FP0050

prime_field_unit_trace_result_bounded

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

Every trace endpoint is a canonical representative, including the empty-history endpoint.

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 n r. (~((p) = 1) /\ forall pfa_factor_left_trace_bounded_domain pfa_factor_right_trace_bounded_domain. (p) = pfa_factor_left_trace_bounded_domain * pfa_factor_right_trace_bounded_domain -> pfa_factor_left_trace_bounded_domain = 1 \/ pfa_factor_right_trace_bounded_domain = 1) -> (((((exists ff_h_pft_trace_bounded_sourcestart. ff_h_pft_trace_bounded_sourcestart + S (0) = S ((S (0)) * c)) /\ exists ff_q_pft_trace_bounded_sourcestart. b = ff_q_pft_trace_bounded_sourcestart * S ((S (0)) * c) + (0))) /\ (((((exists ff_h_pft_trace_bounded_sourceterminal. ff_h_pft_trace_bounded_sourceterminal + S (r) = S ((S (n)) * c)) /\ exists ff_q_pft_trace_bounded_sourceterminal. b = ff_q_pft_trace_bounded_sourceterminal * S ((S (n)) * c) + (r))) /\ ((forall pff_trace_index_trace_bounded_sourcesteps. (exists pfa_gap_trace_bounded_sourcestepsindex. pfa_gap_trace_bounded_sourcestepsindex + S (pff_trace_index_trace_bounded_sourcesteps) = (n)) -> exists pff_trace_before_trace_bounded_sourcesteps pff_trace_after_trace_bounded_sourcesteps. ((((exists ff_h_pft_trace_bounded_sourcestepsbefore. ff_h_pft_trace_bounded_sourcestepsbefore + S (pff_trace_before_trace_bounded_sourcesteps) = S ((S (pff_trace_index_trace_bounded_sourcesteps)) * c)) /\ exists ff_q_pft_trace_bounded_sourcestepsbefore. b = ff_q_pft_trace_bounded_sourcestepsbefore * S ((S (pff_trace_index_trace_bounded_sourcesteps)) * c) + (pff_trace_before_trace_bounded_sourcesteps))) /\ (((((exists ff_h_pft_trace_bounded_sourcestepsafter. ff_h_pft_trace_bounded_sourcestepsafter + S (pff_trace_after_trace_bounded_sourcesteps) = S ((S (S (pff_trace_index_trace_bounded_sourcesteps))) * c)) /\ exists ff_q_pft_trace_bounded_sourcestepsafter. b = ff_q_pft_trace_bounded_sourcestepsafter * S ((S (S (pff_trace_index_trace_bounded_sourcesteps))) * c) + (pff_trace_after_trace_bounded_sourcesteps))) /\ ((((exists pfa_gap_trace_bounded_sourcestepsadditionleft. pfa_gap_trace_bounded_sourcestepsadditionleft + S (pff_trace_before_trace_bounded_sourcesteps) = (p)) /\ (((exists pfa_gap_trace_bounded_sourcestepsadditionright. pfa_gap_trace_bounded_sourcestepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_trace_bounded_sourcestepsadditionresultbound. pfa_gap_trace_bounded_sourcestepsadditionresultbound + S (pff_trace_after_trace_bounded_sourcesteps) = (p)) /\ ((exists pfa_offset_left_trace_bounded_sourcestepsadditionresultcongruence pfa_offset_right_trace_bounded_sourcestepsadditionresultcongruence. ((pff_trace_before_trace_bounded_sourcesteps) + (1)) + (p) * pfa_offset_left_trace_bounded_sourcestepsadditionresultcongruence = (pff_trace_after_trace_bounded_sourcesteps) + (p) * pfa_offset_right_trace_bounded_sourcestepsadditionresultcongruence))))))))))))))))))) -> (exists pfa_gap_trace_bounded_result. pfa_gap_trace_bounded_result + S (r) = (p))

Constructive proof overview

Generated structural guide

Every trace endpoint is a canonical representative, including the empty-history endpoint.

The unchanged tactic script uses 1 declared prerequisite and contains 18 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

FP004F prime_field_unit_trace_residue

Direct dependents

FP0051 prime_field_unit_trace_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

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.

Named ingredients (1)
prime field unit trace residue
01Fix variables and assumptionsL1–7

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 n
  5. L5
    intro r
  6. L6
    intro hp
  7. L7
    intro ht
02Establish hrL8–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime field unit trace residue.

  1. L8
    have hr : ((exists pfa_gap_trace_bounded_residuebound. pfa_gap_trace_bounded_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_trace_bounded_residuecongruence pfa_offset_right_trace_bounded_residuecongruence. (n) + (p) * pfa_offset_left_trace_bounded_residuecongruence = (r) + (p) * pfa_offset_right_trace_bounded_residuecongruence)))
  2. L9
    specialize prime_field_unit_trace_residue (p)
  3. L10
    specialize prime_field_unit_trace_residue (n)
  4. L11
    specialize prime_field_unit_trace_residue (b)
  5. L12
    specialize prime_field_unit_trace_residue (c)
  6. L13
    specialize prime_field_unit_trace_residue (r)
  7. L14
    apply prime_field_unit_trace_residue
  8. L15
    exact hp
  9. L16
    exact ht
03Separate the logical casesL17–17

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

  1. L17
    cases hr
04Use earlier factsL18–18

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

  1. L18
    exact hr_left

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro p
  2. 0002intro b
  3. 0003intro c
  4. 0004intro n
  5. 0005intro r
  6. 0006intro hp
  7. 0007intro ht
  8. 0008have hr : ((exists pfa_gap_trace_bounded_residuebound. pfa_gap_trace_bounded_residuebound + S (r) = (p)) /\ ((exists pfa_offset_left_trace_bounded_residuecongruence pfa_offset_right_trace_bounded_residuecongruence. (n) + (p) * pfa_offset_left_trace_bounded_residuecongruence = (r) + (p) * pfa_offset_right_trace_bounded_residuecongruence)))
  9. 0009specialize prime_field_unit_trace_residue (p)
  10. 0010specialize prime_field_unit_trace_residue (n)
  11. 0011specialize prime_field_unit_trace_residue (b)
  12. 0012specialize prime_field_unit_trace_residue (c)
  13. 0013specialize prime_field_unit_trace_residue (r)
  14. 0014apply prime_field_unit_trace_residue
  15. 0015exact hp
  16. 0016exact ht
  17. 0017cases hr
  18. 0018exact hr_left