FS004J

four_square_complementary_remainders_form_multiple

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

Two square residues whose canonical remainders are predecessor-complements yield an explicit divisor of their sum plus one.

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 p a b q r s t. a = p * q + r -> b = p * s + t -> r + S t = p -> (exists fsri_factor_complementary_multiple. (a + b + 1) = (p) * fsri_factor_complementary_multiple)

Constructive proof overview

Generated structural guide

Two square residues whose canonical remainders are predecessor-complements yield an explicit divisor of their sum plus one.

The unchanged tactic script uses 5 declared prerequisites and contains 21 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

mul_add Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_succ_left Stable theorem; checked-use authorized mul_one Stable theorem; checked-use authorized add_shuffle_middle Alpha theorem; checked-use authorized

Direct dependents

FS000K four_square_odd_prime_half_coordinate_seed FS004K four_square_odd_prime_modular_seed

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

21 script commands · 5 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 assumptionsL1–10

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

  1. L1
    intro p
  2. L2
    intro a
  3. L3
    intro b
  4. L4
    intro q
  5. L5
    intro r
  6. L6
    intro s
  7. L7
    intro t
  8. L8
    intro ha
  9. L9
    intro hb
  10. L10
    intro hcomplement
02Construct an explicit witnessL11–11

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

  1. L11
    exists (q + s + 1)
03Calculate and transport equalitiesL12–13

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

  1. L12
    rewrite ha
  2. L13
    rewrite hb
04Establish hshuffleL14–16

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

  1. L14
    have hshuffle : (p * q + r) + (p * s + t) = (p * q + p * s) + (r + t)
  2. L15
    apply add_shuffle_middle
  3. L16
    rewrite hshuffle
05Establish hleftL17–21

Establish this local claim before using it. It is not an additional assumption.

  1. L17
    have hleft : ((p * q + p * s) + (r + t)) + 1 = (p * q + p * s) + (r + S t)
  2. L18
    simp [add_assoc, add_succ_left]
  3. L19
    rewrite hleft
  4. L20
    rewrite hcomplement
  5. L21
    simp [mul_add, mul_one, add_assoc]

Library-wide reading audit

Original exact command ledger · 21 lines
  1. 0001intro p
  2. 0002intro a
  3. 0003intro b
  4. 0004intro q
  5. 0005intro r
  6. 0006intro s
  7. 0007intro t
  8. 0008intro ha
  9. 0009intro hb
  10. 0010intro hcomplement
  11. 0011exists (q + s + 1)
  12. 0012rewrite ha
  13. 0013rewrite hb
  14. 0014have hshuffle : (p * q + r) + (p * s + t) = (p * q + p * s) + (r + t)
  15. 0015apply add_shuffle_middle
  16. 0016rewrite hshuffle
  17. 0017have hleft : ((p * q + p * s) + (r + t)) + 1 = (p * q + p * s) + (r + S t)
  18. 0018simp [add_assoc, add_succ_left]
  19. 0019rewrite hleft
  20. 0020rewrite hcomplement
  21. 0021simp [mul_add, mul_one, add_assoc]