MX0038

signed_support_incidence_entry_decode

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

Actual signed lookup and beta uniqueness recover the independently stated hit-or-zero cell cases.

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 A r s i j a k z. (exists dst_positive_code_decode_source dst_positive_scale_decode_source dst_negative_code_decode_source dst_negative_scale_decode_source dst_positive_decode_source dst_negative_decode_source. (((A) = (((((dst_positive_code_decode_source) + (dst_positive_scale_decode_source)) * S ((dst_positive_code_decode_source) + (dst_positive_scale_decode_source)) + ((dst_positive_scale_decode_source) + (dst_positive_scale_decode_source))) + (((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) * S ((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) + ((dst_negative_scale_decode_source) + (dst_negative_scale_decode_source)))) * S ((((dst_positive_code_decode_source) + (dst_positive_scale_decode_source)) * S ((dst_positive_code_decode_source) + (dst_positive_scale_decode_source)) + ((dst_positive_scale_decode_source) + (dst_positive_scale_decode_source))) + (((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) * S ((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) + ((dst_negative_scale_decode_source) + (dst_negative_scale_decode_source)))) + ((((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) * S ((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) + ((dst_negative_scale_decode_source) + (dst_negative_scale_decode_source))) + (((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) * S ((dst_negative_code_decode_source) + (dst_negative_scale_decode_source)) + ((dst_negative_scale_decode_source) + (dst_negative_scale_decode_source)))))) /\ (((((exists ff_h_pvs_decode_sourcepositive. ff_h_pvs_decode_sourcepositive + S (dst_positive_decode_source) = S ((S (i)) * dst_positive_scale_decode_source)) /\ exists ff_q_pvs_decode_sourcepositive. dst_positive_code_decode_source = ff_q_pvs_decode_sourcepositive * S ((S (i)) * dst_positive_scale_decode_source) + (dst_positive_decode_source))) /\ (((((exists ff_h_pvs_decode_sourcenegative. ff_h_pvs_decode_sourcenegative + S (dst_negative_decode_source) = S ((S (i)) * dst_negative_scale_decode_source)) /\ exists ff_q_pvs_decode_sourcenegative. dst_negative_code_decode_source = ff_q_pvs_decode_sourcenegative * S ((S (i)) * dst_negative_scale_decode_source) + (dst_negative_decode_source))) /\ (exists ge_balance_positive_decode_sourcevalue ge_balance_negative_decode_sourcevalue. (((((a) = 2 * (ge_balance_positive_decode_sourcevalue) /\ (ge_balance_negative_decode_sourcevalue) = 0) \/ exists ge_signed_half_decode_sourcevaluedecode. (((a) = 2 * ge_signed_half_decode_sourcevaluedecode + 1 /\ (ge_balance_positive_decode_sourcevalue) = 0) /\ (ge_balance_negative_decode_sourcevalue) = S ge_signed_half_decode_sourcevaluedecode))) /\ ((dst_positive_decode_source) + ge_balance_negative_decode_sourcevalue = (dst_negative_decode_source) + ge_balance_positive_decode_sourcevalue))))))))) -> (((exists ff_h_pvs_decode_map. ff_h_pvs_decode_map + S (k) = S ((S (i)) * s)) /\ exists ff_q_pvs_decode_map. r = ff_q_pvs_decode_map * S ((S (i)) * s) + (k))) -> (exists ssr_entry_value_decode_entry ssr_entry_image_decode_entry. ((exists dst_positive_code_decode_entrysource dst_positive_scale_decode_entrysource dst_negative_code_decode_entrysource dst_negative_scale_decode_entrysource dst_positive_decode_entrysource dst_negative_decode_entrysource. (((A) = (((((dst_positive_code_decode_entrysource) + (dst_positive_scale_decode_entrysource)) * S ((dst_positive_code_decode_entrysource) + (dst_positive_scale_decode_entrysource)) + ((dst_positive_scale_decode_entrysource) + (dst_positive_scale_decode_entrysource))) + (((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) * S ((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) + ((dst_negative_scale_decode_entrysource) + (dst_negative_scale_decode_entrysource)))) * S ((((dst_positive_code_decode_entrysource) + (dst_positive_scale_decode_entrysource)) * S ((dst_positive_code_decode_entrysource) + (dst_positive_scale_decode_entrysource)) + ((dst_positive_scale_decode_entrysource) + (dst_positive_scale_decode_entrysource))) + (((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) * S ((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) + ((dst_negative_scale_decode_entrysource) + (dst_negative_scale_decode_entrysource)))) + ((((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) * S ((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) + ((dst_negative_scale_decode_entrysource) + (dst_negative_scale_decode_entrysource))) + (((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) * S ((dst_negative_code_decode_entrysource) + (dst_negative_scale_decode_entrysource)) + ((dst_negative_scale_decode_entrysource) + (dst_negative_scale_decode_entrysource)))))) /\ (((((exists ff_h_pvs_decode_entrysourcepositive. ff_h_pvs_decode_entrysourcepositive + S (dst_positive_decode_entrysource) = S ((S (i)) * dst_positive_scale_decode_entrysource)) /\ exists ff_q_pvs_decode_entrysourcepositive. dst_positive_code_decode_entrysource = ff_q_pvs_decode_entrysourcepositive * S ((S (i)) * dst_positive_scale_decode_entrysource) + (dst_positive_decode_entrysource))) /\ (((((exists ff_h_pvs_decode_entrysourcenegative. ff_h_pvs_decode_entrysourcenegative + S (dst_negative_decode_entrysource) = S ((S (i)) * dst_negative_scale_decode_entrysource)) /\ exists ff_q_pvs_decode_entrysourcenegative. dst_negative_code_decode_entrysource = ff_q_pvs_decode_entrysourcenegative * S ((S (i)) * dst_negative_scale_decode_entrysource) + (dst_negative_decode_entrysource))) /\ (exists ge_balance_positive_decode_entrysourcevalue ge_balance_negative_decode_entrysourcevalue. (((((ssr_entry_value_decode_entry) = 2 * (ge_balance_positive_decode_entrysourcevalue) /\ (ge_balance_negative_decode_entrysourcevalue) = 0) \/ exists ge_signed_half_decode_entrysourcevaluedecode. (((ssr_entry_value_decode_entry) = 2 * ge_signed_half_decode_entrysourcevaluedecode + 1 /\ (ge_balance_positive_decode_entrysourcevalue) = 0) /\ (ge_balance_negative_decode_entrysourcevalue) = S ge_signed_half_decode_entrysourcevaluedecode))) /\ ((dst_positive_decode_entrysource) + ge_balance_negative_decode_entrysourcevalue = (dst_negative_decode_entrysource) + ge_balance_positive_decode_entrysourcevalue))))))))) /\ (((((exists ff_h_pvs_decode_entrymap. ff_h_pvs_decode_entrymap + S (ssr_entry_image_decode_entry) = S ((S (i)) * s)) /\ exists ff_q_pvs_decode_entrymap. r = ff_q_pvs_decode_entrymap * S ((S (i)) * s) + (ssr_entry_image_decode_entry))) /\ (((((j)=(ssr_entry_image_decode_entry)) /\ ((z)=(ssr_entry_value_decode_entry)))) \/ (((~((j)=(ssr_entry_image_decode_entry))) /\ ((z)=0)))))))) -> (((((j)=(k)) /\ ((z)=(a)))) \/ (((~((j)=(k))) /\ ((z)=0))))

Constructive proof overview

Generated structural guide

Actual signed lookup and beta uniqueness recover the independently stated hit-or-zero cell cases.

The unchanged tactic script uses 2 declared prerequisites and contains 36 exact native proof lines.

Alpha v34 checked-use · first admitted v32 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

divisor_signed_table_at_functional Alpha theorem; checked-use authorized beta_at_unique Stable theorem; checked-use authorized

Direct dependents

MX0039 signed_support_incidence_entry_functional

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

36 script commands · 7 reading checkpoints · 2 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 assumptionsL1–10

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

  1. L1
    intro A
  2. L2
    intro r
  3. L3
    intro s
  4. L4
    intro i
  5. L5
    intro j
  6. L6
    intro a
  7. L7
    intro k
  8. L8
    intro z
  9. L9
    intro ha
  10. L10
    intro hm
02Fix variables and assumptionsL11–11

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

  1. L11
    intro he
03Separate the logical casesL12–15

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

  1. L12
    cases he
  2. L13
    cases he_witness
  3. L14
    cases he_witness_witness
  4. L15
    cases he_witness_witness_right
04Establish hvalueL16–23

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

  1. L16
    have hvalue : x=a
  2. L17
    specialize divisor_signed_table_at_functional (A)
  3. L18
    specialize divisor_signed_table_at_functional (i)
  4. L19
    specialize divisor_signed_table_at_functional (x)
  5. L20
    specialize divisor_signed_table_at_functional (a)
  6. L21
    apply divisor_signed_table_at_functional
  7. L22
    exact he_witness_witness_left
  8. L23
    exact ha
05Establish himageL24–33

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

  1. L24
    have himage : x1=k
  2. L25
    specialize beta_at_unique (r)
  3. L26
    specialize beta_at_unique (s)
  4. L27
    specialize beta_at_unique (i)
  5. L28
    specialize beta_at_unique (x1)
  6. L29
    specialize beta_at_unique (k)
  7. L30
    apply beta_at_unique
  8. L31
    exact he_witness_witness_right_left
  9. L32
    exact hm
  10. L33
    rewrite hvalue at he_witness_witness_right_right
06Calculate and transport equalitiesL34–35

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

  1. L34
    rewrite himage at he_witness_witness_right_right
  2. L35
    rewrite himage at he_witness_witness_right_right
07Use earlier factsL36–36

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

  1. L36
    exact he_witness_witness_right_right

Library-wide reading audit

Original exact command ledger · 36 lines
  1. 0001intro A
  2. 0002intro r
  3. 0003intro s
  4. 0004intro i
  5. 0005intro j
  6. 0006intro a
  7. 0007intro k
  8. 0008intro z
  9. 0009intro ha
  10. 0010intro hm
  11. 0011intro he
  12. 0012cases he
  13. 0013cases he_witness
  14. 0014cases he_witness_witness
  15. 0015cases he_witness_witness_right
  16. 0016have hvalue : x=a
  17. 0017specialize divisor_signed_table_at_functional (A)
  18. 0018specialize divisor_signed_table_at_functional (i)
  19. 0019specialize divisor_signed_table_at_functional (x)
  20. 0020specialize divisor_signed_table_at_functional (a)
  21. 0021apply divisor_signed_table_at_functional
  22. 0022exact he_witness_witness_left
  23. 0023exact ha
  24. 0024have himage : x1=k
  25. 0025specialize beta_at_unique (r)
  26. 0026specialize beta_at_unique (s)
  27. 0027specialize beta_at_unique (i)
  28. 0028specialize beta_at_unique (x1)
  29. 0029specialize beta_at_unique (k)
  30. 0030apply beta_at_unique
  31. 0031exact he_witness_witness_right_left
  32. 0032exact hm
  33. 0033rewrite hvalue at he_witness_witness_right_right
  34. 0034rewrite himage at he_witness_witness_right_right
  35. 0035rewrite himage at he_witness_witness_right_right
  36. 0036exact he_witness_witness_right_right