Lean 4 → MLIR → GPU

The book: Verified Deep Learning with Lean 4 (PDF) — the interactive proof blueprint is the book: every theorem clickable, from the pdiv primitives to the whole-network backward passes.

Lean 4 as a specification language for neural networks. Declare the architecture in Lean, render one StableHLO graph — forward, loss, backward and optimizer fused — from proofs that the backward is the Jacobian-transpose of the forward, hand the graph to a trusted lowerer, train end to end. No Python at run time and no autograd library: the gradients are derived at codegen time, in Lean, and machine-checked over the reals.

Companion code for Verified Deep Learning with Lean 4, forthcoming from Apress (Springer Nature) as the follow-up to Convolutional Neural Networks with Swift for TensorFlow (2021).

Current version: v0.7.2. Release history in CHANGELOG.md.

Three ways in

to	do this	needs
read it	the book (PDF), or the API docs	a browser
check the proofs	lake exe cache get && lake build ProofsMinimal (~1 min), then lake build Certs for all of them; The proofs below	Lean 4, no GPU
train a net	the four setup lines in the tour, then lake run mnist (~1 min)	an NVIDIA, AMD or Apple Silicon GPU

The tour

Four commands, one per scale, in the order the book meets the nets, then the demos. The numbers are the book's, from the verified XLA path on one RTX 4060 Ti unless the row says otherwise; the two Imagenette side quests the book does not quote (ResNet-50, MobileNetV4) are medians of five seeds. Setup is the book's Getting started, one track per tier; the short form is

curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh   # Lean 4
python3 -m venv .venv && . .venv/bin/activate && pip install jax-cuda12-pjrt          # the XLA plugin only (jax-rocm7-pjrt on AMD)
gcc -fPIC -O2 -shared ffi/pjrt_ffi.c -ldl -o ffi/libpjrt_ffi.so                        # the shim; needs nothing but libc
lake exe cache get && ./scripts/datasets/download_mnist.sh && lake run mnist           # Mathlib oleans, MNIST, tier 1

(On an Apple Silicon Mac, deploy/mac_setup.sh is those four lines on Metal; the platforms and their pins are PLATFORMS.md.)

tier	command	trains	the number	chapter
1	lake run mnist	linear, MLP and CNN on MNIST, 12 epochs each (~1 min)	92.10 · 97.81 · 98.77 %	1 · 2 · 3
2	lake run cifar	the wide 8-conv net on CIFAR-10: SGD / momentum / AdamW × no-BN / BN, 40 epochs at a constant lr (~5 min)	77.4 % — BN + momentum, median of five	4
3	lake run imagenette	seven nets on Imagenette at 224², 80 epochs AdamW, book order (~6 h)	R34 89.99 (mean of five seeds) · R50 89.71 · MNv2 89.25 · MNv4-Conv-M 86.24 · B0 90.04 · ConvNeXt-T 82.27 · ViT-Tiny 68.74 %	5 · 6 · 7 · 8 · 9
4	lake run imagenet	the same nets on ImageNet-1k at the paper recipes, 4× 3060 / 4060 Ti, weeks of wall-clock; bare it prints the plan and every row's estimate, start runs it	R34 74.14 · R50 78.08 on RSB-A3 and 77.09 on the 2018 recipe · MNv2 71.12 · MNv4-Conv-M 77.42 (side quest) · B0 76.88 · ConvNeXt-T 81.30 · ViT-Tiny 71.61 % on the verified path, none more than 0.51 from its JAX reference	Track 4

The demos ride on the chapter nets, in the order the bestiary (Chapters 11–13) meets them — recognition, then beyond it, then the games; demos/README.md has the command, the figure and the reasoning for each.

demo	command	the number	book
object detection	lake exe yolov1-visdrone-fpn	VisDrone mAP@0.5 0.2363 — ResNet-34 + FPN at 448	11.2
industrial inspection	lake exe yolov1-neudet-fpn (and yolov1-neudet448)	NEU-DET mAP@0.5 0.623 with the VisDrone detector unchanged; the single 14×14 grid gets 0.607 here and 0.0391 on the drones	11.3
people watching	lake exe arasl-signs	ArASL 98.62 % under the published random split, 77.94 % once each hand's frames stay together	11.4
agriculture	lake exe plant-leaf	PlantVillage 99.57 % on lab leaves, 17.80 % on PlantDoc's field leaves — the CAM and an exact Shapley value say why	11.5
remote sensing	lake exe rs-bands	thirteen Sentinel-2 bands as the knob: 97–98 % on EuroSAT whatever the net sees, but in the Amazon the ten-band net keeps 97.0 % where RGB drops to 87.4 (pasture recall 95 % against 74), and in the Cerrado every arm fails on the seven-class map	11.6
segmentation	lake exe unet-brats-r34, then brats-predict	BraTS mIoU 0.741 — a ResNet-34 encoder under a UNet	11.7
language	lake exe tinygpt-shakespeare (also bigram-shakespeare)	2.28 bits/char held-out	12.1
diffusion	lake exe mnist-ddpm-train, then mnist-ddpm-sample	energy distance to MNIST 0.0067 at 50 network evaluations — ancestral DDIM, ahead of the deterministic sampler at 200	12.3
physics	lake exe diffusion-2d muller_brown flow	a flow-matching Boltzmann generator on Müller–Brown; reweighted by its own density it returns the exact well populations to the last digit	12.4
signal processing	lake exe gw-detect	LIGO O3a strain: the chapter-4 CNN reaches half detection at network SNR 6.91 (false-alarm rate 10⁻²), where the matched filter needs 10.32	12.5
beyond vision	lake exe nqs-ising gpt	the transverse-field Ising ground state at N = 12: a GPT wavefunction within 3.3 × 10⁻⁶ of the exact energy	12.6
structure	lake exe distogram-casp train …, then predict and scripts/demos/casp16_fold.py	CASP16: a 5.4M-parameter distogram ResNet on frozen ESM-2 3B folds 78 evaluation units to TM-score 0.646, against ESMFold's 0.778 on the same language model and the field's median model at 0.892	12.7
Pong	lake exe pong-dqn mode=pixels k=4 (pong-env for the baselines)	DQN from four frames on a Pong written in Lean: +15.5 points a game over three seeds, above its own six-number state twin's +13.5 and the scripted tracker's +11.2	13.1
poker	lake exe deep-cfr-leduc (leduc-env for the exact instrument)	Deep CFR on Leduc hold'em, scored by the exact best response: exploitability 0.11–0.23 chips a hand over three seeds and two readouts, against 2.37 for the uniform policy, 1.05 for an honest script and 0.0002 for CFR+	13.2
tic-tac-toe	lake exe alphazero-ttt n=3 (ttt-env for the solved game)	AlphaZero's network alone agrees with the solved game on 97.3 % of 3×3 and 99.4 % of 4×4 positions and never loses to a perfect player at either; at 5×5, four in a row, 87.4 % and unbeaten as X	13.3
tic-tac-toe without the rules	lake exe muzero-ttt n=4 iters=120 steps=300 sims=50 qnorm=0 sweepN=20000 (mode=score for the model's report card)	MuZero's learned model, scored against the solved game at every position and imagined depth: 98.7 % root agreement at 4×4 and unbeaten both sides, the board linearly decodable from the imagined latent on the lines the agent plays (99.7 % of boards after five moves) and lost on random ones (55 %)	13.4

Everything else in the repository is the lab — the ablation and robustness exes, the tests, the Bestiary — the evidence behind these numbers, one level down. lakefile.lean is grouped the same way: the tour first, then the lab by home directory.

The proofs

Every layer's backward is proven to be the Jacobian-transpose of its forward over the exact reals (Mathlib's fderiv), composed up to whole-network VJPs for ResNet-34, MobileNetV2, EfficientNet-B0, ConvNeXt-T and ViT-Tiny, with zero project axioms. For every chapter net the committed train-step render in verified_mlir/ is tied to those proofs at the denotational level: each emitted parameter-update node denotes the certified descent step, and the tiers train on exactly those bytes. What stays trusted is the ℝ→Float32 numerics, the per-op text printing, and the lowerer with its runtime. TRUST.md puts that on one page: which kind of check stands behind each claim, and which hypotheses are discharged. The book's On Verification appendix is the full argument, gap by gap; LeanMlir/Proofs/README.md is the file-level map, and the API docs open on the same map with every name linked.

Check them without a GPU:

lake exe cache get           # Mathlib oleans, ~30 s
lake build ProofsMinimal     # the smallest end-to-end tie, ~1 min
lake build Certs             # every certificate CI checks (the long one)

tests/comparator/run.sh re-runs Lean's kernel typechecker over the headline theorems independently, and tests/comparator/Challenge.lean imports Mathlib and nothing else, so those can be read and checked without reading a line of this project.

Two lowerers, one graph

Training runs through XLA/PJRT, and every number above comes from it. IREE is the second trusted lowerer: it is what the differential oracle (tests/vjp_oracle/) lowers the Lean side through, so that agreement with the JAX reference is evidence from two independent compilers, and each lake run tier has an -iree twin. Building it is historical/IREE_BUILD.md.

Where things are

Each directory has a README that says what is in it and where to start.

• LeanMlir/ — the library; LeanMlir/Proofs/ — the proofs, chapter by chapter; verified_mlir/ — the committed renders the tiers train on
• apps/ — one Main per exe, by tier; demos/ — the bestiary's fifteen demos, in its order; Bestiary/ — 46 read-only NetSpec catalogue entries, Part 2 of the book
• jax/ — the JAX reference implementations the ImageNet path is ported from, and the oracle's ground truth
• tests/ — the gates: render ties, data-parallel checks, the comparator and the VJP oracle; ffi/ — the C runtime the lowerers sit behind
• scripts/ — gates, figures and the ImageNet jobs (scripts/jobs/, lake run <job> runs one); runs/ — the logs and READMEs behind the numbers
• historical/ — the two earlier phases (mnist-lean4/, pure Lean 4 with a C BLAS; mlir_poc/, the Python exporters), the cross-backend traces, the old logs and the upstream bug reproducers, the survey this README used to be (README_survey.md), and the reference notes: RESULTS.md (per-epoch histories), BENCHMARK.md, CUDA.md, ROCM.md, IREE_BUILD.md
• blueprint/ — the book's source; planning/ — working notes

Acknowledgements

Built on Lean 4 (de Moura & Ullrich 2021) and Mathlib (The Lean Mathematical Library, mathlib Community 2020). The graphs are StableHLO, lowered by XLA / PJRT and IREE; the reference implementations are JAX, and the ImageNet nets are held to timm (Wightman) and torchvision. The book is built with leanblueprint (Massot) on plasTeX, and the API docs with doc-gen4.

Citing this work

@software{koonce2026,
  author  = {Brett Koonce},
  title   = {Verified Deep Learning with Lean 4: Formal Backpropagation from MLP to Attention, via MLIR},
  url     = {https://github.com/brettkoonce/lean4-mlir},
  doi     = {10.5281/zenodo.20402133},
  version = {0.7.2},
  year    = {2026},
}