GF0013

gaussian_ring_raw_multiply_commutative

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

Actual signed-coordinate Gaussian multiply is commutative by ordinary natural identities.

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 ap an bp bn cp cn dp dn. (((((((((((ap) * (cp))) + (((an) * (cn))))) + (((((bp) * (dn))) + (((bn) * (dp))))))) + (((((((cp) * (an))) + (((cn) * (ap))))) + (((((dp) * (bp))) + (((dn) * (bn)))))))) = ((((((((cp) * (ap))) + (((cn) * (an))))) + (((((dp) * (bn))) + (((dn) * (bp))))))) + (((((((ap) * (cn))) + (((an) * (cp))))) + (((((bp) * (dp))) + (((bn) * (dn))))))))) /\ (((((((((ap) * (dp))) + (((an) * (dn))))) + (((((bp) * (cp))) + (((bn) * (cn))))))) + (((((((cp) * (bn))) + (((cn) * (bp))))) + (((((dp) * (an))) + (((dn) * (ap)))))))) = ((((((((cp) * (bp))) + (((cn) * (bn))))) + (((((dp) * (ap))) + (((dn) * (an))))))) + (((((((ap) * (dn))) + (((an) * (dp))))) + (((((bp) * (cn))) + (((bn) * (cp)))))))))))

Constructive proof overview

Generated structural guide

Actual signed-coordinate Gaussian multiply is commutative by ordinary natural identities.

The unchanged tactic script uses 2 declared prerequisites and contains 11 exact native proof lines.

Alpha v34 checked-use · first admitted v30 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

mul_comm Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized

Direct dependents

GF0014 gaussian_multiply_commutative

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

11 script commands · 3 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 assumptionsL1–8

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

  1. L1
    intro ap
  2. L2
    intro an
  3. L3
    intro bp
  4. L4
    intro bn
  5. L5
    intro cp
  6. L6
    intro cn
  7. L7
    intro dp
  8. L8
    intro dn
02Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    split
03Calculate and transport equalitiesL10–11

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

  1. L10
    simp [mul_comm, add_comm]
  2. L11
    simp [mul_comm, add_comm]

Library-wide reading audit

Original exact command ledger · 11 lines
  1. 0001intro ap
  2. 0002intro an
  3. 0003intro bp
  4. 0004intro bn
  5. 0005intro cp
  6. 0006intro cn
  7. 0007intro dp
  8. 0008intro dn
  9. 0009split
  10. 0010simp [mul_comm, add_comm]
  11. 0011simp [mul_comm, add_comm]