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import Mathlib
/-!
# 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
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/--
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.
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class
Integrable: {f : ℝ → ℝ} → (integral : ℝ → ℝ → ℝ) → (∀ (a b c : ℝ), integralac=integralab+integralbc) → Integrablef
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As an exercise, prove that flip ends of an interval gives the negative of the integral.
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## 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.
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/--
A _one-jet_ is a value and a derivative at a point.
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structure
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## Constructions of smooth functions
To use the above we need to construct a few smooth functions
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namespace SmoothFunction
/--
Constant functions as smooth functions.
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def
scalarProd⟩
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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.
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/-- The coordinate function -/
def