Interactive Theorem Proving in Coq and the Curry
... to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). On the other hand, a definition gives a value to an identifier by associating a well-formed term. Since this term should have a type, a definition also gives a ...
... to attach a type to an identifier, without giving the value. For example, the declaration of an identifier x with type A is written (x : A). On the other hand, a definition gives a value to an identifier by associating a well-formed term. Since this term should have a type, a definition also gives a ...
the role of logic in teaching, learning and analyzing proof
... (∃ )(∀ ) ( , ) as the same as an AE statement (∀ )(∃ ) ( , ). Transfer problems were also identified from real-life situations to mathematical ones. Hence the authors’ suggestion to help students not by analogies and explanations from natural language but by remaining in the mathematical realm espec ...
... (∃ )(∀ ) ( , ) as the same as an AE statement (∀ )(∃ ) ( , ). Transfer problems were also identified from real-life situations to mathematical ones. Hence the authors’ suggestion to help students not by analogies and explanations from natural language but by remaining in the mathematical realm espec ...
Syntax and Semantics of Propositional and Predicate Logic
... • Function symbols: f , g, etc. Each has an associated arity (number of arguments); to denote that f has arity k, we write f (k) . (Arity 0 is allowed; a function symbol of arity 0 is a constant.) • Relation symbols: A, R, etc. Each has an associated arity; to denote that R has arity k, we write R(k ...
... • Function symbols: f , g, etc. Each has an associated arity (number of arguments); to denote that f has arity k, we write f (k) . (Arity 0 is allowed; a function symbol of arity 0 is a constant.) • Relation symbols: A, R, etc. Each has an associated arity; to denote that R has arity k, we write R(k ...
Lect5-CombinationalLogic
... A PLA can directly implement the truth table of a set of logic functions with multiple inputs and outputs. Each ...
... A PLA can directly implement the truth table of a set of logic functions with multiple inputs and outputs. Each ...
function
... 1. If the free variables of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. 2. Otherwise, assume that the variable y is free in N and bound in M. Consistently replace the binding and the corresponding bound occurrences of y in M by a ...
... 1. If the free variables of N have no bound occurrences in M, then the term M[N/x] is formed by replacing all free occurrences of x in M by N. 2. Otherwise, assume that the variable y is free in N and bound in M. Consistently replace the binding and the corresponding bound occurrences of y in M by a ...
Predicate logic, motivation
... (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone) Even if we don’t yet have a way of proving this argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female philosopher, then there has to exist at least one ...
... (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone) Even if we don’t yet have a way of proving this argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female philosopher, then there has to exist at least one ...
The Discovery of the Computer
... expressed in a logical system is true or false, without producing all the possible theorems of the system. This so-called “decision problem” was answered by Alan Turing, who showed that it is not possible to decide if any theorem is true or false. In doing this, he discovered the computer. The story ...
... expressed in a logical system is true or false, without producing all the possible theorems of the system. This so-called “decision problem” was answered by Alan Turing, who showed that it is not possible to decide if any theorem is true or false. In doing this, he discovered the computer. The story ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.