PQ0034

prime_field_polynomial_monic_constant

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

The entire represented degree-zero monic prefix is the constant one, not an empty prefix.

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 b c. (((~((1) = 0)) /\ (((forall fom_index_pfp_monic_constant_sourcecoefficients. (exists fom_gap_pfp_monic_constant_sourcecoefficients_index_bound. fom_gap_pfp_monic_constant_sourcecoefficients_index_bound + S (fom_index_pfp_monic_constant_sourcecoefficients) = 1) -> exists fom_value_pfp_monic_constant_sourcecoefficients. ((((exists fom_beta_height_pfp_monic_constant_sourcecoefficients_entry. fom_beta_height_pfp_monic_constant_sourcecoefficients_entry + S (fom_value_pfp_monic_constant_sourcecoefficients) = S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c)) /\ exists fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry. b = fom_beta_quotient_pfp_monic_constant_sourcecoefficients_entry * S ((S (fom_index_pfp_monic_constant_sourcecoefficients)) * c) + (fom_value_pfp_monic_constant_sourcecoefficients))) /\ (exists fom_gap_pfp_monic_constant_sourcecoefficients_value_bound. fom_gap_pfp_monic_constant_sourcecoefficients_value_bound + S (fom_value_pfp_monic_constant_sourcecoefficients) = p))) /\ ((((exists ff_h_pfp_monic_constant_sourceleading. ff_h_pfp_monic_constant_sourceleading + S (1) = S ((S (0)) * c)) /\ exists ff_q_pfp_monic_constant_sourceleading. b = ff_q_pfp_monic_constant_sourceleading * S ((S (0)) * c) + (1)))))))) -> (forall pfp_repeat_index_monic_constant_value. (exists pfa_gap_monic_constant_valueindex. pfa_gap_monic_constant_valueindex + S (pfp_repeat_index_monic_constant_value) = (1)) -> (((exists ff_h_pfp_monic_constant_valueentry. ff_h_pfp_monic_constant_valueentry + S (1) = S ((S (pfp_repeat_index_monic_constant_value)) * c)) /\ exists ff_q_pfp_monic_constant_valueentry. b = ff_q_pfp_monic_constant_valueentry * S ((S (pfp_repeat_index_monic_constant_value)) * c) + (1))))

Constructive proof overview

Generated structural guide

The entire represented degree-zero monic prefix is the constant one, not an empty prefix.

The unchanged tactic script uses 2 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

le_zero Stable theorem; checked-use authorized le_of_succ_le_succ Stable theorem; checked-use authorized

Direct dependents

PQ0042 prime_field_polynomial_monic_normalization_constant

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

18 script commands · 4 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 assumptionsL1–4

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

  1. L1
    intro p
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro h
02Separate the logical casesL5–6

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

  1. L5
    cases h
  2. L6
    cases h_right
03Fix variables and assumptionsL7–8

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

  1. L7
    intro i
  2. L8
    intro hi
04Establish hi0L9–18

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le zero.

  1. L9
    have hi0 : i=0
  2. L10
    specialize le_zero (i)
  3. L11
    apply le_zero
  4. L12
    specialize le_of_succ_le_succ (i)
  5. L13
    specialize le_of_succ_le_succ (0)
  6. L14
    apply le_of_succ_le_succ
  7. L15
    exact hi
  8. L16
    rewrite hi0
  9. L17
    rewrite hi0
  10. L18
    exact h_right_right

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro h
  5. 0005cases h
  6. 0006cases h_right
  7. 0007intro i
  8. 0008intro hi
  9. 0009have hi0 : i=0
  10. 0010specialize le_zero (i)
  11. 0011apply le_zero
  12. 0012specialize le_of_succ_le_succ (i)
  13. 0013specialize le_of_succ_le_succ (0)
  14. 0014apply le_of_succ_le_succ
  15. 0015exact hi
  16. 0016rewrite hi0
  17. 0017rewrite hi0
  18. 0018exact h_right_right