WS0002

signed_table_lookup_any

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

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.

Operation tables contain actual beta-coded entries and compare represented signed values, not encodings. The strict sum window is i<l and the separately certified endpoint i=l is unused. Rectangular Fubini and full finite signed Möbius inversion are separate, now-admitted families.

Exact theorem in conservative defined notation

∀ N. ∀ F. ∀ i. ArithTable(N,F) → ∃ x. ArithAt(F,i,x)

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

Definition DAG

ArithTable(N,F) · 1ArithAt(F,i,z) · 1

Actual proof prerequisites

signed_table_domain_resizedivisor_signed_table_lookup · checked external prerequisitele_refl · checked external prerequisite
Original expanded first-order 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)))))))))

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)
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 defined 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