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... A binary relation is a set of pairs. Let rel range over binary relations. If hx, yi ∈ rel then we sometimes write it x rel y. Let dom(rel ) = {x | hx, yi ∈ rel} and ran(rel ) = {y | hx, yi ∈ rel}. We write rel ∗ for the reflexive and transitive closure of the relation rel (see the first line of Figu ...
... A binary relation is a set of pairs. Let rel range over binary relations. If hx, yi ∈ rel then we sometimes write it x rel y. Let dom(rel ) = {x | hx, yi ∈ rel} and ran(rel ) = {y | hx, yi ∈ rel}. We write rel ∗ for the reflexive and transitive closure of the relation rel (see the first line of Figu ...
A Calculus for Type Predicates and Type Coercion
... wherever the syntax calls for a term of a supertype. To make it clear that we mean this notion of type, we will talk about the static type of a term. – When we evaluate a term using some interpretation, we get an element of the domain. Every element of the domain has exactly one type. Our semantics ...
... wherever the syntax calls for a term of a supertype. To make it clear that we mean this notion of type, we will talk about the static type of a term. – When we evaluate a term using some interpretation, we get an element of the domain. Every element of the domain has exactly one type. Our semantics ...
Dependent Types In Lambda Cube
... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...
... If we have ∀x : N at.P (x), we can read it (intuitionistic way) as ”I have a method for constructing an object of the type P (x) using any given object x of the type N at”. So here, it is more like a generalization of the ordinary function type. And that is the reason, why the term ”Dependent functi ...
Document
... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
... of the same type in the given implicit typing. This last is not a serious restriction as we can always replace other constants by terms h(c) using a fresh function symbol h. ...
12 Towards a Theory of Document Structure
... in an unrestricted natural language), or not feasible (for instance if we present a proof of a mathematical proposition with the word true, then it would be necessary to find the proof in order to build its abstract syntax tree). If the document is built using a WYSIWYG structure editor, one is temp ...
... in an unrestricted natural language), or not feasible (for instance if we present a proof of a mathematical proposition with the word true, then it would be necessary to find the proof in order to build its abstract syntax tree). If the document is built using a WYSIWYG structure editor, one is temp ...
PDF
... that we can think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning ou ...
... that we can think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning ou ...
Propositions as types
... think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...
... think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...
PDF
... that we can think of ¬φ as corresponding to a function τ → void. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning ...
... that we can think of ¬φ as corresponding to a function τ → void. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning ...
Lecture 10 Notes
... Understanding the atomic propositions and their “evidence semantics” is more subtle. One way it will be clarified is by looking at first-order and higher-order logic and type theory. Those logics refine and explicate our notion or a proposition. Type theory also shows promise in providing a semantic ...
... Understanding the atomic propositions and their “evidence semantics” is more subtle. One way it will be clarified is by looking at first-order and higher-order logic and type theory. Those logics refine and explicate our notion or a proposition. Type theory also shows promise in providing a semantic ...