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Substitution, written M[x := N], is the process of replacing all free occurrences of the variable x in the expression M with expression N. Substitution on terms of the lambda calculus is defined by recursion on the structure of terms, as follows (note: x and y are only variables while M and N are any lambda expression): x[x := N] = N
Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial:
A substitution is called a ground substitution if it maps all variables of its domain to ground, i.e. variable-free, terms. The substitution instance tσ of a ground substitution is a ground term if all of t ' s variables are in σ ' s domain, i.e. if vars(t) ⊆ dom(σ).
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution, [6] and also known by variant names such as half-tangent substitution or half-angle substitution.
In programming languages such as Ada, [20] Fortran, [21] Perl, [22] Python [23] and Ruby, [24] a double asterisk is used, so is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, [ 25 ] and it must be explicitly used, for example, 3 x {\displaystyle 3x} is written "3*x".
Proper substitution, in Metamath databases that support it, is a derived construct instead of one built into the Metamath language itself. The substitution rule makes no assumption about the logic system in use and only requires that the substitutions of variables are correctly done. Here is a detailed example of how this algorithm works.
In cryptography, an S-box (substitution-box) is a basic component of symmetric key algorithms which performs substitution. In block ciphers , they are typically used to obscure the relationship between the key and the ciphertext , thus ensuring Shannon's property of confusion .