Concepts inProofs by structural induction using partial evaluation
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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Structural induction is a proof method that is used in mathematical logic (e.g. , in the proof of ¿o¿' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction. Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.
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Higher-order function
In mathematics and computer science, a higher-order function (also functional form, functional or functor) is a function that does at least one of the following: take one or more functions as an input output a function All other functions are first order functions. In mathematics higher-order functions are also known as operators or functionals. The derivative in calculus is a common example, since it maps a function to another function.
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Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules.
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Program transformation
A program transformation is any operation that takes a computer program and generates another program. In many cases the transformed program is required to be semantically equivalent to the original, relative to a particular formal semantics and in fewer cases the transformations result in programs that semantically differ from the original in predictable ways.
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