FS000D

four_square_product_shuffle

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

Two natural product factors interchange their middle terms; this is the bounded Euler cross-term cancellation primitive.

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 * c) * (b * d)

Constructive proof overview

Generated structural guide

Two natural product factors interchange their middle terms; this is the bounded Euler cross-term cancellation primitive.

The unchanged tactic script uses 2 declared prerequisites and contains 23 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_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized

Direct dependents

FS000E four_square_product_square

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

23 script commands · 11 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 * (b * (c * d))
03Use earlier factsL6–6

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

  1. L6
    apply mul_assoc
04Calculate and transport equalitiesL7–10

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

  1. L7
    trans a * ((b * c) * d)
  2. L8
    congr
  3. L9
    refl
  4. L10
    symm
05Use earlier factsL11–11

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

  1. L11
    apply mul_assoc
06Calculate and transport equalitiesL12–15

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

  1. L12
    trans a * ((c * b) * d)
  2. L13
    congr
  3. L14
    refl
  4. L15
    congr
07Use earlier factsL16–16

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

  1. L16
    apply mul_comm
08Calculate and transport equalitiesL17–20

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

  1. L17
    refl
  2. L18
    trans a * (c * (b * d))
  3. L19
    congr
  4. L20
    refl
09Use earlier factsL21–21

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

  1. L21
    apply mul_assoc
10Calculate and transport equalitiesL22–22

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

  1. L22
    symm
11Use earlier factsL23–23

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

  1. L23
    apply mul_assoc

Library-wide reading audit

Original exact command ledger · 23 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans a * (b * (c * d))
  6. 0006apply mul_assoc
  7. 0007trans a * ((b * c) * d)
  8. 0008congr
  9. 0009refl
  10. 0010symm
  11. 0011apply mul_assoc
  12. 0012trans a * ((c * b) * d)
  13. 0013congr
  14. 0014refl
  15. 0015congr
  16. 0016apply mul_comm
  17. 0017refl
  18. 0018trans a * (c * (b * d))
  19. 0019congr
  20. 0020refl
  21. 0021apply mul_assoc
  22. 0022symm
  23. 0023apply mul_assoc