Math 2.0

Mathematical logic and foundations Concept

Calculus of inductive constructions

Builds on Calculus of constructionsInductive type

Summary

The calculus of inductive constructions is the calculus of constructions extended with inductive types and a hierarchy of universes. Thierry Coquand and Christine Paulin-Mohring added the inductive definitions. They give natural numbers, lists, equality and logical connectives as primitive types with their own induction principles. It is the logic of the proof assistant Coq, now called Rocq, and a variant of it underlies Lean. It keeps the impredicative sort of propositions of the calculus of constructions and adds predicative universes for data.

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptLambda cube, concept, no evidence levelLambda cubeconceptLeast fixed point, concept, no evidence levelLeast fixed pointconceptMathematical induction, concept, no evidence levelMathematical inductionconceptProposition, 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…conceptInductive type, concept, no evidence levelInductive typeconceptDependent type, concept, no evidence levelDependent typeconceptSimply typed lambda calculus, concept, no evidence levelSimply typed lambda c…conceptSystem F, concept, no evidence levelSystem FconceptCalculus of constructions, concept, no evidence levelCalculus of construct…conceptCalculus of inductive constructions, concept, no evidence levelCalculus of inductive…concept
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. Lambda cube concept
  5. Least fixed point concept
  6. Mathematical induction concept
  7. Proposition concept
  8. Set concept
  9. Set definition
  10. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Negation definition
  14. Biconditional definition
  15. Truth table concept
  16. Modus ponens concept
  17. Mathematical proof concept
  18. Propositional logic concept
  19. Tautology definition
  20. Logical equivalence definition
  21. Double negation law statement
  22. Proof by contradiction concept
  23. Russell's paradox statement
  24. Type theory concept
  25. Curry–Howard correspondence concept
  26. Inductive type concept
  27. Dependent type concept
  28. Simply typed lambda calculus concept
  29. System F concept
  30. Calculus of constructions concept
  31. Calculus of inductive constructions concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Calculus_of_constructions
About this page
TagJCX8X7VSQB
URIm2://JCX8X7VSQB
Kindconcept
WritingW0 stub
MSC03B38, 68V15, 03B40
UsesCalculus of constructions, Inductive type
Changed3 November 2026

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