PQ0044

prime_field_polynomial_monic_normalization_degree_zero_exists

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

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

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 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))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 30 exact native proof lines.

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

Proof neighborhood

Direct dependencies

PQ003C prime_field_polynomial_monic_normalization_exists PQ0042 prime_field_polynomial_monic_normalization_constant

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

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.

Named ingredients (2)
prime field polynomial monic normalization existsprime field polynomial monic normalization constant

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 assumptionsL1–5

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
  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 exact command ledger · 30 lines
  1. 0001intro p
  2. 0002intro ab
  3. 0003intro ac
  4. 0004intro hp
  5. 0005intro hd
  6. 0006have h : exists k bb bc. (((~((1) = 0)) /\ (((exists pfm_leading_normalization_zero_choice. ((((exists ff_h_pfp_normalization_zero_choicesource. ff_h_pfp_normalization_zero_choicesource + S (pfm_leading_normalization_zero_choice) = S ((S (0)) * ac)) /\ exists ff_q_pfp_normalization_zero_choicesource. ab = ff_q_pfp_normalization_zero_choicesource * S ((S (0)) * ac) + (pfm_leading_normalization_zero_choice))) /\ ((((~((pfm_leading_normalization_zero_choice) = 0)) /\ ((((exists pfa_gap_normalization_zero_choiceinversemultiplicationleft. pfa_gap_normalization_zero_choiceinversemultiplicationleft + S (pfm_leading_normalization_zero_choice) = (p)) /\ (((exists pfa_gap_normalization_zero_choiceinversemultiplicationright. pfa_gap_normalization_zero_choiceinversemultiplicationright + S (k) = (p)) /\ ((((exists pfa_gap_normalization_zero_choiceinversemultiplicationresultbound. pfa_gap_normalization_zero_choiceinversemultiplicationresultbound + S (1) = (p)) /\ ((exists pfa_offset_left_normalization_zero_choiceinversemultiplicationresultcongruence pfa_offset_right_normalization_zero_choiceinversemultiplicationresultcongruence. ((pfm_leading_normalization_zero_choice) * (k)) + (p) * pfa_offset_left_normalization_zero_choiceinversemultiplicationresultcongruence = (1) + (p) * pfa_offset_right_normalization_zero_choiceinversemultiplicationresultcongruence))))))))))))))) /\ ((((exists pfa_gap_normalization_zero_choicescalescalar. pfa_gap_normalization_zero_choicescalescalar + S (k) = (p)) /\ ((forall pfp_index_normalization_zero_choicescale. (exists pfa_gap_normalization_zero_choicescaleindex. pfa_gap_normalization_zero_choicescaleindex + S (pfp_index_normalization_zero_choicescale) = (1)) -> exists pfp_source_normalization_zero_choicescale pfp_value_normalization_zero_choicescale. ((((exists ff_h_pfp_normalization_zero_choicescalesource. ff_h_pfp_normalization_zero_choicescalesource + S (pfp_source_normalization_zero_choicescale) = S ((S (pfp_index_normalization_zero_choicescale)) * ac)) /\ exists ff_q_pfp_normalization_zero_choicescalesource. ab = ff_q_pfp_normalization_zero_choicescalesource * S ((S (pfp_index_normalization_zero_choicescale)) * ac) + (pfp_source_normalization_zero_choicescale))) /\ (((((exists ff_h_pfp_normalization_zero_choicescaletarget. ff_h_pfp_normalization_zero_choicescaletarget + S (pfp_value_normalization_zero_choicescale) = S ((S (pfp_index_normalization_zero_choicescale)) * bc)) /\ exists ff_q_pfp_normalization_zero_choicescaletarget. bb = ff_q_pfp_normalization_zero_choicescaletarget * S ((S (pfp_index_normalization_zero_choicescale)) * bc) + (pfp_value_normalization_zero_choicescale))) /\ ((((exists pfa_gap_normalization_zero_choicescaleoperationleft. pfa_gap_normalization_zero_choicescaleoperationleft + S (k) = (p)) /\ (((exists pfa_gap_normalization_zero_choicescaleoperationright. pfa_gap_normalization_zero_choicescaleoperationright + S (pfp_source_normalization_zero_choicescale) = (p)) /\ ((((exists pfa_gap_normalization_zero_choicescaleoperationresultbound. pfa_gap_normalization_zero_choicescaleoperationresultbound + S (pfp_value_normalization_zero_choicescale) = (p)) /\ ((exists pfa_offset_left_normalization_zero_choicescaleoperationresultcongruence pfa_offset_right_normalization_zero_choicescaleoperationresultcongruence. ((k) * (pfp_source_normalization_zero_choicescale)) + (p) * pfa_offset_left_normalization_zero_choicescaleoperationresultcongruence = (pfp_value_normalization_zero_choicescale) + (p) * pfa_offset_right_normalization_zero_choicescaleoperationresultcongruence))))))))))))))))))))))
  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