SS0005

divisor_signed_table_components

A valid finite signed table always supplies its actual nested natural-pair packing.

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

∀ N. ∀ F. ArithTable(N,F) → ∃ x. ∃ y. ∃ z. ∃ n. MatrixMinorFourCode(F,x,y,z,n)

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

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 1ArithTable(N,F) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall N F. (exists dst_positive_code_extract_table dst_positive_scale_extract_table dst_negative_code_extract_table dst_negative_scale_extract_table. (((F) = (((((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) * S ((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) + ((dst_positive_scale_extract_table) + (dst_positive_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))) * S ((((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) * S ((dst_positive_code_extract_table) + (dst_positive_scale_extract_table)) + ((dst_positive_scale_extract_table) + (dst_positive_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))) + ((((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table))) + (((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) * S ((dst_negative_code_extract_table) + (dst_negative_scale_extract_table)) + ((dst_negative_scale_extract_table) + (dst_negative_scale_extract_table)))))) /\ (forall dst_index_extract_table. (exists pvs_le_gap_extract_tabledomain. pvs_le_gap_extract_tabledomain + (dst_index_extract_table) = (N)) -> exists dst_positive_extract_table dst_negative_extract_table dst_value_extract_table. ((((exists ff_h_pvs_extract_tableentrypositive. ff_h_pvs_extract_tableentrypositive + S (dst_positive_extract_table) = S ((S (dst_index_extract_table)) * dst_positive_scale_extract_table)) /\ exists ff_q_pvs_extract_tableentrypositive. dst_positive_code_extract_table = ff_q_pvs_extract_tableentrypositive * S ((S (dst_index_extract_table)) * dst_positive_scale_extract_table) + (dst_positive_extract_table))) /\ (((((exists ff_h_pvs_extract_tableentrynegative. ff_h_pvs_extract_tableentrynegative + S (dst_negative_extract_table) = S ((S (dst_index_extract_table)) * dst_negative_scale_extract_table)) /\ exists ff_q_pvs_extract_tableentrynegative. dst_negative_code_extract_table = ff_q_pvs_extract_tableentrynegative * S ((S (dst_index_extract_table)) * dst_negative_scale_extract_table) + (dst_negative_extract_table))) /\ (exists ge_balance_positive_extract_tableentryvalue ge_balance_negative_extract_tableentryvalue. (((((dst_value_extract_table) = 2 * (ge_balance_positive_extract_tableentryvalue) /\ (ge_balance_negative_extract_tableentryvalue) = 0) \/ exists ge_signed_half_extract_tableentryvaluedecode. (((dst_value_extract_table) = 2 * ge_signed_half_extract_tableentryvaluedecode + 1 /\ (ge_balance_positive_extract_tableentryvalue) = 0) /\ (ge_balance_negative_extract_tableentryvalue) = S ge_signed_half_extract_tableentryvaluedecode))) /\ ((dst_positive_extract_table) + ge_balance_negative_extract_tableentryvalue = (dst_negative_extract_table) + ge_balance_positive_extract_tableentryvalue))))))))) -> exists pb pc nb nc. ((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))))))

Complete tactic proof in conservative notation

All 13 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

13 script commands · 4 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.

01Fix variables and assumptionsL1–3

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

  1. L1
    intro N
  2. L2
    intro F
  3. L3
    intro h
02Separate the logical casesL4–8

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L4
    cases h
  2. L5
    cases h_witness
  3. L6
    cases h_witness_witness
  4. L7
    cases h_witness_witness_witness
  5. L8
    cases h_witness_witness_witness_witness
03Construct an explicit witnessL9–12

Supply the displayed value, then prove that it has the required property.

  1. L9
    exists x
  2. L10
    exists x1
  3. L11
    exists x2
  4. L12
    exists x3
04Use earlier factsL13–13

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

  1. L13
    exact h_witness_witness_witness_witness_left

Library-wide reading audit

Original defined command ledger · 13 lines
  1. 0001intro N
  2. 0002intro F
  3. 0003intro h
  4. 0004cases h
  5. 0005cases h_witness
  6. 0006cases h_witness_witness
  7. 0007cases h_witness_witness_witness
  8. 0008cases h_witness_witness_witness_witness
  9. 0009exists x
  10. 0010exists x1
  11. 0011exists x2
  12. 0012exists x3
  13. 0013exact h_witness_witness_witness_witness_left