FS0025

four_square_euler_cross_swap

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

Two products with fixed outer factors exchange their crossed middle factors.

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 a b c d. (a * b) * (c * d) = (a * d) * (c * b)

Constructive proof overview

Generated structural guide

Two products with fixed outer factors exchange their crossed middle factors.

The unchanged tactic script uses 2 declared prerequisites and contains 12 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_shuffle_four Alpha theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

FS0026 four_square_euler_mixed_ab FS0027 four_square_euler_mixed_ac FS0028 four_square_euler_mixed_ad FS0029 four_square_euler_mixed_bc FS002A four_square_euler_mixed_bd FS002B four_square_euler_mixed_cd FS002Q four_square_euler_double_cross_swap

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

12 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 a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Calculate and transport equalitiesL5–5

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

  1. L5
    trans (a * c) * (b * d)
03Use earlier factsL6–6

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

  1. L6
    apply mul_shuffle_four
04Calculate and transport equalitiesL7–9

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

  1. L7
    trans (a * c) * (d * b)
  2. L8
    congr
  3. L9
    refl
05Use earlier factsL10–10

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

  1. L10
    apply mul_comm
06Calculate and transport equalitiesL11–11

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

  1. L11
    symm
07Use earlier factsL12–12

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

  1. L12
    apply mul_shuffle_four

Library-wide reading audit

Original exact command ledger · 12 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans (a * c) * (b * d)
  6. 0006apply mul_shuffle_four
  7. 0007trans (a * c) * (d * b)
  8. 0008congr
  9. 0009refl
  10. 0010apply mul_comm
  11. 0011symm
  12. 0012apply mul_shuffle_four