SS0018

divisor_signed_table_lookup_from_components

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

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

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 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)))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 3 declared prerequisites and contains 21 exact native proof lines.

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

Proof neighborhood

Direct dependencies

SS0003 divisor_signed_table_from_components SS0006 divisor_signed_table_lookup le_refl Stable theorem; checked-use authorized

Direct dependents

SS001C divisor_signed_table_reindex_functional

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

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.

Named ingredients (2)
divisor signed table from componentsdivisor signed table lookup
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 exact 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