Foundations

Learn Type Theory

A language where every term carries a type, and where the types turn out to be propositions and the terms their proofs. Runs from the simply typed lambda calculus and type inference through System F, dependent types and Martin-Lof's theory, to the homotopy reading where equalities are paths.

Free to start · adaptive placement finds your level · reviews timed to your own forgetting.

What you'll learn

30 lessons in Type Theory

Judgments and contextsThe simply typed lambda calculusPropositions as typesNormalizationProducts, sums and the empty typeType inference and unificationSystem F and polymorphismParametricity and free theoremsDependent typesMartin-Löf type theoryIdentity types and the J eliminatorUniverses and Girard's paradoxInductive types and W-typesThe calculus of constructions and proof assistantsDefinitional against propositional equalityDecidable checking and canonicityHomotopy type theoryHigher inductive types and cubical type theoryProgress & preservationBidirectional type checkingNormalization by evaluationChurch encodingsSubtypingType classes & elaborationClassical logic & controlLinear typesMonads & algebraic effectsRefinement typesGradual typingTermination checking
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Type Theory exactly where it's useful.

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