FS005P

four_square_signed_mod_zero_add

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

Two independently vanishing signed blocks have vanishing sum modulo the multiplier.

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 original first-admission records.

Exact expanded first-order arithmetic statement

forall k a b. (exists ftcn_left_fssq_zero_add_left ftcn_right_fssq_zero_add_left. (a) + (k) * ftcn_left_fssq_zero_add_left = (0) + (k) * ftcn_right_fssq_zero_add_left) -> (exists ftcn_left_fssq_zero_add_right ftcn_right_fssq_zero_add_right. (b) + (k) * ftcn_left_fssq_zero_add_right = (0) + (k) * ftcn_right_fssq_zero_add_right) -> (exists ftcn_left_fssq_zero_add_result ftcn_right_fssq_zero_add_result. (a + b) + (k) * ftcn_left_fssq_zero_add_result = (0) + (k) * ftcn_right_fssq_zero_add_result)

Constructive proof overview

Generated structural guide

Two independently vanishing signed blocks have vanishing sum modulo the multiplier.

The unchanged tactic script uses 2 declared prerequisites and contains 18 exact native proof lines.

dependency-curried kernel-checked theorem body; Alpha enrollment and checked-use authority follow separately sealed release evidence; Stable membership remains unchanged

Historical empty-context replay experiment only; that experiment persisted no certificate and granted no release authority. Current checked use follows separately sealed, independently verified proof bundles; there is no Stable promotion.

Proof neighborhood

Direct dependencies

mod_eq_add Stable theorem; checked-use authorized zero_add Stable theorem; checked-use authorized

Direct dependents

FS004N four_square_signed_conjugate_negative_blocks FS004O four_square_signed_natural_positive_first_blocks FS005Y four_square_signed_conjugate_positive_blocks FS005Z four_square_signed_conjugate_mixed_blocks FS0060 four_square_signed_natural_negative_first_blocks

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

18 script commands · 3 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 assumptionsL1–5

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

  1. L1
    intro k
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro ha
  5. L5
    intro hb
02Establish hsumL6–14

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

  1. L6
    have hsum : exists ftcn_left_fssq_zero_add_sum ftcn_right_fssq_zero_add_sum. (a + b) + (k) * ftcn_left_fssq_zero_add_sum = (0 + 0) + (k) * ftcn_right_fssq_zero_add_sum
  2. L7
    specialize mod_eq_add k
  3. L8
    specialize mod_eq_add a
  4. L9
    specialize mod_eq_add 0
  5. L10
    specialize mod_eq_add b
  6. L11
    specialize mod_eq_add 0
  7. L12
    apply mod_eq_add
  8. L13
    exact ha
  9. L14
    exact hb
03Establish hzeroL15–18

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

  1. L15
    have hzero : 0 + 0 = 0
  2. L16
    apply zero_add
  3. L17
    rewrite hzero at hsum
  4. L18
    exact hsum

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro k
  2. 0002intro a
  3. 0003intro b
  4. 0004intro ha
  5. 0005intro hb
  6. 0006have hsum : exists ftcn_left_fssq_zero_add_sum ftcn_right_fssq_zero_add_sum. (a + b) + (k) * ftcn_left_fssq_zero_add_sum = (0 + 0) + (k) * ftcn_right_fssq_zero_add_sum
  7. 0007specialize mod_eq_add k
  8. 0008specialize mod_eq_add a
  9. 0009specialize mod_eq_add 0
  10. 0010specialize mod_eq_add b
  11. 0011specialize mod_eq_add 0
  12. 0012apply mod_eq_add
  13. 0013exact ha
  14. 0014exact hb
  15. 0015have hzero : 0 + 0 = 0
  16. 0016apply zero_add
  17. 0017rewrite hzero at hsum
  18. 0018exact hsum