Documentation

PnP2023.Lec_03_10.FormalCalculus

Formal Calculus #

We introduce formal structures for integration and differentiation. Properties should be added to make these mathematically sound. But correctness can be ensured temporarily by making sure individual definitions are correct.

Formal Integrals #

class Integrable (f : ℝ → ℝ) :

integral : ℝ → ℝ → ℝ
interval_union : ∀ (a b c : ℝ), integral a c = integral a b + integral b c

Integrability of f, i.e., given an interval [a, b], we can compute the integral of f over that interval. Additivity over intervals is also required.

Instances

def integral (f : ℝ → ℝ) [int : Integrable f] (a : ℝ) (b : ℝ) :

The integral of a function, with the typeclass derived

Equations

integral f a b = Integrable.integral f a b

theorem integral_point (f : ℝ → ℝ) [int : Integrable f] (a : ℝ) :

integral f a a = 0

The integral over a single point is zero, proved as an illustration.

As an exercise, prove that flip ends of an interval gives the negative of the integral.

Formal Derivatives #

We define so called one-jets as a value and a derivative at a point. A differentiable function has values a one-jet at each point.

structure OneJet :

value : ℝ
derivative : ℝ

A one-jet is a value and a derivative at a point.

Instances For

structure SmoothFunction :

jet : ℝ → OneJet

A differentiable function is a function that has a one-jet at each point.

Instances For

def SmoothFunction.derivative (f : SmoothFunction) :

Derivative of a smooth function, i.e., the derivative of the one-jet at a point.

Equations

SmoothFunction.derivative f x = (SmoothFunction.jet f x).derivative

def SmoothFunction.value (f : SmoothFunction) :

The value of a smooth function, i.e., the value of the one-jet at a point.

Equations

SmoothFunction.value f x = (SmoothFunction.jet f x).value

instance fundThm (f : SmoothFunction) :

Integrable (SmoothFunction.derivative f)

Integrable functions can be obtained from smooth functions via the fundamental theorem of calculus.

Equations

One or more equations did not get rendered due to their size.

Constructions of smooth functions #

To use the above we need to construct a few smooth functions

def SmoothFunction.constant (c : ℝ) :

Constant functions as smooth functions.

Equations

SmoothFunction.constant c = { jet := fun x => { value := c, derivative := 0 } }

def SmoothFunction.sum (f : SmoothFunction) (g : SmoothFunction) :

Sum of smooth functions.

Equations

One or more equations did not get rendered due to their size.

def SmoothFunction.prod (f : SmoothFunction) (g : SmoothFunction) :

Product of smooth functions using Liebnitz rule.

Equations

One or more equations did not get rendered due to their size.

def SmoothFunction.scalarProd (c : ℝ) (f : SmoothFunction) :

Product of a scalar and a smooth function.

Equations

SmoothFunction.scalarProd c f = { jet := fun x => { value := c * SmoothFunction.value f x, derivative := c * SmoothFunction.derivative f x } }

instance SmoothFunction.instAddSmoothFunction :

Add SmoothFunction

Addition operation on smooth functions

Equations

SmoothFunction.instAddSmoothFunction = { add := SmoothFunction.sum }

instance SmoothFunction.instMulSmoothFunction :

Mul SmoothFunction

Multiplication operation on smooth functions

Equations

SmoothFunction.instMulSmoothFunction = { mul := SmoothFunction.prod }

instance SmoothFunction.instSMulRealSmoothFunction :

SMul ℝ SmoothFunction

Scalar multiplication for smooth functions

Equations

SmoothFunction.instSMulRealSmoothFunction = { smul := SmoothFunction.scalarProd }

This gives polynomial functions as a special case. As an exercise, prove that smooth functions form a Ring (indeed an Algebra over ℝ).

We will define some polynomials as smooth functions as an example.

def SmoothFunction.x :

The coordinate function

Equations

SmoothFunction.x = { jet := fun x => { value := x, derivative := 1 } }

def SmoothFunction.pow (f : SmoothFunction) :

ℕ → SmoothFunction

The power function for a smooth function (automatic if ring is proved)

Equations

SmoothFunction.pow f 0 = SmoothFunction.constant 1
SmoothFunction.pow f (Nat.succ n) = f * SmoothFunction.pow f n

instance SmoothFunction.instHPowSmoothFunctionNat :

HPow SmoothFunction ℕ SmoothFunction

Equations

SmoothFunction.instHPowSmoothFunctionNat = { hPow := SmoothFunction.pow }

instance SmoothFunction.instCoeRealSmoothFunction :

Coe ℝ SmoothFunction

Equations

SmoothFunction.instCoeRealSmoothFunction = { coe := SmoothFunction.constant }

def SmoothFunction.poly :

A polynomial. We can have cleaner notation but the goal is to illustrate the construction

Equations

SmoothFunction.poly = 2 • SmoothFunction.x + 3 • SmoothFunction.x ^ 3 + SmoothFunction.constant 7