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; LeanToTypeTheory and TypeTheoryToLean exchange declarations with the TypeTheory proof engine; 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.

LeanToTypeTheory imports a Lean constant, with everything it depends on, into the Wolfram Language proof engine TypeTheory, whose own kernel checks it again; TypeTheoryToLean writes a TypeTheory proof as Lean source for Lean's kernel. The bridge loads the paclet WolframInstitute/TypeTheory on first use.

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

Paclet Guide

Wolfram`LeanLink`

embeds the

Lean 4

theorem prover in the Wolfram Language through a native

LibraryLink

shim.

LeanImport

and

LeanImportString

load constants from a compiled module (Mathlib included) or a source string into a

LeanEnvironment

LeanTerm

values; each term carries its type and proof as symbolic expression trees built from the CIC heads (

LeanConst

LeanApp

LeanForall

, …), pretty-prints as Lean source, and draws as a

Graph

LeanState

and

LeanTactic

run interactive tactic proofs,

transpiles a Wolfram

into checkable Lean,

and

exchange proofs with the Wolfram Language proof engine TypeTheory, and

LeanCompile

lowers a Lean definition to a

FunctionCompile

function.

Importing and exporting

LeanImport

import constants from a compiled Lean module, returning a

LeanEnvironment

LeanImportString

compile a Lean 4 source string into a

LeanEnvironment

; with

"ProjectDir"

it can import a built project

LeanBuild

build a Lake project's modules with its pinned toolchain

LeanEvaluate

evaluate a Lean expression with

#eval

and return its value

LeanExportString

convert a

LeanEnvironment

back to Lean 4 source

LeanExport

write a

LeanEnvironment

to a

.lean

file

Environments and terms

LeanEnvironment

a collection of named Lean constants; supports

one Lean constant, with

"Type"

"Term"

"TypeForm"

"Axioms"

"Module"

"Source"

"Parameters"

"ExprGraph"

properties

LeanConstant

the raw

name/kind/type/term

record returned by the native shim

Querying a loaded environment

LeanExpr

the type of a constant as a symbolic expression tree

LeanValue

the proof or definition body of a constant

LeanConstantInfo

full constant info as a

LeanConstant

LeanListConstants

every constant in a module as an

Association

LeanLoadEnvironment

load a Lean environment handle for repeated queries

LeanFreeEnvironment

release a loaded environment handle

Expression heads (CIC)

LeanConst

a reference to a declared constant at given universe levels

LeanApp

function application

LeanForall

∀

/ dependent function (arrow) type

LeanLam

-abstraction

LeanLet

let

-binding

LeanBVar

a bound variable by de Bruijn index

LeanFVar

a free variable (local hypothesis or parameter)

LeanMVar

a metavariable (unresolved placeholder)

LeanSort

a universe (

Prop

Type

, …)

LeanLitNat

a natural-number literal

LeanLitStr

a string literal

LeanProj

a structure-field projection

LeanNoValue

marks a constant with no body (axiom, opaque)

LeanTruncated

marks an expression cut off at the depth limit

Universe levels

LeanLevelZero

level 0, the universe of

Prop

LeanLevelSucc

the successor of a level

LeanLevelMax

the maximum of two levels

LeanLevelIMax

the impredicative max (collapses to 0)

LeanLevelParam

a named universe parameter (

)

LeanLevelMVar

a universe metavariable

Interactive proofs

LeanState

open a theorem as a proof goal; holds

"Goals"

"Complete"

"GoalCount"

LeanTactic

a tactic, applied as

LeanTactic[tac][state]

LeanGoal

a single goal, with

"Target"

and

"Context"

Transpiling Wolfram proofs

ProofToLean

transpile a Wolfram

ProofObject

into a checkable

LeanEnvironment

The TypeTheory proof engine

LeanToTypeTheory

import Lean constants into the Wolfram Language proof engine

TypeTheory

, whose own kernel checks them again

TypeTheoryToLean

write a TypeTheory proof as Lean source, for Lean's kernel to check

Compiling Lean to native code

convert a

into a

with

arguments

compile a

(or whole environment) via

FunctionCompile

Out-of-process project queries

LeanExprGraph

expression graph for a constant, computed by a Lean subprocess

LeanCallGraph

call / dependency graph for a constant, via subprocess

LeanListTheorems

list a project's theorems via subprocess

ImportDOT

import a DOT digraph file as a styled

Graph

▪

Lean 4 (lean-lang.org)

▪

Mathlib4 (GitHub)

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, 0]]], List[List[List[44.92900085449219`, 282.59088134765625`], List[181.00001525878906`, 204.0298843383789`], List[181.00001525878906`, 46.90887451171875`], List[44.92900085449219`, 125.46986389160156`], List[44.92900085449219`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6627450980392157`, 0.803921568627451`, 0.5686274509803921`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[355.0710144042969`, 282.59088134765625`], List[355.0710144042969`, 125.46986389160156`], List[219.`, 46.90887451171875`], List[219.`, 204.0298843383789`], List[355.0710144042969`, 282.59088134765625`]]]]], List[FaceForm[List[RGBColor[0.6901960784313725`, 0.5882352941176471`, 0.8117647058823529`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[200.`, 394.0606994628906`], List[336.0710144042969`, 315.4997024536133`], List[200.`, 236.93968200683594`], 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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"]][ PacletObject["Wolfram/LeanLink"]["AssetLocation", "Examples"]]$

Out[1]=

The example environment bundled with the paclet:

In[2]:=

Out[2]=

How many constants:

In[3]:=

Out[3]=

Of what kinds:

In[4]:=

Out[4]=

The example environment bundled with the paclet:

In[5]:=

Out[5]=

A single constant is a LeanTerm summary box:

In[6]:=

Out[6]=

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

In[7]:=

Out[7]=

And the proof term:

In[8]:=

Out[8]=

The example environment bundled with the paclet:

In[9]:=

Out[9]=

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

In[10]:=

Out[10]=

"CallGraph" shows which constants a proof depends on:

In[11]:=

Out[11]=

Scope (2)

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

In[12]:=

$mathlibDir = FileNameJoin[{$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`], 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`, 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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[12]=

The example environment bundled with the paclet:

In[13]:=

Out[13]=

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

In[14]:=

Out[14]=

Applications (3)

The example environment bundled with the paclet:

In[15]:=

Out[15]=

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

In[16]:=

$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[16]=

Introduce the proposition:

In[17]:=

$s1 = 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`], 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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`LeanTactic"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["intro P"][s0]$

Out[17]=

Then the hypothesis:

In[18]:=

$s2 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.0025`], List[FaceForm[List[RGBColor[0.9607843137254902`, 0.5058823529411764`, 0.19607843137254902`], Opacity[1.`]]], FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3], List[0, 1, 0], List[1, 3, 3]]], List[List[List[205.`, 22.863691329956055`], List[205.`, 212.31669425964355`], List[246.01799774169922`, 235.99870109558105`], List[369.0710144042969`, 307.0436840057373`], List[369.0710144042969`, 117.59068870544434`], List[205.`, 22.863691329956055`]], List[List[30.928985595703125`, 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Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram/LeanLink", "Wolfram`LeanLink`LeanTactic"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["intro h"][s1]$

Out[18]=

And close the goal with the hypothesis:

In[19]:=

$s3 = 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[19]=

The proof is complete:

In[20]:=

Out[20]=

Transpile a Wolfram ProofObject into a checkable Lean environment:

In[21]:=

$leanEnv = 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[21]=

The generated constants - premises, derived step, and goal:

In[22]:=

Out[22]=

Opening the final goal confirms the transpiled proof:

In[23]:=

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Out[23]=

A Lean definition, compiled from source:

In[24]:=

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Out[24]=

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

In[25]:=

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Out[25]=

Run it:

In[26]:=

Out[26]=

Properties and Relations (3)

LeanImportString round-trips with LeanExportString. Import a source string:

In[27]:=

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Out[27]=

Query it:

In[28]:=

Out[28]=

And export it back to Lean source:

In[29]:=

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Out[29]=

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:v4.35.0) installed via elan: the ~190 MB Lean runtime is not bundled, so on first use LeanLink copies the shim into the user's cache ($CacheBaseDirectory) and links libleanshared from that toolchain next to the copy. It writes nothing into the paclet's own folder, which can be read-only or shared. Install it with elan toolchain install leanprover/lean4:v4.35.0. 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.

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[30]:=

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Out[30]=

Disclosures

Wolfram Language built-in symbols
Learn More »

Compatibility

Wolfram Language Version 14.0

External Links

Version History

License Information

MIT License

Paclet Source

Source Metadata

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

Wolfram Language Paclet Repository

Wolfram/ LeanLink

Installation Instructions

Details

Paclet Guide

Examples

Basic Examples (4)

Scope (2)

Applications (3)

Properties and Relations (3)

Possible Issues

Neat Examples (1)

Disclosures

Compatibility

External Links

Version History

License Information

Paclet Source

Source Metadata

See Also