Shared canonical integer codes · nearest square quotient · strict norm decrease · Constructive arithmetic

Constructive Gaussian Euclidean division

a,b∈ℤ[i], b≠0 ⇒ ∃q,r. a=bq+r ∧ N(r)<N(b)

Construct actual Gaussian quotient and remainder codes for every nonzero divisor, using witnessed signed rounding and the genuine norm a²+b².

Open the definition-aware map Read the final theorem See campaign milestone G081

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 certificate

Fully expanded arithmetic

Inspect all 4591 native tactic lines and 320 actual proof prerequisites with every definition fully expanded.

Open the exact edition →

Focused route

Final dependency cone

Start at theorem GI005D and follow only the lemmas and conservative definitions supporting gaussian_euclidean_division_exists.

Trace prerequisites →
Zoom between mathematical scales: complete research atlas → research domain → proof family → G081 milestone → theorem and definition dependencies.
Major independently established statements: GI0051 gaussian_norm_exists_unique · GI0055 gaussian_add_exists · GI0059 gaussian_multiply_exists · GI005B gaussian_norm_multiply · GI005D gaussian_euclidean_division_exists.
Independently verified Alpha v34 checked-use theorem family: 93 dependency-curried kernel-checked theorem bodies · 320 proof prerequisites · 21 linked definitions · 25 definition-dependency arrows · 4591 exact tactic lines · first admitted v28 · not Stable. The unchanged intuitionistic kernel and separately compiled Lean verifier independently accept all 862 bundle nodes; SHA-256 e56dda386bf60759d1bacda45417eacd7e6a67fd6e23799f002aac9964253ae1.
Exact mathematical boundary: The natural-code carrier consists of genuine pairs of the existing signed integers; no new primitive arithmetic is trusted. The theorem constructs quotient, remainder, and actual norm witnesses. Gaussian gcd, unique factorization, and prime classification are separate targets.