FP0052

prime_field_unit_multiple_residue

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

An actual repeated sum of n ones has canonical residue n.

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 n r. (~((p) = 1) /\ forall pfa_factor_left_multiple_residue_domain pfa_factor_right_multiple_residue_domain. (p) = pfa_factor_left_multiple_residue_domain * pfa_factor_right_multiple_residue_domain -> pfa_factor_left_multiple_residue_domain = 1 \/ pfa_factor_right_multiple_residue_domain = 1) -> (exists pff_history_code_multiple_residue_source pff_history_scale_multiple_residue_source. (((((exists ff_h_pft_multiple_residue_sourcehistorystart. ff_h_pft_multiple_residue_sourcehistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystart. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystart * S ((S (0)) * pff_history_scale_multiple_residue_source) + (0))) /\ (((((exists ff_h_pft_multiple_residue_sourcehistoryterminal. ff_h_pft_multiple_residue_sourcehistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistoryterminal. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistoryterminal * S ((S (n)) * pff_history_scale_multiple_residue_source) + (r))) /\ ((forall pff_trace_index_multiple_residue_sourcehistorysteps. (exists pfa_gap_multiple_residue_sourcehistorystepsindex. pfa_gap_multiple_residue_sourcehistorystepsindex + S (pff_trace_index_multiple_residue_sourcehistorysteps) = (n)) -> exists pff_trace_before_multiple_residue_sourcehistorysteps pff_trace_after_multiple_residue_sourcehistorysteps. ((((exists ff_h_pft_multiple_residue_sourcehistorystepsbefore. ff_h_pft_multiple_residue_sourcehistorystepsbefore + S (pff_trace_before_multiple_residue_sourcehistorysteps) = S ((S (pff_trace_index_multiple_residue_sourcehistorysteps)) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystepsbefore. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystepsbefore * S ((S (pff_trace_index_multiple_residue_sourcehistorysteps)) * pff_history_scale_multiple_residue_source) + (pff_trace_before_multiple_residue_sourcehistorysteps))) /\ (((((exists ff_h_pft_multiple_residue_sourcehistorystepsafter. ff_h_pft_multiple_residue_sourcehistorystepsafter + S (pff_trace_after_multiple_residue_sourcehistorysteps) = S ((S (S (pff_trace_index_multiple_residue_sourcehistorysteps))) * pff_history_scale_multiple_residue_source)) /\ exists ff_q_pft_multiple_residue_sourcehistorystepsafter. pff_history_code_multiple_residue_source = ff_q_pft_multiple_residue_sourcehistorystepsafter * S ((S (S (pff_trace_index_multiple_residue_sourcehistorysteps))) * pff_history_scale_multiple_residue_source) + (pff_trace_after_multiple_residue_sourcehistorysteps))) /\ ((((exists pfa_gap_multiple_residue_sourcehistorystepsadditionleft. pfa_gap_multiple_residue_sourcehistorystepsadditionleft + S (pff_trace_before_multiple_residue_sourcehistorysteps) = (p)) /\ (((exists pfa_gap_multiple_residue_sourcehistorystepsadditionright. pfa_gap_multiple_residue_sourcehistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_residue_sourcehistorystepsadditionresultbound. pfa_gap_multiple_residue_sourcehistorystepsadditionresultbound + S (pff_trace_after_multiple_residue_sourcehistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_residue_sourcehistorystepsadditionresultcongruence pfa_offset_right_multiple_residue_sourcehistorystepsadditionresultcongruence. ((pff_trace_before_multiple_residue_sourcehistorysteps) + (1)) + (p) * pfa_offset_left_multiple_residue_sourcehistorystepsadditionresultcongruence = (pff_trace_after_multiple_residue_sourcehistorysteps) + (p) * pfa_offset_right_multiple_residue_sourcehistorystepsadditionresultcongruence)))))))))))))))))))) -> (((exists pfa_gap_multiple_residue_resultbound. pfa_gap_multiple_residue_resultbound + S (r) = (p)) /\ ((exists pfa_offset_left_multiple_residue_resultcongruence pfa_offset_right_multiple_residue_resultcongruence. (n) + (p) * pfa_offset_left_multiple_residue_resultcongruence = (r) + (p) * pfa_offset_right_multiple_residue_resultcongruence))))

Constructive proof overview

Generated structural guide

An actual repeated sum of n ones has canonical residue n.

The unchanged tactic script uses 1 declared prerequisite and contains 15 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

FP0054 prime_field_unit_multiple_functional FP0056 prime_field_characteristic_exact

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

15 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.

Named ingredients (1)
prime field unit trace residue
01Fix variables and assumptionsL1–5

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

  1. L1
    intro p
  2. L2
    intro n
  3. L3
    intro r
  4. L4
    intro hp
  5. L5
    intro hm
02Separate the logical casesL6–7

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

  1. L6
    cases hm
  2. L7
    cases hm_witness
03Use earlier factsL8–15

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

  1. L8
    specialize prime_field_unit_trace_residue (p)
  2. L9
    specialize prime_field_unit_trace_residue (n)
  3. L10
    specialize prime_field_unit_trace_residue (x)
  4. L11
    specialize prime_field_unit_trace_residue (x1)
  5. L12
    specialize prime_field_unit_trace_residue (r)
  6. L13
    apply prime_field_unit_trace_residue
  7. L14
    exact hp
  8. L15
    exact hm_witness_witness

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro r
  4. 0004intro hp
  5. 0005intro hm
  6. 0006cases hm
  7. 0007cases hm_witness
  8. 0008specialize prime_field_unit_trace_residue (p)
  9. 0009specialize prime_field_unit_trace_residue (n)
  10. 0010specialize prime_field_unit_trace_residue (x)
  11. 0011specialize prime_field_unit_trace_residue (x1)
  12. 0012specialize prime_field_unit_trace_residue (r)
  13. 0013apply prime_field_unit_trace_residue
  14. 0014exact hp
  15. 0015exact hm_witness_witness