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| {{Expert-subject|Computer science|date=November 2008}}
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| The '''Calculus of Constructions''' ('''CoC''') is a significant [[type theory]] created by [[Thierry Coquand]]. It can serve as both a typed programming language and as [[Constructivism (mathematics)|constructive]] [[Foundations of mathematics|foundation for mathematics]]. For this second reason, the CoC and its derivatives have been the basis for [[Coq]] and other [[proof assistant]]s.
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| Its derivatives include:
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| * Calculus of Inductive Constructions ("CiC") - added [[Intuitionistic_type_theory#Inductive_types|inductive types]]
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| * Calculus of (Co)Inductive Constructions (also "CiC"?) - added [[Coinduction]]
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| * predicative Calculus of Inductive Constructions ("pCiC") - removed some [[impredicativity]]
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| ==General traits==
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| The CoC is a higher-order [[typed lambda calculus]], initially developed by [[Thierry Coquand]]. It is well known for being at the top of [[Henk Barendregt|Barendregt]]'s [[lambda cube]]. It is possible within CoC to define functions from, say, integers to types, types to types as well as functions from integers to integers.
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| The CoC is [[normalization property (lambda-calculus)|strongly normalizing]], although, by [[Gödel's incompleteness theorem]], it is impossible to prove this property within the CoC since it implies consistency.
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| == Usage ==
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| The CoC has developed alongside the [[Coq]] [[proof assistant]]. As features were added (or possible liabilities removed) to the theory, they became available in Coq.
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| Derivatives of the CoC are used in other proof assistants, such as [[Matita]].
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| ==The basics of the calculus of constructions==
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| The Calculus of Constructions can be considered an extension of the [[Curry–Howard isomorphism]]. The Curry–Howard isomorphism associates a term in the [[Typed lambda calculus|simply typed lambda calculus]] with each natural-deduction proof in [[intuitionistic logic|intuitionistic propositional logic]]. The Calculus of Constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
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| ===Terms===
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| A ''term'' in the calculus of constructions is constructed using the following rules:
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| * '''T''' is a term (also called ''Type'')
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| * '''P''' is a term (also called ''Prop'', the type of all propositions)
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| * Variables (''x'', ''y'', ...) are terms
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| * If <math>A</math> and <math>B</math> are terms, then so are
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| ** <math>( A~B )</math>
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| ** <math>(\mathbf{\lambda}x:A . B)</math>
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| ** <math>(\forall x:A . B)</math>
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| The calculus of constructions has five kinds of objects:
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| # ''proofs'', which are terms whose types are ''propositions''
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| # ''propositions'', which are also known as ''small types''
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| # ''predicates'', which are functions that return propositions
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| # ''large types'', which are the types of predicates. ('''P''' is an example of a large type)
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| # '''T''' itself, which is the type of large types.
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| ===Judgments===
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| The calculus of constructions allows proving '''typing judgments''':
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| :<math> x_1:A_1, x_2:A_2, \ldots \vdash t:B</math> | |
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| Which can be read as the implication
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| : If variables <math>x_1, x_2, \ldots</math> have types <math>A_1, A_2, \ldots</math>, then term <math>t</math> has type <math>B</math>.
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| The valid judgments for the calculus of constructions are derivable from a set of inference rules. In the following, we use <math>\Gamma</math> to mean a sequence of type assignments
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| <math> x_1:A_1, x_2:A_2, \ldots </math>, and we use '''K''' to mean either '''P''' or '''T'''. We shall write <math> A : B :C</math> to mean "<math>A</math> has type
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| <math>B</math>, and <math>B</math> has type <math>C</math>". We shall write <math>B(x:=N)</math> to mean the result of substituting the term
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| <math>N</math> for the variable <math>x</math> in
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| the term <math>B</math>.
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| An inference rule is written in the form
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| :<math> {\Gamma \vdash A:B} \over {\Gamma' \vdash C:D} </math> | |
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| which means
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| : If <math> \Gamma \vdash A:B </math> is a valid judgment, then so is <math> \Gamma' \vdash C:D </math>
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| ===Inference rules for the calculus of constructions===
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| '''1'''. <math> {{} \over {} \Gamma \vdash P : T} </math>
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| '''2'''. <math> {\Gamma \vdash A : K \over
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| {\Gamma, x:A \vdash x : A}} </math>
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| '''3'''. <math> {\Gamma, x:A \vdash t : B : K \over
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| {\Gamma \vdash (\lambda x:A . t) : (\forall x:A . B) : K}} </math>
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| '''4'''. <math> {\Gamma \vdash M : (\forall x:A . B)\qquad\qquad\Gamma
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| \vdash N : A \over
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| {\Gamma \vdash M N : B(x := N)}} </math>
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| '''5'''. <math> {\Gamma \vdash M : A \qquad \qquad A =_\beta B \qquad \qquad B : K
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| \over {\Gamma \vdash M : B}} </math>
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| ===Defining logical operators===
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| The calculus of constructions has very few basic operators: the only logical operator for forming propositions is <math>\forall</math>. However, this one operator is sufficient to define all the other logical operators:
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| : <math>
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| \begin{matrix}
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| A \Rightarrow B & \equiv & \forall x:A . B & (x \notin B) \\
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| A \wedge B & \equiv & \forall C:P . (A \Rightarrow B \Rightarrow C) \Rightarrow C & \\
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| A \vee B & \equiv & \forall C:P . (A \Rightarrow C) \Rightarrow (B \Rightarrow C) \Rightarrow C & \\
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| \neg A & \equiv & \forall C:P . (A \Rightarrow C) & \\
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| \exists x:A.B & \equiv & \forall C:P . (\forall x:A.(B \Rightarrow C)) \Rightarrow C &
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| \end{matrix}
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| </math>
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| ===Defining data types===
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| The basic data types used in computer science can be defined
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| within the Calculus of Constructions:
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| ; Booleans : <math>\forall A: P . A \Rightarrow A \Rightarrow A</math>
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| ; Naturals : <math>\forall A:P .
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| (A \Rightarrow A) \Rightarrow (A \Rightarrow A)</math>
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| ; Product <math>A \times B</math> : <math>A \wedge B</math>
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| ; Disjoint union <math>A + B</math> : <math>A \vee B</math>
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| Note that Booleans and Naturals are defined in the same way as in [[Church encoding]]. However additional problems raise from propositional extensionality and proof irrelevance [http://coq.inria.fr/stdlib/Coq.Logic.ClassicalFacts.html].
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| ==See also== | |
| * [[Lambda calculus]]
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| * [[Typed lambda calculus]]
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| * [[Lambda cube]]
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| * [[System F]]
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| * [[Curry–Howard isomorphism]]
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| * [[Intuitionistic logic]]
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| * [[Intuitionistic type theory]]
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| * [[Homotopy type theory]]
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| ===Theorists=== | |
| * [[Thierry Coquand|Coquand, Thierry]]
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| * [[Jean-Yves Girard|Girard, Jean-Yves]]
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| == References ==
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| * Thierry Coquand and Gérard Huet: The Calculus of Constructions. Information and Computation, Vol. 76, Issue 2-3, 1988.
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| * For a source freely accessible online, see Coquand and Huet: [http://hal.inria.fr/inria-00076024/en/ The calculus of constructions]. Technical Report 530, INRIA, Centre de Rocquencourt, 1986. Note terminology is rather different. For instance, (<math>\forall x:A . B</math>) is written [''x'' : ''A''] ''B''.
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| * M. W. Bunder and Jonathan P. Seldin: [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.88.9497 Variants of the Basic Calculus of Constructions]. 2004.
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| * Maria João Frade (2009) [http://www3.di.uminho.pt/~mjf/pub/SFV-CIC-2up.pdf Calculus of Inductive Constructions (talk)].
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| [[Category:Dependently typed programming]]
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| [[Category:Lambda calculus]]
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| [[Category:Type theory]]
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