FS004D

four_square_equal_square_remainders_are_congruent

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

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

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 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)

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 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

add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

FS004E four_square_half_square_residue_prefix_injective

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 · 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.

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 exact 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