SS0008

divisor_signed_table_restrict

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.

The same packed table remains valid on every shorter finite domain, without a new encoding or changed entries.

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

These are genuine signed-table and finite-sum foundations, not full divisor-sum cancellation or Möbius inversion. G007 remains open. The historical MatrixMinorFourCode definition is reused solely as generic nested pairing of four beta parameters; no matrix-specific hypothesis is imported. Equality is equality of represented signed values, not equality of arbitrary component codes.

Exact theorem in conservative defined notation

∀ N. ∀ K. ∀ F. ArithTable(N,F) → Le(K,N) → ArithTable(K,F)

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

Definition DAG

MatrixMinorFourCode(z,up,us,un,ut) · 1ArithTable(N,F) · 2Le(a,b) · 1

Actual proof prerequisites

divisor_signed_table_componentsdivisor_signed_table_from_components
Original expanded first-order statement 
forall N K F. (exists dst_positive_code_restriction_source dst_positive_scale_restriction_source dst_negative_code_restriction_source dst_negative_scale_restriction_source. (((F) = (((((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) * S ((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) + ((dst_positive_scale_restriction_source) + (dst_positive_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))) * S ((((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) * S ((dst_positive_code_restriction_source) + (dst_positive_scale_restriction_source)) + ((dst_positive_scale_restriction_source) + (dst_positive_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))) + ((((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source))) + (((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) * S ((dst_negative_code_restriction_source) + (dst_negative_scale_restriction_source)) + ((dst_negative_scale_restriction_source) + (dst_negative_scale_restriction_source)))))) /\ (forall dst_index_restriction_source. (exists pvs_le_gap_restriction_sourcedomain. pvs_le_gap_restriction_sourcedomain + (dst_index_restriction_source) = (N)) -> exists dst_positive_restriction_source dst_negative_restriction_source dst_value_restriction_source. ((((exists ff_h_pvs_restriction_sourceentrypositive. ff_h_pvs_restriction_sourceentrypositive + S (dst_positive_restriction_source) = S ((S (dst_index_restriction_source)) * dst_positive_scale_restriction_source)) /\ exists ff_q_pvs_restriction_sourceentrypositive. dst_positive_code_restriction_source = ff_q_pvs_restriction_sourceentrypositive * S ((S (dst_index_restriction_source)) * dst_positive_scale_restriction_source) + (dst_positive_restriction_source))) /\ (((((exists ff_h_pvs_restriction_sourceentrynegative. ff_h_pvs_restriction_sourceentrynegative + S (dst_negative_restriction_source) = S ((S (dst_index_restriction_source)) * dst_negative_scale_restriction_source)) /\ exists ff_q_pvs_restriction_sourceentrynegative. dst_negative_code_restriction_source = ff_q_pvs_restriction_sourceentrynegative * S ((S (dst_index_restriction_source)) * dst_negative_scale_restriction_source) + (dst_negative_restriction_source))) /\ (exists ge_balance_positive_restriction_sourceentryvalue ge_balance_negative_restriction_sourceentryvalue. (((((dst_value_restriction_source) = 2 * (ge_balance_positive_restriction_sourceentryvalue) /\ (ge_balance_negative_restriction_sourceentryvalue) = 0) \/ exists ge_signed_half_restriction_sourceentryvaluedecode. (((dst_value_restriction_source) = 2 * ge_signed_half_restriction_sourceentryvaluedecode + 1 /\ (ge_balance_positive_restriction_sourceentryvalue) = 0) /\ (ge_balance_negative_restriction_sourceentryvalue) = S ge_signed_half_restriction_sourceentryvaluedecode))) /\ ((dst_positive_restriction_source) + ge_balance_negative_restriction_sourceentryvalue = (dst_negative_restriction_source) + ge_balance_positive_restriction_sourceentryvalue))))))))) -> (exists pvs_le_gap_restriction_bound. pvs_le_gap_restriction_bound + (K) = (N)) -> (exists dst_positive_code_restriction_target dst_positive_scale_restriction_target dst_negative_code_restriction_target dst_negative_scale_restriction_target. (((F) = (((((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) * S ((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) + ((dst_positive_scale_restriction_target) + (dst_positive_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))) * S ((((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) * S ((dst_positive_code_restriction_target) + (dst_positive_scale_restriction_target)) + ((dst_positive_scale_restriction_target) + (dst_positive_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))) + ((((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target))) + (((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) * S ((dst_negative_code_restriction_target) + (dst_negative_scale_restriction_target)) + ((dst_negative_scale_restriction_target) + (dst_negative_scale_restriction_target)))))) /\ (forall dst_index_restriction_target. (exists pvs_le_gap_restriction_targetdomain. pvs_le_gap_restriction_targetdomain + (dst_index_restriction_target) = (K)) -> exists dst_positive_restriction_target dst_negative_restriction_target dst_value_restriction_target. ((((exists ff_h_pvs_restriction_targetentrypositive. ff_h_pvs_restriction_targetentrypositive + S (dst_positive_restriction_target) = S ((S (dst_index_restriction_target)) * dst_positive_scale_restriction_target)) /\ exists ff_q_pvs_restriction_targetentrypositive. dst_positive_code_restriction_target = ff_q_pvs_restriction_targetentrypositive * S ((S (dst_index_restriction_target)) * dst_positive_scale_restriction_target) + (dst_positive_restriction_target))) /\ (((((exists ff_h_pvs_restriction_targetentrynegative. ff_h_pvs_restriction_targetentrynegative + S (dst_negative_restriction_target) = S ((S (dst_index_restriction_target)) * dst_negative_scale_restriction_target)) /\ exists ff_q_pvs_restriction_targetentrynegative. dst_negative_code_restriction_target = ff_q_pvs_restriction_targetentrynegative * S ((S (dst_index_restriction_target)) * dst_negative_scale_restriction_target) + (dst_negative_restriction_target))) /\ (exists ge_balance_positive_restriction_targetentryvalue ge_balance_negative_restriction_targetentryvalue. (((((dst_value_restriction_target) = 2 * (ge_balance_positive_restriction_targetentryvalue) /\ (ge_balance_negative_restriction_targetentryvalue) = 0) \/ exists ge_signed_half_restriction_targetentryvaluedecode. (((dst_value_restriction_target) = 2 * ge_signed_half_restriction_targetentryvaluedecode + 1 /\ (ge_balance_positive_restriction_targetentryvalue) = 0) /\ (ge_balance_negative_restriction_targetentryvalue) = S ge_signed_half_restriction_targetentryvaluedecode))) /\ ((dst_positive_restriction_target) + ge_balance_negative_restriction_targetentryvalue = (dst_negative_restriction_target) + ge_balance_positive_restriction_targetentryvalue)))))))))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 4 reading checkpoints · 1 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 componentsdivisor signed table from components
01Fix variables and assumptionsL1–5

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

  1. L1
    intro N
  2. L2
    intro K
  3. L3
    intro F
  4. L4
    intro ht
  5. L5
    intro hbound
02Establish hrepL6–10

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply divisor signed table components.

  1. L6
    have hrep : ∃ pb. ∃ pc. ∃ nb. ∃ nc. MatrixMinorFourCode(F,pb,pc,nb,nc)Definitions: MatrixMinorFourCode(F,pb,pc,nb,nc)Original native command in the exact edition
  2. L7
    specialize divisor_signed_table_components (N)
  3. L8
    specialize divisor_signed_table_components (F)
  4. L9
    apply divisor_signed_table_components
  5. L10
    exact ht
03Separate the logical casesL11–14

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

  1. L11
    cases hrep
  2. L12
    cases hrep_witness
  3. L13
    cases hrep_witness_witness
  4. L14
    cases hrep_witness_witness_witness
04Use earlier factsL15–22

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

  1. L15
    specialize divisor_signed_table_from_components (K)
  2. L16
    specialize divisor_signed_table_from_components (F)
  3. L17
    specialize divisor_signed_table_from_components (x)
  4. L18
    specialize divisor_signed_table_from_components (x1)
  5. L19
    specialize divisor_signed_table_from_components (x2)
  6. L20
    specialize divisor_signed_table_from_components (x3)
  7. L21
    apply divisor_signed_table_from_components
  8. L22
    exact hrep_witness_witness_witness_witness

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro N
  2. 0002intro K
  3. 0003intro F
  4. 0004intro ht
  5. 0005intro hbound
  6. 0006have hrep : ∃ pb. ∃ pc. ∃ nb. ∃ nc. MatrixMinorFourCode(F,pb,pc,nb,nc)
  7. 0007specialize divisor_signed_table_components (N)
  8. 0008specialize divisor_signed_table_components (F)
  9. 0009apply divisor_signed_table_components
  10. 0010exact ht
  11. 0011cases hrep
  12. 0012cases hrep_witness
  13. 0013cases hrep_witness_witness
  14. 0014cases hrep_witness_witness_witness
  15. 0015specialize divisor_signed_table_from_components (K)
  16. 0016specialize divisor_signed_table_from_components (F)
  17. 0017specialize divisor_signed_table_from_components (x)
  18. 0018specialize divisor_signed_table_from_components (x1)
  19. 0019specialize divisor_signed_table_from_components (x2)
  20. 0020specialize divisor_signed_table_from_components (x3)
  21. 0021apply divisor_signed_table_from_components
  22. 0022exact hrep_witness_witness_witness_witness