GF0023 — gaussian_ring_raw_add_cancel_left

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.

Inputs are genuine canonical signed-pair codes, not arbitrary naturals. Products start at the actual Gaussian identity, whose code is six. The factor list uses the proved prime-divisor property; irreducibility alone is not silently renamed primality. Uniqueness supplies equal lengths, a bounded bijection, and an actual unit at each match, including repeated factors. Units have empty factorizations and zero is excluded. Sorted primary representatives, Gaussian prime classification, and Eisenstein factorization are separate targets.

Exact theorem in conservative defined notation

∀ ap. ∀ an. ∀ bp. ∀ bn. ∀ cp. ∀ cn. ∀ dp. ∀ dn. ∀ ep. ∀ en. ∀ fp. ∀ fn. ap + cp + (an + en) = ap + ep + (an + cn) ∧ bp + dp + (bn + fn) = bp + fp + (bn + dn) → cp + en = ep + cn ∧ dp + fn = fp + dn

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

none

Actual proof prerequisites

gaussian_ring_pair_sum_cancel

Original expanded first-order statement

forall ap an bp bn cp cn dp dn ep en fp fn. (((((((ap) + (cp))) + (((an) + (en)))) = ((((ap) + (ep))) + (((an) + (cn))))) /\ (((((bp) + (dp))) + (((bn) + (fn)))) = ((((bp) + (fp))) + (((bn) + (dn))))))) -> (((((cp) + (en)) = ((ep) + (cn))) /\ (((dp) + (fn)) = ((fp) + (dn)))))

Complete tactic proof in conservative notation

All 31 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

31 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.

Named ingredients (1)

gaussian ring pair sum cancel

01Fix variables and assumptionsL1–10

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