FP0052

prime_field_unit_multiple_residue

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

Alpha v34 checked-use · first admitted v31 · 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.

This checkpoint constructs prime-order fields (k=1), with genuine finite arithmetic tables, cardinality, and characteristic. Inversion is proved only for nonzero elements; the table's zero entry is a zero-to-zero convention. G091 for every prime power p^k, with an irreducible polynomial of degree k, remains open. No extension-field construction or G091 closure is claimed.

Exact theorem in conservative defined notation

∀ p. ∀ n. ∀ r. Prime(p) → FpUnitMultiple(p,n,r) → CanonicalModularResidue(p,n,r)

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

Definition DAG

CanonicalModularResidue(m,a,r) · 1FpUnitMultiple(p,n,r) · 1Prime(p) · 1

Actual proof prerequisites

prime_field_unit_trace_residue
Original expanded first-order 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))))

Complete tactic proof in conservative notation

All 15 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

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.

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 (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 defined 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