WS0002

signed_table_lookup_any

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Every actual table supplies an actual signed value at any requested index, rather than a totality oracle.

Exact expanded first-order arithmetic statement

forall N F i. (exists dst_positive_code_any_lookup_table dst_positive_scale_any_lookup_table dst_negative_code_any_lookup_table dst_negative_scale_any_lookup_table. (((F) = (((((dst_positive_code_any_lookup_table) + (dst_positive_scale_any_lookup_table)) * S ((dst_positive_code_any_lookup_table) + (dst_positive_scale_any_lookup_table)) + ((dst_positive_scale_any_lookup_table) + (dst_positive_scale_any_lookup_table))) + (((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) * S ((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) + ((dst_negative_scale_any_lookup_table) + (dst_negative_scale_any_lookup_table)))) * S ((((dst_positive_code_any_lookup_table) + (dst_positive_scale_any_lookup_table)) * S ((dst_positive_code_any_lookup_table) + (dst_positive_scale_any_lookup_table)) + ((dst_positive_scale_any_lookup_table) + (dst_positive_scale_any_lookup_table))) + (((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) * S ((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) + ((dst_negative_scale_any_lookup_table) + (dst_negative_scale_any_lookup_table)))) + ((((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) * S ((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) + ((dst_negative_scale_any_lookup_table) + (dst_negative_scale_any_lookup_table))) + (((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) * S ((dst_negative_code_any_lookup_table) + (dst_negative_scale_any_lookup_table)) + ((dst_negative_scale_any_lookup_table) + (dst_negative_scale_any_lookup_table)))))) /\ (forall dst_index_any_lookup_table. (exists pvs_le_gap_any_lookup_tabledomain. pvs_le_gap_any_lookup_tabledomain + (dst_index_any_lookup_table) = (N)) -> exists dst_positive_any_lookup_table dst_negative_any_lookup_table dst_value_any_lookup_table. ((((exists ff_h_pvs_any_lookup_tableentrypositive. ff_h_pvs_any_lookup_tableentrypositive + S (dst_positive_any_lookup_table) = S ((S (dst_index_any_lookup_table)) * dst_positive_scale_any_lookup_table)) /\ exists ff_q_pvs_any_lookup_tableentrypositive. dst_positive_code_any_lookup_table = ff_q_pvs_any_lookup_tableentrypositive * S ((S (dst_index_any_lookup_table)) * dst_positive_scale_any_lookup_table) + (dst_positive_any_lookup_table))) /\ (((((exists ff_h_pvs_any_lookup_tableentrynegative. ff_h_pvs_any_lookup_tableentrynegative + S (dst_negative_any_lookup_table) = S ((S (dst_index_any_lookup_table)) * dst_negative_scale_any_lookup_table)) /\ exists ff_q_pvs_any_lookup_tableentrynegative. dst_negative_code_any_lookup_table = ff_q_pvs_any_lookup_tableentrynegative * S ((S (dst_index_any_lookup_table)) * dst_negative_scale_any_lookup_table) + (dst_negative_any_lookup_table))) /\ (exists ge_balance_positive_any_lookup_tableentryvalue ge_balance_negative_any_lookup_tableentryvalue. (((((dst_value_any_lookup_table) = 2 * (ge_balance_positive_any_lookup_tableentryvalue) /\ (ge_balance_negative_any_lookup_tableentryvalue) = 0) \/ exists ge_signed_half_any_lookup_tableentryvaluedecode. (((dst_value_any_lookup_table) = 2 * ge_signed_half_any_lookup_tableentryvaluedecode + 1 /\ (ge_balance_positive_any_lookup_tableentryvalue) = 0) /\ (ge_balance_negative_any_lookup_tableentryvalue) = S ge_signed_half_any_lookup_tableentryvaluedecode))) /\ ((dst_positive_any_lookup_table) + ge_balance_negative_any_lookup_tableentryvalue = (dst_negative_any_lookup_table) + ge_balance_positive_any_lookup_tableentryvalue))))))))) -> exists z. (exists dst_positive_code_any_lookup_result dst_positive_scale_any_lookup_result dst_negative_code_any_lookup_result dst_negative_scale_any_lookup_result dst_positive_any_lookup_result dst_negative_any_lookup_result. (((F) = (((((dst_positive_code_any_lookup_result) + (dst_positive_scale_any_lookup_result)) * S ((dst_positive_code_any_lookup_result) + (dst_positive_scale_any_lookup_result)) + ((dst_positive_scale_any_lookup_result) + (dst_positive_scale_any_lookup_result))) + (((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) * S ((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) + ((dst_negative_scale_any_lookup_result) + (dst_negative_scale_any_lookup_result)))) * S ((((dst_positive_code_any_lookup_result) + (dst_positive_scale_any_lookup_result)) * S ((dst_positive_code_any_lookup_result) + (dst_positive_scale_any_lookup_result)) + ((dst_positive_scale_any_lookup_result) + (dst_positive_scale_any_lookup_result))) + (((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) * S ((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) + ((dst_negative_scale_any_lookup_result) + (dst_negative_scale_any_lookup_result)))) + ((((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) * S ((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) + ((dst_negative_scale_any_lookup_result) + (dst_negative_scale_any_lookup_result))) + (((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) * S ((dst_negative_code_any_lookup_result) + (dst_negative_scale_any_lookup_result)) + ((dst_negative_scale_any_lookup_result) + (dst_negative_scale_any_lookup_result)))))) /\ (((((exists ff_h_pvs_any_lookup_resultpositive. ff_h_pvs_any_lookup_resultpositive + S (dst_positive_any_lookup_result) = S ((S (i)) * dst_positive_scale_any_lookup_result)) /\ exists ff_q_pvs_any_lookup_resultpositive. dst_positive_code_any_lookup_result = ff_q_pvs_any_lookup_resultpositive * S ((S (i)) * dst_positive_scale_any_lookup_result) + (dst_positive_any_lookup_result))) /\ (((((exists ff_h_pvs_any_lookup_resultnegative. ff_h_pvs_any_lookup_resultnegative + S (dst_negative_any_lookup_result) = S ((S (i)) * dst_negative_scale_any_lookup_result)) /\ exists ff_q_pvs_any_lookup_resultnegative. dst_negative_code_any_lookup_result = ff_q_pvs_any_lookup_resultnegative * S ((S (i)) * dst_negative_scale_any_lookup_result) + (dst_negative_any_lookup_result))) /\ (exists ge_balance_positive_any_lookup_resultvalue ge_balance_negative_any_lookup_resultvalue. (((((z) = 2 * (ge_balance_positive_any_lookup_resultvalue) /\ (ge_balance_negative_any_lookup_resultvalue) = 0) \/ exists ge_signed_half_any_lookup_resultvaluedecode. (((z) = 2 * ge_signed_half_any_lookup_resultvaluedecode + 1 /\ (ge_balance_positive_any_lookup_resultvalue) = 0) /\ (ge_balance_negative_any_lookup_resultvalue) = S ge_signed_half_any_lookup_resultvaluedecode))) /\ ((dst_positive_any_lookup_result) + ge_balance_negative_any_lookup_resultvalue = (dst_negative_any_lookup_result) + ge_balance_positive_any_lookup_resultvalue)))))))))

Constructive proof overview

Generated structural guide

Every actual table supplies an actual signed value at any requested index, rather than a totality oracle.

The unchanged tactic script uses 3 declared prerequisites and contains 15 exact native proof lines.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Proof neighborhood

Direct dependencies

WS0001 signed_table_domain_resize divisor_signed_table_lookup external actual_inherited_body_freshly_checked_in_complete_bundle; no checked-use authority le_refl Alpha theorem; checked-use authorized

Direct dependents

WS0006 signed_table_add_extensional_unique WS000A signed_table_multiply_extensional_unique WS000E signed_table_scalar_extensional_unique WS0010 signed_table_add_exists WS0013 signed_table_multiply_exists WS0016 signed_table_scalar_exists WS0024 signed_table_weighted_add_distributive WS0026 signed_table_weighted_scalar_commute

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or 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)
signed table domain resize
01Fix variables and assumptionsL1–4

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

  1. L1
    intro N
  2. L2
    intro F
  3. L3
    intro i
  4. L4
    intro ht
02Use earlier factsL5–14

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

  1. L5
    specialize divisor_signed_table_lookup (i)
  2. L6
    specialize divisor_signed_table_lookup (F)
  3. L7
    specialize divisor_signed_table_lookup (i)
  4. L8
    apply divisor_signed_table_lookup
  5. L9
    specialize signed_table_domain_resize (N)
  6. L10
    specialize signed_table_domain_resize (i)
  7. L11
    specialize signed_table_domain_resize (F)
  8. L12
    apply signed_table_domain_resize
  9. L13
    exact ht
  10. L14
    specialize le_refl (i)
03Use earlier factsL15–15

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

  1. L15
    apply le_refl

Library-wide reading audit

Original exact command ledger · 15 lines
  1. 0001intro N
  2. 0002intro F
  3. 0003intro i
  4. 0004intro ht
  5. 0005specialize divisor_signed_table_lookup (i)
  6. 0006specialize divisor_signed_table_lookup (F)
  7. 0007specialize divisor_signed_table_lookup (i)
  8. 0008apply divisor_signed_table_lookup
  9. 0009specialize signed_table_domain_resize (N)
  10. 0010specialize signed_table_domain_resize (i)
  11. 0011specialize signed_table_domain_resize (F)
  12. 0012apply signed_table_domain_resize
  13. 0013exact ht
  14. 0014specialize le_refl (i)
  15. 0015apply le_refl