FS004D · theorem body

four_square_equal_square_remainders_are_congruent

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

Equal canonical remainders of two natural squares give their balanced modular congruence.

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. ∀ x. ∀ y. ∀ q. ∀ r. ∀ t. x · x = p · q + t → y · y = p · r + t → ModEq(p,x · x,y · y)

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

PD0008 ModEq

In local proof propositions

none
Exact expanded first-order statement
forall p x y q r t. x * x = p * q + t -> y * y = p * r + t -> (exists fsri_left_equal_remainders fsri_right_equal_remainders. (x * x) + (p) * fsri_left_equal_remainders = (y * y) + (p) * fsri_right_equal_remainders)

Proof neighborhood

Direct theorem prerequisites

add_assoc · Stable closed add_comm · Stable closed

Direct theorem dependents

FS004E four_square_half_square_residue_prefix_injective

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 · 8 reading checkpoints · 0 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 assumptionsL1–8

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

  1. L1
    intro p
  2. L2
    intro x
  3. L3
    intro y
  4. L4
    intro q
  5. L5
    intro r
  6. L6
    intro t
  7. L7
    intro hx
  8. L8
    intro hy
02Construct an explicit witnessL9–10

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

  1. L9
    exists r
  2. L10
    exists q
03Calculate and transport equalitiesL11–15

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

  1. L11
    rewrite hx
  2. L12
    rewrite hy
  3. L13
    simp [add_assoc, add_comm]
  4. L14
    trans (t + p * r) + p * q
  5. L15
    symm
04Use earlier factsL16–16

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

  1. L16
    apply add_assoc
05Calculate and transport equalitiesL17–18

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

  1. L17
    trans (p * r + t) + p * q
  2. L18
    congr
06Use earlier factsL19–19

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

  1. L19
    apply add_comm
07Calculate and transport equalitiesL20–20

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

  1. L20
    refl
08Use earlier factsL21–21

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

  1. L21
    apply add_assoc

Library-wide reading audit

Original defined command ledger · 21 lines
  1. 0001intro p
  2. 0002intro x
  3. 0003intro y
  4. 0004intro q
  5. 0005intro r
  6. 0006intro t
  7. 0007intro hx
  8. 0008intro hy
  9. 0009exists r
  10. 0010exists q
  11. 0011rewrite hx
  12. 0012rewrite hy
  13. 0013simp [add_assoc, add_comm]
  14. 0014trans (t + p * r) + p * q
  15. 0015symm
  16. 0016apply add_assoc
  17. 0017trans (p * r + t) + p * q
  18. 0018congr
  19. 0019apply add_comm
  20. 0020refl
  21. 0021apply add_assoc