WS0001

signed_table_domain_resize

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

An actual packed beta table has canonical entries on every finite domain; resizing the certificate never changes its streams.

Exact expanded first-order arithmetic statement

forall N M F. (exists dst_positive_code_resize_input dst_positive_scale_resize_input dst_negative_code_resize_input dst_negative_scale_resize_input. (((F) = (((((dst_positive_code_resize_input) + (dst_positive_scale_resize_input)) * S ((dst_positive_code_resize_input) + (dst_positive_scale_resize_input)) + ((dst_positive_scale_resize_input) + (dst_positive_scale_resize_input))) + (((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) * S ((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) + ((dst_negative_scale_resize_input) + (dst_negative_scale_resize_input)))) * S ((((dst_positive_code_resize_input) + (dst_positive_scale_resize_input)) * S ((dst_positive_code_resize_input) + (dst_positive_scale_resize_input)) + ((dst_positive_scale_resize_input) + (dst_positive_scale_resize_input))) + (((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) * S ((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) + ((dst_negative_scale_resize_input) + (dst_negative_scale_resize_input)))) + ((((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) * S ((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) + ((dst_negative_scale_resize_input) + (dst_negative_scale_resize_input))) + (((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) * S ((dst_negative_code_resize_input) + (dst_negative_scale_resize_input)) + ((dst_negative_scale_resize_input) + (dst_negative_scale_resize_input)))))) /\ (forall dst_index_resize_input. (exists pvs_le_gap_resize_inputdomain. pvs_le_gap_resize_inputdomain + (dst_index_resize_input) = (N)) -> exists dst_positive_resize_input dst_negative_resize_input dst_value_resize_input. ((((exists ff_h_pvs_resize_inputentrypositive. ff_h_pvs_resize_inputentrypositive + S (dst_positive_resize_input) = S ((S (dst_index_resize_input)) * dst_positive_scale_resize_input)) /\ exists ff_q_pvs_resize_inputentrypositive. dst_positive_code_resize_input = ff_q_pvs_resize_inputentrypositive * S ((S (dst_index_resize_input)) * dst_positive_scale_resize_input) + (dst_positive_resize_input))) /\ (((((exists ff_h_pvs_resize_inputentrynegative. ff_h_pvs_resize_inputentrynegative + S (dst_negative_resize_input) = S ((S (dst_index_resize_input)) * dst_negative_scale_resize_input)) /\ exists ff_q_pvs_resize_inputentrynegative. dst_negative_code_resize_input = ff_q_pvs_resize_inputentrynegative * S ((S (dst_index_resize_input)) * dst_negative_scale_resize_input) + (dst_negative_resize_input))) /\ (exists ge_balance_positive_resize_inputentryvalue ge_balance_negative_resize_inputentryvalue. (((((dst_value_resize_input) = 2 * (ge_balance_positive_resize_inputentryvalue) /\ (ge_balance_negative_resize_inputentryvalue) = 0) \/ exists ge_signed_half_resize_inputentryvaluedecode. (((dst_value_resize_input) = 2 * ge_signed_half_resize_inputentryvaluedecode + 1 /\ (ge_balance_positive_resize_inputentryvalue) = 0) /\ (ge_balance_negative_resize_inputentryvalue) = S ge_signed_half_resize_inputentryvaluedecode))) /\ ((dst_positive_resize_input) + ge_balance_negative_resize_inputentryvalue = (dst_negative_resize_input) + ge_balance_positive_resize_inputentryvalue))))))))) -> (exists dst_positive_code_resize_output dst_positive_scale_resize_output dst_negative_code_resize_output dst_negative_scale_resize_output. (((F) = (((((dst_positive_code_resize_output) + (dst_positive_scale_resize_output)) * S ((dst_positive_code_resize_output) + (dst_positive_scale_resize_output)) + ((dst_positive_scale_resize_output) + (dst_positive_scale_resize_output))) + (((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) * S ((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) + ((dst_negative_scale_resize_output) + (dst_negative_scale_resize_output)))) * S ((((dst_positive_code_resize_output) + (dst_positive_scale_resize_output)) * S ((dst_positive_code_resize_output) + (dst_positive_scale_resize_output)) + ((dst_positive_scale_resize_output) + (dst_positive_scale_resize_output))) + (((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) * S ((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) + ((dst_negative_scale_resize_output) + (dst_negative_scale_resize_output)))) + ((((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) * S ((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) + ((dst_negative_scale_resize_output) + (dst_negative_scale_resize_output))) + (((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) * S ((dst_negative_code_resize_output) + (dst_negative_scale_resize_output)) + ((dst_negative_scale_resize_output) + (dst_negative_scale_resize_output)))))) /\ (forall dst_index_resize_output. (exists pvs_le_gap_resize_outputdomain. pvs_le_gap_resize_outputdomain + (dst_index_resize_output) = (M)) -> exists dst_positive_resize_output dst_negative_resize_output dst_value_resize_output. ((((exists ff_h_pvs_resize_outputentrypositive. ff_h_pvs_resize_outputentrypositive + S (dst_positive_resize_output) = S ((S (dst_index_resize_output)) * dst_positive_scale_resize_output)) /\ exists ff_q_pvs_resize_outputentrypositive. dst_positive_code_resize_output = ff_q_pvs_resize_outputentrypositive * S ((S (dst_index_resize_output)) * dst_positive_scale_resize_output) + (dst_positive_resize_output))) /\ (((((exists ff_h_pvs_resize_outputentrynegative. ff_h_pvs_resize_outputentrynegative + S (dst_negative_resize_output) = S ((S (dst_index_resize_output)) * dst_negative_scale_resize_output)) /\ exists ff_q_pvs_resize_outputentrynegative. dst_negative_code_resize_output = ff_q_pvs_resize_outputentrynegative * S ((S (dst_index_resize_output)) * dst_negative_scale_resize_output) + (dst_negative_resize_output))) /\ (exists ge_balance_positive_resize_outputentryvalue ge_balance_negative_resize_outputentryvalue. (((((dst_value_resize_output) = 2 * (ge_balance_positive_resize_outputentryvalue) /\ (ge_balance_negative_resize_outputentryvalue) = 0) \/ exists ge_signed_half_resize_outputentryvaluedecode. (((dst_value_resize_output) = 2 * ge_signed_half_resize_outputentryvaluedecode + 1 /\ (ge_balance_positive_resize_outputentryvalue) = 0) /\ (ge_balance_negative_resize_outputentryvalue) = S ge_signed_half_resize_outputentryvaluedecode))) /\ ((dst_positive_resize_output) + ge_balance_negative_resize_outputentryvalue = (dst_negative_resize_output) + ge_balance_positive_resize_outputentryvalue)))))))))

Constructive proof overview

Generated structural guide

An actual packed beta table has canonical entries on every finite domain; resizing the certificate never changes its streams.

The unchanged tactic script uses 2 declared prerequisites and contains 21 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

divisor_signed_table_components external actual_inherited_body_freshly_checked_in_complete_bundle; no checked-use authority divisor_signed_table_from_components external actual_inherited_body_freshly_checked_in_complete_bundle; no checked-use authority

Direct dependents

WS0002 signed_table_lookup_any WS0004 signed_table_add_restrict WS0008 signed_table_multiply_restrict WS000C signed_table_scalar_restrict WS000F signed_table_add_extend WS0010 signed_table_add_exists WS0012 signed_table_multiply_extend WS0013 signed_table_multiply_exists WS0015 signed_table_scalar_extend WS0016 signed_table_scalar_exists

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

21 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.

01Fix variables and assumptionsL1–4

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

  1. L1
    intro N
  2. L2
    intro M
  3. L3
    intro F
  4. L4
    intro ht
02Establish hrepL5–9

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

  1. L5
    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. L6
    specialize divisor_signed_table_components (N)
  3. L7
    specialize divisor_signed_table_components (F)
  4. L8
    apply divisor_signed_table_components
  5. L9
    exact ht
03Separate the logical casesL10–13

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

  1. L10
    cases hrep
  2. L11
    cases hrep_witness
  3. L12
    cases hrep_witness_witness
  4. L13
    cases hrep_witness_witness_witness
04Use earlier factsL14–21

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

  1. L14
    specialize divisor_signed_table_from_components (M)
  2. L15
    specialize divisor_signed_table_from_components (F)
  3. L16
    specialize divisor_signed_table_from_components (x)
  4. L17
    specialize divisor_signed_table_from_components (x1)
  5. L18
    specialize divisor_signed_table_from_components (x2)
  6. L19
    specialize divisor_signed_table_from_components (x3)
  7. L20
    apply divisor_signed_table_from_components
  8. L21
    exact hrep_witness_witness_witness_witness

Library-wide reading audit

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