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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).
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.
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
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.
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Free substitution or rolling substitution is a rule in some sports that allows players to enter and leave the game for other players many times during the course of a game, generally during a time-out or other break in live play; and for coaches to bring in and take out players an unlimited number of times.
And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x + 1 = 0 , if x is substituted with 1 , this implies 1 + 1 = 2 = 0 , which is false, which implies that if x + 1 = 0 then x cannot be 1 .