Into to Lean 4 and

its use in scientific computing

Tomáš Skřivan

Carnegie Mellon University

Hoskinson Center for Formal Mathematics

15.5.2024

Natural numbers

define natural numbers and

prove commutativity of addition

Symbolic and automatic

differentiation

def foo (x : R) : R := x^2

def foo_deriv (x : R) : R := 2*x

automatic differentiation (AD) synthesizes derivatives of a program

application in machine learning, optimization, and scientific computing

Symbolic and automatic

differentiation

def foo (x : R) : R := x^2

noncomputable
def foo_deriv (x : R) : R := fderiv R (fun x' => foo x') x 1

noncomputable
def fderiv (f : X → Y) (x : X) : X →L[𝕜] Y :=
  if h : ∃ (f' : X →L[𝕜] Y), (fun x' => f x' - f x - f' (x' - x)) =o[𝓝 x] fun x' => x' - x then
    choose h
  else 
  	0

definition of derivative in mathlib

using mathlib derivative

def foo_deriv (x : R) : R := (fderiv R (fun x' => foo x') x 1) rewrite_by fun_trans

fun_trans is a general tactic for function transformation

General function transformation

setting up fderiv as function transformation
definition:

@[fun_trans]
noncomputable
def fderiv (f : X → Y) (x : X) : X →L[𝕜] Y := ...

@[fun_trans]
theorem fderiv_add :
    (fderiv K fun x => f x + g x)
    =
    fun x => fderiv K f x + fderiv K g x := ...

@[fun_trans]
theorem fderiv_comp :
    fderiv 𝕜 (fun x => (f (g x))) 
    = 
    fun x => fun dx =>L[K] fderiv K f (g x) (fderiv K g x dx) := ...

@[fun_trans]
theorem fderiv_id :
    (fderiv K fun x : X => x) = fun _ => fun dx =>L[K] dx := ...

transformation rules:

code demo

Array expression optimization

custom interactive optimizations
example on basic linear algebra operation saxpy

def saxpy (a : Float) (x y : Float^[n]) := a•x+y

def saxpy_optimized (a : Float) (x y : Float^[n]) := x.mapIdx (fun i xi => a*xi + y[i])

def_optimize saxpy by
  simp only [HAdd.hAdd,Add.add,HSMul.hSMul,SMul.smul]
  simp only [mapMono_mapIdxMono]

semi-automatic with def_optimize macro

code demo

Lean 4

purely functional programming language with dependent types
- interactive theorem prover
- general purpose programming language
mathlib
- large library of formalized mathematics
- active community
easily extensible - Lean is written in Leanb
support for imperative code with mutable variables
support for FFI through C

Copy of Scientific Computing in Lean

By lecopivo

Copy of Scientific Computing in Lean

Into to Lean 4 and

its use in scientific computing

Natural numbers

Symbolic and automatic

differentiation

Symbolic and automatic

differentiation

General function transformation

Array expression optimization

Lean 4

Copy of Scientific Computing in Lean

More from lecopivo