SS0007

divisor_signed_table_at_functional

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

Actual beta functionality and canonical signed balance make each lookup code literally unique.

Exact expanded first-order arithmetic statement

forall F i a b. (exists dst_positive_code_functional_first dst_positive_scale_functional_first dst_negative_code_functional_first dst_negative_scale_functional_first dst_positive_functional_first dst_negative_functional_first. (((F) = (((((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) * S ((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) + ((dst_positive_scale_functional_first) + (dst_positive_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))) * S ((((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) * S ((dst_positive_code_functional_first) + (dst_positive_scale_functional_first)) + ((dst_positive_scale_functional_first) + (dst_positive_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))) + ((((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first))) + (((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) * S ((dst_negative_code_functional_first) + (dst_negative_scale_functional_first)) + ((dst_negative_scale_functional_first) + (dst_negative_scale_functional_first)))))) /\ (((((exists ff_h_pvs_functional_firstpositive. ff_h_pvs_functional_firstpositive + S (dst_positive_functional_first) = S ((S (i)) * dst_positive_scale_functional_first)) /\ exists ff_q_pvs_functional_firstpositive. dst_positive_code_functional_first = ff_q_pvs_functional_firstpositive * S ((S (i)) * dst_positive_scale_functional_first) + (dst_positive_functional_first))) /\ (((((exists ff_h_pvs_functional_firstnegative. ff_h_pvs_functional_firstnegative + S (dst_negative_functional_first) = S ((S (i)) * dst_negative_scale_functional_first)) /\ exists ff_q_pvs_functional_firstnegative. dst_negative_code_functional_first = ff_q_pvs_functional_firstnegative * S ((S (i)) * dst_negative_scale_functional_first) + (dst_negative_functional_first))) /\ (exists ge_balance_positive_functional_firstvalue ge_balance_negative_functional_firstvalue. (((((a) = 2 * (ge_balance_positive_functional_firstvalue) /\ (ge_balance_negative_functional_firstvalue) = 0) \/ exists ge_signed_half_functional_firstvaluedecode. (((a) = 2 * ge_signed_half_functional_firstvaluedecode + 1 /\ (ge_balance_positive_functional_firstvalue) = 0) /\ (ge_balance_negative_functional_firstvalue) = S ge_signed_half_functional_firstvaluedecode))) /\ ((dst_positive_functional_first) + ge_balance_negative_functional_firstvalue = (dst_negative_functional_first) + ge_balance_positive_functional_firstvalue))))))))) -> (exists dst_positive_code_functional_second dst_positive_scale_functional_second dst_negative_code_functional_second dst_negative_scale_functional_second dst_positive_functional_second dst_negative_functional_second. (((F) = (((((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) * S ((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) + ((dst_positive_scale_functional_second) + (dst_positive_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))) * S ((((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) * S ((dst_positive_code_functional_second) + (dst_positive_scale_functional_second)) + ((dst_positive_scale_functional_second) + (dst_positive_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))) + ((((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second))) + (((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) * S ((dst_negative_code_functional_second) + (dst_negative_scale_functional_second)) + ((dst_negative_scale_functional_second) + (dst_negative_scale_functional_second)))))) /\ (((((exists ff_h_pvs_functional_secondpositive. ff_h_pvs_functional_secondpositive + S (dst_positive_functional_second) = S ((S (i)) * dst_positive_scale_functional_second)) /\ exists ff_q_pvs_functional_secondpositive. dst_positive_code_functional_second = ff_q_pvs_functional_secondpositive * S ((S (i)) * dst_positive_scale_functional_second) + (dst_positive_functional_second))) /\ (((((exists ff_h_pvs_functional_secondnegative. ff_h_pvs_functional_secondnegative + S (dst_negative_functional_second) = S ((S (i)) * dst_negative_scale_functional_second)) /\ exists ff_q_pvs_functional_secondnegative. dst_negative_code_functional_second = ff_q_pvs_functional_secondnegative * S ((S (i)) * dst_negative_scale_functional_second) + (dst_negative_functional_second))) /\ (exists ge_balance_positive_functional_secondvalue ge_balance_negative_functional_secondvalue. (((((b) = 2 * (ge_balance_positive_functional_secondvalue) /\ (ge_balance_negative_functional_secondvalue) = 0) \/ exists ge_signed_half_functional_secondvaluedecode. (((b) = 2 * ge_signed_half_functional_secondvaluedecode + 1 /\ (ge_balance_positive_functional_secondvalue) = 0) /\ (ge_balance_negative_functional_secondvalue) = S ge_signed_half_functional_secondvaluedecode))) /\ ((dst_positive_functional_second) + ge_balance_negative_functional_secondvalue = (dst_negative_functional_second) + ge_balance_positive_functional_secondvalue))))))))) -> a = b

Constructive proof overview

Generated structural guide

Actual beta functionality and canonical signed balance make each lookup code literally unique.

The unchanged tactic script uses 3 declared prerequisites and contains 57 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

SS0002 divisor_signed_table_at_to_components beta_at_unique Alpha theorem; checked-use authorized signed_balance_functional Alpha theorem; checked-use authorized

Direct dependents

SS001C divisor_signed_table_reindex_functional

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

57 script commands · 9 reading checkpoints · 3 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 (1)
divisor signed table at to components

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

01Fix variables and assumptionsL1–6

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

  1. L1
    intro F
  2. L2
    intro i
  3. L3
    intro a
  4. L4
    intro b
  5. L5
    intro ha
  6. L6
    intro hb
02Separate the logical casesL7–15

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

  1. L7
    cases ha
  2. L8
    cases ha_witness
  3. L9
    cases ha_witness_witness
  4. L10
    cases ha_witness_witness_witness
  5. L11
    cases ha_witness_witness_witness_witness
  6. L12
    cases ha_witness_witness_witness_witness_witness
  7. L13
    cases ha_witness_witness_witness_witness_witness_witness
  8. L14
    cases ha_witness_witness_witness_witness_witness_witness_right
  9. L15
    cases ha_witness_witness_witness_witness_witness_witness_right_right
03Establish hotherL16–25

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

  1. L16
    have hother : ∃ p. ∃ n. BetaAt(x,x1,i,p) ∧ (BetaAt(x2,x3,i,n) ∧ SignedBalance(b,p,n))Definitions: SignedBalanceBetaAt
  2. L17
    specialize divisor_signed_table_at_to_components (F)
  3. L18
    specialize divisor_signed_table_at_to_components (x)
  4. L19
    specialize divisor_signed_table_at_to_components (x1)
  5. L20
    specialize divisor_signed_table_at_to_components (x2)
  6. L21
    specialize divisor_signed_table_at_to_components (x3)
  7. L22
    specialize divisor_signed_table_at_to_components (i)
  8. L23
    specialize divisor_signed_table_at_to_components (b)
  9. L24
    apply divisor_signed_table_at_to_components
  10. L25
    exact ha_witness_witness_witness_witness_witness_witness_left
04Use earlier factsL26–26

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

  1. L26
    exact hb
05Separate the logical casesL27–30

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

  1. L27
    cases hother
  2. L28
    cases hother_witness
  3. L29
    cases hother_witness_witness
  4. L30
    cases hother_witness_witness_right
06Establish hpL31–39

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L31
    have hp : x6 = x4
  2. L32
    specialize beta_at_unique (x)
  3. L33
    specialize beta_at_unique (x1)
  4. L34
    specialize beta_at_unique (i)
  5. L35
    specialize beta_at_unique (x6)
  6. L36
    specialize beta_at_unique (x4)
  7. L37
    apply beta_at_unique
  8. L38
    exact hother_witness_witness_left
  9. L39
    exact ha_witness_witness_witness_witness_witness_witness_right_left
07Establish hnL40–49

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L40
    have hn : x7 = x5
  2. L41
    specialize beta_at_unique (x2)
  3. L42
    specialize beta_at_unique (x3)
  4. L43
    specialize beta_at_unique (i)
  5. L44
    specialize beta_at_unique (x7)
  6. L45
    specialize beta_at_unique (x5)
  7. L46
    apply beta_at_unique
  8. L47
    exact hother_witness_witness_right_left
  9. L48
    exact ha_witness_witness_witness_witness_witness_witness_right_right_left
  10. L49
    rewrite hp at hother_witness_witness_right_right
08Calculate and transport equalitiesL50–50

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L50
    rewrite hn at hother_witness_witness_right_right
09Use earlier factsL51–57

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

  1. L51
    specialize signed_balance_functional (x4)
  2. L52
    specialize signed_balance_functional (x5)
  3. L53
    specialize signed_balance_functional (a)
  4. L54
    specialize signed_balance_functional (b)
  5. L55
    apply signed_balance_functional
  6. L56
    exact ha_witness_witness_witness_witness_witness_witness_right_right_right
  7. L57
    exact hother_witness_witness_right_right

Library-wide reading audit

Original exact command ledger · 57 lines
  1. 0001intro F
  2. 0002intro i
  3. 0003intro a
  4. 0004intro b
  5. 0005intro ha
  6. 0006intro hb
  7. 0007cases ha
  8. 0008cases ha_witness
  9. 0009cases ha_witness_witness
  10. 0010cases ha_witness_witness_witness
  11. 0011cases ha_witness_witness_witness_witness
  12. 0012cases ha_witness_witness_witness_witness_witness
  13. 0013cases ha_witness_witness_witness_witness_witness_witness
  14. 0014cases ha_witness_witness_witness_witness_witness_witness_right
  15. 0015cases ha_witness_witness_witness_witness_witness_witness_right_right
  16. 0016have hother : exists p n. (((((exists ff_h_pvs_functional_otherpositive. ff_h_pvs_functional_otherpositive + S (p) = S ((S (i)) * x1)) /\ exists ff_q_pvs_functional_otherpositive. x = ff_q_pvs_functional_otherpositive * S ((S (i)) * x1) + (p))) /\ (((((exists ff_h_pvs_functional_othernegative. ff_h_pvs_functional_othernegative + S (n) = S ((S (i)) * x3)) /\ exists ff_q_pvs_functional_othernegative. x2 = ff_q_pvs_functional_othernegative * S ((S (i)) * x3) + (n))) /\ (exists ge_balance_positive_functional_othervalue ge_balance_negative_functional_othervalue. (((((b) = 2 * (ge_balance_positive_functional_othervalue) /\ (ge_balance_negative_functional_othervalue) = 0) \/ exists ge_signed_half_functional_othervaluedecode. (((b) = 2 * ge_signed_half_functional_othervaluedecode + 1 /\ (ge_balance_positive_functional_othervalue) = 0) /\ (ge_balance_negative_functional_othervalue) = S ge_signed_half_functional_othervaluedecode))) /\ ((p) + ge_balance_negative_functional_othervalue = (n) + ge_balance_positive_functional_othervalue)))))))
  17. 0017specialize divisor_signed_table_at_to_components (F)
  18. 0018specialize divisor_signed_table_at_to_components (x)
  19. 0019specialize divisor_signed_table_at_to_components (x1)
  20. 0020specialize divisor_signed_table_at_to_components (x2)
  21. 0021specialize divisor_signed_table_at_to_components (x3)
  22. 0022specialize divisor_signed_table_at_to_components (i)
  23. 0023specialize divisor_signed_table_at_to_components (b)
  24. 0024apply divisor_signed_table_at_to_components
  25. 0025exact ha_witness_witness_witness_witness_witness_witness_left
  26. 0026exact hb
  27. 0027cases hother
  28. 0028cases hother_witness
  29. 0029cases hother_witness_witness
  30. 0030cases hother_witness_witness_right
  31. 0031have hp : x6 = x4
  32. 0032specialize beta_at_unique (x)
  33. 0033specialize beta_at_unique (x1)
  34. 0034specialize beta_at_unique (i)
  35. 0035specialize beta_at_unique (x6)
  36. 0036specialize beta_at_unique (x4)
  37. 0037apply beta_at_unique
  38. 0038exact hother_witness_witness_left
  39. 0039exact ha_witness_witness_witness_witness_witness_witness_right_left
  40. 0040have hn : x7 = x5
  41. 0041specialize beta_at_unique (x2)
  42. 0042specialize beta_at_unique (x3)
  43. 0043specialize beta_at_unique (i)
  44. 0044specialize beta_at_unique (x7)
  45. 0045specialize beta_at_unique (x5)
  46. 0046apply beta_at_unique
  47. 0047exact hother_witness_witness_right_left
  48. 0048exact ha_witness_witness_witness_witness_witness_witness_right_right_left
  49. 0049rewrite hp at hother_witness_witness_right_right
  50. 0050rewrite hn at hother_witness_witness_right_right
  51. 0051specialize signed_balance_functional (x4)
  52. 0052specialize signed_balance_functional (x5)
  53. 0053specialize signed_balance_functional (a)
  54. 0054specialize signed_balance_functional (b)
  55. 0055apply signed_balance_functional
  56. 0056exact ha_witness_witness_witness_witness_witness_witness_right_right_right
  57. 0057exact hother_witness_witness_right_right