FS0047

four_square_ordered_square_difference_factor

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

The ordered difference of two natural squares factors subtraction-free as their gap times their sum.

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 x y d. y = x + d -> x * x + d * (y + x) = y * y

Constructive proof overview

Generated structural guide

The ordered difference of two natural squares factors subtraction-free as their gap times their sum.

The unchanged tactic script uses 5 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

mul_add Stable theorem; checked-use authorized add_mul Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

FS0048 four_square_ordered_square_congruence_factors

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 · 7 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 assumptionsL1–4

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

  1. L1
    intro x
  2. L2
    intro y
  3. L3
    intro d
  4. L4
    intro horder
02Calculate and transport equalitiesL5–12

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

  1. L5
    rewrite horder
  2. L6
    rewrite horder
  3. L7
    rewrite horder
  4. L8
    simp [mul_add, add_mul, mul_comm, add_assoc, add_comm]
  5. L9
    congr
  6. L10
    refl
  7. L11
    trans (d * d + d * x) + x * x
  8. L12
    symm
03Use earlier factsL13–13

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

  1. L13
    apply add_assoc
04Calculate and transport equalitiesL14–15

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

  1. L14
    trans (d * x + d * d) + x * x
  2. L15
    congr
05Use earlier factsL16–16

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

  1. L16
    apply add_comm
06Calculate and transport equalitiesL17–17

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

  1. L17
    refl
07Use earlier factsL18–18

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

  1. L18
    apply add_assoc

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro x
  2. 0002intro y
  3. 0003intro d
  4. 0004intro horder
  5. 0005rewrite horder
  6. 0006rewrite horder
  7. 0007rewrite horder
  8. 0008simp [mul_add, add_mul, mul_comm, add_assoc, add_comm]
  9. 0009congr
  10. 0010refl
  11. 0011trans (d * d + d * x) + x * x
  12. 0012symm
  13. 0013apply add_assoc
  14. 0014trans (d * x + d * d) + x * x
  15. 0015congr
  16. 0016apply add_comm
  17. 0017refl
  18. 0018apply add_assoc