GI0014

gaussian_signed_product_interchange_positive

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

Actual signed-product positive components permit interchange of the first two factors, by ordinary semiring certificates.

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 e f. ((((a) * (((((c) * (e))) + (((d) * (f))))))) + (((b) * (((((c) * (f))) + (((d) * (e)))))))) = ((((c) * (((((a) * (e))) + (((b) * (f))))))) + (((d) * (((((a) * (f))) + (((b) * (e))))))))

Constructive proof overview

Generated structural guide

Actual signed-product positive components permit interchange of the first two factors, by ordinary semiring certificates.

The unchanged tactic script uses 8 declared prerequisites and contains 7 exact native proof lines.

Alpha v34 checked-use · first admitted v28 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized mul_assoc 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 four_square_add_swap_right_tail Alpha theorem; checked-use authorized GI0001 natural_mul_swap_right_tail

Direct dependents

GI0017 gaussian_signed_product_shuffle

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

7 script commands · 2 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 assumptionsL1–6

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
  5. L5
    intro e
  6. L6
    intro f
02Calculate and transport equalitiesL7–7

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

  1. L7
    simp [add_mul, mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]

Library-wide reading audit

Original exact command ledger · 7 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005intro e
  6. 0006intro f
  7. 0007simp [add_mul, mul_add, mul_assoc, mul_comm, add_assoc, add_comm, four_square_add_swap_right_tail, natural_mul_swap_right_tail]