Math 2.0

Mathematical logic and foundations Concept

Dependent type

Builds on Type theoryCurry–Howard correspondence

Summary

A dependent type is a type that depends on a value. An example is the type of lists of length exactly nn, which changes with the number nn. Dependent types give two new type formers. In a dependent function type Π\Pi, the result type depends on the argument. In a dependent pair type Σ\Sigma, the type of the second component depends on the first. Under the Curry–Howard correspondence, Π\Pi types are universal statements and Σ\Sigma types are existential statements. They are the basis of type theories used as foundations, and of proof assistants such as Agda, Coq and Lean.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementType theory, concept, no evidence levelType theoryconceptCurry–Howard correspondence, concept, no evidence levelCurry–Howard correspo…conceptDependent type, concept, no evidence levelDependent typeconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Intuitionistic logic concept
  3. Lambda calculus concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Mathematical proof concept
  15. Propositional logic concept
  16. Tautology definition
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Russell's paradox statement
  21. Type theory concept
  22. Curry–Howard correspondence concept
  23. Dependent type concept

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dependent_type
About this page
TagM6W6ZB4PPX
URIm2://M6W6ZB4PPX
Kindconcept
WritingW0 stub
MSC03B38, 68N15, 03F55
UsesType theory, Curry–Howard correspondence
Same as
Changed3 November 2026

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