WS0019

signed_table_add_medial

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

The actual four canonical signed summands may be regrouped across two prefix/last-entry pairs, with a genuinely constructed intermediate sum.

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 b c d ab cd ac bd out. (exists dsa_ap_medial_ab dsa_an_medial_ab dsa_bp_medial_ab dsa_bn_medial_ab dsa_cp_medial_ab dsa_cn_medial_ab. (((((a) = 2 * (dsa_ap_medial_ab) /\ (dsa_an_medial_ab) = 0) \/ exists ge_signed_half_medial_ableft. (((a) = 2 * ge_signed_half_medial_ableft + 1 /\ (dsa_ap_medial_ab) = 0) /\ (dsa_an_medial_ab) = S ge_signed_half_medial_ableft))) /\ ((((((b) = 2 * (dsa_bp_medial_ab) /\ (dsa_bn_medial_ab) = 0) \/ exists ge_signed_half_medial_abright. (((b) = 2 * ge_signed_half_medial_abright + 1 /\ (dsa_bp_medial_ab) = 0) /\ (dsa_bn_medial_ab) = S ge_signed_half_medial_abright))) /\ ((((((ab) = 2 * (dsa_cp_medial_ab) /\ (dsa_cn_medial_ab) = 0) \/ exists ge_signed_half_medial_aboutput. (((ab) = 2 * ge_signed_half_medial_aboutput + 1 /\ (dsa_cp_medial_ab) = 0) /\ (dsa_cn_medial_ab) = S ge_signed_half_medial_aboutput))) /\ ((dsa_ap_medial_ab + dsa_bp_medial_ab) + dsa_cn_medial_ab = (dsa_an_medial_ab + dsa_bn_medial_ab) + dsa_cp_medial_ab))))))) -> (exists dsa_ap_medial_cd dsa_an_medial_cd dsa_bp_medial_cd dsa_bn_medial_cd dsa_cp_medial_cd dsa_cn_medial_cd. (((((c) = 2 * (dsa_ap_medial_cd) /\ (dsa_an_medial_cd) = 0) \/ exists ge_signed_half_medial_cdleft. (((c) = 2 * ge_signed_half_medial_cdleft + 1 /\ (dsa_ap_medial_cd) = 0) /\ (dsa_an_medial_cd) = S ge_signed_half_medial_cdleft))) /\ ((((((d) = 2 * (dsa_bp_medial_cd) /\ (dsa_bn_medial_cd) = 0) \/ exists ge_signed_half_medial_cdright. (((d) = 2 * ge_signed_half_medial_cdright + 1 /\ (dsa_bp_medial_cd) = 0) /\ (dsa_bn_medial_cd) = S ge_signed_half_medial_cdright))) /\ ((((((cd) = 2 * (dsa_cp_medial_cd) /\ (dsa_cn_medial_cd) = 0) \/ exists ge_signed_half_medial_cdoutput. (((cd) = 2 * ge_signed_half_medial_cdoutput + 1 /\ (dsa_cp_medial_cd) = 0) /\ (dsa_cn_medial_cd) = S ge_signed_half_medial_cdoutput))) /\ ((dsa_ap_medial_cd + dsa_bp_medial_cd) + dsa_cn_medial_cd = (dsa_an_medial_cd + dsa_bn_medial_cd) + dsa_cp_medial_cd))))))) -> (exists dsa_ap_medial_ac dsa_an_medial_ac dsa_bp_medial_ac dsa_bn_medial_ac dsa_cp_medial_ac dsa_cn_medial_ac. (((((a) = 2 * (dsa_ap_medial_ac) /\ (dsa_an_medial_ac) = 0) \/ exists ge_signed_half_medial_acleft. (((a) = 2 * ge_signed_half_medial_acleft + 1 /\ (dsa_ap_medial_ac) = 0) /\ (dsa_an_medial_ac) = S ge_signed_half_medial_acleft))) /\ ((((((c) = 2 * (dsa_bp_medial_ac) /\ (dsa_bn_medial_ac) = 0) \/ exists ge_signed_half_medial_acright. (((c) = 2 * ge_signed_half_medial_acright + 1 /\ (dsa_bp_medial_ac) = 0) /\ (dsa_bn_medial_ac) = S ge_signed_half_medial_acright))) /\ ((((((ac) = 2 * (dsa_cp_medial_ac) /\ (dsa_cn_medial_ac) = 0) \/ exists ge_signed_half_medial_acoutput. (((ac) = 2 * ge_signed_half_medial_acoutput + 1 /\ (dsa_cp_medial_ac) = 0) /\ (dsa_cn_medial_ac) = S ge_signed_half_medial_acoutput))) /\ ((dsa_ap_medial_ac + dsa_bp_medial_ac) + dsa_cn_medial_ac = (dsa_an_medial_ac + dsa_bn_medial_ac) + dsa_cp_medial_ac))))))) -> (exists dsa_ap_medial_bd dsa_an_medial_bd dsa_bp_medial_bd dsa_bn_medial_bd dsa_cp_medial_bd dsa_cn_medial_bd. (((((b) = 2 * (dsa_ap_medial_bd) /\ (dsa_an_medial_bd) = 0) \/ exists ge_signed_half_medial_bdleft. (((b) = 2 * ge_signed_half_medial_bdleft + 1 /\ (dsa_ap_medial_bd) = 0) /\ (dsa_an_medial_bd) = S ge_signed_half_medial_bdleft))) /\ ((((((d) = 2 * (dsa_bp_medial_bd) /\ (dsa_bn_medial_bd) = 0) \/ exists ge_signed_half_medial_bdright. (((d) = 2 * ge_signed_half_medial_bdright + 1 /\ (dsa_bp_medial_bd) = 0) /\ (dsa_bn_medial_bd) = S ge_signed_half_medial_bdright))) /\ ((((((bd) = 2 * (dsa_cp_medial_bd) /\ (dsa_cn_medial_bd) = 0) \/ exists ge_signed_half_medial_bdoutput. (((bd) = 2 * ge_signed_half_medial_bdoutput + 1 /\ (dsa_cp_medial_bd) = 0) /\ (dsa_cn_medial_bd) = S ge_signed_half_medial_bdoutput))) /\ ((dsa_ap_medial_bd + dsa_bp_medial_bd) + dsa_cn_medial_bd = (dsa_an_medial_bd + dsa_bn_medial_bd) + dsa_cp_medial_bd))))))) -> (exists dsa_ap_medial_out dsa_an_medial_out dsa_bp_medial_out dsa_bn_medial_out dsa_cp_medial_out dsa_cn_medial_out. (((((ac) = 2 * (dsa_ap_medial_out) /\ (dsa_an_medial_out) = 0) \/ exists ge_signed_half_medial_outleft. (((ac) = 2 * ge_signed_half_medial_outleft + 1 /\ (dsa_ap_medial_out) = 0) /\ (dsa_an_medial_out) = S ge_signed_half_medial_outleft))) /\ ((((((bd) = 2 * (dsa_bp_medial_out) /\ (dsa_bn_medial_out) = 0) \/ exists ge_signed_half_medial_outright. (((bd) = 2 * ge_signed_half_medial_outright + 1 /\ (dsa_bp_medial_out) = 0) /\ (dsa_bn_medial_out) = S ge_signed_half_medial_outright))) /\ ((((((out) = 2 * (dsa_cp_medial_out) /\ (dsa_cn_medial_out) = 0) \/ exists ge_signed_half_medial_outoutput. (((out) = 2 * ge_signed_half_medial_outoutput + 1 /\ (dsa_cp_medial_out) = 0) /\ (dsa_cn_medial_out) = S ge_signed_half_medial_outoutput))) /\ ((dsa_ap_medial_out + dsa_bp_medial_out) + dsa_cn_medial_out = (dsa_an_medial_out + dsa_bn_medial_out) + dsa_cp_medial_out))))))) -> (exists dsa_ap_medial_target dsa_an_medial_target dsa_bp_medial_target dsa_bn_medial_target dsa_cp_medial_target dsa_cn_medial_target. (((((ab) = 2 * (dsa_ap_medial_target) /\ (dsa_an_medial_target) = 0) \/ exists ge_signed_half_medial_targetleft. (((ab) = 2 * ge_signed_half_medial_targetleft + 1 /\ (dsa_ap_medial_target) = 0) /\ (dsa_an_medial_target) = S ge_signed_half_medial_targetleft))) /\ ((((((cd) = 2 * (dsa_bp_medial_target) /\ (dsa_bn_medial_target) = 0) \/ exists ge_signed_half_medial_targetright. (((cd) = 2 * ge_signed_half_medial_targetright + 1 /\ (dsa_bp_medial_target) = 0) /\ (dsa_bn_medial_target) = S ge_signed_half_medial_targetright))) /\ ((((((out) = 2 * (dsa_cp_medial_target) /\ (dsa_cn_medial_target) = 0) \/ exists ge_signed_half_medial_targetoutput. (((out) = 2 * ge_signed_half_medial_targetoutput + 1 /\ (dsa_cp_medial_target) = 0) /\ (dsa_cn_medial_target) = S ge_signed_half_medial_targetoutput))) /\ ((dsa_ap_medial_target + dsa_bp_medial_target) + dsa_cn_medial_target = (dsa_an_medial_target + dsa_bn_medial_target) + dsa_cp_medial_target)))))))

Constructive proof overview

Generated structural guide

The actual four canonical signed summands may be regrouped across two prefix/last-entry pairs, with a genuinely constructed intermediate sum.

The unchanged tactic script uses 4 declared prerequisites and contains 59 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

signed_add_total Alpha theorem; checked-use authorized signed_add_associative Alpha theorem; checked-use authorized signed_add_commutative Alpha theorem; checked-use authorized WS0018 signed_table_add_reassociate

Direct dependents

WS001B signed_prefix_sum_pointwise_add

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

59 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)
signed table add reassociate

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 assumptionsL1–10

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

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
  5. L5
    intro ab
  6. L6
    intro cd
  7. L7
    intro ac
  8. L8
    intro bd
  9. L9
    intro out
  10. L10
    intro hab
02Fix variables and assumptionsL11–14

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

  1. L11
    intro hcd
  2. L12
    intro hac
  3. L13
    intro hbd
  4. L14
    intro hout
03Establish hcyL15–18

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

  1. L15
    have hcy : ∃ w. SignedAdd(c,bd,w)Definitions: SignedAdd
  2. L16
    specialize signed_add_total (c)
  3. L17
    specialize signed_add_total (bd)
  4. L18
    apply signed_add_total
04Separate the logical casesL19–19

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

  1. L19
    cases hcy
05Establish hrightL20–29

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

  1. L20
    have hright : SignedAdd(b,cd,x)Definitions: SignedAdd
  2. L21
    specialize signed_add_associative (b)
  3. L22
    specialize signed_add_associative (d)
  4. L23
    specialize signed_add_associative (c)
  5. L24
    specialize signed_add_associative (bd)
  6. L25
    specialize signed_add_associative (cd)
  7. L26
    specialize signed_add_associative (x)
  8. L27
    apply signed_add_associative
  9. L28
    exact hbd
  10. L29
    specialize signed_add_commutative (c)
06Use earlier factsL30–38

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

  1. L30
    specialize signed_add_commutative (bd)
  2. L31
    specialize signed_add_commutative (x)
  3. L32
    apply signed_add_commutative
  4. L33
    exact hcy_witness
  5. L34
    specialize signed_add_commutative (c)
  6. L35
    specialize signed_add_commutative (d)
  7. L36
    specialize signed_add_commutative (cd)
  8. L37
    apply signed_add_commutative
  9. L38
    exact hcd
07Establish hfullL39–48

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

  1. L39
    have hfull : SignedAdd(a,x,out)Definitions: SignedAdd
  2. L40
    specialize signed_add_associative (a)
  3. L41
    specialize signed_add_associative (c)
  4. L42
    specialize signed_add_associative (bd)
  5. L43
    specialize signed_add_associative (ac)
  6. L44
    specialize signed_add_associative (x)
  7. L45
    specialize signed_add_associative (out)
  8. L46
    apply signed_add_associative
  9. L47
    exact hac
  10. L48
    exact hout
08Use earlier factsL49–58

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

  1. L49
    exact hcy_witness
  2. L50
    specialize signed_table_add_reassociate (a)
  3. L51
    specialize signed_table_add_reassociate (b)
  4. L52
    specialize signed_table_add_reassociate (cd)
  5. L53
    specialize signed_table_add_reassociate (ab)
  6. L54
    specialize signed_table_add_reassociate (x)
  7. L55
    specialize signed_table_add_reassociate (out)
  8. L56
    apply signed_table_add_reassociate
  9. L57
    exact hab
  10. L58
    exact hright
09Use earlier factsL59–59

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

  1. L59
    exact hfull

Library-wide reading audit

Original exact command ledger · 59 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro ab
  6. 0006intro cd
  7. 0007intro ac
  8. 0008intro bd
  9. 0009intro out
  10. 0010intro hab
  11. 0011intro hcd
  12. 0012intro hac
  13. 0013intro hbd
  14. 0014intro hout
  15. 0015have hcy : exists w. (exists dsa_ap_medial_middle dsa_an_medial_middle dsa_bp_medial_middle dsa_bn_medial_middle dsa_cp_medial_middle dsa_cn_medial_middle. (((((c) = 2 * (dsa_ap_medial_middle) /\ (dsa_an_medial_middle) = 0) \/ exists ge_signed_half_medial_middleleft. (((c) = 2 * ge_signed_half_medial_middleleft + 1 /\ (dsa_ap_medial_middle) = 0) /\ (dsa_an_medial_middle) = S ge_signed_half_medial_middleleft))) /\ ((((((bd) = 2 * (dsa_bp_medial_middle) /\ (dsa_bn_medial_middle) = 0) \/ exists ge_signed_half_medial_middleright. (((bd) = 2 * ge_signed_half_medial_middleright + 1 /\ (dsa_bp_medial_middle) = 0) /\ (dsa_bn_medial_middle) = S ge_signed_half_medial_middleright))) /\ ((((((w) = 2 * (dsa_cp_medial_middle) /\ (dsa_cn_medial_middle) = 0) \/ exists ge_signed_half_medial_middleoutput. (((w) = 2 * ge_signed_half_medial_middleoutput + 1 /\ (dsa_cp_medial_middle) = 0) /\ (dsa_cn_medial_middle) = S ge_signed_half_medial_middleoutput))) /\ ((dsa_ap_medial_middle + dsa_bp_medial_middle) + dsa_cn_medial_middle = (dsa_an_medial_middle + dsa_bn_medial_middle) + dsa_cp_medial_middle)))))))
  16. 0016specialize signed_add_total (c)
  17. 0017specialize signed_add_total (bd)
  18. 0018apply signed_add_total
  19. 0019cases hcy
  20. 0020have hright : exists dsa_ap_medial_right dsa_an_medial_right dsa_bp_medial_right dsa_bn_medial_right dsa_cp_medial_right dsa_cn_medial_right. (((((b) = 2 * (dsa_ap_medial_right) /\ (dsa_an_medial_right) = 0) \/ exists ge_signed_half_medial_rightleft. (((b) = 2 * ge_signed_half_medial_rightleft + 1 /\ (dsa_ap_medial_right) = 0) /\ (dsa_an_medial_right) = S ge_signed_half_medial_rightleft))) /\ ((((((cd) = 2 * (dsa_bp_medial_right) /\ (dsa_bn_medial_right) = 0) \/ exists ge_signed_half_medial_rightright. (((cd) = 2 * ge_signed_half_medial_rightright + 1 /\ (dsa_bp_medial_right) = 0) /\ (dsa_bn_medial_right) = S ge_signed_half_medial_rightright))) /\ ((((((x) = 2 * (dsa_cp_medial_right) /\ (dsa_cn_medial_right) = 0) \/ exists ge_signed_half_medial_rightoutput. (((x) = 2 * ge_signed_half_medial_rightoutput + 1 /\ (dsa_cp_medial_right) = 0) /\ (dsa_cn_medial_right) = S ge_signed_half_medial_rightoutput))) /\ ((dsa_ap_medial_right + dsa_bp_medial_right) + dsa_cn_medial_right = (dsa_an_medial_right + dsa_bn_medial_right) + dsa_cp_medial_right))))))
  21. 0021specialize signed_add_associative (b)
  22. 0022specialize signed_add_associative (d)
  23. 0023specialize signed_add_associative (c)
  24. 0024specialize signed_add_associative (bd)
  25. 0025specialize signed_add_associative (cd)
  26. 0026specialize signed_add_associative (x)
  27. 0027apply signed_add_associative
  28. 0028exact hbd
  29. 0029specialize signed_add_commutative (c)
  30. 0030specialize signed_add_commutative (bd)
  31. 0031specialize signed_add_commutative (x)
  32. 0032apply signed_add_commutative
  33. 0033exact hcy_witness
  34. 0034specialize signed_add_commutative (c)
  35. 0035specialize signed_add_commutative (d)
  36. 0036specialize signed_add_commutative (cd)
  37. 0037apply signed_add_commutative
  38. 0038exact hcd
  39. 0039have hfull : exists dsa_ap_medial_full dsa_an_medial_full dsa_bp_medial_full dsa_bn_medial_full dsa_cp_medial_full dsa_cn_medial_full. (((((a) = 2 * (dsa_ap_medial_full) /\ (dsa_an_medial_full) = 0) \/ exists ge_signed_half_medial_fullleft. (((a) = 2 * ge_signed_half_medial_fullleft + 1 /\ (dsa_ap_medial_full) = 0) /\ (dsa_an_medial_full) = S ge_signed_half_medial_fullleft))) /\ ((((((x) = 2 * (dsa_bp_medial_full) /\ (dsa_bn_medial_full) = 0) \/ exists ge_signed_half_medial_fullright. (((x) = 2 * ge_signed_half_medial_fullright + 1 /\ (dsa_bp_medial_full) = 0) /\ (dsa_bn_medial_full) = S ge_signed_half_medial_fullright))) /\ ((((((out) = 2 * (dsa_cp_medial_full) /\ (dsa_cn_medial_full) = 0) \/ exists ge_signed_half_medial_fulloutput. (((out) = 2 * ge_signed_half_medial_fulloutput + 1 /\ (dsa_cp_medial_full) = 0) /\ (dsa_cn_medial_full) = S ge_signed_half_medial_fulloutput))) /\ ((dsa_ap_medial_full + dsa_bp_medial_full) + dsa_cn_medial_full = (dsa_an_medial_full + dsa_bn_medial_full) + dsa_cp_medial_full))))))
  40. 0040specialize signed_add_associative (a)
  41. 0041specialize signed_add_associative (c)
  42. 0042specialize signed_add_associative (bd)
  43. 0043specialize signed_add_associative (ac)
  44. 0044specialize signed_add_associative (x)
  45. 0045specialize signed_add_associative (out)
  46. 0046apply signed_add_associative
  47. 0047exact hac
  48. 0048exact hout
  49. 0049exact hcy_witness
  50. 0050specialize signed_table_add_reassociate (a)
  51. 0051specialize signed_table_add_reassociate (b)
  52. 0052specialize signed_table_add_reassociate (cd)
  53. 0053specialize signed_table_add_reassociate (ab)
  54. 0054specialize signed_table_add_reassociate (x)
  55. 0055specialize signed_table_add_reassociate (out)
  56. 0056apply signed_table_add_reassociate
  57. 0057exact hab
  58. 0058exact hright
  59. 0059exact hfull