FP0054

prime_field_unit_multiple_functional

Any two actual histories of the same number of additions of one have identical endpoints.

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. ∀ s. Prime(p) → FpUnitMultiple(p,n,r) → FpUnitMultiple(p,n,s) → r = s

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

Definition DAG

FpUnitMultiple(p,n,r) · 2Prime(p) · 1

Actual proof prerequisites

prime_field_unit_multiple_residuebinary_canonical_residue_functional · checked external prerequisite
Original expanded first-order statement 
forall p n r s. (~((p) = 1) /\ forall pfa_factor_left_multiple_unique_domain pfa_factor_right_multiple_unique_domain. (p) = pfa_factor_left_multiple_unique_domain * pfa_factor_right_multiple_unique_domain -> pfa_factor_left_multiple_unique_domain = 1 \/ pfa_factor_right_multiple_unique_domain = 1) -> (exists pff_history_code_multiple_unique_first pff_history_scale_multiple_unique_first. (((((exists ff_h_pft_multiple_unique_firsthistorystart. ff_h_pft_multiple_unique_firsthistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystart. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystart * S ((S (0)) * pff_history_scale_multiple_unique_first) + (0))) /\ (((((exists ff_h_pft_multiple_unique_firsthistoryterminal. ff_h_pft_multiple_unique_firsthistoryterminal + S (r) = S ((S (n)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistoryterminal. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistoryterminal * S ((S (n)) * pff_history_scale_multiple_unique_first) + (r))) /\ ((forall pff_trace_index_multiple_unique_firsthistorysteps. (exists pfa_gap_multiple_unique_firsthistorystepsindex. pfa_gap_multiple_unique_firsthistorystepsindex + S (pff_trace_index_multiple_unique_firsthistorysteps) = (n)) -> exists pff_trace_before_multiple_unique_firsthistorysteps pff_trace_after_multiple_unique_firsthistorysteps. ((((exists ff_h_pft_multiple_unique_firsthistorystepsbefore. ff_h_pft_multiple_unique_firsthistorystepsbefore + S (pff_trace_before_multiple_unique_firsthistorysteps) = S ((S (pff_trace_index_multiple_unique_firsthistorysteps)) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystepsbefore. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystepsbefore * S ((S (pff_trace_index_multiple_unique_firsthistorysteps)) * pff_history_scale_multiple_unique_first) + (pff_trace_before_multiple_unique_firsthistorysteps))) /\ (((((exists ff_h_pft_multiple_unique_firsthistorystepsafter. ff_h_pft_multiple_unique_firsthistorystepsafter + S (pff_trace_after_multiple_unique_firsthistorysteps) = S ((S (S (pff_trace_index_multiple_unique_firsthistorysteps))) * pff_history_scale_multiple_unique_first)) /\ exists ff_q_pft_multiple_unique_firsthistorystepsafter. pff_history_code_multiple_unique_first = ff_q_pft_multiple_unique_firsthistorystepsafter * S ((S (S (pff_trace_index_multiple_unique_firsthistorysteps))) * pff_history_scale_multiple_unique_first) + (pff_trace_after_multiple_unique_firsthistorysteps))) /\ ((((exists pfa_gap_multiple_unique_firsthistorystepsadditionleft. pfa_gap_multiple_unique_firsthistorystepsadditionleft + S (pff_trace_before_multiple_unique_firsthistorysteps) = (p)) /\ (((exists pfa_gap_multiple_unique_firsthistorystepsadditionright. pfa_gap_multiple_unique_firsthistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_unique_firsthistorystepsadditionresultbound. pfa_gap_multiple_unique_firsthistorystepsadditionresultbound + S (pff_trace_after_multiple_unique_firsthistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_unique_firsthistorystepsadditionresultcongruence pfa_offset_right_multiple_unique_firsthistorystepsadditionresultcongruence. ((pff_trace_before_multiple_unique_firsthistorysteps) + (1)) + (p) * pfa_offset_left_multiple_unique_firsthistorystepsadditionresultcongruence = (pff_trace_after_multiple_unique_firsthistorysteps) + (p) * pfa_offset_right_multiple_unique_firsthistorystepsadditionresultcongruence)))))))))))))))))))) -> (exists pff_history_code_multiple_unique_second pff_history_scale_multiple_unique_second. (((((exists ff_h_pft_multiple_unique_secondhistorystart. ff_h_pft_multiple_unique_secondhistorystart + S (0) = S ((S (0)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystart. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystart * S ((S (0)) * pff_history_scale_multiple_unique_second) + (0))) /\ (((((exists ff_h_pft_multiple_unique_secondhistoryterminal. ff_h_pft_multiple_unique_secondhistoryterminal + S (s) = S ((S (n)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistoryterminal. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistoryterminal * S ((S (n)) * pff_history_scale_multiple_unique_second) + (s))) /\ ((forall pff_trace_index_multiple_unique_secondhistorysteps. (exists pfa_gap_multiple_unique_secondhistorystepsindex. pfa_gap_multiple_unique_secondhistorystepsindex + S (pff_trace_index_multiple_unique_secondhistorysteps) = (n)) -> exists pff_trace_before_multiple_unique_secondhistorysteps pff_trace_after_multiple_unique_secondhistorysteps. ((((exists ff_h_pft_multiple_unique_secondhistorystepsbefore. ff_h_pft_multiple_unique_secondhistorystepsbefore + S (pff_trace_before_multiple_unique_secondhistorysteps) = S ((S (pff_trace_index_multiple_unique_secondhistorysteps)) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystepsbefore. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystepsbefore * S ((S (pff_trace_index_multiple_unique_secondhistorysteps)) * pff_history_scale_multiple_unique_second) + (pff_trace_before_multiple_unique_secondhistorysteps))) /\ (((((exists ff_h_pft_multiple_unique_secondhistorystepsafter. ff_h_pft_multiple_unique_secondhistorystepsafter + S (pff_trace_after_multiple_unique_secondhistorysteps) = S ((S (S (pff_trace_index_multiple_unique_secondhistorysteps))) * pff_history_scale_multiple_unique_second)) /\ exists ff_q_pft_multiple_unique_secondhistorystepsafter. pff_history_code_multiple_unique_second = ff_q_pft_multiple_unique_secondhistorystepsafter * S ((S (S (pff_trace_index_multiple_unique_secondhistorysteps))) * pff_history_scale_multiple_unique_second) + (pff_trace_after_multiple_unique_secondhistorysteps))) /\ ((((exists pfa_gap_multiple_unique_secondhistorystepsadditionleft. pfa_gap_multiple_unique_secondhistorystepsadditionleft + S (pff_trace_before_multiple_unique_secondhistorysteps) = (p)) /\ (((exists pfa_gap_multiple_unique_secondhistorystepsadditionright. pfa_gap_multiple_unique_secondhistorystepsadditionright + S (1) = (p)) /\ ((((exists pfa_gap_multiple_unique_secondhistorystepsadditionresultbound. pfa_gap_multiple_unique_secondhistorystepsadditionresultbound + S (pff_trace_after_multiple_unique_secondhistorysteps) = (p)) /\ ((exists pfa_offset_left_multiple_unique_secondhistorystepsadditionresultcongruence pfa_offset_right_multiple_unique_secondhistorystepsadditionresultcongruence. ((pff_trace_before_multiple_unique_secondhistorysteps) + (1)) + (p) * pfa_offset_left_multiple_unique_secondhistorystepsadditionresultcongruence = (pff_trace_after_multiple_unique_secondhistorysteps) + (p) * pfa_offset_right_multiple_unique_secondhistorystepsadditionresultcongruence)))))))))))))))))))) -> r = s

Complete tactic proof in conservative notation

All 24 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

24 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 multiple residue
01Fix variables and assumptionsL1–7

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 s
  5. L5
    intro hp
  6. L6
    intro hr
  7. L7
    intro hs
02Use earlier factsL8–17

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

  1. L8
    specialize binary_canonical_residue_functional (p)
  2. L9
    specialize binary_canonical_residue_functional (n)
  3. L10
    specialize binary_canonical_residue_functional (r)
  4. L11
    specialize binary_canonical_residue_functional (s)
  5. L12
    apply binary_canonical_residue_functional
  6. L13
    specialize prime_field_unit_multiple_residue (p)
  7. L14
    specialize prime_field_unit_multiple_residue (n)
  8. L15
    specialize prime_field_unit_multiple_residue (r)
  9. L16
    apply prime_field_unit_multiple_residue
  10. L17
    exact hp
03Use earlier factsL18–24

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

  1. L18
    exact hr
  2. L19
    specialize prime_field_unit_multiple_residue (p)
  3. L20
    specialize prime_field_unit_multiple_residue (n)
  4. L21
    specialize prime_field_unit_multiple_residue (s)
  5. L22
    apply prime_field_unit_multiple_residue
  6. L23
    exact hp
  7. L24
    exact hs

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro p
  2. 0002intro n
  3. 0003intro r
  4. 0004intro s
  5. 0005intro hp
  6. 0006intro hr
  7. 0007intro hs
  8. 0008specialize binary_canonical_residue_functional (p)
  9. 0009specialize binary_canonical_residue_functional (n)
  10. 0010specialize binary_canonical_residue_functional (r)
  11. 0011specialize binary_canonical_residue_functional (s)
  12. 0012apply binary_canonical_residue_functional
  13. 0013specialize prime_field_unit_multiple_residue (p)
  14. 0014specialize prime_field_unit_multiple_residue (n)
  15. 0015specialize prime_field_unit_multiple_residue (r)
  16. 0016apply prime_field_unit_multiple_residue
  17. 0017exact hp
  18. 0018exact hr
  19. 0019specialize prime_field_unit_multiple_residue (p)
  20. 0020specialize prime_field_unit_multiple_residue (n)
  21. 0021specialize prime_field_unit_multiple_residue (s)
  22. 0022apply prime_field_unit_multiple_residue
  23. 0023exact hp
  24. 0024exact hs