TS001S · theorem body

two_square_cross_product_interchange

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

The cross products ac·bd and ad·bc coincide by associativity and commutativity.

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

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

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

none

In local proof propositions

none
Exact expanded first-order statement
forall a b c d. (a * c) * (b * d) = (a * d) * (b * c)

Proof neighborhood

Direct theorem prerequisites

mul_shuffle_four · Alpha closed mul_comm · Stable closed

Direct theorem dependents

TS001W brahmagupta_fibonacci_two_square_identity

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

12 script commands · 5 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 assumptionsL1–4

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 equalitiesL5–6

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

  1. L5
    trans (a * b) * (c * d)
  2. L6
    symm
03Use earlier factsL7–7

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

  1. L7
    apply mul_shuffle_four
04Calculate and transport equalitiesL8–10

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

  1. L8
    trans (a * b) * (d * c)
  2. L9
    congr
  3. L10
    refl
05Use earlier factsL11–12

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

  1. L11
    apply mul_comm
  2. L12
    apply mul_shuffle_four

Library-wide reading audit

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