7.9.1. The Working Pattern🔗

The main workflow has five steps.

1. Write the model or artifact in a typed form: tensors, parameters, graph nodes, certificates, datasets, or runtime logs.

2. Choose the semantics relevant to the question: real tensors, FP32, IEEE32Exec, graph denotation, autograd tapes, verifier domains, or external artifact formats.

3. Run the computation or importer: training loop, CUDA path, PyTorch bridge, graph compiler, verifier pass, or external producer.

4. Inspect the result with ordinary outputs, widgets, logs, graph views, or certificate reports.

5. Cite the strongest available support: executable run, Lean checker, theorem, or explicit assumption boundary.

This is why TorchLean is organized as a framework rather than a collection of isolated examples. The same model may appear as user-facing API code, an eager runtime computation, a graph artifact, a verification input, and a theorem target.

7.9.2. Levels Of Support🔗

The guide uses different words for different levels of support:

• Execution: a command runs and produces concrete tensors, logs, samples, bounds, or reports.

• Inspection: the produced object is visible enough to debug, through printed output, widgets, graph views, training curves, or certificate diagnostics.

• Checking: Lean parses or recomputes an artifact and accepts or rejects a stated contract.

• Theorem support: a Lean theorem connects the checked contract to the intended semantic claim.

• External assumption: a Python exporter, CUDA kernel, dataset converter, solver, or native library remains outside the proved fragment and is named as part of the claim.

This distinction is practical. A loss curve can show that a run behaved sensibly. A certificate checker can validate a bound artifact. A theorem can state why an accepted artifact implies a semantic property. These are related, but they are not interchangeable.

7.9.3. What The Chapters Contribute🔗

Each part of the guide contributes one layer of that pattern:

• Introduction explains the problem TorchLean is built to address.

• Building Models gives the typed tensor, model, data, and training surface.

• Runtime, Autograd, and Interop explains how models execute, train, and cross the PyTorch boundary.

• Semantics and Graphs gives a shared IR and semantic target for runtimes, exporters, widgets, and verifiers.

• Floating Point and Native Boundaries separates real specifications, proof-side float models, executable IEEE behavior, CUDA, and external tools.

• Verification and Certificates explains how bounds, autograd proofs, runtime approximation, scientific certificates, and imported artifacts become checked claims.

• Examples and Applications shows the system in use: model zoo runs, generative models, RL examples, widgets, BugZoo case studies, and command-line workflows.

Together they give a path from ordinary model code to a precise statement about what has been run, checked, proved, or assumed.

7.9.4. How To State A TorchLean Result🔗

When writing about TorchLean, the strongest statements have a simple form:

• name the object: model, graph, tensor, tape, certificate, parameter store, dataset, or external artifact;

• name the semantics: real, FP32, IEEE32Exec, graph denotation, verifier abstraction, or runtime execution path;

• name the support: command output, widget inspection, checker acceptance, theorem, or assumption;

• state the property: shape agreement, gradient correctness, robustness margin, approximation bound, certificate validity, or runtime behavior.

For example, a model-zoo run can establish that a particular training script executes and writes a loss trace. A verifier command can establish that a graph artifact passes a bound checker. A soundness theorem can establish that accepted bounds enclose the graph semantics for a supported fragment. A CUDA or Python path can be used, but the external part remains part of the statement unless a bridge theorem or replay checker covers it.

7.9.5. The Extension Rule🔗

The same rule applies when extending the project. A new feature is strongest when it adds one clear bridge:

• a new model should expose its data boundary, parameter shapes, training loop, and logs;

• a new operator should name its spec and runtime behavior;

• a new graph pass should state what semantics it preserves or approximates;

• a new verifier should identify the abstract domain, checker, and soundness theorem surface;

• a new external integration should specify the artifact format and the trust boundary;

• a new widget should render an existing Lean object rather than define a new semantic meaning.

That rule keeps TorchLean extensible without making the system vague. Every addition should make at least one object easier to run, inspect, check, or prove.

7.9.6. Closing🔗

TorchLean is useful because it keeps two activities connected: building neural-network systems and stating mathematical claims about them. The code can still look like ML code. The examples can still train, log, save, load, and call native or external tools. The difference is that the important objects have names in Lean, and claims about those objects can be tied to checkers, theorems, or explicit assumptions.

The closing habit is simple: identify the object, identify the semantics, and say exactly what has been checked. That is the standard the rest of the project should preserve.