Automated proof checking is the process of using software for checking proofs for correctness. It is one of the most developed fields in automated reasoning. Automated proof checking differs from automated theorem proving in that automated proof checking simply mechanically checks the formal workings of an existing proof, instead of trying to develop new proofs or theorems itself.
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Logical framework
In logic, a logical framework provides a means to define (or present) a logic as a signature in a higher-order type theory in such a way that provability of a formula in the original logic reduces to a type inhabitation problem in the framework type theory. This approach has been used successfully for (interactive) automated theorem proving. The first logical framework was Automath, however the name of the idea comes from the more widely known Edinburgh Logical Framework, LF.
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Dependent type
In computer science and logic, a dependent type is a type that depends on a value. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like ATS, Agda and Epigram. An example is the type of n-tuples of real numbers. This is a dependent type because the type depends on the value n. Deciding equality of dependent types in a program may require computations.
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Higher-order logic
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic.
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Rule of inference
In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion. For example, the rule of inference modus ponens takes two premises, one in the form of "If p then q" and another in the form of "p" and returns the conclusion "q".
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Formal system
A formal system is loosely speaking, any well defined system of abstract thought, on the model of mathematics. Technically, Euclid's elements, with a model consisting of 23 definitions and 10 postulates/axioms followed by 13 books of theorems with proof, is often held to be the first formal system and displays the characteristic of a formal system.
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Mathematical proof
In mathematics, a proof is a demonstration that if some fundamental statements are assumed to be true, then some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases.
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Arity
In logic, mathematics, and computer science, the arity Listen/¿ær¿ti/ of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product. The term springs from such words as unary, binary, ternary, etc. The term "arity" is primarily used with reference to functions of the form f : V ¿ S, where V ¿ S, and S is some set.
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