SS0018

divisor_signed_table_lookup_from_components

Actual component streams construct a canonical lookup at any specified index; finite consumers retain their explicit index bounds.

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.

These are actual signed-table and finite-sum foundations. Equality compares represented signed values, not arbitrary encodings. MatrixMinorFourCode is reused solely as generic nested pairing, without a matrix hypothesis. Full finite signed G007 is established separately in the Möbius-inversion family.

Exact theorem in conservative defined notation

∀ F. ∀ pb. ∀ pc. ∀ nb. ∀ nc. ∀ i. MatrixMinorFourCode(F,pb,pc,nb,nc) → ∃ x. ArithAt(F,i,x)

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

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 1ArithAt(F,i,z) · 1

Actual proof prerequisites

divisor_signed_table_from_componentsdivisor_signed_table_lookuple_refl · checked external prerequisite
Original expanded first-order statement 
forall F pb pc nb nc i. ((F) = (((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) * S ((((pb) + (pc)) * S ((pb) + (pc)) + ((pc) + (pc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))) + ((((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc))) + (((nb) + (nc)) * S ((nb) + (nc)) + ((nc) + (nc)))))) -> exists z. (exists dst_positive_code_lookup_result dst_positive_scale_lookup_result dst_negative_code_lookup_result dst_negative_scale_lookup_result dst_positive_lookup_result dst_negative_lookup_result. (((F) = (((((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) * S ((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) + ((dst_positive_scale_lookup_result) + (dst_positive_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))) * S ((((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) * S ((dst_positive_code_lookup_result) + (dst_positive_scale_lookup_result)) + ((dst_positive_scale_lookup_result) + (dst_positive_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))) + ((((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result))) + (((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) * S ((dst_negative_code_lookup_result) + (dst_negative_scale_lookup_result)) + ((dst_negative_scale_lookup_result) + (dst_negative_scale_lookup_result)))))) /\ (((((exists ff_h_pvs_lookup_resultpositive. ff_h_pvs_lookup_resultpositive + S (dst_positive_lookup_result) = S ((S (i)) * dst_positive_scale_lookup_result)) /\ exists ff_q_pvs_lookup_resultpositive. dst_positive_code_lookup_result = ff_q_pvs_lookup_resultpositive * S ((S (i)) * dst_positive_scale_lookup_result) + (dst_positive_lookup_result))) /\ (((((exists ff_h_pvs_lookup_resultnegative. ff_h_pvs_lookup_resultnegative + S (dst_negative_lookup_result) = S ((S (i)) * dst_negative_scale_lookup_result)) /\ exists ff_q_pvs_lookup_resultnegative. dst_negative_code_lookup_result = ff_q_pvs_lookup_resultnegative * S ((S (i)) * dst_negative_scale_lookup_result) + (dst_negative_lookup_result))) /\ (exists ge_balance_positive_lookup_resultvalue ge_balance_negative_lookup_resultvalue. (((((z) = 2 * (ge_balance_positive_lookup_resultvalue) /\ (ge_balance_negative_lookup_resultvalue) = 0) \/ exists ge_signed_half_lookup_resultvaluedecode. (((z) = 2 * ge_signed_half_lookup_resultvaluedecode + 1 /\ (ge_balance_positive_lookup_resultvalue) = 0) /\ (ge_balance_negative_lookup_resultvalue) = S ge_signed_half_lookup_resultvaluedecode))) /\ ((dst_positive_lookup_result) + ge_balance_negative_lookup_resultvalue = (dst_negative_lookup_result) + ge_balance_positive_lookup_resultvalue)))))))))

Complete tactic proof in conservative notation

All 21 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

21 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 (2)
divisor signed table lookupdivisor signed table from components
01Fix variables and assumptionsL1–7

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

  1. L1
    intro F
  2. L2
    intro pb
  3. L3
    intro pc
  4. L4
    intro nb
  5. L5
    intro nc
  6. L6
    intro i
  7. L7
    intro hrep
02Use earlier factsL8–17

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

  1. L8
    specialize divisor_signed_table_lookup (i)
  2. L9
    specialize divisor_signed_table_lookup (F)
  3. L10
    specialize divisor_signed_table_lookup (i)
  4. L11
    apply divisor_signed_table_lookup
  5. L12
    specialize divisor_signed_table_from_components (i)
  6. L13
    specialize divisor_signed_table_from_components (F)
  7. L14
    specialize divisor_signed_table_from_components (pb)
  8. L15
    specialize divisor_signed_table_from_components (pc)
  9. L16
    specialize divisor_signed_table_from_components (nb)
  10. L17
    specialize divisor_signed_table_from_components (nc)
03Use earlier factsL18–21

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

  1. L18
    apply divisor_signed_table_from_components
  2. L19
    exact hrep
  3. L20
    specialize le_refl (i)
  4. L21
    apply le_refl

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro F
  2. 0002intro pb
  3. 0003intro pc
  4. 0004intro nb
  5. 0005intro nc
  6. 0006intro i
  7. 0007intro hrep
  8. 0008specialize divisor_signed_table_lookup (i)
  9. 0009specialize divisor_signed_table_lookup (F)
  10. 0010specialize divisor_signed_table_lookup (i)
  11. 0011apply divisor_signed_table_lookup
  12. 0012specialize divisor_signed_table_from_components (i)
  13. 0013specialize divisor_signed_table_from_components (F)
  14. 0014specialize divisor_signed_table_from_components (pb)
  15. 0015specialize divisor_signed_table_from_components (pc)
  16. 0016specialize divisor_signed_table_from_components (nb)
  17. 0017specialize divisor_signed_table_from_components (nc)
  18. 0018apply divisor_signed_table_from_components
  19. 0019exact hrep
  20. 0020specialize le_refl (i)
  21. 0021apply le_refl