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.
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.
Exact theorem in conservative defined notation
∀ a. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ i. ∀ j. ∀ k. ∀ l. (a · e + b · f + (c · h + d · g)) · i + (a · f + b · e + (c · g + d · h)) · j + ((a · g + b · h + (c · e + d · f)) · l + (a · h + b · g + (c · f + d · e)) · k) + (a · (e · j + f · i + (g · k + h · l)) + b · (e · i + f · j + (g · l + h · k)) + (c · (e · k + f · l + (g · i + h · j)) + d · (e · l + f · k + (g · j + h · i)))) = a · (e · i + f · j + (g · l + h · k)) + b · (e · j + f · i + (g · k + h · l)) + (c · (e · l + f · k + (g · j + h · i)) + d · (e · k + f · l + (g · i + h · j))) + ((a · e + b · f + (c · h + d · g)) · j + (a · f + b · e + (c · g + d · h)) · i + ((a · g + b · h + (c · e + d · f)) · k + (a · h + b · g + (c · f + d · e)) · l)) ∧ (a · e + b · f + (c · h + d · g)) · k + (a · f + b · e + (c · g + d · h)) · l + ((a · g + b · h + (c · e + d · f)) · i + (a · h + b · g + (c · f + d · e)) · j) + (a · (e · l + f · k + (g · j + h · i)) + b · (e · k + f · l + (g · i + h · j)) + (c · (e · j + f · i + (g · k + h · l)) + d · (e · i + f · j + (g · l + h · k)))) = a · (e · k + f · l + (g · i + h · j)) + b · (e · l + f · k + (g · j + h · i)) + (c · (e · i + f · j + (g · l + h · k)) + d · (e · j + f · i + (g · k + h · l))) + ((a · e + b · f + (c · h + d · g)) · l + (a · f + b · e + (c · g + d · h)) · k + ((a · g + b · h + (c · e + d · f)) · j + (a · h + b · g + (c · f + d · e)) · i))Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Complete tactic proof in conservative notation
All 21 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
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.
Named ingredients (4)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–12
03Separate the logical casesL13–13
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L13
split
04Calculate and transport equalitiesL14–14
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L14
congr
05Use earlier factsL15–15
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L15
apply gaussian_product_associate_real_positive
06Calculate and transport equalitiesL16–16
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L16
symm
07Use earlier factsL17–17
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L17
apply gaussian_product_associate_real_negative
08Calculate and transport equalitiesL18–18
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L18
congr
09Use earlier factsL19–19
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L19
apply gaussian_product_associate_imaginary_positive
10Calculate and transport equalitiesL20–20
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L20
symm
11Use earlier factsL21–21
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L21
apply gaussian_product_associate_imaginary_negative
Original defined command ledger · 21 lines
- 0001
intro a - 0002
intro b - 0003
intro c - 0004
intro d - 0005
intro e - 0006
intro f - 0007
intro g - 0008
intro h - 0009
intro i - 0010
intro j - 0011
intro k - 0012
intro l - 0013
split - 0014
congr - 0015
apply gaussian_product_associate_real_positive - 0016
symm - 0017
apply gaussian_product_associate_real_negative - 0018
congr - 0019
apply gaussian_product_associate_imaginary_positive - 0020
symm - 0021
apply gaussian_product_associate_imaginary_negative