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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
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). A substitution σ is called a linear substitution if tσ is a linear term for some (and hence every) linear term t containing precisely the variables of σ ' s domain, i.e. with vars(t) = dom(σ).
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:
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."
A unification problem is a finite set E={ l 1 ≐ r 1, ..., l n ≐ r n} of equations to solve, where l i, r i are in the set of terms or expressions.Depending on which expressions or terms are allowed to occur in an equation set or unification problem, and which expressions are considered equal, several frameworks of unification are distinguished.
The term redex, short for reducible expression, refers to subterms that can be reduced by one of the reduction rules. For example, ( λ x . M ) N {\displaystyle (\lambda x.M)\ N} is a β-redex in expressing the substitution of N {\displaystyle N} for x {\displaystyle x} in M {\displaystyle M} ; if x {\displaystyle x} is not free in M ...
Term vs. whole life insurance. With term life insurance, the policyholder chooses a period during which their policy is active — usually somewhere between 10 and 30 years. The policyholder pays ...
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.