6136:
5647:
4492:
9102:
8476:
8848:
8724:
6131:{\displaystyle {\begin{aligned}\operatorname {eval} &=\operatorname {eval} \ \operatorname {strategy} ]]\\\operatorname {apply} &=\operatorname {canonym} ,x]\\\operatorname {apply} &=x{\text{ if x does match the above.}}\\\operatorname {eval} &=\operatorname {eval} ]\\\operatorname {eval} &=L\\\operatorname {lazy} &=X\\\operatorname {eager} &=\operatorname {eval} \end{aligned}}}
4126:
9463:
9344:
6398:
8988:
8362:
2440:
8735:
8611:
43:
6624:
4487:{\displaystyle {\begin{aligned}\operatorname {canonym} &=\operatorname {canonym} \\\operatorname {canonym} &=\lambda \operatorname {name} (Q).\operatorname {canonym} ,Q+N]\\\operatorname {canonym} &=\operatorname {canonym} \ \operatorname {canonym} \\\operatorname {canonym} &=\operatorname {name} (M)\end{aligned}}}
9355:
9236:
6217:
3592:
The problem of how variables may be renamed is difficult. This definition avoids the problem by substituting all names with canonical names, which are constructed based on the position of the definition of the name in the expression. The approach is analogous to what a compiler does, but has been
6197:
9097:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .(\lambda \operatorname {PNF} .(\lambda \operatorname {PNFNF} .\operatorname {PNF} \operatorname {PNFNF} )(\lambda \operatorname {PNFNS} .\operatorname {PNF} \operatorname {PNFNS} ))(\operatorname {P} \operatorname {PN} ))}
8471:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .(\lambda \operatorname {PNF} .(\lambda \operatorname {PNFNF} .\operatorname {PNF} \operatorname {PNFNF} )(\lambda \operatorname {PNFNS} .\operatorname {PNF} \operatorname {PNFNS} ))(\operatorname {P} \operatorname {PN} ))}
2131:
8843:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .((\lambda \operatorname {PNF} .({\color {Red}\operatorname {PNF} }\operatorname {PN} )\operatorname {PNF} )(\lambda \operatorname {PNS} .(\operatorname {P} {\color {Red}\operatorname {PNS} )}\operatorname {PNS} )))}
5017:
of a lambda expression, M, is denoted as FV(M). This is the set of variable names that have instances not bound (used) in a lambda abstraction, within the lambda expression. They are the variable names that may be bound to formal parameter variables from outside the lambda expression.
8719:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .((\lambda \operatorname {PNF} .({\color {Blue}\operatorname {P} }\operatorname {PN} )\operatorname {PNF} )(\lambda \operatorname {PNS} .(\operatorname {P} {\color {Blue}\operatorname {PN} })\operatorname {PNS} )))}
2919:
A lambda expression that cannot be reduced further, by either β-redex, or η-redex is in normal form. Note that alpha-conversion may convert functions. All normal forms that can be converted into each other by α-conversion are defined to be equal. See the main article on
7944:
6461:
9666:
6202:
The result may be different depending on the strategy used. Eager evaluation will apply all reductions possible, leaving the result in normal form, while lazy evaluation will omit some reductions in parameters, leaving the result in "weak head normal form".
9458:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .((\lambda \operatorname {PNF} .(\operatorname {P} \operatorname {PN} )\operatorname {PNF} )(\lambda \operatorname {PNS} .(\operatorname {P} \operatorname {PN} )\operatorname {PNS} )))}
9339:{\displaystyle (\lambda \operatorname {P} .\lambda \operatorname {PN} .((\lambda \operatorname {PNF} .(\operatorname {P} \operatorname {PN} )\operatorname {PNF} )(\lambda \operatorname {PNS} .(\operatorname {P} \operatorname {PN} )\operatorname {PNS} )))}
7792:
7682:
5569:
The definitions given here assume that the first definition that matches the lambda expression will be used. This convention is used to make the definition more readable. Otherwise some if conditions would be required to make the definition precise.
3721:
1663:
The precise rules for alpha-conversion are not completely trivial. First, when alpha-converting an abstraction, the only variable occurrences that are renamed are those that are bound by the same abstraction. For example, an alpha-conversion of
7534:
5565:
This mathematical definition is structured so that it represents the result, and not the way it gets calculated. However the result may be different between lazy and eager evaluation. This difference is described in the evaluation formulas.
2123:
8150:
7978:β-reduction captures the idea of function application (also called a function call), and implements the substitution of the actual parameter expression for the formal parameter variable. β-reduction is defined in terms of substitution.
1285:
6393:{\displaystyle {\begin{aligned}\operatorname {normal} &=\operatorname {false} \\\operatorname {normal} &=\operatorname {false} \\\operatorname {normal} &=\operatorname {normal} \land \operatorname {normal} \end{aligned}}}
8982:
8356:
1210:
6452:(The definition below is flawed, it is in contradiction with the definition saying that weak head normal form is either head normal form or the term is an abstraction. The notion has been introduced by Simon Peyton Jones.)
6147:
3931:
9223:
8598:
9992:
9829:
2435:{\displaystyle {\begin{aligned}(M_{1}\ M_{2})&\equiv (M_{1})\ (M_{2})\\(\lambda x.M)&\equiv \lambda x.M\\(\lambda y.M)&\equiv \lambda y.(M){\text{, if }}x\neq y{\text{, provided }}y\notin FV(N)\end{aligned}}}
5282:
5531:
5447:
6685:
The canonical naming definition deals with the problem of variable identity by constructing a unique name for each variable based on the position of the lambda abstraction for the variable name in the expression.
5361:
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10100:
6442:
6668:
7070:
6619:{\displaystyle {\begin{aligned}\operatorname {whnf} &=\operatorname {false} \\\operatorname {whnf} &=\operatorname {false} \\\operatorname {whnf} &=\operatorname {whnf} \end{aligned}}}
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6222:
6152:
5652:
4131:
3568:
2136:
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6703:
The formal parameter variable is said to bind the variable name wherever it occurs free in the body. Variable (names) that have already been matched to formal parameter variable are said to be
9895:
9732:
7251:
7162:
6697:
The lambda abstraction operator, λ, takes a formal parameter variable and a body expression. When evaluated the formal parameter variable is identified with the value of the actual parameter.
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2023:. 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 λ expression).
1899:
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1780:
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The naive β-redex changed the meaning of the expression because x and y from the actual parameter became captured when the expressions were substituted in the inner abstractions.
464:
6753:
6700:
Variables in a lambda expression may either be "bound" or "free". Bound variables are variable names that are already attached to formal parameter variables in the expression.
3503:
939:
7840:
6755:. Also note that a variable is bound by its "nearest" lambda abstraction. In the following example the single occurrence of x in the expression is bound by the second lambda:
6455:
Reductions to the function (the head) have been applied, but not all reductions to the parameter have been applied. This is the result obtained from applying lazy evaluation.
5104:
5065:
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2916:
The purpose of β-reduction is to calculate a value. A value in lambda calculus is a function. So β-reduction continues until the expression looks like a function abstraction.
1398:
738:
697:
9838:
The alpha renaming removed the problem by changing the names of x and y in the inner abstraction so that they are distinct from the names of x and y in the actual parameter.
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of a lambda expression, M, is denoted as BV(M). This is the set of variable names that have instances bound (used) in a lambda abstraction, within the lambda expression.
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1654:
1625:
615:
4105:
Execution is performing β-reductions and η-reductions on subexpressions in the canonym of a lambda expression until the result is a lambda function (abstraction) in the
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Canonym is a function that takes a lambda expression and renames all names canonically, based on their positions in the expression. This might be implemented as,
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Mathematicians will sometimes restrict a variable to be a single alphabetic character. When using this convention the comma is omitted from the variable list.
8868:
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1144:
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1535:
is an η-redex. The expression to which a redex reduces is called its reduct; using the previous example, the reducts of these expressions are respectively
6192:{\displaystyle {\begin{aligned}\operatorname {strategy} &=\operatorname {lazy} \\\operatorname {strategy} &=\operatorname {eager} \end{aligned}}}
852:, is said to bind its variable wherever it occurs in the body of the abstraction. Variables that fall within the scope of an abstraction are said to be
3827:
9115:
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9902:
9739:
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1905:
1785:
Second, alpha-conversion is not possible if it would result in a variable getting captured by a different abstraction. For example, if we replace
5212:
7075:
This is to stop bound variables with the same name being substituted. This would not have occurred in a canonically renamed lambda expression.
5456:
5372:
6689:
This definition introduces the rules used in the standard definition and relates explains them in terms of the canonical renaming definition.
10583:
10535:
10372:
10024:
5291:
7387:
In an α-conversion, names may be substituted for new names if the new name is not free in the body, as this would lead to the capture of
10384:
6678:
The standard definition of lambda calculus uses some definitions which may be considered as theorems, which can be proved based on the
2723:
79:
6409:
4497:
Where, N is the string "N", F is the string "F", S is the string "S", + is concatenation, and "name" converts a string into a name
2445:
To substitute into a lambda abstraction, it is sometimes necessary to α-convert the expression. For example, it is not correct for
7322:
Alpha-conversion, sometimes known as alpha-renaming, allows bound variable names to be changed. For example, alpha-conversion of
6635:
1598:
Alpha-conversion, sometimes known as alpha-renaming, allows bound variable names to be changed. For example, alpha-conversion of
3943:
is a renaming of a lambda expression to give the expression standard names, based on the position of the name in the expression.
7939:{\displaystyle \lambda \operatorname {P} .\lambda \operatorname {PN} .({\color {Red}\operatorname {PN} }\operatorname {PN} )}
6980:
2593:β-reduction captures the idea of function application. β-reduction is defined in terms of substitution: the β-reduction of
3514:
10126:
9844:
9681:
7173:
7084:
96:. Two definitions of the language are given here: a standard definition, and a definition using mathematical formulas.
3384:
941:. Also note that a variable is bound by its "nearest" abstraction. In the following example the single occurrence of
745:
10367:, Studies in Logic and the Foundations of Mathematics, vol. 103 (Revised ed.), North Holland, Amsterdam.,
9661:{\displaystyle {\begin{array}{r}((\lambda x.z\ x)(\lambda y.z\ y))\\((\lambda a.z\ a)(\lambda b.z\ b))\end{array}}}
6758:
3728:
6211:
All reductions that can be applied have been applied. This is the result obtained from applying eager evaluation.
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7787:{\displaystyle \lambda \operatorname {P} .\lambda \operatorname {PN} .(\operatorname {P} \operatorname {PN} )}
7677:{\displaystyle \lambda \operatorname {P} .\lambda \operatorname {PN} .(\operatorname {P} \operatorname {PN} )}
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Note that rule 1 must be modified if it is to be used on non canonically renamed lambda expressions. See
2596:
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3452:
903:
477:
To keep the notation of lambda expressions uncluttered, the following conventions are usually applied.
7804:
7539:
Note that the substitution will not recurse into the body of lambda expressions with formal parameter
5075:
5036:
4942:
3716:{\displaystyle \operatorname {alpha-conv} (a)\to \operatorname {canonym} =\operatorname {canonym} ,P]}
2968:
Lambda
Calculus has a simple syntax. A lambda calculus program has the syntax of an expression where,
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7529:{\displaystyle (y\not \in FV(b)\land a(\lambda x.b)=\lambda y.b)\to \operatorname {alpha-con} (a)}
7303:, if they can be β-converted into the same expression, and α/η-equivalence are defined similarly.
4882:
3601:
The execution of a lambda expression proceeds using the following reductions and transformations,
1348:, if they can be β-converted into the same expression, and α/η-equivalence are defined similarly.
10427:
7354:
7325:
4796:
3508:
An abstraction with multiple parameters is equivalent to multiple abstractions of one parameter.
2691:
2118:{\displaystyle {\begin{aligned}x&\equiv N\\y&\equiv y{\text{, if }}x\neq y\end{aligned}}}
1909:
1630:
1601:
620:
576:
469:
Instances of rule 2 are known as abstractions and instances of rule 3 are known as applications.
8145:{\displaystyle (\forall z:z\not \in FV(y)\lor z\not \in BV(b))\to \operatorname {beta-redex} =b}
7989:
in the body, β-reduction may be performed on the lambda abstraction without canonical renaming.
10018:η-reduction may be used without change on lambda expressions that are not canonically renamed.
7261:
The meaning of lambda expressions is defined by how expressions can be transformed or reduced.
10579:
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835:
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207:
10426:, vol. 0804, Department of Mathematics and Statistics, University of Ottawa, p. 9,
1660:. Frequently in uses of lambda calculus, α-equivalent terms are considered to be equivalent.
544:
4106:
2921:
2911:
510:
2550:
was supposed to be free but ended up being bound. The correct substitution in this case is
1280:{\displaystyle \operatorname {FV} (M\ N)=\operatorname {FV} (M)\cup \operatorname {FV} (N)}
177:
150:
123:
10454:
10396:
484:
31:
4547:
4511:
2585:, up to α-equivalence. Notice that substitution is defined uniquely up to α-equivalence.
1294:. Closed lambda expressions are also known as combinators and are equivalent to terms in
10503:
10441:
8977:{\displaystyle (\lambda x.\lambda y.(\lambda z.(\lambda a.z\ a)(\lambda b.z\ b))(x\ y))}
8351:{\displaystyle (\lambda x.\lambda y.(\lambda z.(\lambda x.z\ x)(\lambda y.z\ y))(x\ y))}
10499:
10012:
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1403:
1205:{\displaystyle \operatorname {FV} (\lambda x.M)=\operatorname {FV} (M)\backslash \{x\}}
1122:
1023:
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883:
863:
310:
262:
10105:
The problem with using an η-redex when f has free variables is shown in this example,
10599:
1359:, refers to subterms that can be reduced by one of the reduction rules. For example,
105:
17:
1904:
In programming languages with static scope, alpha-conversion can be used to make
3926:{\displaystyle x\not \in \operatorname {FV} (f)\to \operatorname {eta-redex} =f}
1306:
The meaning of lambda expressions is defined by how expressions can be reduced.
92:
Lambda calculus is a formal mathematical system based on lambda abstraction and
9218:{\displaystyle (\lambda x.\lambda y.((\lambda a.(x\ y)a)(\lambda b.(x\ y)b)))}
8593:{\displaystyle (\lambda x.\lambda y.((\lambda x.(x\ y)x)(\lambda y.(x\ y)y)))}
6714:
For example, in the following expression y is a bound variable and x is free:
4505:
Map from one value to another if the value is in the map. O is the empty map.
9987:{\displaystyle \operatorname {BV} ((\lambda a.z\ a)(\lambda b.z\ b))=\{a,b\}}
9824:{\displaystyle \operatorname {BV} ((\lambda x.z\ x)(\lambda y.z\ y))=\{x,y\}}
9485:
5277:{\displaystyle \mathrm {FV} (\lambda x.M)=\mathrm {FV} (M)\setminus \{x\}}
2804:
they give the same result for all arguments. η-reduction converts between
10504:
A Proof-Theoretic
Account of Programming and the Role of Reduction Rules.
5526:{\displaystyle \mathrm {BV} (M\ N)=\mathrm {BV} (M)\cup \mathrm {BV} (N)}
5442:{\displaystyle \mathrm {FV} (M\ N)=\mathrm {FV} (M)\cup \mathrm {FV} (N)}
7314:, refers to subterms that can be reduced by one of the reduction rules.
4674:
If L is a lambda expression, x is a name, and y is a lambda expression;
1072:
and is defined by recursion on the structure of the terms, as follows:
4081:
is the set of variables that do not belong to a lambda abstraction in
5356:{\displaystyle \mathrm {BV} (\lambda x.M)=\mathrm {BV} (M)\cup \{x\}}
4112:
All α-conversions of a λ-expression are considered to be equivalent.
3378:
A lambda abstraction has a lower precedence than an application, so;
7559:
because of the change to the substitution operator described above.
1963:, is the process of replacing all free occurrences of the variable
10432:
7299:
We also speak of the resulting equivalences: two expressions are
2949:
In the form of a lambda abstraction whose body is not reducible.
2781:{\displaystyle ((\lambda n.\ n\times 2)\ 7)\rightarrow 7\times 2}
1344:
We also speak of the resulting equivalences: two expressions are
10307:
This improper use of η-reduction changes the meaning by leaving
10095:{\displaystyle x\not \in \mathrm {FV} (f)\to {\text{η-redex}}=f}
5535:
Combine the bound variables from the function and the parameter
5451:
Combine the free variables from the function and the parameter
7078:
For example the previous rules would have wrongly translated,
36:
7691:
correctly rename y to k, (because k is not used in the body)
7167:
The new rules block this substitution so that it remains as,
6437:{\displaystyle \operatorname {normal} =\operatorname {true} }
10551:"Notes on evaluating λ-calculus terms and abstract machines"
1290:
An expression that contains no free variables is said to be
1190:
10011:, which in this context is that two functions are the same
6663:{\displaystyle \operatorname {whnf} =\operatorname {true} }
2800:, which in this context is that two functions are the same
3264:
A variable as used by computer scientists has the syntax,
8856:
x (P) and y (PN) have been captured in the substitution.
5002:
7380:. Terms that differ only by alpha-conversion are called
6679:
1782:. The latter has a different meaning from the original.
1656:. Terms that differ only by alpha-conversion are called
3593:
adapted to work within the constraints of mathematics.
57:
10576:
The implementation of functional programming languages
7065:{\displaystyle z\neq x\ \to (\lambda z.b)=\lambda z.b}
6141:
Then the evaluation strategy may be chosen as either,
5553:
10578:. Englewood Cliffs, NJ: Prentice/Hill International.
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The β-redex then proceeded with the intended meaning.
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8215:, to meet the pre-condition for this transformation.
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The Bound
Variable Set, BV, is used in the rule for
6707:. All other variables in the expression are called
3563:{\displaystyle \lambda x.y.z=\lambda x.\lambda y.z}
52:
may be too technical for most readers to understand
10336:
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10211:
10178:{\displaystyle (\lambda x.(\lambda y.y\,x)\,x)\,a}
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9890:{\displaystyle \operatorname {FV} (x\ y)=\{x,y\}}
9727:{\displaystyle \operatorname {FV} (x\ y)=\{x,y\}}
7986:
7982:
7388:
7246:{\displaystyle (\lambda x.x\ z)=(\lambda x.x\ z)}
7157:{\displaystyle (\lambda x.x\ z)=(\lambda x.y\ z)}
4047:
961:in the expression is bound by the second lambda:
541:Applications are assumed to be left-associative:
10478:Barendregt, Henk; Barendsen, Erik (March 2000),
7801:naively rename y to z, (wrong because z free in
6674:Derivation of standard from the math definition
5545:The Free Variable Set, FV is used above in the
3436:{\displaystyle \lambda x.y\ z=\lambda x.(y\ z)}
1400:is a β-redex in expressing the substitution of
782:{\displaystyle \lambda x.\lambda y.\lambda z.N}
5637:is a name prefix possibly an empty string and
3940:
1918:alpha renaming to make name resolution trivial
10364:The Lambda Calculus: Its Syntax and Semantics
10015:they give the same result for all arguments.
8862:Alpha rename inner, x → a, y → b
7291:: which captures a notion of extensionality (
6804:{\displaystyle \lambda x.y\ (\lambda x.z\ x)}
4709:means substitute x by y in L. The rules are,
3815:{\displaystyle \operatorname {beta-redex} =b}
1336:: which captures a notion of extensionality (
8:
10495:
10493:
9981:
9969:
9884:
9872:
9818:
9806:
9721:
9709:
5546:
5350:
5344:
5271:
5265:
5146:
5140:
5028:The rules for the two sets are given below.
1199:
1193:
1112:{\displaystyle \operatorname {FV} (x)=\{x\}}
1106:
1100:
10521:
10519:
7796:No change to canonical renamed expression.
5196:{\displaystyle \mathrm {BV} (x)=\emptyset }
1007:{\displaystyle \lambda x.y(\lambda x.z\ x)}
860:. For example, in the following expression
7887:{\displaystyle \lambda z.\lambda z.(z\ z)}
7736:{\displaystyle \lambda z.\lambda k.(z\ k)}
7626:{\displaystyle \lambda z.\lambda y.(z\ y)}
6819:
5573:Running or evaluating a lambda expression
3946:
1908:simpler by ensuring that no variable name
742:A sequence of abstractions is contracted:
10526:Turbak, Franklyn; Gifford, David (2008),
10431:
10330:
10316:
10288:
10274:
10254:
10247:
10230:
10205:
10191:
10171:
10164:
10157:
10128:
10054:
10034:
10026:
9904:
9846:
9741:
9683:
9484:
9482:
9357:
9238:
9117:
8990:
8870:
8818:
8778:
8737:
8694:
8654:
8613:
8492:
8364:
8244:
8200:
8180:
8160:
8062:
7997:
7952:
7923:
7900:
7849:
7806:
7749:
7698:
7639:
7588:
7544:
7485:
7399:
7356:
7327:
7281:: applying functions to their arguments (
7264:There are three kinds of transformation:
7175:
7086:
6982:
6969:{\displaystyle (\lambda x.b)=\lambda x.b}
6916:
6898:{\displaystyle (\lambda p.b)=\lambda p.b}
6830:
6760:
6719:
6637:
6465:
6463:
6411:
6221:
6219:
6151:
6149:
5960:
5899:
5796:
5651:
5649:
5585:
5506:
5486:
5460:
5458:
5422:
5402:
5376:
5374:
5324:
5295:
5293:
5245:
5216:
5214:
5170:
5168:
5120:
5118:
5079:
5077:
5040:
5038:
4944:
4884:
4849:
4806:
4798:
4785:{\displaystyle (\lambda p.b)=\lambda p.b}
4717:
4679:
4600:
4549:
4513:
4130:
4128:
4086:
4054:
4029:
4009:
3989:
3954:
3855:
3829:
3732:
3730:
3612:
3610:
3516:
3454:
3386:
2887:
2867:
2847:
2809:
2725:
2693:
2650:
2598:
2555:
2535:
2500:
2450:
2402:
2388:
2231:
2197:
2159:
2146:
2135:
2133:
2097:
2033:
2031:
2008:
1988:
1968:
1933:
1868:
1830:
1810:
1790:
1749:
1707:
1669:
1632:
1603:
1575:
1540:
1505:
1485:
1465:
1445:
1425:
1405:
1364:
1326:: applying functions to their arguments (
1218:
1146:
1124:
1080:
1045:
1025:
966:
946:
905:
885:
865:
837:
794:
747:
704:
663:
628:
578:
546:
512:
486:
433:
401:
387:{\displaystyle (\lambda x.M)\in \Lambda }
358:
332:
312:
284:
264:
236:
209:
185:
179:
158:
152:
131:
125:
80:Learn how and when to remove this message
64:, without removing the technical details.
10528:Design concepts in programming languages
10107:
8220:
7564:
5030:
2970:
2926:
10353:
5262:
2688:For example, assuming some encoding of
5623:{\displaystyle \operatorname {eval} ]}
5152:{\displaystyle \mathrm {FV} (x)=\{x\}}
4074:{\displaystyle \operatorname {FV} (f)}
1065:{\displaystyle \operatorname {FV} (M)}
8819:
8779:
8695:
8655:
7924:
2957:In the form of a lambda abstraction.
2720:, we have the following β-reduction:
1894:{\displaystyle \lambda y.\lambda y.y}
1856:{\displaystyle \lambda x.\lambda y.x}
1775:{\displaystyle \lambda y.\lambda x.y}
1733:{\displaystyle \lambda y.\lambda x.x}
1695:{\displaystyle \lambda x.\lambda x.x}
62:make it understandable to non-experts
7:
10420:Lecture Notes on the Lambda Calculus
10397:"Example for Rules of Associativity"
6814:Changes to the substitution operator
5003:Changes to the substitution operator
2941:No β- or η-reductions are possible.
1309:There are three kinds of reduction:
104:This formal definition was given by
10455:"Example for Rule of Associativity"
10361:Barendregt, Hendrik Pieter (1984),
10261:{\displaystyle (\lambda y.y\,x)\,a}
6680:definition as mathematical formulas
5556:of non canonical lambda expression.
3588:Definition as mathematical formulas
3446:Applications are left associative;
2638:{\displaystyle ((\lambda V.E)\ E')}
619:The body of an abstraction extends
481:Outermost parentheses are dropped:
116:Lambda expressions are composed of
10038:
10035:
10007:η-reduction expresses the idea of
9431:
9398:
9365:
9312:
9279:
9246:
9079:
8998:
8812:
8745:
8688:
8656:
8621:
8453:
8372:
8090:
8087:
8084:
8081:
8078:
8072:
8069:
8066:
8063:
8002:
7905:
7772:
7754:
7662:
7644:
7510:
7507:
7504:
7498:
7495:
7492:
7489:
7486:
5985:
5982:
5979:
5976:
5973:
5967:
5964:
5961:
5824:
5821:
5818:
5815:
5812:
5806:
5803:
5800:
5797:
5510:
5507:
5490:
5487:
5464:
5461:
5426:
5423:
5406:
5403:
5380:
5377:
5328:
5325:
5299:
5296:
5249:
5246:
5220:
5217:
5190:
5174:
5171:
5124:
5121:
5083:
5080:
5044:
5041:
3880:
3877:
3874:
3871:
3868:
3862:
3859:
3856:
3760:
3757:
3754:
3751:
3748:
3742:
3739:
3736:
3733:
3640:
3637:
3634:
3631:
3625:
3622:
3619:
3616:
3613:
2796:η-reduction expresses the idea of
459:{\displaystyle (M\ N)\in \Lambda }
453:
415:
381:
340:
292:
238:
25:
8175:to rename names that are free in
6748:{\displaystyle \lambda y.x\ x\ y}
3498:{\displaystyle x\ y\ z=(x\ y)\ z}
3216:The variable list is defined as,
934:{\displaystyle \lambda y.x\ x\ y}
856:. All other variables are called
7835:{\displaystyle \lambda y.(z\ y)}
5901: if x does match the above.
5286:Free variables of M excluding x
5099:{\displaystyle \mathrm {BV} (M)}
5060:{\displaystyle \mathrm {FV} (M)}
4991:{\displaystyle z\neq x\to (z)=z}
3984:is the substitution of the name
1901:, which is not at all the same.
1393:{\displaystyle (\lambda x.M)\ N}
733:{\displaystyle (\lambda x.M)\ N}
692:{\displaystyle \lambda x.(M\ N)}
204:the abstraction symbols lambda '
41:
30:For a general introduction, see
10574:Peyton Jones, Simon L. (1987).
10481:Introduction to Lambda Calculus
3026:Anonymous function definition.
421:{\displaystyle M,N\in \Lambda }
231:The set of lambda expressions,
10337:{\displaystyle \lambda y.y\,x}
10295:{\displaystyle \lambda a.a\,x}
10251:
10232:
10212:{\displaystyle \lambda a.a\,a}
10168:
10161:
10142:
10130:
10083:
10080:
10071:
10059:
10051:
10048:
10042:
9963:
9960:
9939:
9936:
9915:
9912:
9866:
9854:
9800:
9797:
9776:
9773:
9752:
9749:
9703:
9691:
9651:
9648:
9636:
9627:
9624:
9621:
9600:
9597:
9576:
9573:
9566:
9563:
9551:
9542:
9539:
9536:
9515:
9512:
9491:
9488:
9452:
9449:
9446:
9440:
9428:
9416:
9413:
9407:
9395:
9383:
9380:
9359:
9333:
9330:
9327:
9321:
9309:
9297:
9294:
9288:
9276:
9264:
9261:
9240:
9212:
9209:
9206:
9200:
9188:
9176:
9173:
9167:
9155:
9143:
9140:
9119:
9091:
9088:
9076:
9073:
9070:
9049:
9046:
9025:
9013:
8992:
8971:
8968:
8956:
8953:
8950:
8929:
8926:
8905:
8893:
8872:
8837:
8834:
8831:
8823:
8809:
8797:
8794:
8788:
8775:
8763:
8760:
8739:
8713:
8710:
8707:
8701:
8685:
8673:
8670:
8664:
8651:
8639:
8636:
8615:
8587:
8584:
8581:
8575:
8563:
8551:
8548:
8542:
8530:
8518:
8515:
8494:
8465:
8462:
8450:
8447:
8444:
8423:
8420:
8399:
8387:
8366:
8345:
8342:
8330:
8327:
8324:
8303:
8300:
8279:
8267:
8246:
8155:Alpha renaming may be used on
8139:
8127:
8118:
8097:
8059:
8056:
8053:
8047:
8029:
8023:
7999:
7974:β-reduction (capture avoiding)
7933:
7920:
7881:
7869:
7829:
7817:
7781:
7769:
7730:
7718:
7671:
7659:
7620:
7608:
7523:
7517:
7482:
7479:
7476:
7464:
7446:
7431:
7422:
7416:
7401:
7240:
7219:
7213:
7201:
7198:
7177:
7151:
7130:
7124:
7112:
7109:
7088:
7059:
7047:
7029:
7017:
7014:
6999:
6996:
6948:
6936:
6933:
6918:
6892:
6880:
6862:
6850:
6847:
6832:
6798:
6777:
6651:
6645:
6609:
6603:
6587:
6575:
6552:
6549:
6537:
6525:
6502:
6493:
6478:
6475:
6425:
6419:
6383:
6377:
6365:
6359:
6343:
6331:
6308:
6305:
6293:
6281:
6258:
6249:
6234:
6231:
6121:
6115:
6099:
6093:
6070:
6064:
6041:
6035:
6022:
6019:
6016:
6004:
5992:
5957:
5941:
5938:
5926:
5914:
5886:
5880:
5867:
5858:
5849:
5834:
5831:
5793:
5777:
5768:
5753:
5750:
5737:
5734:
5731:
5725:
5713:
5707:
5698:
5689:
5673:
5661:
5617:
5614:
5602:
5593:
5520:
5514:
5500:
5494:
5480:
5468:
5436:
5430:
5416:
5410:
5396:
5384:
5338:
5332:
5318:
5303:
5259:
5253:
5239:
5224:
5184:
5178:
5134:
5128:
5093:
5087:
5054:
5048:
4979:
4967:
4964:
4958:
4955:
4919:
4907:
4904:
4898:
4895:
4865:
4853:
4846:
4834:
4825:
4813:
4810:
4800:
4779:
4767:
4749:
4737:
4734:
4719:
4696:
4684:
4659:{\displaystyle x\neq z\to M=M}
4653:
4647:
4638:
4632:
4629:
4617:
4611:
4575:
4569:
4566:
4554:
4524:
4518:
4477:
4474:
4468:
4462:
4446:
4428:
4415:
4391:
4379:
4355:
4339:
4315:
4302:
4287:
4275:
4263:
4251:
4245:
4226:
4199:
4186:
4168:
4152:
4140:
4068:
4062:
3971:
3959:
3914:
3911:
3899:
3887:
3852:
3849:
3843:
3809:
3797:
3788:
3767:
3710:
3701:
3695:
3689:
3677:
3665:
3656:
3653:
3647:
3486:
3474:
3430:
3418:
2835:{\displaystyle \lambda x.(fx)}
2829:
2820:
2766:
2763:
2754:
2730:
2727:
2672:
2655:
2632:
2618:
2603:
2600:
2572:
2557:
2517:
2502:
2482:
2470:
2467:
2452:
2425:
2419:
2385:
2382:
2370:
2364:
2345:
2333:
2330:
2315:
2289:
2277:
2274:
2259:
2252:
2249:
2237:
2224:
2218:
2215:
2203:
2190:
2180:
2168:
2165:
2139:
2084:
2072:
2052:
2040:
1950:
1938:
1557:
1545:
1528:{\displaystyle \lambda x.M\ x}
1381:
1366:
1274:
1268:
1256:
1250:
1238:
1226:
1187:
1181:
1169:
1154:
1094:
1088:
1059:
1053:
1001:
980:
721:
706:
686:
674:
651:{\displaystyle \lambda x.M\ N}
604:
595:
583:
580:
526:
514:
447:
435:
375:
360:
1:
5547:definition of the η-reduction
5365:Bound variables of M plus x.
2578:{\displaystyle (\lambda z.x)}
2523:{\displaystyle (\lambda x.x)}
2488:{\displaystyle (\lambda x.y)}
817:{\displaystyle \lambda xyz.N}
346:{\displaystyle M\in \Lambda }
298:{\displaystyle x\in \Lambda }
7985:in the actual parameter and
7271:: changing bound variables (
5009:Free and bound variable sets
4931:{\displaystyle z=x\to (z)=y}
1316:: changing bound variables (
7373:{\displaystyle \lambda y.y}
7344:{\displaystyle \lambda x.x}
4871:{\displaystyle (X\,Y)=X\,Y}
2713:{\displaystyle 2,7,\times }
1649:{\displaystyle \lambda y.y}
1620:{\displaystyle \lambda x.x}
610:{\displaystyle ((M\ N)\ P)}
10622:
10530:, MIT press, p. 251,
3173:E.g. x, y, fact, sum, ...
2909:
2530:, because the substituted
832:The abstraction operator,
573:may be written instead of
29:
10513:(4), pages 265-282, 1988.
9899:The bound variables are,
9736:The bound variables are,
7981:If no variable names are
6818:In the definition of the
4116:Canonym - Canonical Names
4004:by the lambda expression
10417:Selinger, Peter (2008),
9841:The free variables are,
9678:The free variables are,
6693:Free and bound variables
3266:
3218:
3181:
3146:
3112:
3068:
3034:
2988:
2964:Syntax definition in BNF
2882:does not appear free in
1020:of a lambda expression,
880:is a bound variable and
845:{\displaystyle \lambda }
828:Free and bound variables
621:as far right as possible
244:{\displaystyle \Lambda }
217:{\displaystyle \lambda }
6908:must be replaced with,
1912:a name in a containing
566:{\displaystyle M\ N\ P}
10500:de Queiroz, Ruy J.G.B.
10338:
10296:
10262:
10213:
10179:
10096:
9988:
9891:
9825:
9728:
9662:
9471:x and y not captured.
9459:
9340:
9219:
9098:
8978:
8844:
8720:
8594:
8480:Original expressions.
8472:
8352:
8209:
8189:
8169:
8146:
7961:
7940:
7888:
7836:
7788:
7737:
7686:Original expressions.
7678:
7627:
7553:
7530:
7374:
7345:
7247:
7158:
7066:
6970:
6899:
6805:
6749:
6664:
6620:
6438:
6394:
6193:
6132:
5624:
5527:
5443:
5357:
5278:
5205:where x is a variable
5197:
5161:where x is a variable
5153:
5100:
5061:
4992:
4932:
4872:
4786:
4703:
4660:
4588:
4537:
4488:
4095:
4075:
4038:
4018:
3998:
3978:
3927:
3816:
3717:
3564:
3499:
3437:
3210:Bracketed expression.
2896:
2876:
2856:
2836:
2782:
2714:
2679:
2639:
2579:
2544:
2524:
2489:
2436:
2119:
2017:
1997:
1977:
1957:
1928:Substitution, written
1895:
1857:
1819:
1799:
1776:
1734:
1696:
1650:
1621:
1584:
1564:
1529:
1494:
1474:
1454:
1434:
1414:
1394:
1281:
1206:
1133:
1113:
1066:
1034:
1008:
955:
935:
894:
874:
846:
818:
783:
734:
693:
652:
611:
567:
533:
532:{\displaystyle (M\ N)}
501:
460:
422:
388:
347:
321:
299:
273:
245:
218:
195:
168:
141:
27:Mathematical formalism
10339:
10297:
10263:
10214:
10180:
10097:
9989:
9892:
9826:
9729:
9663:
9460:
9341:
9220:
9099:
8979:
8845:
8721:
8595:
8473:
8353:
8210:
8190:
8170:
8147:
7962:
7941:
7889:
7837:
7789:
7738:
7679:
7628:
7554:
7531:
7375:
7346:
7285:), calling functions;
7248:
7159:
7067:
6971:
6900:
6820:Substitution Operator
6806:
6750:
6665:
6621:
6448:Weak head normal form
6439:
6395:
6194:
6133:
5625:
5528:
5444:
5358:
5279:
5198:
5154:
5106:- Bound Variable Set
5101:
5062:
4993:
4933:
4873:
4787:
4704:
4670:Substitution operator
4661:
4589:
4538:
4489:
4096:
4076:
4039:
4024:in lambda expression
4019:
3999:
3979:
3947:Substitution Operator
3928:
3817:
3718:
3565:
3500:
3438:
2954:Weak Head Normal Form
2897:
2877:
2857:
2837:
2783:
2715:
2680:
2640:
2580:
2545:
2525:
2490:
2437:
2120:
2018:
1998:
1978:
1958:
1896:
1858:
1820:
1800:
1777:
1735:
1697:
1651:
1622:
1585:
1565:
1530:
1495:
1475:
1455:
1435:
1415:
1395:
1282:
1207:
1134:
1114:
1067:
1035:
1009:
956:
936:
895:
875:
847:
819:
784:
735:
694:
653:
612:
568:
534:
502:
461:
423:
389:
348:
322:
300:
274:
246:
219:
196:
194:{\displaystyle v_{n}}
169:
167:{\displaystyle v_{2}}
142:
140:{\displaystyle v_{1}}
18:Weak head normal form
10315:
10273:
10229:
10190:
10127:
10025:
9903:
9845:
9740:
9682:
9481:
9356:
9237:
9116:
8989:
8869:
8736:
8612:
8491:
8363:
8243:
8199:
8179:
8159:
7996:
7951:
7899:
7848:
7805:
7748:
7697:
7638:
7587:
7543:
7398:
7355:
7326:
7312:reducible expression
7174:
7085:
6981:
6915:
6829:
6759:
6718:
6636:
6629:In all other cases,
6462:
6410:
6403:In all other cases,
6218:
6148:
5648:
5584:
5457:
5373:
5292:
5213:
5167:
5117:
5076:
5067:- Free Variable Set
5037:
4943:
4883:
4797:
4716:
4678:
4599:
4548:
4512:
4127:
4085:
4053:
4028:
4008:
3988:
3953:
3828:
3729:
3609:
3580:y is a variable list
3515:
3453:
3385:
2886:
2866:
2846:
2808:
2724:
2692:
2649:
2597:
2554:
2534:
2499:
2449:
2132:
2030:
2007:
1987:
1967:
1932:
1867:
1829:
1809:
1789:
1748:
1706:
1668:
1631:
1602:
1574:
1539:
1504:
1484:
1464:
1444:
1424:
1404:
1363:
1357:reducible expression
1217:
1145:
1123:
1079:
1044:
1024:
965:
945:
904:
884:
864:
836:
793:
746:
703:
662:
627:
577:
545:
511:
500:{\displaystyle M\ N}
485:
432:
400:
357:
331:
311:
283:
279:is a variable, then
263:
235:
208:
178:
151:
124:
94:function application
10442:2008arXiv0804.3434S
5561:Evaluation strategy
4587:{\displaystyle M=y}
4536:{\displaystyle O=x}
253:defined inductively
100:Standard definition
10457:. Lambda-bound.com
10399:. Lambda-bound.com
10334:
10292:
10258:
10223:Naive η-reduction
10209:
10175:
10092:
9984:
9887:
9821:
9724:
9671:In this example,
9658:
9656:
9455:
9336:
9215:
9094:
8974:
8840:
8826:
8783:
8716:
8699:
8659:
8590:
8468:
8348:
8230:Canonically named
8205:
8185:
8165:
8142:
7957:
7936:
7928:
7884:
7832:
7784:
7733:
7674:
7623:
7574:Canonically named
7549:
7526:
7370:
7341:
7243:
7154:
7062:
6966:
6895:
6801:
6745:
6660:
6616:
6614:
6434:
6390:
6388:
6189:
6187:
6128:
6126:
5620:
5523:
5439:
5353:
5274:
5193:
5149:
5096:
5057:
4988:
4928:
4868:
4782:
4699:
4656:
4584:
4533:
4484:
4482:
4091:
4071:
4034:
4014:
3994:
3974:
3923:
3812:
3713:
3583:z is an expression
3560:
3495:
3433:
2892:
2872:
2852:
2832:
2778:
2710:
2675:
2635:
2575:
2540:
2520:
2485:
2432:
2430:
2115:
2113:
2013:
1993:
1983:in the expression
1973:
1953:
1891:
1853:
1815:
1795:
1772:
1730:
1692:
1646:
1617:
1580:
1560:
1525:
1490:
1470:
1450:
1430:
1410:
1390:
1277:
1202:
1129:
1109:
1062:
1030:
1004:
951:
931:
890:
870:
842:
814:
789:is abbreviated as
779:
730:
689:
648:
607:
563:
529:
497:
456:
418:
384:
343:
327:is a variable and
317:
295:
269:
241:
214:
191:
164:
137:
10585:978-0-13-453333-9
10555:scholar.google.ca
10537:978-0-262-20175-9
10374:978-0-444-87508-2
10305:
10304:
10114:Lambda expression
10057:
9956:
9932:
9862:
9793:
9769:
9699:
9644:
9617:
9593:
9559:
9532:
9508:
9475:
9474:
9468:
9467:
9196:
9163:
8964:
8946:
8922:
8853:
8852:
8571:
8538:
8338:
8320:
8296:
8208:{\displaystyle b}
8188:{\displaystyle y}
8168:{\displaystyle b}
8114:
8077:
7971:
7970:
7960:{\displaystyle z}
7877:
7825:
7726:
7616:
7552:{\displaystyle x}
7503:
7236:
7194:
7147:
7105:
6995:
6794:
6776:
6741:
6735:
6583:
6545:
6498:
6339:
6301:
6254:
6012:
5972:
5934:
5902:
5854:
5811:
5773:
5718:
5669:
5539:
5538:
5476:
5392:
4702:{\displaystyle L}
4384:
4323:
4094:{\displaystyle f}
4048:Free Variable Set
4037:{\displaystyle b}
4017:{\displaystyle v}
3997:{\displaystyle p}
3977:{\displaystyle b}
3907:
3867:
3784:
3747:
3630:
3491:
3482:
3467:
3461:
3426:
3402:
3214:
3213:
3104:A function call.
2961:
2960:
2895:{\displaystyle f}
2875:{\displaystyle x}
2855:{\displaystyle f}
2759:
2744:
2678:{\displaystyle E}
2623:
2543:{\displaystyle x}
2405:
2391:
2223:
2154:
2100:
2016:{\displaystyle R}
1996:{\displaystyle E}
1976:{\displaystyle V}
1956:{\displaystyle E}
1818:{\displaystyle y}
1798:{\displaystyle x}
1583:{\displaystyle M}
1563:{\displaystyle M}
1521:
1493:{\displaystyle M}
1473:{\displaystyle x}
1453:{\displaystyle M}
1433:{\displaystyle x}
1413:{\displaystyle N}
1386:
1296:combinatory logic
1234:
1132:{\displaystyle x}
1033:{\displaystyle M}
997:
954:{\displaystyle x}
927:
921:
893:{\displaystyle x}
873:{\displaystyle y}
726:
682:
644:
600:
591:
559:
553:
522:
493:
443:
320:{\displaystyle x}
272:{\displaystyle x}
90:
89:
82:
16:(Redirected from
10613:
10590:
10589:
10571:
10565:
10564:
10562:
10561:
10547:
10541:
10540:
10523:
10514:
10497:
10488:
10487:
10486:
10475:
10466:
10465:
10463:
10462:
10451:
10445:
10444:
10435:
10425:
10414:
10408:
10407:
10405:
10404:
10393:
10387:
10382:
10377:, archived from
10358:
10343:
10341:
10340:
10335:
10310:
10301:
10299:
10298:
10293:
10267:
10265:
10264:
10259:
10218:
10216:
10215:
10210:
10184:
10182:
10181:
10176:
10108:
10101:
10099:
10098:
10093:
10058:
10055:
10041:
9993:
9991:
9990:
9985:
9954:
9930:
9896:
9894:
9893:
9888:
9860:
9830:
9828:
9827:
9822:
9791:
9767:
9733:
9731:
9730:
9725:
9697:
9675:In the β-redex,
9667:
9665:
9664:
9659:
9657:
9642:
9615:
9591:
9557:
9530:
9506:
9464:
9462:
9461:
9456:
9345:
9343:
9342:
9337:
9228:
9227:
9224:
9222:
9221:
9216:
9194:
9161:
9103:
9101:
9100:
9095:
8983:
8981:
8980:
8975:
8962:
8944:
8920:
8849:
8847:
8846:
8841:
8827:
8784:
8725:
8723:
8722:
8717:
8700:
8660:
8603:
8602:
8599:
8597:
8596:
8591:
8569:
8536:
8477:
8475:
8474:
8469:
8357:
8355:
8354:
8349:
8336:
8318:
8294:
8221:
8214:
8212:
8211:
8206:
8194:
8192:
8191:
8186:
8174:
8172:
8171:
8166:
8151:
8149:
8148:
8143:
8112:
8093:
8075:
7966:
7964:
7963:
7958:
7945:
7943:
7942:
7937:
7929:
7893:
7891:
7890:
7885:
7875:
7841:
7839:
7838:
7833:
7823:
7793:
7791:
7790:
7785:
7742:
7740:
7739:
7734:
7724:
7683:
7681:
7680:
7675:
7632:
7630:
7629:
7624:
7614:
7565:
7558:
7556:
7555:
7550:
7535:
7533:
7532:
7527:
7513:
7501:
7379:
7377:
7376:
7371:
7350:
7348:
7347:
7342:
7252:
7250:
7249:
7244:
7234:
7192:
7163:
7161:
7160:
7155:
7145:
7103:
7071:
7069:
7068:
7063:
6993:
6975:
6973:
6972:
6967:
6904:
6902:
6901:
6896:
6810:
6808:
6807:
6802:
6792:
6774:
6754:
6752:
6751:
6746:
6739:
6733:
6669:
6667:
6666:
6661:
6625:
6623:
6622:
6617:
6615:
6581:
6543:
6496:
6443:
6441:
6440:
6435:
6399:
6397:
6396:
6391:
6389:
6337:
6299:
6252:
6198:
6196:
6195:
6190:
6188:
6137:
6135:
6134:
6129:
6127:
6010:
5988:
5970:
5932:
5903:
5900:
5852:
5827:
5809:
5771:
5716:
5667:
5629:
5627:
5626:
5621:
5532:
5530:
5529:
5524:
5513:
5493:
5474:
5467:
5448:
5446:
5445:
5440:
5429:
5409:
5390:
5383:
5362:
5360:
5359:
5354:
5331:
5302:
5283:
5281:
5280:
5275:
5252:
5223:
5202:
5200:
5199:
5194:
5177:
5158:
5156:
5155:
5150:
5127:
5105:
5103:
5102:
5097:
5086:
5066:
5064:
5063:
5058:
5047:
5031:
4997:
4995:
4994:
4989:
4937:
4935:
4934:
4929:
4877:
4875:
4874:
4869:
4791:
4789:
4788:
4783:
4708:
4706:
4705:
4700:
4665:
4663:
4662:
4657:
4593:
4591:
4590:
4585:
4542:
4540:
4539:
4534:
4493:
4491:
4490:
4485:
4483:
4382:
4321:
4100:
4098:
4097:
4092:
4080:
4078:
4077:
4072:
4043:
4041:
4040:
4035:
4023:
4021:
4020:
4015:
4003:
4001:
4000:
3995:
3983:
3981:
3980:
3975:
3932:
3930:
3929:
3924:
3905:
3883:
3865:
3821:
3819:
3818:
3813:
3782:
3763:
3745:
3722:
3720:
3719:
3714:
3643:
3628:
3569:
3567:
3566:
3561:
3504:
3502:
3501:
3496:
3489:
3480:
3465:
3459:
3442:
3440:
3439:
3434:
3424:
3400:
3370:
3367:
3364:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3260:
3257:
3254:
3250:
3247:
3244:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3204:
3201:
3198:
3194:
3191:
3188:
3185:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3134:
3131:
3128:
3125:
3122:
3119:
3118:application-term
3116:
3099:
3096:
3093:
3090:
3087:
3086:application-term
3084:
3081:
3078:
3075:
3074:application-term
3072:
3056:
3053:
3052:application-term
3050:
3047:
3044:
3041:
3038:
3031:Application term
3021:
3018:
3015:
3011:
3008:
3005:
3001:
2998:
2995:
2992:
2971:
2946:Head Normal Form
2930:Normal Form Type
2927:
2922:Beta normal form
2912:Beta normal form
2901:
2899:
2898:
2893:
2881:
2879:
2878:
2873:
2861:
2859:
2858:
2853:
2841:
2839:
2838:
2833:
2787:
2785:
2784:
2779:
2757:
2742:
2719:
2717:
2716:
2711:
2684:
2682:
2681:
2676:
2671:
2644:
2642:
2641:
2636:
2631:
2621:
2584:
2582:
2581:
2576:
2549:
2547:
2546:
2541:
2529:
2527:
2526:
2521:
2494:
2492:
2491:
2486:
2441:
2439:
2438:
2433:
2431:
2406:
2404:, provided
2403:
2392:
2389:
2236:
2235:
2221:
2202:
2201:
2164:
2163:
2152:
2151:
2150:
2124:
2122:
2121:
2116:
2114:
2101:
2098:
2022:
2020:
2019:
2014:
2003:with expression
2002:
2000:
1999:
1994:
1982:
1980:
1979:
1974:
1962:
1960:
1959:
1954:
1900:
1898:
1897:
1892:
1862:
1860:
1859:
1854:
1824:
1822:
1821:
1816:
1804:
1802:
1801:
1796:
1781:
1779:
1778:
1773:
1739:
1737:
1736:
1731:
1702:could result in
1701:
1699:
1698:
1693:
1655:
1653:
1652:
1647:
1626:
1624:
1623:
1618:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1534:
1532:
1531:
1526:
1519:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1439:
1437:
1436:
1431:
1419:
1417:
1416:
1411:
1399:
1397:
1396:
1391:
1384:
1286:
1284:
1283:
1278:
1232:
1211:
1209:
1208:
1203:
1138:
1136:
1135:
1130:
1118:
1116:
1115:
1110:
1071:
1069:
1068:
1063:
1040:, is denoted as
1039:
1037:
1036:
1031:
1013:
1011:
1010:
1005:
995:
960:
958:
957:
952:
940:
938:
937:
932:
925:
919:
899:
897:
896:
891:
879:
877:
876:
871:
851:
849:
848:
843:
823:
821:
820:
815:
788:
786:
785:
780:
739:
737:
736:
731:
724:
698:
696:
695:
690:
680:
657:
655:
654:
649:
642:
616:
614:
613:
608:
598:
589:
572:
570:
569:
564:
557:
551:
538:
536:
535:
530:
520:
506:
504:
503:
498:
491:
465:
463:
462:
457:
441:
427:
425:
424:
419:
393:
391:
390:
385:
352:
350:
349:
344:
326:
324:
323:
318:
304:
302:
301:
296:
278:
276:
275:
270:
250:
248:
247:
242:
223:
221:
220:
215:
200:
198:
197:
192:
190:
189:
173:
171:
170:
165:
163:
162:
146:
144:
143:
138:
136:
135:
85:
78:
74:
71:
65:
45:
44:
37:
21:
10621:
10620:
10616:
10615:
10614:
10612:
10611:
10610:
10606:Lambda calculus
10596:
10595:
10594:
10593:
10586:
10573:
10572:
10568:
10559:
10557:
10549:
10548:
10544:
10538:
10525:
10524:
10517:
10498:
10491:
10484:
10477:
10476:
10469:
10460:
10458:
10453:
10452:
10448:
10423:
10416:
10415:
10411:
10402:
10400:
10395:
10394:
10390:
10375:
10360:
10359:
10355:
10350:
10344:unsubstituted.
10313:
10312:
10308:
10271:
10270:
10227:
10226:
10188:
10187:
10125:
10124:
10023:
10022:
10005:
9901:
9900:
9843:
9842:
9738:
9737:
9680:
9679:
9655:
9654:
9570:
9569:
9479:
9478:
9354:
9353:
9235:
9234:
9114:
9113:
8987:
8986:
8867:
8866:
8734:
8733:
8610:
8609:
8489:
8488:
8361:
8360:
8241:
8240:
8197:
8196:
8177:
8176:
8157:
8156:
7994:
7993:
7976:
7949:
7948:
7897:
7896:
7846:
7845:
7803:
7802:
7746:
7745:
7695:
7694:
7636:
7635:
7585:
7584:
7541:
7540:
7396:
7395:
7353:
7352:
7324:
7323:
7320:
7259:
7172:
7171:
7083:
7082:
6979:
6978:
6913:
6912:
6827:
6826:
6816:
6757:
6756:
6716:
6715:
6695:
6676:
6634:
6633:
6613:
6612:
6590:
6566:
6565:
6555:
6516:
6515:
6505:
6460:
6459:
6450:
6408:
6407:
6387:
6386:
6346:
6322:
6321:
6311:
6272:
6271:
6261:
6216:
6215:
6209:
6186:
6185:
6175:
6169:
6168:
6158:
6146:
6145:
6125:
6124:
6102:
6084:
6083:
6073:
6055:
6054:
6044:
6026:
6025:
5944:
5905:
5904:
5889:
5871:
5870:
5780:
5741:
5740:
5676:
5646:
5645:
5641:is defined by,
5582:
5581:
5563:
5455:
5454:
5371:
5370:
5290:
5289:
5211:
5210:
5165:
5164:
5115:
5114:
5074:
5073:
5035:
5034:
5023:bound variables
5011:
4941:
4940:
4881:
4880:
4795:
4794:
4714:
4713:
4676:
4675:
4672:
4597:
4596:
4546:
4545:
4510:
4509:
4503:
4481:
4480:
4449:
4419:
4418:
4342:
4306:
4305:
4229:
4190:
4189:
4155:
4125:
4124:
4118:
4083:
4082:
4051:
4050:
4026:
4025:
4006:
4005:
3986:
3985:
3951:
3950:
3826:
3825:
3727:
3726:
3607:
3606:
3605:α-conversion -
3599:
3590:
3577:x is a variable
3513:
3512:
3451:
3450:
3383:
3382:
3373:
3372:
3368:
3365:
3362:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3262:
3261:
3258:
3255:
3252:
3248:
3245:
3242:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3207:
3206:
3202:
3199:
3196:
3192:
3189:
3186:
3183:
3170:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3136:
3135:
3132:
3129:
3126:
3123:
3120:
3117:
3114:
3101:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3058:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3023:
3022:
3019:
3016:
3013:
3009:
3006:
3003:
2999:
2996:
2993:
2990:
2966:
2914:
2908:
2884:
2883:
2864:
2863:
2844:
2843:
2806:
2805:
2794:
2722:
2721:
2690:
2689:
2664:
2647:
2646:
2624:
2595:
2594:
2591:
2552:
2551:
2532:
2531:
2497:
2496:
2447:
2446:
2429:
2428:
2348:
2312:
2311:
2292:
2256:
2255:
2227:
2193:
2183:
2155:
2142:
2130:
2129:
2112:
2111:
2087:
2066:
2065:
2055:
2028:
2027:
2005:
2004:
1985:
1984:
1965:
1964:
1930:
1929:
1926:
1906:name resolution
1865:
1864:
1827:
1826:
1807:
1806:
1787:
1786:
1746:
1745:
1740:, but it could
1704:
1703:
1666:
1665:
1629:
1628:
1600:
1599:
1596:
1572:
1571:
1537:
1536:
1502:
1501:
1482:
1481:
1480:is not free in
1462:
1461:
1442:
1441:
1422:
1421:
1402:
1401:
1361:
1360:
1304:
1215:
1214:
1143:
1142:
1121:
1120:
1077:
1076:
1042:
1041:
1022:
1021:
963:
962:
943:
942:
902:
901:
882:
881:
862:
861:
834:
833:
830:
791:
790:
744:
743:
701:
700:
660:
659:
625:
624:
575:
574:
543:
542:
509:
508:
483:
482:
475:
430:
429:
398:
397:
355:
354:
329:
328:
309:
308:
281:
280:
261:
260:
233:
232:
227:parentheses ( )
206:
205:
181:
176:
175:
154:
149:
148:
127:
122:
121:
114:
102:
86:
75:
69:
66:
58:help improve it
55:
46:
42:
35:
32:Lambda calculus
28:
23:
22:
15:
12:
11:
5:
10619:
10617:
10609:
10608:
10598:
10597:
10592:
10591:
10584:
10566:
10542:
10536:
10515:
10489:
10467:
10446:
10409:
10388:
10373:
10352:
10351:
10349:
10346:
10333:
10329:
10326:
10323:
10320:
10303:
10302:
10291:
10287:
10284:
10281:
10278:
10268:
10257:
10253:
10250:
10246:
10243:
10240:
10237:
10234:
10224:
10220:
10219:
10208:
10204:
10201:
10198:
10195:
10185:
10174:
10170:
10167:
10163:
10160:
10156:
10153:
10150:
10147:
10144:
10141:
10138:
10135:
10132:
10122:
10119:
10118:
10115:
10112:
10103:
10102:
10091:
10088:
10085:
10082:
10079:
10076:
10073:
10070:
10067:
10064:
10061:
10053:
10050:
10047:
10044:
10040:
10037:
10033:
10030:
10013:if and only if
10009:extensionality
10004:
10001:
10000:
9999:
9996:
9995:
9994:
9983:
9980:
9977:
9974:
9971:
9968:
9965:
9962:
9959:
9953:
9950:
9947:
9944:
9941:
9938:
9935:
9929:
9926:
9923:
9920:
9917:
9914:
9911:
9908:
9897:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9865:
9859:
9856:
9853:
9850:
9836:
9833:
9832:
9831:
9820:
9817:
9814:
9811:
9808:
9805:
9802:
9799:
9796:
9790:
9787:
9784:
9781:
9778:
9775:
9772:
9766:
9763:
9760:
9757:
9754:
9751:
9748:
9745:
9734:
9723:
9720:
9717:
9714:
9711:
9708:
9705:
9702:
9696:
9693:
9690:
9687:
9669:
9668:
9653:
9650:
9647:
9641:
9638:
9635:
9632:
9629:
9626:
9623:
9620:
9614:
9611:
9608:
9605:
9602:
9599:
9596:
9590:
9587:
9584:
9581:
9578:
9575:
9572:
9571:
9568:
9565:
9562:
9556:
9553:
9550:
9547:
9544:
9541:
9538:
9535:
9529:
9526:
9523:
9520:
9517:
9514:
9511:
9505:
9502:
9499:
9496:
9493:
9490:
9487:
9486:
9473:
9472:
9469:
9466:
9465:
9454:
9451:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9412:
9409:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9385:
9382:
9379:
9376:
9373:
9370:
9367:
9364:
9361:
9351:
9347:
9346:
9335:
9332:
9329:
9326:
9323:
9320:
9317:
9314:
9311:
9308:
9305:
9302:
9299:
9296:
9293:
9290:
9287:
9284:
9281:
9278:
9275:
9272:
9269:
9266:
9263:
9260:
9257:
9254:
9251:
9248:
9245:
9242:
9232:
9225:
9214:
9211:
9208:
9205:
9202:
9199:
9193:
9190:
9187:
9184:
9181:
9178:
9175:
9172:
9169:
9166:
9160:
9157:
9154:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9111:
9107:
9106:
9104:
9093:
9090:
9087:
9084:
9081:
9078:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9054:
9051:
9048:
9045:
9042:
9039:
9036:
9033:
9030:
9027:
9024:
9021:
9018:
9015:
9012:
9009:
9006:
9003:
9000:
8997:
8994:
8984:
8973:
8970:
8967:
8961:
8958:
8955:
8952:
8949:
8943:
8940:
8937:
8934:
8931:
8928:
8925:
8919:
8916:
8913:
8910:
8907:
8904:
8901:
8898:
8895:
8892:
8889:
8886:
8883:
8880:
8877:
8874:
8864:
8858:
8857:
8854:
8851:
8850:
8839:
8836:
8833:
8830:
8825:
8822:
8817:
8814:
8811:
8808:
8805:
8802:
8799:
8796:
8793:
8790:
8787:
8782:
8777:
8774:
8771:
8768:
8765:
8762:
8759:
8756:
8753:
8750:
8747:
8744:
8741:
8731:
8727:
8726:
8715:
8712:
8709:
8706:
8703:
8698:
8693:
8690:
8687:
8684:
8681:
8678:
8675:
8672:
8669:
8666:
8663:
8658:
8653:
8650:
8647:
8644:
8641:
8638:
8635:
8632:
8629:
8626:
8623:
8620:
8617:
8607:
8600:
8589:
8586:
8583:
8580:
8577:
8574:
8568:
8565:
8562:
8559:
8556:
8553:
8550:
8547:
8544:
8541:
8535:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8511:
8508:
8505:
8502:
8499:
8496:
8486:
8485:Naive beta 1,
8482:
8481:
8478:
8467:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8389:
8386:
8383:
8380:
8377:
8374:
8371:
8368:
8358:
8347:
8344:
8341:
8335:
8332:
8329:
8326:
8323:
8317:
8314:
8311:
8308:
8305:
8302:
8299:
8293:
8290:
8287:
8284:
8281:
8278:
8275:
8272:
8269:
8266:
8263:
8260:
8257:
8254:
8251:
8248:
8238:
8235:
8234:
8231:
8228:
8225:
8204:
8184:
8164:
8153:
8152:
8141:
8138:
8135:
8132:
8129:
8126:
8123:
8120:
8117:
8111:
8108:
8105:
8102:
8099:
8096:
8092:
8089:
8086:
8083:
8080:
8074:
8071:
8068:
8065:
8061:
8058:
8055:
8052:
8049:
8046:
8043:
8040:
8037:
8034:
8031:
8028:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
8004:
8001:
7975:
7972:
7969:
7968:
7956:
7946:
7935:
7932:
7927:
7922:
7919:
7916:
7913:
7910:
7907:
7904:
7894:
7883:
7880:
7874:
7871:
7868:
7865:
7862:
7859:
7856:
7853:
7843:
7831:
7828:
7822:
7819:
7816:
7813:
7810:
7798:
7797:
7794:
7783:
7780:
7777:
7774:
7771:
7768:
7765:
7762:
7759:
7756:
7753:
7743:
7732:
7729:
7723:
7720:
7717:
7714:
7711:
7708:
7705:
7702:
7692:
7688:
7687:
7684:
7673:
7670:
7667:
7664:
7661:
7658:
7655:
7652:
7649:
7646:
7643:
7633:
7622:
7619:
7613:
7610:
7607:
7604:
7601:
7598:
7595:
7592:
7582:
7579:
7578:
7575:
7572:
7569:
7548:
7537:
7536:
7525:
7522:
7519:
7516:
7512:
7509:
7506:
7500:
7497:
7494:
7491:
7488:
7484:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7460:
7457:
7454:
7451:
7448:
7445:
7442:
7439:
7436:
7433:
7430:
7427:
7424:
7421:
7418:
7415:
7412:
7409:
7406:
7403:
7389:free variables
7369:
7366:
7363:
7360:
7340:
7337:
7334:
7331:
7319:
7316:
7297:
7296:
7286:
7276:
7258:
7257:Transformation
7255:
7254:
7253:
7242:
7239:
7233:
7230:
7227:
7224:
7221:
7218:
7215:
7212:
7209:
7206:
7203:
7200:
7197:
7191:
7188:
7185:
7182:
7179:
7165:
7164:
7153:
7150:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7102:
7099:
7096:
7093:
7090:
7073:
7072:
7061:
7058:
7055:
7052:
7049:
7046:
7043:
7040:
7037:
7034:
7031:
7028:
7025:
7022:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6998:
6992:
6989:
6986:
6976:
6965:
6962:
6959:
6956:
6953:
6950:
6947:
6944:
6941:
6938:
6935:
6932:
6929:
6926:
6923:
6920:
6906:
6905:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6858:
6855:
6852:
6849:
6846:
6843:
6840:
6837:
6834:
6815:
6812:
6800:
6797:
6791:
6788:
6785:
6782:
6779:
6773:
6770:
6767:
6764:
6744:
6738:
6732:
6729:
6726:
6723:
6694:
6691:
6675:
6672:
6671:
6670:
6659:
6656:
6653:
6650:
6647:
6644:
6641:
6627:
6626:
6611:
6608:
6605:
6602:
6599:
6596:
6593:
6591:
6589:
6586:
6580:
6577:
6574:
6571:
6568:
6567:
6564:
6561:
6558:
6556:
6554:
6551:
6548:
6542:
6539:
6536:
6533:
6530:
6527:
6524:
6521:
6518:
6517:
6514:
6511:
6508:
6506:
6504:
6501:
6495:
6492:
6489:
6486:
6483:
6480:
6477:
6474:
6471:
6468:
6467:
6449:
6446:
6445:
6444:
6433:
6430:
6427:
6424:
6421:
6418:
6415:
6401:
6400:
6385:
6382:
6379:
6376:
6373:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6347:
6345:
6342:
6336:
6333:
6330:
6327:
6324:
6323:
6320:
6317:
6314:
6312:
6310:
6307:
6304:
6298:
6295:
6292:
6289:
6286:
6283:
6280:
6277:
6274:
6273:
6270:
6267:
6264:
6262:
6260:
6257:
6251:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6224:
6223:
6208:
6205:
6200:
6199:
6184:
6181:
6178:
6176:
6174:
6171:
6170:
6167:
6164:
6161:
6159:
6157:
6154:
6153:
6139:
6138:
6123:
6120:
6117:
6114:
6111:
6108:
6105:
6103:
6101:
6098:
6095:
6092:
6089:
6086:
6085:
6082:
6079:
6076:
6074:
6072:
6069:
6066:
6063:
6060:
6057:
6056:
6053:
6050:
6047:
6045:
6043:
6040:
6037:
6034:
6031:
6028:
6027:
6024:
6021:
6018:
6015:
6009:
6006:
6003:
6000:
5997:
5994:
5991:
5987:
5984:
5981:
5978:
5975:
5969:
5966:
5963:
5959:
5956:
5953:
5950:
5947:
5945:
5943:
5940:
5937:
5931:
5928:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5906:
5898:
5895:
5892:
5890:
5888:
5885:
5882:
5879:
5876:
5873:
5872:
5869:
5866:
5863:
5860:
5857:
5851:
5848:
5845:
5842:
5839:
5836:
5833:
5830:
5826:
5823:
5820:
5817:
5814:
5808:
5805:
5802:
5799:
5795:
5792:
5789:
5786:
5783:
5781:
5779:
5776:
5770:
5767:
5764:
5761:
5758:
5755:
5752:
5749:
5746:
5743:
5742:
5739:
5736:
5733:
5730:
5727:
5724:
5721:
5715:
5712:
5709:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5682:
5679:
5677:
5675:
5672:
5666:
5663:
5660:
5657:
5654:
5653:
5631:
5630:
5619:
5616:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5562:
5559:
5558:
5557:
5550:
5537:
5536:
5533:
5522:
5519:
5516:
5512:
5509:
5505:
5502:
5499:
5496:
5492:
5489:
5485:
5482:
5479:
5473:
5470:
5466:
5463:
5452:
5449:
5438:
5435:
5432:
5428:
5425:
5421:
5418:
5415:
5412:
5408:
5405:
5401:
5398:
5395:
5389:
5386:
5382:
5379:
5367:
5366:
5363:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5330:
5327:
5323:
5320:
5317:
5314:
5311:
5308:
5305:
5301:
5298:
5287:
5284:
5273:
5270:
5267:
5264:
5261:
5258:
5255:
5251:
5248:
5244:
5241:
5238:
5235:
5232:
5229:
5226:
5222:
5219:
5207:
5206:
5203:
5192:
5189:
5186:
5183:
5180:
5176:
5173:
5162:
5159:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5126:
5123:
5111:
5110:
5107:
5095:
5092:
5089:
5085:
5082:
5071:
5068:
5056:
5053:
5050:
5046:
5043:
5015:free variables
5010:
5007:
4999:
4998:
4987:
4984:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4957:
4954:
4951:
4948:
4938:
4927:
4924:
4921:
4918:
4915:
4912:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4888:
4878:
4867:
4864:
4861:
4858:
4855:
4852:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4805:
4802:
4792:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4760:
4757:
4754:
4751:
4748:
4745:
4742:
4739:
4736:
4733:
4730:
4727:
4724:
4721:
4698:
4695:
4692:
4689:
4686:
4683:
4671:
4668:
4667:
4666:
4655:
4652:
4649:
4646:
4643:
4640:
4637:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4604:
4594:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4543:
4532:
4529:
4526:
4523:
4520:
4517:
4502:
4499:
4495:
4494:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4450:
4448:
4445:
4442:
4439:
4436:
4433:
4430:
4427:
4424:
4421:
4420:
4417:
4414:
4411:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4348:
4345:
4343:
4341:
4338:
4335:
4332:
4329:
4326:
4320:
4317:
4314:
4311:
4308:
4307:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4230:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4156:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4132:
4117:
4114:
4103:
4102:
4090:
4070:
4067:
4064:
4061:
4058:
4045:
4033:
4013:
3993:
3973:
3970:
3967:
3964:
3961:
3958:
3944:
3934:
3933:
3922:
3919:
3916:
3913:
3910:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3882:
3879:
3876:
3873:
3870:
3864:
3861:
3858:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3824:η-reduction -
3822:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3781:
3778:
3775:
3772:
3769:
3766:
3762:
3759:
3756:
3753:
3750:
3744:
3741:
3738:
3735:
3725:β-reduction -
3723:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3642:
3639:
3636:
3633:
3627:
3624:
3621:
3618:
3615:
3598:
3595:
3589:
3586:
3585:
3584:
3581:
3578:
3571:
3570:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3506:
3505:
3494:
3488:
3485:
3479:
3476:
3473:
3470:
3464:
3458:
3444:
3443:
3432:
3429:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3399:
3396:
3393:
3390:
3344:extension-char
3326:extension-char
3267:
3219:
3212:
3211:
3208:
3182:
3179:
3175:
3174:
3171:
3147:
3144:
3140:
3139:
3137:
3113:
3110:
3106:
3105:
3102:
3069:
3066:
3062:
3061:
3059:
3035:
3032:
3028:
3027:
3024:
2989:
2986:
2982:
2981:
2978:
2975:
2965:
2962:
2959:
2958:
2955:
2951:
2950:
2947:
2943:
2942:
2939:
2935:
2934:
2931:
2910:Main article:
2907:
2904:
2891:
2871:
2851:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2802:if and only if
2798:extensionality
2793:
2790:
2777:
2774:
2771:
2768:
2765:
2762:
2756:
2753:
2750:
2747:
2741:
2738:
2735:
2732:
2729:
2709:
2706:
2703:
2700:
2697:
2674:
2670:
2667:
2663:
2660:
2657:
2654:
2634:
2630:
2627:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2590:
2587:
2574:
2571:
2568:
2565:
2562:
2559:
2539:
2519:
2516:
2513:
2510:
2507:
2504:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2443:
2442:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2401:
2398:
2395:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2349:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2293:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2257:
2254:
2251:
2248:
2245:
2242:
2239:
2234:
2230:
2226:
2220:
2217:
2214:
2211:
2208:
2205:
2200:
2196:
2192:
2189:
2186:
2184:
2182:
2179:
2176:
2173:
2170:
2167:
2162:
2158:
2149:
2145:
2141:
2138:
2137:
2126:
2125:
2110:
2107:
2104:
2096:
2093:
2090:
2088:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2067:
2064:
2061:
2058:
2056:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2035:
2012:
1992:
1972:
1952:
1949:
1946:
1943:
1940:
1937:
1925:
1922:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1814:
1794:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1645:
1642:
1639:
1636:
1616:
1613:
1610:
1607:
1595:
1592:
1579:
1559:
1556:
1553:
1550:
1547:
1544:
1524:
1518:
1515:
1512:
1509:
1489:
1469:
1449:
1429:
1409:
1389:
1383:
1380:
1377:
1374:
1371:
1368:
1342:
1341:
1331:
1321:
1303:
1300:
1288:
1287:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1231:
1228:
1225:
1222:
1212:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1140:
1128:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1061:
1058:
1055:
1052:
1049:
1029:
1018:free variables
1003:
1000:
994:
991:
988:
985:
982:
979:
976:
973:
970:
950:
930:
924:
918:
915:
912:
909:
889:
869:
841:
829:
826:
825:
824:
813:
810:
807:
804:
801:
798:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
740:
729:
723:
720:
717:
714:
711:
708:
688:
685:
679:
676:
673:
670:
667:
647:
641:
638:
635:
632:
617:
606:
603:
597:
594:
588:
585:
582:
562:
556:
550:
539:
528:
525:
519:
516:
496:
490:
474:
471:
467:
466:
455:
452:
449:
446:
440:
437:
417:
414:
411:
408:
405:
394:
383:
380:
377:
374:
371:
368:
365:
362:
342:
339:
336:
316:
305:
294:
291:
288:
268:
240:
229:
228:
225:
213:
202:
188:
184:
161:
157:
134:
130:
113:
110:
101:
98:
88:
87:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10618:
10607:
10604:
10603:
10601:
10587:
10581:
10577:
10570:
10567:
10556:
10552:
10546:
10543:
10539:
10533:
10529:
10522:
10520:
10516:
10512:
10509:
10505:
10501:
10496:
10494:
10490:
10483:
10482:
10474:
10472:
10468:
10456:
10450:
10447:
10443:
10439:
10434:
10429:
10422:
10421:
10413:
10410:
10398:
10392:
10389:
10386:
10381:on 2004-08-23
10380:
10376:
10370:
10366:
10365:
10357:
10354:
10347:
10345:
10331:
10327:
10324:
10321:
10318:
10289:
10285:
10282:
10279:
10276:
10269:
10255:
10248:
10244:
10241:
10238:
10235:
10225:
10222:
10221:
10206:
10202:
10199:
10196:
10193:
10186:
10172:
10165:
10158:
10154:
10151:
10148:
10145:
10139:
10136:
10133:
10123:
10121:
10120:
10116:
10113:
10110:
10109:
10106:
10089:
10086:
10077:
10074:
10068:
10065:
10062:
10056:η-redex
10045:
10031:
10028:
10021:
10020:
10019:
10016:
10014:
10010:
10002:
9997:
9978:
9975:
9972:
9966:
9957:
9951:
9948:
9945:
9942:
9933:
9927:
9924:
9921:
9918:
9909:
9906:
9898:
9881:
9878:
9875:
9869:
9863:
9857:
9851:
9848:
9840:
9839:
9837:
9834:
9815:
9812:
9809:
9803:
9794:
9788:
9785:
9782:
9779:
9770:
9764:
9761:
9758:
9755:
9746:
9743:
9735:
9718:
9715:
9712:
9706:
9700:
9694:
9688:
9685:
9677:
9676:
9674:
9673:
9672:
9645:
9639:
9633:
9630:
9618:
9612:
9609:
9606:
9603:
9594:
9588:
9585:
9582:
9579:
9560:
9554:
9548:
9545:
9533:
9527:
9524:
9521:
9518:
9509:
9503:
9500:
9497:
9494:
9477:
9476:
9470:
9443:
9437:
9434:
9425:
9422:
9419:
9410:
9404:
9401:
9392:
9389:
9386:
9377:
9374:
9371:
9368:
9362:
9352:
9349:
9348:
9324:
9318:
9315:
9306:
9303:
9300:
9291:
9285:
9282:
9273:
9270:
9267:
9258:
9255:
9252:
9249:
9243:
9233:
9230:
9229:
9226:
9203:
9197:
9191:
9185:
9182:
9179:
9170:
9164:
9158:
9152:
9149:
9146:
9137:
9134:
9131:
9128:
9125:
9122:
9112:
9109:
9108:
9105:
9085:
9082:
9067:
9064:
9061:
9058:
9055:
9052:
9043:
9040:
9037:
9034:
9031:
9028:
9022:
9019:
9016:
9010:
9007:
9004:
9001:
8995:
8985:
8965:
8959:
8947:
8941:
8938:
8935:
8932:
8923:
8917:
8914:
8911:
8908:
8902:
8899:
8896:
8890:
8887:
8884:
8881:
8878:
8875:
8865:
8863:
8860:
8859:
8855:
8828:
8820:
8815:
8806:
8803:
8800:
8791:
8785:
8780:
8772:
8769:
8766:
8757:
8754:
8751:
8748:
8742:
8732:
8729:
8728:
8704:
8696:
8691:
8682:
8679:
8676:
8667:
8661:
8648:
8645:
8642:
8633:
8630:
8627:
8624:
8618:
8608:
8605:
8604:
8601:
8578:
8572:
8566:
8560:
8557:
8554:
8545:
8539:
8533:
8527:
8524:
8521:
8512:
8509:
8506:
8503:
8500:
8497:
8487:
8484:
8483:
8479:
8459:
8456:
8441:
8438:
8435:
8432:
8429:
8426:
8417:
8414:
8411:
8408:
8405:
8402:
8396:
8393:
8390:
8384:
8381:
8378:
8375:
8369:
8359:
8339:
8333:
8321:
8315:
8312:
8309:
8306:
8297:
8291:
8288:
8285:
8282:
8276:
8273:
8270:
8264:
8261:
8258:
8255:
8252:
8249:
8239:
8237:
8236:
8232:
8229:
8227:λ-expression
8226:
8223:
8222:
8219:
8218:See example;
8216:
8202:
8195:but bound in
8182:
8162:
8136:
8133:
8130:
8124:
8121:
8115:
8109:
8106:
8103:
8100:
8094:
8050:
8044:
8041:
8038:
8035:
8032:
8026:
8020:
8017:
8014:
8011:
8008:
8005:
7992:
7991:
7990:
7988:
7984:
7979:
7973:
7967:is captured.
7954:
7947:
7930:
7925:
7917:
7914:
7911:
7908:
7902:
7895:
7878:
7872:
7866:
7863:
7860:
7857:
7854:
7851:
7844:
7826:
7820:
7814:
7811:
7808:
7800:
7799:
7795:
7778:
7775:
7766:
7763:
7760:
7757:
7751:
7744:
7727:
7721:
7715:
7712:
7709:
7706:
7703:
7700:
7693:
7690:
7689:
7685:
7668:
7665:
7656:
7653:
7650:
7647:
7641:
7634:
7617:
7611:
7605:
7602:
7599:
7596:
7593:
7590:
7583:
7581:
7580:
7576:
7573:
7571:λ-expression
7570:
7568:α-conversion
7567:
7566:
7563:
7562:See example;
7560:
7546:
7520:
7514:
7473:
7470:
7467:
7461:
7458:
7455:
7452:
7449:
7443:
7440:
7437:
7434:
7428:
7425:
7419:
7413:
7410:
7407:
7404:
7394:
7393:
7392:
7390:
7385:
7383:
7367:
7364:
7361:
7358:
7338:
7335:
7332:
7329:
7317:
7315:
7313:
7309:
7304:
7302:
7294:
7290:
7287:
7284:
7280:
7277:
7274:
7270:
7267:
7266:
7265:
7262:
7256:
7237:
7231:
7228:
7225:
7222:
7216:
7210:
7207:
7204:
7195:
7189:
7186:
7183:
7180:
7170:
7169:
7168:
7148:
7142:
7139:
7136:
7133:
7127:
7121:
7118:
7115:
7106:
7100:
7097:
7094:
7091:
7081:
7080:
7079:
7076:
7056:
7053:
7050:
7044:
7041:
7038:
7035:
7032:
7026:
7023:
7020:
7011:
7008:
7005:
7002:
6990:
6987:
6984:
6977:
6963:
6960:
6957:
6954:
6951:
6945:
6942:
6939:
6930:
6927:
6924:
6921:
6911:
6910:
6909:
6889:
6886:
6883:
6877:
6874:
6871:
6868:
6865:
6859:
6856:
6853:
6844:
6841:
6838:
6835:
6825:
6824:
6823:
6821:
6813:
6811:
6795:
6789:
6786:
6783:
6780:
6771:
6768:
6765:
6762:
6742:
6736:
6730:
6727:
6724:
6721:
6712:
6710:
6706:
6701:
6698:
6692:
6690:
6687:
6683:
6681:
6673:
6657:
6654:
6648:
6642:
6639:
6632:
6631:
6630:
6606:
6600:
6597:
6594:
6592:
6584:
6578:
6572:
6569:
6562:
6559:
6557:
6546:
6540:
6534:
6531:
6528:
6522:
6519:
6512:
6509:
6507:
6499:
6490:
6487:
6484:
6481:
6472:
6469:
6458:
6457:
6456:
6453:
6447:
6431:
6428:
6422:
6416:
6413:
6406:
6405:
6404:
6380:
6374:
6371:
6368:
6362:
6356:
6353:
6350:
6348:
6340:
6334:
6328:
6325:
6318:
6315:
6313:
6302:
6296:
6290:
6287:
6284:
6278:
6275:
6268:
6265:
6263:
6255:
6246:
6243:
6240:
6237:
6228:
6225:
6214:
6213:
6212:
6206:
6204:
6182:
6179:
6177:
6172:
6165:
6162:
6160:
6155:
6144:
6143:
6142:
6118:
6112:
6109:
6106:
6104:
6096:
6090:
6087:
6080:
6077:
6075:
6067:
6061:
6058:
6051:
6048:
6046:
6038:
6032:
6029:
6013:
6007:
6001:
5998:
5995:
5989:
5954:
5951:
5948:
5946:
5935:
5929:
5923:
5920:
5917:
5911:
5908:
5896:
5893:
5891:
5883:
5877:
5874:
5864:
5861:
5855:
5846:
5843:
5840:
5837:
5828:
5790:
5787:
5784:
5782:
5774:
5765:
5762:
5759:
5756:
5747:
5744:
5728:
5722:
5719:
5710:
5704:
5701:
5695:
5692:
5686:
5683:
5680:
5678:
5670:
5664:
5658:
5655:
5644:
5643:
5642:
5640:
5636:
5611:
5608:
5605:
5599:
5596:
5590:
5587:
5580:
5579:
5578:
5576:
5571:
5567:
5560:
5555:
5551:
5548:
5544:
5543:
5542:
5534:
5517:
5503:
5497:
5483:
5477:
5471:
5453:
5450:
5433:
5419:
5413:
5399:
5393:
5387:
5369:
5368:
5364:
5347:
5341:
5335:
5321:
5315:
5312:
5309:
5306:
5288:
5285:
5268:
5256:
5242:
5236:
5233:
5230:
5227:
5209:
5208:
5204:
5187:
5181:
5163:
5160:
5143:
5137:
5131:
5113:
5112:
5108:
5090:
5072:
5069:
5051:
5033:
5032:
5029:
5026:
5024:
5019:
5016:
5008:
5006:
5004:
4985:
4982:
4976:
4973:
4970:
4961:
4952:
4949:
4946:
4939:
4925:
4922:
4916:
4913:
4910:
4901:
4892:
4889:
4886:
4879:
4862:
4859:
4856:
4850:
4843:
4840:
4837:
4831:
4828:
4822:
4819:
4816:
4807:
4803:
4793:
4776:
4773:
4770:
4764:
4761:
4758:
4755:
4752:
4746:
4743:
4740:
4731:
4728:
4725:
4722:
4712:
4711:
4710:
4693:
4690:
4687:
4681:
4669:
4650:
4644:
4641:
4635:
4626:
4623:
4620:
4614:
4608:
4605:
4602:
4595:
4581:
4578:
4572:
4563:
4560:
4557:
4551:
4544:
4530:
4527:
4521:
4515:
4508:
4507:
4506:
4501:Map operators
4500:
4498:
4471:
4465:
4459:
4456:
4453:
4451:
4443:
4440:
4437:
4434:
4431:
4425:
4422:
4412:
4409:
4406:
4403:
4400:
4397:
4394:
4388:
4385:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4352:
4349:
4346:
4344:
4336:
4333:
4330:
4327:
4324:
4318:
4312:
4309:
4299:
4296:
4293:
4290:
4284:
4281:
4278:
4272:
4269:
4266:
4260:
4257:
4254:
4248:
4242:
4239:
4236:
4233:
4231:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4196:
4193:
4183:
4180:
4177:
4174:
4171:
4165:
4162:
4159:
4157:
4149:
4146:
4143:
4137:
4134:
4123:
4122:
4121:
4115:
4113:
4110:
4108:
4088:
4065:
4059:
4056:
4049:
4046:
4031:
4011:
3991:
3968:
3965:
3962:
3956:
3948:
3945:
3942:
3939:
3938:
3937:
3920:
3917:
3908:
3902:
3896:
3893:
3890:
3884:
3846:
3840:
3837:
3834:
3831:
3823:
3806:
3803:
3800:
3794:
3791:
3785:
3779:
3776:
3773:
3770:
3764:
3724:
3707:
3704:
3698:
3692:
3686:
3683:
3680:
3674:
3671:
3668:
3662:
3659:
3650:
3644:
3604:
3603:
3602:
3596:
3594:
3587:
3582:
3579:
3576:
3575:
3574:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3511:
3510:
3509:
3492:
3483:
3477:
3471:
3468:
3462:
3456:
3449:
3448:
3447:
3427:
3421:
3415:
3412:
3409:
3406:
3403:
3397:
3394:
3391:
3388:
3381:
3380:
3379:
3376:
3265:
3256:variable-list
3224:variable-list
3217:
3209:
3180:
3177:
3176:
3172:
3145:
3142:
3141:
3138:
3111:
3108:
3107:
3103:
3067:
3064:
3063:
3060:
3033:
3030:
3029:
3025:
3007:variable-list
2987:
2984:
2983:
2979:
2976:
2973:
2972:
2969:
2963:
2956:
2953:
2952:
2948:
2945:
2944:
2940:
2937:
2936:
2932:
2929:
2928:
2925:
2924:for details.
2923:
2917:
2913:
2906:Normalization
2905:
2903:
2889:
2869:
2849:
2826:
2823:
2817:
2814:
2811:
2803:
2799:
2791:
2789:
2775:
2772:
2769:
2760:
2751:
2748:
2745:
2739:
2736:
2733:
2707:
2704:
2701:
2698:
2695:
2686:
2668:
2665:
2661:
2658:
2652:
2628:
2625:
2615:
2612:
2609:
2606:
2588:
2586:
2569:
2566:
2563:
2560:
2537:
2514:
2511:
2508:
2505:
2495:to result in
2479:
2476:
2473:
2464:
2461:
2458:
2455:
2422:
2416:
2413:
2410:
2407:
2399:
2396:
2393:
2379:
2376:
2373:
2367:
2361:
2358:
2355:
2352:
2350:
2342:
2339:
2336:
2327:
2324:
2321:
2318:
2308:
2305:
2302:
2299:
2296:
2294:
2286:
2283:
2280:
2271:
2268:
2265:
2262:
2246:
2243:
2240:
2232:
2228:
2212:
2209:
2206:
2198:
2194:
2187:
2185:
2177:
2174:
2171:
2160:
2156:
2147:
2143:
2128:
2127:
2108:
2105:
2102:
2094:
2091:
2089:
2081:
2078:
2075:
2069:
2062:
2059:
2057:
2049:
2046:
2043:
2037:
2026:
2025:
2024:
2010:
1990:
1970:
1947:
1944:
1941:
1935:
1923:
1921:
1919:
1915:
1911:
1907:
1902:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1812:
1792:
1783:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1743:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1661:
1659:
1643:
1640:
1637:
1634:
1614:
1611:
1608:
1605:
1593:
1591:
1577:
1554:
1551:
1548:
1542:
1522:
1516:
1513:
1510:
1507:
1487:
1467:
1447:
1427:
1407:
1387:
1378:
1375:
1372:
1369:
1358:
1354:
1349:
1347:
1339:
1335:
1332:
1329:
1325:
1322:
1319:
1315:
1312:
1311:
1310:
1307:
1301:
1299:
1297:
1293:
1271:
1265:
1262:
1259:
1253:
1247:
1244:
1241:
1235:
1229:
1223:
1220:
1213:
1196:
1184:
1178:
1175:
1172:
1166:
1163:
1160:
1157:
1151:
1148:
1141:
1139:is a variable
1126:
1103:
1097:
1091:
1085:
1082:
1075:
1074:
1073:
1056:
1050:
1047:
1027:
1019:
1014:
998:
992:
989:
986:
983:
977:
974:
971:
968:
948:
928:
922:
916:
913:
910:
907:
887:
867:
859:
855:
839:
827:
811:
808:
805:
802:
799:
796:
776:
773:
770:
767:
764:
761:
758:
755:
752:
749:
741:
727:
718:
715:
712:
709:
683:
677:
671:
668:
665:
645:
639:
636:
633:
630:
622:
618:
601:
592:
586:
560:
554:
548:
540:
523:
517:
494:
488:
480:
479:
478:
472:
470:
450:
444:
438:
412:
409:
406:
403:
395:
378:
372:
369:
366:
363:
337:
334:
314:
306:
289:
286:
266:
258:
257:
256:
254:
226:
224:' and dot '.'
211:
203:
186:
182:
159:
155:
132:
128:
119:
118:
117:
111:
109:
107:
106:Alonzo Church
99:
97:
95:
84:
81:
73:
70:November 2014
63:
59:
53:
50:This article
48:
39:
38:
33:
19:
10575:
10569:
10558:. Retrieved
10554:
10545:
10527:
10510:
10507:
10480:
10459:. Retrieved
10449:
10419:
10412:
10401:. Retrieved
10391:
10379:the original
10363:
10356:
10306:
10117:β-reduction
10104:
10017:
10006:
9670:
8861:
8224:β-reduction
8217:
8154:
7980:
7977:
7561:
7538:
7386:
7382:α-equivalent
7381:
7321:
7318:α-conversion
7311:
7310:, short for
7307:
7305:
7301:β-equivalent
7300:
7298:
7292:
7288:
7282:
7278:
7272:
7269:α-conversion
7268:
7263:
7260:
7166:
7077:
7074:
6907:
6817:
6713:
6708:
6704:
6702:
6699:
6696:
6688:
6684:
6677:
6628:
6454:
6451:
6402:
6210:
6201:
6140:
5638:
5634:
5632:
5574:
5572:
5568:
5564:
5540:
5027:
5022:
5020:
5014:
5012:
5000:
4673:
4504:
4496:
4119:
4111:
4104:
3935:
3600:
3591:
3572:
3507:
3445:
3377:
3374:
3263:
3215:
2980:Description
2967:
2933:Definition.
2918:
2915:
2795:
2687:
2592:
2444:
1927:
1924:Substitution
1903:
1784:
1741:
1662:
1658:α-equivalent
1657:
1627:might yield
1597:
1594:α-conversion
1356:
1355:, short for
1352:
1350:
1346:β-equivalent
1345:
1343:
1337:
1333:
1327:
1323:
1317:
1314:α-conversion
1313:
1308:
1305:
1291:
1289:
1017:
1015:
857:
853:
831:
476:
468:
230:
115:
103:
91:
76:
67:
51:
10385:Corrections
10003:η-reduction
7351:might give
7289:η-reduction
7279:β-reduction
6207:Normal form
5021:The set of
5013:The set of
4107:normal form
3065:Application
2985:Abstraction
2938:Normal Form
2792:η-reduction
2589:β-reduction
1334:η-reduction
1324:β-reduction
1016:The set of
507:instead of
10560:2024-05-14
10508:Dialectica
10461:2012-06-18
10403:2012-06-18
10348:References
9231:Canonical
8606:Canonical
6822:the rule,
3200:expression
3040:expression
3017:expression
2994:expression
2390:, if
2099:, if
1744:result in
120:variables
112:Definition
10433:0804.3434
10319:λ
10277:λ
10236:λ
10194:λ
10146:λ
10134:λ
10111:Reduction
10063:λ
10052:→
9943:λ
9919:λ
9910:
9852:
9780:λ
9756:λ
9747:
9689:
9604:λ
9580:λ
9519:λ
9495:λ
9435:
9420:λ
9402:
9387:λ
9372:λ
9363:λ
9316:
9301:λ
9283:
9268:λ
9253:λ
9244:λ
9180:λ
9147:λ
9132:λ
9123:λ
9083:
9065:
9053:λ
9041:
9029:λ
9017:λ
9005:λ
8996:λ
8933:λ
8909:λ
8897:λ
8885:λ
8876:λ
8816:
8801:λ
8767:λ
8752:λ
8743:λ
8692:
8677:λ
8643:λ
8628:λ
8619:λ
8555:λ
8522:λ
8507:λ
8498:λ
8457:
8439:
8427:λ
8415:
8403:λ
8391:λ
8379:λ
8370:λ
8307:λ
8283:λ
8271:λ
8259:λ
8250:λ
8101:λ
8095:
8060:→
8033:∨
8003:∀
7912:λ
7903:λ
7861:λ
7852:λ
7809:λ
7776:
7761:λ
7752:λ
7710:λ
7701:λ
7666:
7651:λ
7642:λ
7600:λ
7591:λ
7515:
7483:→
7453:λ
7435:λ
7426:∧
7359:λ
7330:λ
7306:The term
7223:λ
7181:λ
7134:λ
7092:λ
7036:λ
7003:λ
6997:→
6988:≠
6955:λ
6922:λ
6869:λ
6836:λ
6781:λ
6763:λ
6722:λ
6643:
6601:
6573:
6529:λ
6523:
6482:λ
6473:
6417:
6375:
6369:∧
6357:
6329:
6285:λ
6279:
6238:λ
6229:
6113:
6091:
6062:
6033:
5996:λ
5990:
5955:
5918:λ
5912:
5878:
5838:λ
5829:
5791:
5757:λ
5748:
5723:
5705:
5696:
5687:
5659:
5600:
5591:
5504:∪
5420:∪
5342:∪
5307:λ
5263:∖
5228:λ
5191:∅
4956:→
4950:≠
4896:→
4756:λ
4723:λ
4612:→
4606:≠
4460:
4426:
4389:
4353:
4313:
4261:
4243:
4237:λ
4203:λ
4197:
4166:
4138:
4060:
3891:λ
3885:
3853:→
3841:
3771:λ
3765:
3687:
3663:
3657:→
3645:
3597:Semantics
3549:λ
3540:λ
3519:λ
3410:λ
3389:λ
3335:extension
3314:extension
3302:extension
3293:extension
2862:whenever
2812:λ
2773:×
2767:→
2749:×
2734:λ
2708:×
2607:λ
2561:λ
2506:λ
2456:λ
2411:∉
2397:≠
2356:λ
2353:≡
2319:λ
2300:λ
2297:≡
2263:λ
2188:≡
2106:≠
2092:≡
2060:≡
1880:λ
1871:λ
1863:, we get
1842:λ
1833:λ
1761:λ
1752:λ
1719:λ
1710:λ
1681:λ
1672:λ
1635:λ
1606:λ
1508:λ
1370:λ
1351:The term
1302:Reduction
1266:
1260:∪
1248:
1224:
1191:∖
1179:
1158:λ
1152:
1086:
1051:
984:λ
969:λ
908:λ
900:is free:
840:λ
797:λ
768:λ
759:λ
750:λ
710:λ
666:λ
631:λ
454:Λ
451:∈
416:Λ
413:∈
382:Λ
379:∈
364:λ
341:Λ
338:∈
293:Λ
290:∈
251:, can be
239:Λ
212:λ
10600:Category
10383:—
10032:∉
9350:Natural
9110:Beta 2,
8730:Natural
8233:Comment
8039:∉
8015:∉
7577:Comment
7408:∉
6173:strategy
6156:strategy
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