Wolfram/ LeanLink

Native link between Wolfram Language and Lean 4

Contributed by: Nik Murzin

The entry points are LeanImport and LeanImportString (load an environment) and LeanExport / LeanExportString (write Lean source back). LeanTerm and LeanEnvironment are the two computable wrappers; the Parse*-style CIC heads - LeanConst, LeanApp, LeanForall, LeanLam, LeanLet, LeanBVar, LeanFVar, LeanSort, LeanLitNat, LeanLitStr, LeanProj - build expressions, with universe levels LeanLevelZero, LeanLevelSucc, LeanLevelMax. LeanState, LeanTactic, and LeanGoal run interactive proofs; ProofToLean transpiles a ProofObject; LeanCompile, LeanToFunction compile to native code; LeanExprGraph, LeanCallGraph, LeanListTheorems query a project out of process.

Installation Instructions

To install this paclet in your Wolfram Language environment, evaluate this code:
PacletInstall["Wolfram/LeanLink"]

To load the code after installation, evaluate this code:
Needs["Wolfram`LeanLink`"]

Details

LeanLink is a high-performance native bridge between the Wolfram Language and the Lean 4 theorem prover. It embeds the Lean runtime directly through a compiled LibraryLink shim - there is no subprocess per call and no text-protocol round-trip for the core API.

A loaded environment is a LeanEnvironment, a typed wrapper over Association[name → LeanTerm, …]. LeanImport loads constants from a compiled Lean module (Mathlib included); LeanImportString compiles a Lean source string on the fly.

Every Lean expression is a symbolic tree built from a small set of CIC heads - LeanConst, LeanApp, LeanForall, LeanLam, LeanBVar, LeanSort, LeanLitNat, … - each carrying box formatting so it displays in Lean-source notation.

A LeanTerm exposes "Type" / "Term" (expression trees), "TypeForm" / "TermForm" (pretty-printed source), "Parameters" (the unfolded binder chain), and "ExprGraph" / "CallGraph" (native Graph visualizations).

LeanState and LeanTactic drive interactive tactic proofs step by step; ProofToLean transpiles a Wolfram ProofObject into a checkable LeanEnvironment; LeanCompile and LeanToFunction lower a Lean definition to a FunctionCompile function.

The serialization backend is a compact binary (WXF) format, so goal states and expression trees cross the boundary without a parsing pass.

Paclet Guide

Examples

Basic Examples (4)

Load the bundled example environment:

In[1]:=

$env = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], List[20.928985595703125`, 317.4896984100342`], List[20.928985595703125`, 315.7036876678467`], List[20.928985595703125`, 114.70369529724121`], List[20.928985595703125`, 112.91769218444824`], List[21.882003784179688`, 111.26669120788574`], List[23.428985595703125`, 110.37369346618652`], List[197.5`, 9.87369155883789`], List[198.27300024032593`, 9.426692008972168`], List[199.13700008392334`, 9.203690528869629`], List[200.`, 9.203690528869629`], List[200.86299991607666`, 9.203690528869629`], List[201.72699999809265`, 9.426692008972168`], List[202.5`, 9.87369155883789`], List[376.5710144042969`, 110.37369346618652`], List[378.1179962158203`, 111.26669120788574`], List[379.0710144042969`, 112.91769218444824`], List[379.0710144042969`, 114.70369529724121`], List[379.0710144042969`, 315.7036876678467`], List[379.0710144042969`, 317.4896984100342`], List[378.1179962158203`, 319.1406993865967`], List[376.5710144042969`, 320.03370475769043`]]]]], List[FaceForm[List[RGBColor[0.5529411764705883`, 0.6745098039215687`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 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Out[1]=

How many constants, and of what kinds:

In[2]:=

Out[2]=

In[3]:=

Out[3]=

A single constant is a LeanTerm summary box:

In[4]:=

Out[4]=

Its type, pretty-printed as Lean source - the proposition it proves:

In[5]:=

Out[5]=

And the proof term:

In[6]:=

Out[6]=

Every expression is a tree; "ExprGraph" draws it:

In[7]:=

Out[7]=

"CallGraph" shows which constants a proof depends on:

In[8]:=

Out[8]=

Scope (2)

Import from Mathlib by pointing "ProjectDir" at a built mathlib4 checkout (gated so the page still builds without one):

In[9]:=

$mathlibDir = FileNameJoin[{$HomeDirectory, "src", "mathlib4"}]; If[ DirectoryQ[FileNameJoin[{mathlibDir, ".lake", "build"}]], mulComm = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 307.0436840057373`], List[153.98200225830078`, 235.99870109558105`], List[195.`, 212.31669425964355`], List[195.`, 22.863691329956055`], List[30.928985595703125`, 117.59068870544434`], List[30.928985595703125`, 307.0436840057373`]], List[List[200.`, 410.42970085144043`], List[364.0710144042969`, 315.7036876678467`], List[241.01799774169922`, 244.65868949890137`], List[200.`, 220.97669792175293`], List[158.98200225830078`, 244.65868949890137`], List[35.928985595703125`, 315.7036876678467`], List[200.`, 410.42970085144043`]], List[List[376.5710144042969`, 320.03370475769043`], List[202.5`, 420.53370475769043`], List[200.95300006866455`, 421.42667961120605`], List[199.04699993133545`, 421.42667961120605`], List[197.5`, 420.53370475769043`], List[23.428985595703125`, 320.03370475769043`], List[21.882003784179688`, 319.1406993865967`], 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Style[Defer[PacletSymbol["Wolfram/LeanLink", "Wolfram`LeanLink`LeanImport"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram/LeanLink", "Wolfram`LeanLink`LeanImport"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][ "Mathlib.Algebra.Group.Basic", "ProjectDir" -> mathlibDir, "Filter" -> "mul_comm"]; mulComm["mul_comm"]["TypeForm"], "Mathlib not built - run: lake exe cache get && lake build Mathlib.Algebra.Group.Basic" ]$

Out[9]=

The "Parameters" property unfolds a theorem's binder chain - explicit, implicit, and instance arguments - as a Dataset:

In[10]:=

Out[10]=

Applications (3)

Open any theorem as a proof goal and step through it with tactics:

In[11]:=

$s0 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 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Out[11]=

In[12]:=

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Out[12]=

Transpile a Wolfram ProofObject into a checkable Lean environment:

In[13]:=

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Out[13]=

In[14]:=

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Out[14]=

Lower a (non-dependent) Lean definition to native code via FunctionCompile:

In[15]:=

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Out[15]=

Properties and Relations (1)

LeanImportString round-trips with LeanExportString: import a source string, query it, and export an environment back to Lean source:

In[16]:=

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Out[16]=

ProofToLean complements MetamathImport - both bring an external formal-proof corpus into the Wolfram Language; LeanLink targets Lean 4 / Mathlib and additionally drives the prover interactively.

Possible Issues

The native bridge needs the platform shim under LeanLink/LibraryResources/ and the Lean toolchain it was built against (leanprover/lean4:v4.30.0) installed via elan: the ~190 MB Lean runtime is not bundled, so on first use LeanLink links libleanshared from that toolchain next to the shim. Install it with elan toolchain install leanprover/lean4:v4.30.0. Mathlib import and the out-of-process graph functions (LeanExprGraph, LeanCallGraph) additionally need lake on PATH and a built project - LeanImport of an unbuilt module returns an empty environment.

LeanCompile's dependent-type path (Vector-sized signatures via TypePi) requires a forked Wolfram compiler; the non-dependent path works against the stock compiler.

A type shown in isolation can print an unresolved bound variable as #0 / #1; the full-theorem "TypeForm" resolves binder names.

Neat Examples (1)

The type of a Mathlib theorem, drawn as its expression tree (gated on a built checkout):

In[17]:=

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Out[17]=

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Citation:
- Leonardo de Moura, Sebastian Ullrich, *The Lean 4 Theorem Prover and Programming Language*, CADE 28, 2021

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