formalizeddiffer

Formalizeddiffer is a formal framework for representing and manipulating differentiation operators within machine-checked mathematics. It provides a typed language for functions, derivatives, and differential forms, enabling rigorous reasoning about calculus inside interactive proof systems and computer algebra tools.

Origin and scope: The concept arises from efforts to formalize calculus in proof assistants, drawing on ideas from type theory, algebra, and differential geometry. It aims to unify differentiation rules with formal inference so that derivative identities can be proven and verified automatically.

Key features: Core syntax for functions and derivatives, including higher-order derivatives; rules for differentiation such as constant, sum, product, and chain rule; support for differential forms and exterior derivative; algebraic structures like rings and vector spaces; interfaces for automated proof tactics in proof assistants.

Applications: Formal verification of calculus theorems, formalization of differential equations and physical models, and educational tools that illustrate derivative reasoning with machine-checked correctness.

Example: In formalizeddiffer, the derivative of f is written Df(x); the chain rule is D(f ∘ g)(x) = Df(g(x)) · Dg(x).

Relation to related fields: Connects calculus, differential geometry, and formal methods in computer science; differs from purely symbolic computer algebra systems by emphasizing proof-oriented semantics and machine checkability.

See also: Calculus, Differential geometry, Formal methods, Theorem proving.

References: General texts on formalized mathematics and proof assistants provide context; related papers discuss formalized differentiation within proof systems.