MX003B

signed_support_incidence_zero_source_value

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

Every incidence cell of a represented zero source is zero, even at an out-of-window image.

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 z. (exists dst_positive_code_zero_source dst_positive_scale_zero_source dst_negative_code_zero_source dst_negative_scale_zero_source dst_positive_zero_source dst_negative_zero_source. (((A) = (((((dst_positive_code_zero_source) + (dst_positive_scale_zero_source)) * S ((dst_positive_code_zero_source) + (dst_positive_scale_zero_source)) + ((dst_positive_scale_zero_source) + (dst_positive_scale_zero_source))) + (((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) * S ((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) + ((dst_negative_scale_zero_source) + (dst_negative_scale_zero_source)))) * S ((((dst_positive_code_zero_source) + (dst_positive_scale_zero_source)) * S ((dst_positive_code_zero_source) + (dst_positive_scale_zero_source)) + ((dst_positive_scale_zero_source) + (dst_positive_scale_zero_source))) + (((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) * S ((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) + ((dst_negative_scale_zero_source) + (dst_negative_scale_zero_source)))) + ((((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) * S ((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) + ((dst_negative_scale_zero_source) + (dst_negative_scale_zero_source))) + (((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) * S ((dst_negative_code_zero_source) + (dst_negative_scale_zero_source)) + ((dst_negative_scale_zero_source) + (dst_negative_scale_zero_source)))))) /\ (((((exists ff_h_pvs_zero_sourcepositive. ff_h_pvs_zero_sourcepositive + S (dst_positive_zero_source) = S ((S (i)) * dst_positive_scale_zero_source)) /\ exists ff_q_pvs_zero_sourcepositive. dst_positive_code_zero_source = ff_q_pvs_zero_sourcepositive * S ((S (i)) * dst_positive_scale_zero_source) + (dst_positive_zero_source))) /\ (((((exists ff_h_pvs_zero_sourcenegative. ff_h_pvs_zero_sourcenegative + S (dst_negative_zero_source) = S ((S (i)) * dst_negative_scale_zero_source)) /\ exists ff_q_pvs_zero_sourcenegative. dst_negative_code_zero_source = ff_q_pvs_zero_sourcenegative * S ((S (i)) * dst_negative_scale_zero_source) + (dst_negative_zero_source))) /\ (exists ge_balance_positive_zero_sourcevalue ge_balance_negative_zero_sourcevalue. (((((0) = 2 * (ge_balance_positive_zero_sourcevalue) /\ (ge_balance_negative_zero_sourcevalue) = 0) \/ exists ge_signed_half_zero_sourcevaluedecode. (((0) = 2 * ge_signed_half_zero_sourcevaluedecode + 1 /\ (ge_balance_positive_zero_sourcevalue) = 0) /\ (ge_balance_negative_zero_sourcevalue) = S ge_signed_half_zero_sourcevaluedecode))) /\ ((dst_positive_zero_source) + ge_balance_negative_zero_sourcevalue = (dst_negative_zero_source) + ge_balance_positive_zero_sourcevalue))))))))) -> (exists ssr_entry_value_zero_entry ssr_entry_image_zero_entry. ((exists dst_positive_code_zero_entrysource dst_positive_scale_zero_entrysource dst_negative_code_zero_entrysource dst_negative_scale_zero_entrysource dst_positive_zero_entrysource dst_negative_zero_entrysource. (((A) = (((((dst_positive_code_zero_entrysource) + (dst_positive_scale_zero_entrysource)) * S ((dst_positive_code_zero_entrysource) + (dst_positive_scale_zero_entrysource)) + ((dst_positive_scale_zero_entrysource) + (dst_positive_scale_zero_entrysource))) + (((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) * S ((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) + ((dst_negative_scale_zero_entrysource) + (dst_negative_scale_zero_entrysource)))) * S ((((dst_positive_code_zero_entrysource) + (dst_positive_scale_zero_entrysource)) * S ((dst_positive_code_zero_entrysource) + (dst_positive_scale_zero_entrysource)) + ((dst_positive_scale_zero_entrysource) + (dst_positive_scale_zero_entrysource))) + (((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) * S ((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) + ((dst_negative_scale_zero_entrysource) + (dst_negative_scale_zero_entrysource)))) + ((((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) * S ((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) + ((dst_negative_scale_zero_entrysource) + (dst_negative_scale_zero_entrysource))) + (((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) * S ((dst_negative_code_zero_entrysource) + (dst_negative_scale_zero_entrysource)) + ((dst_negative_scale_zero_entrysource) + (dst_negative_scale_zero_entrysource)))))) /\ (((((exists ff_h_pvs_zero_entrysourcepositive. ff_h_pvs_zero_entrysourcepositive + S (dst_positive_zero_entrysource) = S ((S (i)) * dst_positive_scale_zero_entrysource)) /\ exists ff_q_pvs_zero_entrysourcepositive. dst_positive_code_zero_entrysource = ff_q_pvs_zero_entrysourcepositive * S ((S (i)) * dst_positive_scale_zero_entrysource) + (dst_positive_zero_entrysource))) /\ (((((exists ff_h_pvs_zero_entrysourcenegative. ff_h_pvs_zero_entrysourcenegative + S (dst_negative_zero_entrysource) = S ((S (i)) * dst_negative_scale_zero_entrysource)) /\ exists ff_q_pvs_zero_entrysourcenegative. dst_negative_code_zero_entrysource = ff_q_pvs_zero_entrysourcenegative * S ((S (i)) * dst_negative_scale_zero_entrysource) + (dst_negative_zero_entrysource))) /\ (exists ge_balance_positive_zero_entrysourcevalue ge_balance_negative_zero_entrysourcevalue. (((((ssr_entry_value_zero_entry) = 2 * (ge_balance_positive_zero_entrysourcevalue) /\ (ge_balance_negative_zero_entrysourcevalue) = 0) \/ exists ge_signed_half_zero_entrysourcevaluedecode. (((ssr_entry_value_zero_entry) = 2 * ge_signed_half_zero_entrysourcevaluedecode + 1 /\ (ge_balance_positive_zero_entrysourcevalue) = 0) /\ (ge_balance_negative_zero_entrysourcevalue) = S ge_signed_half_zero_entrysourcevaluedecode))) /\ ((dst_positive_zero_entrysource) + ge_balance_negative_zero_entrysourcevalue = (dst_negative_zero_entrysource) + ge_balance_positive_zero_entrysourcevalue))))))))) /\ (((((exists ff_h_pvs_zero_entrymap. ff_h_pvs_zero_entrymap + S (ssr_entry_image_zero_entry) = S ((S (i)) * s)) /\ exists ff_q_pvs_zero_entrymap. r = ff_q_pvs_zero_entrymap * S ((S (i)) * s) + (ssr_entry_image_zero_entry))) /\ (((((j)=(ssr_entry_image_zero_entry)) /\ ((z)=(ssr_entry_value_zero_entry)))) \/ (((~((j)=(ssr_entry_image_zero_entry))) /\ ((z)=0)))))))) -> z=0

Constructive proof overview

Generated structural guide

Every incidence cell of a represented zero source is zero, even at an out-of-window image.

The unchanged tactic script uses 1 declared prerequisite and contains 27 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

Direct dependents

MX0046 signed_support_incidence_row_sum_value

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

27 script commands · 8 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 assumptionsL1–8

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 z
  7. L7
    intro ha
  8. L8
    intro he
02Separate the logical casesL9–12

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

  1. L9
    cases he
  2. L10
    cases he_witness
  3. L11
    cases he_witness_witness
  4. L12
    cases he_witness_witness_right
03Establish hvL13–20

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

  1. L13
    have hv : x=0
  2. L14
    specialize divisor_signed_table_at_functional (A)
  3. L15
    specialize divisor_signed_table_at_functional (i)
  4. L16
    specialize divisor_signed_table_at_functional (x)
  5. L17
    specialize divisor_signed_table_at_functional (0)
  6. L18
    apply divisor_signed_table_at_functional
  7. L19
    exact he_witness_witness_left
  8. L20
    exact ha
04Separate the logical casesL21–22

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

  1. L21
    cases he_witness_witness_right_right
  2. L22
    cases he_witness_witness_right_right_left
05Calculate and transport equalitiesL23–23

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

  1. L23
    trans x
06Use earlier factsL24–25

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

  1. L24
    exact he_witness_witness_right_right_left_right
  2. L25
    exact hv
07Separate the logical casesL26–26

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

  1. L26
    cases he_witness_witness_right_right_right
08Use earlier factsL27–27

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

  1. L27
    exact he_witness_witness_right_right_right_right

Library-wide reading audit

Original exact command ledger · 27 lines
  1. 0001intro A
  2. 0002intro r
  3. 0003intro s
  4. 0004intro i
  5. 0005intro j
  6. 0006intro z
  7. 0007intro ha
  8. 0008intro he
  9. 0009cases he
  10. 0010cases he_witness
  11. 0011cases he_witness_witness
  12. 0012cases he_witness_witness_right
  13. 0013have hv : x=0
  14. 0014specialize divisor_signed_table_at_functional (A)
  15. 0015specialize divisor_signed_table_at_functional (i)
  16. 0016specialize divisor_signed_table_at_functional (x)
  17. 0017specialize divisor_signed_table_at_functional (0)
  18. 0018apply divisor_signed_table_at_functional
  19. 0019exact he_witness_witness_left
  20. 0020exact ha
  21. 0021cases he_witness_witness_right_right
  22. 0022cases he_witness_witness_right_right_left
  23. 0023trans x
  24. 0024exact he_witness_witness_right_right_left_right
  25. 0025exact hv
  26. 0026cases he_witness_witness_right_right_right
  27. 0027exact he_witness_witness_right_right_right_right