MX003D

signed_support_incidence_flat_entry_exists

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

Division by the positive padded width S M and actual incidence lookup construct every flat value, including M=0.

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 M k. (exists dst_positive_code_flat_source_table dst_positive_scale_flat_source_table dst_negative_code_flat_source_table dst_negative_scale_flat_source_table. (((A) = (((((dst_positive_code_flat_source_table) + (dst_positive_scale_flat_source_table)) * S ((dst_positive_code_flat_source_table) + (dst_positive_scale_flat_source_table)) + ((dst_positive_scale_flat_source_table) + (dst_positive_scale_flat_source_table))) + (((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) * S ((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) + ((dst_negative_scale_flat_source_table) + (dst_negative_scale_flat_source_table)))) * S ((((dst_positive_code_flat_source_table) + (dst_positive_scale_flat_source_table)) * S ((dst_positive_code_flat_source_table) + (dst_positive_scale_flat_source_table)) + ((dst_positive_scale_flat_source_table) + (dst_positive_scale_flat_source_table))) + (((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) * S ((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) + ((dst_negative_scale_flat_source_table) + (dst_negative_scale_flat_source_table)))) + ((((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) * S ((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) + ((dst_negative_scale_flat_source_table) + (dst_negative_scale_flat_source_table))) + (((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) * S ((dst_negative_code_flat_source_table) + (dst_negative_scale_flat_source_table)) + ((dst_negative_scale_flat_source_table) + (dst_negative_scale_flat_source_table)))))) /\ (forall dst_index_flat_source_table. (exists pvs_le_gap_flat_source_tabledomain. pvs_le_gap_flat_source_tabledomain + (dst_index_flat_source_table) = (0)) -> exists dst_positive_flat_source_table dst_negative_flat_source_table dst_value_flat_source_table. ((((exists ff_h_pvs_flat_source_tableentrypositive. ff_h_pvs_flat_source_tableentrypositive + S (dst_positive_flat_source_table) = S ((S (dst_index_flat_source_table)) * dst_positive_scale_flat_source_table)) /\ exists ff_q_pvs_flat_source_tableentrypositive. dst_positive_code_flat_source_table = ff_q_pvs_flat_source_tableentrypositive * S ((S (dst_index_flat_source_table)) * dst_positive_scale_flat_source_table) + (dst_positive_flat_source_table))) /\ (((((exists ff_h_pvs_flat_source_tableentrynegative. ff_h_pvs_flat_source_tableentrynegative + S (dst_negative_flat_source_table) = S ((S (dst_index_flat_source_table)) * dst_negative_scale_flat_source_table)) /\ exists ff_q_pvs_flat_source_tableentrynegative. dst_negative_code_flat_source_table = ff_q_pvs_flat_source_tableentrynegative * S ((S (dst_index_flat_source_table)) * dst_negative_scale_flat_source_table) + (dst_negative_flat_source_table))) /\ (exists ge_balance_positive_flat_source_tableentryvalue ge_balance_negative_flat_source_tableentryvalue. (((((dst_value_flat_source_table) = 2 * (ge_balance_positive_flat_source_tableentryvalue) /\ (ge_balance_negative_flat_source_tableentryvalue) = 0) \/ exists ge_signed_half_flat_source_tableentryvaluedecode. (((dst_value_flat_source_table) = 2 * ge_signed_half_flat_source_tableentryvaluedecode + 1 /\ (ge_balance_positive_flat_source_tableentryvalue) = 0) /\ (ge_balance_negative_flat_source_tableentryvalue) = S ge_signed_half_flat_source_tableentryvaluedecode))) /\ ((dst_positive_flat_source_table) + ge_balance_negative_flat_source_tableentryvalue = (dst_negative_flat_source_table) + ge_balance_positive_flat_source_tableentryvalue))))))))) -> exists z. (exists ssr_flat_row_flat_total ssr_flat_column_flat_total. (((k)=(((S (M))*(ssr_flat_row_flat_total)+(ssr_flat_column_flat_total)))) /\ (((exists pvs_gap_flat_totalremainder. pvs_gap_flat_totalremainder + S (ssr_flat_column_flat_total) = (S (M))) /\ (exists ssr_entry_value_flat_totalentry ssr_entry_image_flat_totalentry. ((exists dst_positive_code_flat_totalentrysource dst_positive_scale_flat_totalentrysource dst_negative_code_flat_totalentrysource dst_negative_scale_flat_totalentrysource dst_positive_flat_totalentrysource dst_negative_flat_totalentrysource. (((A) = (((((dst_positive_code_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource)) * S ((dst_positive_code_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource)) + ((dst_positive_scale_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource))) + (((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) * S ((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) + ((dst_negative_scale_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)))) * S ((((dst_positive_code_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource)) * S ((dst_positive_code_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource)) + ((dst_positive_scale_flat_totalentrysource) + (dst_positive_scale_flat_totalentrysource))) + (((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) * S ((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) + ((dst_negative_scale_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)))) + ((((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) * S ((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) + ((dst_negative_scale_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource))) + (((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) * S ((dst_negative_code_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)) + ((dst_negative_scale_flat_totalentrysource) + (dst_negative_scale_flat_totalentrysource)))))) /\ (((((exists ff_h_pvs_flat_totalentrysourcepositive. ff_h_pvs_flat_totalentrysourcepositive + S (dst_positive_flat_totalentrysource) = S ((S (ssr_flat_row_flat_total)) * dst_positive_scale_flat_totalentrysource)) /\ exists ff_q_pvs_flat_totalentrysourcepositive. dst_positive_code_flat_totalentrysource = ff_q_pvs_flat_totalentrysourcepositive * S ((S (ssr_flat_row_flat_total)) * dst_positive_scale_flat_totalentrysource) + (dst_positive_flat_totalentrysource))) /\ (((((exists ff_h_pvs_flat_totalentrysourcenegative. ff_h_pvs_flat_totalentrysourcenegative + S (dst_negative_flat_totalentrysource) = S ((S (ssr_flat_row_flat_total)) * dst_negative_scale_flat_totalentrysource)) /\ exists ff_q_pvs_flat_totalentrysourcenegative. dst_negative_code_flat_totalentrysource = ff_q_pvs_flat_totalentrysourcenegative * S ((S (ssr_flat_row_flat_total)) * dst_negative_scale_flat_totalentrysource) + (dst_negative_flat_totalentrysource))) /\ (exists ge_balance_positive_flat_totalentrysourcevalue ge_balance_negative_flat_totalentrysourcevalue. (((((ssr_entry_value_flat_totalentry) = 2 * (ge_balance_positive_flat_totalentrysourcevalue) /\ (ge_balance_negative_flat_totalentrysourcevalue) = 0) \/ exists ge_signed_half_flat_totalentrysourcevaluedecode. (((ssr_entry_value_flat_totalentry) = 2 * ge_signed_half_flat_totalentrysourcevaluedecode + 1 /\ (ge_balance_positive_flat_totalentrysourcevalue) = 0) /\ (ge_balance_negative_flat_totalentrysourcevalue) = S ge_signed_half_flat_totalentrysourcevaluedecode))) /\ ((dst_positive_flat_totalentrysource) + ge_balance_negative_flat_totalentrysourcevalue = (dst_negative_flat_totalentrysource) + ge_balance_positive_flat_totalentrysourcevalue))))))))) /\ (((((exists ff_h_pvs_flat_totalentrymap. ff_h_pvs_flat_totalentrymap + S (ssr_entry_image_flat_totalentry) = S ((S (ssr_flat_row_flat_total)) * s)) /\ exists ff_q_pvs_flat_totalentrymap. r = ff_q_pvs_flat_totalentrymap * S ((S (ssr_flat_row_flat_total)) * s) + (ssr_entry_image_flat_totalentry))) /\ (((((ssr_flat_column_flat_total)=(ssr_entry_image_flat_totalentry)) /\ ((z)=(ssr_entry_value_flat_totalentry)))) \/ (((~((ssr_flat_column_flat_total)=(ssr_entry_image_flat_totalentry))) /\ ((z)=0))))))))))))

Constructive proof overview

Generated structural guide

Division by the positive padded width S M and actual incidence lookup construct every flat value, including M=0.

The unchanged tactic script uses 3 declared prerequisites and contains 32 exact native proof lines.

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

Proof neighborhood

Direct dependencies

division_remainder_exists Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized MX003A signed_support_incidence_entry_exists

Direct dependents

MX0041 signed_support_incidence_flat_prefix_exists

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

32 script commands · 10 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.

Named ingredients (1)
signed support incidence entry exists

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 A
  2. L2
    intro r
  3. L3
    intro s
  4. L4
    intro M
  5. L5
    intro k
  6. L6
    intro hA
02Establish hdL7–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply division remainder exists.

  1. L7
    have hd : exists i j. ((k=(S M)*i+j) /\ (exists pvs_gap_flat_division. pvs_gap_flat_division + S (j) = (S M)))
  2. L8
    specialize division_remainder_exists (S M)
  3. L9
    specialize division_remainder_exists (k)
  4. L10
    apply division_remainder_exists
  5. L11
    specialize succ_ne_zero (M)
  6. L12
    apply succ_ne_zero
03Separate the logical casesL13–15

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

  1. L13
    cases hd
  2. L14
    cases hd_witness
  3. L15
    cases hd_witness_witness
04Establish hvL16–23

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply signed support incidence entry exists.

  1. L16
    have hv : ∃ z. SignedIncidenceEntry(A,r,s,x,x1,z)Definitions: SignedIncidenceEntry
  2. L17
    specialize signed_support_incidence_entry_exists (A)
  3. L18
    specialize signed_support_incidence_entry_exists (r)
  4. L19
    specialize signed_support_incidence_entry_exists (s)
  5. L20
    specialize signed_support_incidence_entry_exists (x)
  6. L21
    specialize signed_support_incidence_entry_exists (x1)
  7. L22
    apply signed_support_incidence_entry_exists
  8. L23
    exact hA
05Separate the logical casesL24–24

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

  1. L24
    cases hv
06Construct an explicit witnessL25–27

Supply the displayed value, then prove that it has the required property.

  1. L25
    exists x2
  2. L26
    exists x
  3. L27
    exists x1
07Separate the logical casesL28–28

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

  1. L28
    split
08Use earlier factsL29–29

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

  1. L29
    exact hd_witness_witness_left
09Separate the logical casesL30–30

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

  1. L30
    split
10Use earlier factsL31–32

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

  1. L31
    exact hd_witness_witness_right
  2. L32
    exact hv_witness

Library-wide reading audit

Original exact command ledger · 32 lines
  1. 0001intro A
  2. 0002intro r
  3. 0003intro s
  4. 0004intro M
  5. 0005intro k
  6. 0006intro hA
  7. 0007have hd : exists i j. ((k=(S M)*i+j) /\ (exists pvs_gap_flat_division. pvs_gap_flat_division + S (j) = (S M)))
  8. 0008specialize division_remainder_exists (S M)
  9. 0009specialize division_remainder_exists (k)
  10. 0010apply division_remainder_exists
  11. 0011specialize succ_ne_zero (M)
  12. 0012apply succ_ne_zero
  13. 0013cases hd
  14. 0014cases hd_witness
  15. 0015cases hd_witness_witness
  16. 0016have hv : exists z. (exists ssr_entry_value_flat_value ssr_entry_image_flat_value. ((exists dst_positive_code_flat_valuesource dst_positive_scale_flat_valuesource dst_negative_code_flat_valuesource dst_negative_scale_flat_valuesource dst_positive_flat_valuesource dst_negative_flat_valuesource. (((A) = (((((dst_positive_code_flat_valuesource) + (dst_positive_scale_flat_valuesource)) * S ((dst_positive_code_flat_valuesource) + (dst_positive_scale_flat_valuesource)) + ((dst_positive_scale_flat_valuesource) + (dst_positive_scale_flat_valuesource))) + (((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) * S ((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) + ((dst_negative_scale_flat_valuesource) + (dst_negative_scale_flat_valuesource)))) * S ((((dst_positive_code_flat_valuesource) + (dst_positive_scale_flat_valuesource)) * S ((dst_positive_code_flat_valuesource) + (dst_positive_scale_flat_valuesource)) + ((dst_positive_scale_flat_valuesource) + (dst_positive_scale_flat_valuesource))) + (((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) * S ((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) + ((dst_negative_scale_flat_valuesource) + (dst_negative_scale_flat_valuesource)))) + ((((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) * S ((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) + ((dst_negative_scale_flat_valuesource) + (dst_negative_scale_flat_valuesource))) + (((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) * S ((dst_negative_code_flat_valuesource) + (dst_negative_scale_flat_valuesource)) + ((dst_negative_scale_flat_valuesource) + (dst_negative_scale_flat_valuesource)))))) /\ (((((exists ff_h_pvs_flat_valuesourcepositive. ff_h_pvs_flat_valuesourcepositive + S (dst_positive_flat_valuesource) = S ((S (x)) * dst_positive_scale_flat_valuesource)) /\ exists ff_q_pvs_flat_valuesourcepositive. dst_positive_code_flat_valuesource = ff_q_pvs_flat_valuesourcepositive * S ((S (x)) * dst_positive_scale_flat_valuesource) + (dst_positive_flat_valuesource))) /\ (((((exists ff_h_pvs_flat_valuesourcenegative. ff_h_pvs_flat_valuesourcenegative + S (dst_negative_flat_valuesource) = S ((S (x)) * dst_negative_scale_flat_valuesource)) /\ exists ff_q_pvs_flat_valuesourcenegative. dst_negative_code_flat_valuesource = ff_q_pvs_flat_valuesourcenegative * S ((S (x)) * dst_negative_scale_flat_valuesource) + (dst_negative_flat_valuesource))) /\ (exists ge_balance_positive_flat_valuesourcevalue ge_balance_negative_flat_valuesourcevalue. (((((ssr_entry_value_flat_value) = 2 * (ge_balance_positive_flat_valuesourcevalue) /\ (ge_balance_negative_flat_valuesourcevalue) = 0) \/ exists ge_signed_half_flat_valuesourcevaluedecode. (((ssr_entry_value_flat_value) = 2 * ge_signed_half_flat_valuesourcevaluedecode + 1 /\ (ge_balance_positive_flat_valuesourcevalue) = 0) /\ (ge_balance_negative_flat_valuesourcevalue) = S ge_signed_half_flat_valuesourcevaluedecode))) /\ ((dst_positive_flat_valuesource) + ge_balance_negative_flat_valuesourcevalue = (dst_negative_flat_valuesource) + ge_balance_positive_flat_valuesourcevalue))))))))) /\ (((((exists ff_h_pvs_flat_valuemap. ff_h_pvs_flat_valuemap + S (ssr_entry_image_flat_value) = S ((S (x)) * s)) /\ exists ff_q_pvs_flat_valuemap. r = ff_q_pvs_flat_valuemap * S ((S (x)) * s) + (ssr_entry_image_flat_value))) /\ (((((x1)=(ssr_entry_image_flat_value)) /\ ((z)=(ssr_entry_value_flat_value)))) \/ (((~((x1)=(ssr_entry_image_flat_value))) /\ ((z)=0))))))))
  17. 0017specialize signed_support_incidence_entry_exists (A)
  18. 0018specialize signed_support_incidence_entry_exists (r)
  19. 0019specialize signed_support_incidence_entry_exists (s)
  20. 0020specialize signed_support_incidence_entry_exists (x)
  21. 0021specialize signed_support_incidence_entry_exists (x1)
  22. 0022apply signed_support_incidence_entry_exists
  23. 0023exact hA
  24. 0024cases hv
  25. 0025exists x2
  26. 0026exists x
  27. 0027exists x1
  28. 0028split
  29. 0029exact hd_witness_witness_left
  30. 0030split
  31. 0031exact hd_witness_witness_right
  32. 0032exact hv_witness