FS004J · theorem body

four_square_complementary_remainders_form_multiple

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

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

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

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.

Statement with defined notation

∀ p. ∀ a. ∀ b. ∀ q. ∀ r. ∀ s. ∀ t. a = p · q + r → b = p · s + t → r + S t = p → Dvd(p,a + b + 1)

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

PD0003 Dvd

In local proof propositions

none
Exact expanded first-order 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)

Proof neighborhood

Direct theorem prerequisites

mul_add · Stable closed add_assoc · Stable closed add_succ_left · Stable closed mul_one · Stable closed add_shuffle_middle · Alpha closed

Direct theorem dependents

FS000K four_square_odd_prime_half_coordinate_seed FS004K four_square_odd_prime_modular_seed

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

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 defined 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]