https://github.com/leanprover/hex-gram-schmidt-mathlib
Mathlib correspondence proofs for hex-gram-schmidt
https://github.com/leanprover/hex-gram-schmidt-mathlib
Last synced: about 1 month ago
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Mathlib correspondence proofs for hex-gram-schmidt
- Host: GitHub
- URL: https://github.com/leanprover/hex-gram-schmidt-mathlib
- Owner: leanprover
- License: apache-2.0
- Created: 2026-06-24T12:38:05.000Z (about 2 months ago)
- Default Branch: main
- Last Pushed: 2026-07-08T04:24:37.000Z (about 1 month ago)
- Last Synced: 2026-07-08T06:09:52.207Z (about 1 month ago)
- Language: Lean
- Homepage: https://kim-em.github.io/hex-dev/find/?domain=Verso.Genre.Manual.section&name=hex-gram-schmidt
- Size: 308 KB
- Stars: 0
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Agents: AGENTS.md
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README
# hex-gram-schmidt-mathlib
Part of [`hex`](https://github.com/kim-em/hex-dev), a computer algebra
library for Lean 4. The aim is fast executable code, fully verified, built
with spec-driven development.
`hex-gram-schmidt-mathlib` is the Mathlib bridge for
[`hex-gram-schmidt`](https://github.com/leanprover/hex-gram-schmidt). It
identifies the executable dense-matrix Gram-Schmidt basis with Mathlib's
`InnerProductSpace.gramSchmidt`, and the fraction-free integer
coefficients with Bareiss determinants. It depends on
[`hex-gram-schmidt`](https://github.com/leanprover/hex-gram-schmidt) and on
Mathlib.
# Quickstart
Add to your `lakefile.toml`:
```toml
[[require]]
name = "hex-gram-schmidt-mathlib"
git = "https://github.com/leanprover/hex-gram-schmidt-mathlib.git"
rev = "main"
```
```lean
import HexGramSchmidtMathlib
open Hex
-- Each row of the executable integer Gram-Schmidt basis matches Mathlib's
-- `gramSchmidt` after casting the row into Euclidean space.
#check @GramSchmidtMathlib.int_basis_row_eq_gramSchmidt
-- Mathlib's real inner product agrees with the executable rational dot product.
#check @GramSchmidtMathlib.rowToEuclidean_inner
-- The fraction-free integer coefficients are Bareiss determinants.
#check @GramSchmidt.Int.scaledCoeffs_eq_scaledCoeffMatrix_bareiss
```
# Functionality
The proof-facing API splits into two parts.
- The view `GramSchmidtMathlib.rowToEuclidean` from a rational dense row
into `EuclideanSpace ℝ (Fin m)`, with its additivity, scaling, and
inner-product transfer lemmas (`rowToEuclidean_add`,
`rowToEuclidean_smul`, `rowToEuclidean_inner`).
- The rowwise correspondences `rat_basis_row_eq_gramSchmidt` and
`int_basis_row_eq_gramSchmidt` between the executable basis and Mathlib's
`InnerProductSpace.gramSchmidt`.
- The integer determinantal layer: `gramDet_eq_prod_normSq`,
`scaledCoeffs_eq_scaledCoeffMatrix_bareiss`, and `scaledCoeffs_eq`,
relating the leading Gram determinants and the scaled coefficients to
Bareiss determinants of the [`hex-matrix`](https://github.com/leanprover/hex-matrix)
Gram matrices.
# Verification
The rowwise correspondence with Mathlib's orthogonalization is fully
proven. Over the rationals:
```lean
theorem rat_basis_row_eq_gramSchmidt (b : Matrix Rat n m) (i : Fin n) :
rowToEuclidean ((Hex.GramSchmidt.Rat.basis b).row i) =
InnerProductSpace.gramSchmidt ℝ (ratRowFamily b) i
```
and the integer version, casting the integer rows first:
```lean
theorem int_basis_row_eq_gramSchmidt (b : Matrix Int n m) (i : Fin n) :
rowToEuclidean ((Hex.GramSchmidt.Int.basis b).row i) =
InnerProductSpace.gramSchmidt ℝ (intRowFamily b) i
```
The fraction-free integer surface is identified with Bareiss
determinants. Below the diagonal the scaled coefficients are exactly:
```lean
theorem scaledCoeffs_eq_scaledCoeffMatrix_bareiss
(b : Matrix Int n m) (i j : Fin n) (hji : j.val < i.val) :
GramSchmidt.entry (scaledCoeffs b) i j =
Matrix.bareiss (GramSchmidt.scaledCoeffMatrix b i j hji)
```
and for an independent matrix the leading Gram determinant is the product
of squared Gram-Schmidt norms:
```lean
theorem gramDet_eq_prod_normSq (b : Matrix Int n m)
(hli : independent b) (k : Nat) (hk : k ≤ n) :
(gramDet b k hk : Rat) = gramSchmidtNormProduct b k hk
```
The executable Gram-Schmidt algorithm and its own algebraic API live in
[`hex-gram-schmidt`](https://github.com/leanprover/hex-gram-schmidt).
# Reference manual
The hex reference manual covers this library and its computational base at
.
# Contributing
Development happens in the [`hex-dev`](https://github.com/kim-em/hex-dev)
monorepo, not in this published mirror. Contributions are welcome as pull
requests to the `SPEC/` directory: describe the behaviour you want, and
leave the implementation to the maintainer.