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) : Type

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.

• integral : ℝ → ℝ → ℝ
• interval_union : ∀ (a b c : ℝ), Integrable.integral f a c = Integrable.integral f a b + Integrable.integral f b c

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

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

• value : ℝ
• derivative : ℝ

structure SmoothFunction : Type

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

• jet : ℝ → OneJet

def SmoothFunction.derivative (f : SmoothFunction) : ℝ → ℝ

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

Equations

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

def SmoothFunction.value (f : SmoothFunction) : ℝ → ℝ

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

Equations

• SmoothFunction.value f x = (f.jet 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

• fundThm f = { integral := fun (a b : ℝ) => SmoothFunction.value f b - SmoothFunction.value f a, interval_union := ⋯ }

Constructions of smooth functions

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

def SmoothFunction.constant (c : ℝ) : SmoothFunction

Constant functions as smooth functions.

Equations

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

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

Product of a scalar and a smooth function.

Equations

• SmoothFunction.scalarProd c f = { jet := fun (x : ℝ) => { 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 : SmoothFunction

The coordinate function

Equations

• SmoothFunction.x = { jet := fun (x : ℝ) => { 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 : SmoothFunction

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