FS005Q

four_square_signed_mod_zero_equivalent

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

Any two natural signed blocks that both vanish modulo the multiplier are congruent.

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_equiv_left ftcn_right_fssq_zero_equiv_left. (a) + (k) * ftcn_left_fssq_zero_equiv_left = (0) + (k) * ftcn_right_fssq_zero_equiv_left) -> (exists ftcn_left_fssq_zero_equiv_right ftcn_right_fssq_zero_equiv_right. (b) + (k) * ftcn_left_fssq_zero_equiv_right = (0) + (k) * ftcn_right_fssq_zero_equiv_right) -> (exists ftcn_left_fssq_zero_equiv_result ftcn_right_fssq_zero_equiv_result. (a) + (k) * ftcn_left_fssq_zero_equiv_result = (b) + (k) * ftcn_right_fssq_zero_equiv_result)

Constructive proof overview

Generated structural guide

Any two natural signed blocks that both vanish modulo the multiplier are congruent.

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_symm Stable theorem; checked-use authorized mod_eq_trans Stable theorem; checked-use authorized

Direct dependents

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 · 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 assumptionsL1–5

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 hreverseL6–15

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

  1. L6
    have hreverse : exists ftcn_left_fssq_zero_equiv_reverse ftcn_right_fssq_zero_equiv_reverse. (0) + (k) * ftcn_left_fssq_zero_equiv_reverse = (b) + (k) * ftcn_right_fssq_zero_equiv_reverse
  2. L7
    specialize mod_eq_symm k
  3. L8
    specialize mod_eq_symm b
  4. L9
    specialize mod_eq_symm 0
  5. L10
    apply mod_eq_symm
  6. L11
    exact hb
  7. L12
    specialize mod_eq_trans k
  8. L13
    specialize mod_eq_trans a
  9. L14
    specialize mod_eq_trans 0
  10. L15
    specialize mod_eq_trans b
03Use earlier factsL16–18

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

  1. L16
    apply mod_eq_trans
  2. L17
    exact ha
  3. L18
    exact hreverse

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 hreverse : exists ftcn_left_fssq_zero_equiv_reverse ftcn_right_fssq_zero_equiv_reverse. (0) + (k) * ftcn_left_fssq_zero_equiv_reverse = (b) + (k) * ftcn_right_fssq_zero_equiv_reverse
  7. 0007specialize mod_eq_symm k
  8. 0008specialize mod_eq_symm b
  9. 0009specialize mod_eq_symm 0
  10. 0010apply mod_eq_symm
  11. 0011exact hb
  12. 0012specialize mod_eq_trans k
  13. 0013specialize mod_eq_trans a
  14. 0014specialize mod_eq_trans 0
  15. 0015specialize mod_eq_trans b
  16. 0016apply mod_eq_trans
  17. 0017exact ha
  18. 0018exact hreverse