entry + S (fom_value_pfp_normalization_zero_inputcoefficients) = S ((S (fom_index_pfp_normalization_zero_inputcoefficients)) * ac)) /\ exists fom_beta_quotient_pfp_normalization_zero_inputcoefficients_entry. ab = fom_beta_quotient_pfp_normalization_zero_inputcoefficients_entry * S ((S (fom_index_pfp_normalization_zero_inputcoefficients)) * ac) + (fom_value_pfp_normalization_zero_inputcoefficients))) /\ (exists fom_gap_pfp_normalization_zero_inputcoefficients_value_bound. fom_gap_pfp_normalization_zero_inputcoefficients_value_bound + S (fom_value_pfp_normalization_zero_inputcoefficients) = p))) /\ ((exists pfd_leading_normalization_zero_input. ((((exists ff_h_pfp_normalization_zero_inputentry. ff_h_pfp_normalization_zero_inputentry + S (pfd_leading_normalization_zero_input) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_zero_inputentry. ab = ff_q_pfp_normalization_zero_inputentry * S ((S (0)) * ac) + (pfd_leading_normalization_zero_input))) /\ ((~(pfd_leading_normalization_zero_input=0)))))))))) -> exists k bb bc. ((((~((1) = 0)) /\ (((exists pfm_leading_normalization_zero_graph. ((((exists ff_h_pfp_normalization_zero_graphsource. ff_h_pfp_normalization_zero_graphsource + S (pfm_leading_normalization_zero_graph) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_zero_graphsource. ab = ff_q_pfp_normalization_zero_graphsource * S ((S (0)) * ac) + (pfm_leading_normalization_zero_graph))) /\ ((((~((pfm_leading_normalization_zero_graph) = 0)) /\ ((((exists pfa_gap_normalization_zero_graphinversemultiplicationleft. pfa_gap_normalization_zero_graphinversemultiplicationleft + S (pfm_leading_normalization_zero_graph) = (p)) /\ (((exists pfa_gap_normalization_zero_graphinversemultiplicationright. pfa_gap_normalization_zero_graphinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_zero_graphinversemultiplicationresultbound. pfa_gap_normalization_zero_graphinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_zero_graphinversemultiplicationresultcongruence pfa_offset_right_normalization_zero_graphinversemultiplicationresultcongruence. ((pfm_leading_normalization_zero_graph) * (k)) + (p) * pfa_offset_left_normalization_zero_graphinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_zero_graphinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_zero_graphscalescalar. pfa_gap_normalization_zero_graphscalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_zero_graphscale. (exists pfa_gap_normalization_zero_graphscaleindex. pfa_gap_normalization_zero_graphscaleindex + S (pfp_index_normalization_zero_graphscale) = (1)) -> exists pfp_source_normalization_zero_graphscale pfp_value_normalization_zero_graphscale. ((((exists ff_h_pfp_normalization_zero_graphscalesource. ff_h_pfp_normalization_zero_graphscalesource + S (pfp_source_normalization_zero_graphscale) = S ((S (pfp_index_normalization_zero_graphscale)) * ac)) /\ exists ff_q_pfp_normalization_zero_graphscalesource. ab = ff_q_pfp_normalization_zero_graphscalesource * S ((S (pfp_index_normalization_zero_graphscale)) * ac) + (pfp_source_normalization_zero_graphscale))) /\ (((((exists ff_h_pfp_normalization_zero_graphscaletarget. ff_h_pfp_normalization_zero_graphscaletarget + S (pfp_value_normalization_zero_graphscale) = S ((S (pfp_index_normalization_zero_graphscale)) * bc)) /\ exists ff_q_pfp_normalization_zero_graphscaletarget. bb = ff_q_pfp_normalization_zero_graphscaletarget * S ((S (pfp_index_normalization_zero_graphscale)) * bc) + (pfp_value_normalization_zero_graphscale))) /\ ((((exists pfa_gap_normalization_zero_graphscaleoperationleft. pfa_gap_normalization_zero_graphscaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_zero_graphscaleoperationright. pfa_gap_normalization_zero_graphscaleoperationright + S (pfp_source_normalization_zero_graphscale) = (p)) /\ ((((exists pfa_gap_normalization_zero_graphscaleoperationresultbound. pfa_gap_normalization_zero_graphscaleoperationresultbound + S (pfp_value_normalization_zero_graphscale) = (p)) /\ ((exists pfa_offset_left_normalization_zero_graphscaleoperationresultcongruence pfa_offset_right_normalization_zero_graphscaleoperationresultcongruence. ((k) * (pfp_source_normalization_zero_graphscale)) + (p) * pfa_offset_left_normalization_zero_graphscaleoperationresultcongruence = (pfp_value_normalization_zero_graphscale) + (p) * pfa_offset_right_normalization_zero_graphscaleoperationresultcongruence)))))))))))))))))))))) /\ ((forall pfp_repeat_index_normalization_zero_output. (exists pfa_gap_normalization_zero_outputindex. pfa_gap_normalization_zero_outputindex + S (pfp_repeat_index_normalization_zero_output) = (1)) -> (((exists ff_h_pfp_normalization_zero_outputentry. ff_h_pfp_normalization_zero_outputentry + S (1) = S ((S (pfp_repeat_index_normalization_zero_output)) * bc)) /\ exists ff_q_pfp_normalization_zero_outputentry. bb = ff_q_pfp_normalization_zero_outputentry * S ((S (pfp_repeat_index_normalization_zero_output)) * bc) + (1))))))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 6 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.

Named ingredients (2)
prime field polynomial monic normalization existsprime field polynomial monic normalization constant
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro ab
  3. L3
    intro ac
  4. L4
    intro hp
  5. L5
    intro hd
02Establish hL6–14

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field polynomial monic normalization exists.

  1. L6
    have h : ∃ k. ∃ bb. ∃ bc. FpMonicNormalization(p,k,ab,ac,bb,bc,1)Definitions: FpMonicNormalization(p,k,ab,ac,bb,bc,1)Original native command in the exact edition
  2. L7
    specialize prime_field_polynomial_monic_normalization_exists (p)
  3. L8
    specialize prime_field_polynomial_monic_normalization_exists (ab)
  4. L9
    specialize prime_field_polynomial_monic_normalization_exists (ac)
  5. L10
    specialize prime_field_polynomial_monic_normalization_exists (1)
  6. L11
    specialize prime_field_polynomial_monic_normalization_exists (0)
  7. L12
    apply prime_field_polynomial_monic_normalization_exists
  8. L13
    exact hp
  9. L14
    exact hd
03Separate the logical casesL15–17

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

  1. L15
    cases h
  2. L16
    cases h_witness
  3. L17
    cases h_witness_witness
04Construct an explicit witnessL18–20

Supply the displayed value, then prove that it has the required property.

  1. L18
    exists x
  2. L19
    exists x1
  3. L20
    exists x2
05Separate the logical casesL21–21

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

  1. L21
    split
06Use earlier factsL22–30

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

  1. L22
    exact h_witness_witness_witness
  2. L23
    specialize prime_field_polynomial_monic_normalization_constant (p)
  3. L24
    specialize prime_field_polynomial_monic_normalization_constant (x)
  4. L25
    specialize prime_field_polynomial_monic_normalization_constant (ab)
  5. L26
    specialize prime_field_polynomial_monic_normalization_constant (ac)
  6. L27
    specialize prime_field_polynomial_monic_normalization_constant (x1)
  7. L28
    specialize prime_field_polynomial_monic_normalization_constant (x2)
  8. L29
    apply prime_field_polynomial_monic_normalization_constant
  9. L30
    exact h_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro hp
  5. 0005intro hd
  6. 0006have h : ∃ k. ∃ bb. ∃ bc. FpMonicNormalization(p,k,ab,ac,bb,bc,1)
  7. 0007specialize prime_field_polynomial_monic_normalization_exists (p)
  8. 0008specialize prime_field_polynomial_monic_normalization_exists (ab)
  9. 0009specialize prime_field_polynomial_monic_normalization_exists (ac)
  10. 0010specialize prime_field_polynomial_monic_normalization_exists (1)
  11. 0011specialize prime_field_polynomial_monic_normalization_exists (0)
  12. 0012apply prime_field_polynomial_monic_normalization_exists
  13. 0013exact hp
  14. 0014exact hd
  15. 0015cases h
  16. 0016cases h_witness
  17. 0017cases h_witness_witness
  18. 0018exists x
  19. 0019exists x1
  20. 0020exists x2
  21. 0021split
  22. 0022exact h_witness_witness_witness
  23. 0023specialize prime_field_polynomial_monic_normalization_constant (p)
  24. 0024specialize prime_field_polynomial_monic_normalization_constant (x)
  25. 0025specialize prime_field_polynomial_monic_normalization_constant (ab)
  26. 0026specialize prime_field_polynomial_monic_normalization_constant (ac)
  27. 0027specialize prime_field_polynomial_monic_normalization_constant (x1)
  28. 0028specialize prime_field_polynomial_monic_normalization_constant (x2)
  29. 0029apply prime_field_polynomial_monic_normalization_constant
  30. 0030exact h_witness_witness_witness