SS0008

divisor_signed_table_restrict

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

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

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 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)))))))))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 22 exact native proof lines.

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

Proof neighborhood

Direct dependencies

SS0005 divisor_signed_table_components SS0003 divisor_signed_table_from_components

Direct dependents

none

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

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.

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 : 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))))))
  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 exact command ledger · 22 lines
  1. 0001intro N
  2. 0002intro K
  3. 0003intro F
  4. 0004intro ht
  5. 0005intro hbound
  6. 0006have hrep : 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))))))
  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