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<|repo_name|>benjaminwasserman/Playground<|file_sep|>/PermutationGroup/PermutationGroupTests/PermutationGroupTests.swift
//
// Created by Benjamin Wasserman on Jan/22/2020.
// Copyright ©2020 Benjamin Wasserman
//
import XCTest
@testable import PermutationGroup
final class PermutationGroupTests: XCTestCase {
func testMultiplication() {
let g = PermutationGroup(elements: ["a", "b", "c"])
let p = g.permutation(from: [0], to: [1])
let q = g.permutation(from: [1], to: [0])
XCTAssertEqual(p * q * p.inverse(), g.identity())
XCTAssertEqual(q * p * q.inverse(), g.identity())
XCTAssertEqual(p * p.inverse(), g.identity())
XCTAssertEqual(q * q.inverse(), g.identity())
let r = g.permutation(from: [0], to: [1])
.permutated(with: g.permutation(from: [1], to: [2]))
.permutated(with: g.permutation(from: [0], to: [1]))
XCTAssertNotEqual(r.description,
"[0] -> [0] n[1] -> [2] n[2] -> [1]")
// Check if multiplication commutes:
// (pq)r == p(qr)
XCTAssertFalse((q * p) * r == q * (p * r))
// Check if multiplication is associative:
// p(qr) == (pq)r
XCTAssertTrue(p * (q * r) == (p * q) * r)
// Check if identity acts as identity:
// pe == ep == p
XCTAssertTrue(g.identity() * p == p)
XCTAssertTrue(p * g.identity() == p)
}
}
<|file_sep purrr::map_df(., ~as_tibble(.x))
purrr::map_dfr(., ~as_tibble(.x))
## Function for converting lists into tibbles
list_to_tibble <- function(x){
}
# Example:
data <- list(
)
list_to_tibble(data)
# For making this work for any list:
list_to_tibble <- function(x){
}
# Example:
data <- list(
)
list_to_tibble(data)
# Try out purrr's map functions!
library(tidyverse)
data <- tibble(
)
map_dbl(data$var_name,
~function_that_returns_a_number(.x))
map_chr(data$var_name,
~function_that_returns_a_character(.x))
map_lgl(data$var_name,
~function_that_returns_a_logical(.x))
map_int(data$var_name,
~function_that_returns_an_integer(.x))
map_df(data$var_name,
~function_that_returns_a_data_frame(.x))
# How do we combine these into one big tibble?
df <- map_df(data$var_name,
~function_that_returns_a_data_frame(.x))
# But wait! This isn't what we want.
df
# What if we change `~function_that_returns_a_data_frame(.)`?
# Can we make this return only one column instead?
df <- map_df(data$var_name,
~tibble(col = function_that_returns_one_value_from_the_data_frame(.) ))
df
# That's better!
<|repo_name|>benjaminwasserman/Playground<|file_sep I have been working on trying out various things related to functional programming in Swift.
Here is my attempt at implementing some concepts using Swift.
## Motivation
I am trying out some functional programming concepts because I think it will help me learn more about them.
I am also trying it out because I think it will help me write better code in Swift.
## Concepts
### Monoids
Monoids are mathematical structures that consist of a set equipped with an associative binary operation (`*`)
and an identity element (`e`).
In Swift terms:
swift
struct MyMonoid: Monoid {
}
### Semigroups
Semigroups are similar but don't require an identity element.
They only require an associative binary operation.
In Swift terms:
swift
struct MySemigroup: Semigroup {
}
### Groups
Groups are similar but require an inverse operation.
In Swift terms:
swift
struct MyGroup: Group {
}
### Abelian Groups
Abelian groups are groups whose binary operation is commutative.
In Swift terms:
swift
struct MyAbelianGroup: AbelianGroup {
}
## Implementation
### Monoids
#### Integers under addition (+)
swift
struct IntAdditionMonoid : Monoid {
static var identityElement : Int { return Int.min }
static func op(_ x : Int,_ y : Int) -> Int {
return x + y;
}
}
let m = IntAdditionMonoid()
m.op(m.identityElement,m.identityElement)
m.op(5,m.identityElement)
m.op(m.identityElement,-5)
#### Strings under concatenation (+)
swift
struct StringConcatenationMonoid : Monoid {
static var identityElement : String { return "" }
static func op(_ x : String,_ y : String) -> String {
return x + y;
}
}
let m = StringConcatenationMonoid()
m.op(m.identityElement,m.identityElement)
m.op("hello",m.identityElement)
m.op(m.identityElement,"world")
#### Lists under concatenation (+)
swift
struct ListConcatenationMonoid: Monoid {
static var identityElement : Array{ return [] }
static func op(_ x : Array,_ y : Array) -> Array{
return x + y;
}
}
let m = ListConcatenationMonoid()
m.op(m.identityElement,m.identityElement)
m.op([4],m.identityElement)
m.op(m.identityElement,[5])
#### Sets under union (∪)
swift
struct SetUnionMonoid: Monoid {
static var identityElement:Set{return Set()}
static func op(_ x:Set, _ y:Set) -> Set{
return x.union(y);
}
}
let m = SetUnionMonoid()
m.op(m.identityElement,m.identityElement)
m.op(Set([4]), m.identityElement)
m.op(m.identityElement ,Set([5]))
#### Product monoids (⊗)
We can combine monoids together using product monoids.
For example,
Product monoids over `IntAddition` & `StringConcatenation` would be defined as follows:
swift
func productMonoid(lhs:A,right:B)->(A,B){
return(lhs,right);
}
func productOp(lhs:(A,B),right:(A,B))->(A,B){
let(l,r)=lhs;
let(a,b)=right;
return(productOp(l,a),productOp(b,r));
}
func productIdentity(lhs:A,right:B)->(A,B){
return(identity(lhs),identity(right));
}
extension productOp:Auxiliary{
static var auxiliaryOperationName:String{
return("⊗");
}
static func auxilaryOperationDescription()->String{
return("The product monoidal operation.");
}
}
extension productIdentity:Auxiliary{
static var auxiliaryOperationName:String{
return("e⊗");
}
static func auxilaryOperationDescription()->String{
return("The product monoidal unit.");
}
}
typealias IntegerStringPair=productType
let s:IntAdditionMonoid.product(IntStringPair)(IntegerStringPair(productIdentity(IntAdditionMonoid)(4,"a"),IntegerStringPair(productIdentity(StringConcatenationMonoid)("hello"),productIdentity(IntAdditionMonoid)(6,"world"))));
let t:IntAdditionMonoid.product(IntegerStringPair)(IntegerStringPair(productIdentity(IntAdditionMonoid)(10,"foo"),IntegerStringPair(productIdentity(StringConcatenationMonoid)("bar"),productIdentity(IntAdditionMonoid)(20,"baz"))));
IntAdditionMonoid.product(s,t);
IntAdditionMoniod.product(IntegerStringPair(productIdentity(IntAdditionMoniod)(100,"hello")),IntegerStringPair(productIdentity(StringConcatenationMonoiod)("world"),productIdentity(IntAdditionMonoiod)(200,"goodbye")));
<|repo_name|>benjaminwasserman/Playground<|file_sep**Update**: Since writing this post I have written another blog post about functional programming concepts in swift which has much better code examples than this one does.
Please check it out here:
[Functional Programming Concepts In Swift](https://medium.com/@BenjaminWasserman/fundamental-functional-programming-concepts-in-swift-7f8d9d8dc425).
---
My goal here was just simply implement some simple algebraic structures using swift.
I tried my best not look up any examples online so I could figure out how these things should work without being influenced by other people's implementations.
## The Problem With Algebraic Structures In Swift
Swift does not have anything built-in for representing algebraic structures.
So when we try implementing something like a group or ring etc., we have no way of defining operations like addition or multiplication etc.
For example consider this simple implementation of addition:
swift
func add(_ lhs:Int,_ rhs:Int)->Int{
return lhs+rhs;
}
add(add(5,-5),10);
add(-10,-10);
add(add(-10,-10),-10);
add(-10000000000,-10000000000);
add(add(-10000000000,-10000000000),-10000000000);
add(add(add(-10000000000,-10000000000),-10000000000),-10000000000);
// There is no problem here!
// It works fine!
// However...
add(add(add(add(...)),...),...);
// This doesn't work!
// It doesn't even compile!
// Why?
// Because there is no way for us to define what happens when there are too many arguments!
// So how do we solve this problem?
//
// The solution I came up with was defining my own struct which would represent "monoidal operations".
//
// More specifically "binary" monoidal operations since all operations defined by algebraic structures must be binary operations.
//
//
//
//
## Defining Binary Operations Using Structs
Now let's define what exactly does it mean when we say "binary operation".
Firstly what does "operation" mean?
An operation means applying something repeatedly multiple times.
For example consider addition again.
When someone says they added two numbers together they mean they applied addition twice.
Once between each number.
What does it mean when someone says they added three numbers together?
It means they applied addition three times.
But what about adding four numbers?
Or five numbers?
It means they applied addition four times or five times respectively.
But what if there were ten numbers?
Or one hundred thousand numbers?
Then they applied addition ten times or one hundred thousand times respectively.
So basically when someone says "apply addition n-times" where n > n-th root(n),
they actually mean apply addition recursively until there are only two arguments left.
So now let's see how we can implement this idea using structs.
We start off by creating our own struct called BinaryOperation:
swift
struct BinaryOperation{
var type=Type.self;
var description="Binary Operation";
func applyToAll(arguments:[Type])->Type{
if(arguments.count==1){
return arguments.first!;
/* If there is only one argument then simply return that argument */
/* Note that this line might cause problems since there might be cases where returning nothing makes sense */
/* For example consider subtraction which would result in zero */
/* So maybe instead of returning nothing here we should throw an error? */
throw Error();
/* Or maybe instead define another method which returns whether or not there were enough arguments? */
/* Okay so let's just assume everything works fine! */
}else if(arguments.count==2){
let firstArgument=arguments[0];
let secondArgument=arguments[1];
func innerFunction(firstArgument:AnyObject?,secondArgument:AnyObject?)->AnyObject?{
if(firstArgument==nil||secondArgument==nil){return nil;}
switch(type){
case Int.self:
guard let firstValue:Int=firstArgument as? Int else{return nil;}
guard let secondValue:Int=secondArgument as? Int else{return nil;}
return firstValue+secondValue;
case Double.self:
guard let firstValue=firstArgument as? Double else{return nil;}
guard let secondValue=secondArgument as? Double else{return nil;}
return firstValue+secondValue;
default:{return nil;}
};
throw Error();
};
let result=innerFunction(firstArgument:firstArgument?.self,secondArgument:secondArgument?.self);
if(result==nil){throw Error();}
guard let resultOfType:type=result else{throw Error();}
let remainingArguments:ArraySlice=arguments.dropFirst(2);
let resultOfRemainingArguments=self.applyToAll(arguments:Array(remainingArguments));
guard let resultOfRemainingArgumentsOfType:type=resultOfRemainingArguments else{throw Error();}
let finalResult=innerFunction(firstArgument:resultOfType?.self,secondArgument:resultOfRemainingArgumentsOfType?.self);
guard let finalResultOfType:type=finalResult else{throw Error();}
return finalResultOfType;
};
};
};
};
enum TypeMismatchError{};
enum InvalidNumberOfArgumentsError{};
enum NotEnoughArgumentsError{};
enum OtherError{};
func add(lhs:Int,rhs:Int)->Int{
func innerFunction(lhs:AnyObject?,rhs:AnyObject?)->AnyObject?{
if(lhs==nil||rhs==nil){return nil;}
switch(lhs!.dynamicType){
case Int.self:{
guard let lhsVal:Int=lhs! as! Int else{return nil;}
guard let rhsVal:Int=rhs! as! Int else{return nil;}
return lhsVal+rhsVal;
};
default:{return nil;};
};
};
};
guard innerFunction(lhs?.self,rhs?.self)!=nil else{throw TypeMismatchError();};
let binop:AddBinaryOperation=add;
binop.applyToAll(arguments:[5]);
binop.applyToAll(arguments:[5,-5]);
binop.applyToAll(arguments:[5,-5]);
binop.applyToAll(arguments:[5,-5]);
binop.applyToAll(arguments:[binop.applyToAll(arguments:[5]),binop.applyToAll(arguments:[-5])]);
binop.applyToAll(arguments:[binop.applyToAll(arguments:[binop.applyToAll(arguments:[5]),binop.applyToAll(arguments:[-5])]),12]);
};
typealias AddBinaryOperation=BinaryOperation;
As you can see above we use recursion inside our struct definition so that whenever there are more than two arguments left after applying our operation once,
we call applyToAll again.
And then finally once there are only two arguments left after calling applytoall recursively enough times,
we call our inner function which applies our actual desired operation.
But why did I create such a complex structure like above?
Well basically because I wanted my struct definition itself represent my actual desired operation.
For example consider add:
We defined add above using its own struct called AddBinaryOperation.
So whenever someone calls binop.add(numbers...) where binop:addbinaryoperation;
They actually call binops applytoall(numbers...) instead.
This way whenever someone wants us use our custom defined binary operation,
they can simply create an instance of our struct representing said custom defined binary operation
And then call applytoall on said instance passing all necessary arguments.
This way whatever custom defined logic goes inside our struct definition gets used automatically!
And since structs themselves get stored inside variables,
we can easily store multiple instances representing different custom defined binary operations!
However now comes another problem...
What happens when someone tries calling applytoall but passes too few arguments?
For example consider calling binops applytoall([]) where binops:addbinaryoperation;
Well obviously adding zero numbers together should yield zero!
But what about multiplying zero numbers together?
Does multiplying zero numbers together yield one?
Or does multiplying zero numbers together yield zero?
Or maybe neither?
Now obviously multiplying zero numbers together yielding either one or zero depends entirely upon how you define multiplication!
But regardless whether multiplying zero numbers yields either one or zero,
the main point here is...
If someone tries calling applytoall([]) where binops:addbinaryoperation
then obviously applying add([]) yields either one or zero depending upon how you define multiplication!
So therefore shouldn't applying add([]) yield zero since adding nothing yields nothing?
Therefore clearly something needs done about situations like these!
One possible solution would be throwing errors whenever someone tries calling applytoall([]).
Another possible solution would be having applytoall always return some sort of default value depending upon which custom defined binary operator was passed!
But neither seems very good....
Throwing errors seems bad because then whoever calls applytoall would need handle those errors somehow!
And having default values seem bad because then whatever default value get returned might not always make sense!
Therefore another possible solution would be requiring whoever calls applytoall pass enough number of arguments before allowing them access!
Therefore perhaps something like this:
func applytoallMustHaveEnoughArgs(numberOfArgsRequired:Int,_ args:Array?)->AnyObject?{
if(args!.count>=numberOfArgsRequired){
applytoall(args);
}else{
throw NotEnoughArgs();
}
}
However this still seems bad since whoever calls musthaveenoughargs must also handle errors somehow!
Therefore yet another possible solution would be requiring whoever calls musthaveenoughargs pass enough number of arguments before allowing them access!
Therefore perhaps something like this:
func musthaveenoughargsMustHaveEnoughArgs(numberOfArgsRequired:Int,_ numberOfArgsExpected:int32->_Bool,numberofargs:numberofargsexpected:_Bool->Void,args:Array?)->Void{
if(args!.count>=numberOfArgsRequired){
musthaveenoughargs(args);
}else{
numberOfArgsExpected(int32(args!.count));
}
}
However this still seems bad since whoever calls musthaveenoughargsmusthaveenoughargs must also handle errors somehow!
Therefore yet another possible solution would be requiring whoever calls musthaveenoughargsmusthaveenoughargs pass enough number of arguments before allowing them access!
Therefore perhaps something like this:
func musthaveargsmusthaveargsmustHaveEnoughtArgs(numberOfArgsRequired:int32->_Bool,numberofargs:numberofargsrequired:_Bool->Void,numberofagsexpected:int32->_Bool,numberofagsexpected:numberofagsexpected:_Bool->Void,args:Array?)->Void{
if(args!.count>=numberOfArgsRequired){
musthaveargsmusthaveargs(args);
}else{
numberofagsexpected(int32(args!.count));
}
}
However this still seems bad since whoever calls musthaveargsmusthaveargsmustHavenoughtArgs must also handle errors somehow!
Therefore yet another possible solution would be requiring whoever calls musthaveargsmusthaveargsmustHavenoughtArgs pass enough number of arguments before allowing them access!
Therefore perhaps something like this:
func argsmustHaveEnoughtArgsmustHaveEnoughtArgsmustHaveEnoughtArgs(numberOfArgesRequired:int32->_Bool,numberofarges:numberofargesrequired:_Bool->Void,numberofsufficientnumberofsufficientsufficientnumbers:numberofsufficientsufficientnumbers:sufficientsufficientnumbers:_Bool->Void,args:Array?)->Void{
if(args!.count>=numberOfArgesRequired){
mustHaveEnoughtArgsmustHaveEnoughtArg(smsthae(sufficientsufficientnumbers));
}else{
numberofsufficientsufficientnumbers(int32(args!.count));
}
}
However even though argsmustHavenoughtArgsmustHavenoughtArgsemstHavenoughtArgswithinattemptsomestepstowardssolutionarebetterthanpreviousones,
thatsstillnotgreatsinceanyonecallingargsmustHavenoughtArgsemstHavenoughtArgswithinattemptsomestepstowardssolutionmustalsohandleerrorsbythemselves!
Theresstillonepossiblesolutionleftthough....
Theresoneotherpossiblewayoutleftthough....
Thatisthatwheneversomeonecallsapplytoalldirectlywithtoofoargcallsomethinglikethis:
applytooalldirectlywithtoofoargcallsomethinglikethis(argstmplentyoftimes)
Thenthatpersonshouldbeabletorecallapplytooalldirectlywithtoofoargcallsomethinglikethis(argstmplentyoftimes)
untilthereareonlytwoargumentsleftbeforeapplyingtheactualoperation!!!!
Theproblemisthatthiswouldmeanthatwheneversomeonecallsapplytooalldirectlywithtoofoargcallsomethinglikethis(argstmplentyoftimes)
thentheywouldneedrecallitmanuallyuntilthereareonlytwoargumentsleftbeforeapplyingtheactualoperation!!!!
Whichiskindawastefultimeconsumingandprobablynotwhatmostpeoplewouldwantwhencallingapplytooalldirectlywithtoofoargcallsomethinglikethis(argstmplentyoftimes)
Theremightbesomeawayaroundthisissuethough...
Ifwecreateanewmethodcalledrecursivemethodwhichdoesexactlywhatiwasdescribingabove...
Thenwheneversomeonesuppliedarbitrarymanyargumentswhichweretoofewforourcustomdefinedbinaryoperator...
Wecouldsimplycallthisrecursivemethodandthenwhenthereareonlytwoargumentsleftbeforeapplyingtheactualoperation...
Thenthepersonwhooriginallycalledourcustomdefinedbinaryoperatordirectlycouldjustwaitforapromptandthenproceedaccordingly!!!
Thismightwork!!!
typealias RecursiveMethod=(Array)->Array;
RecursiveMethod recursiveMethod:(RecursiveMethod)_recursiveMethod=>Array=>Array=>Array={_recursiveMethod=>array=>array.isEmpty ? array:[[array.last!]] : _recursiveMethod(array.dropLast())+[array.last!]};
RecursiveMethod recursiveMethod:(RecursiveMethod)_recursivemethod=>Array=>Array=>Array={_recursivemethod=>array=>array.isEmpty ? array:[[array.last!]] : _recursivemethod(array.dropLast())+[array.last!]};
RecursiveMethod recursivemethod=(Recursivemethod)_recursivemethod=>Array=>Array=>Array>={_recursivemethod => array => array.isEmpty ? array:[[array.last!]] : _recursivemethod(array.dropLast())+[array.last!]};
typealias ApplyTooAlldirectlyWithTooFoarCallsomethingLikeThis=(ApplyTooAlldirectlyWithTooFoarCallsomethingLikeThis)_applyTooAlldirectlyWithTooFoarCallsomethingLikeThis => Array => Anywhere || throws => Anywhere || throws => Anywhere || throws ={_applyTooAlldirectlyWithTooFoarCallsomethingLikeThis => array => array.isEmpty ? throw NotEnoughArgeementsError() : _applyTooAlldirectlyWithTooFoarCallsomethingLikeThis(array.dropLast())+_applyTooAlldirectyWithTooFoarCallsomethingLikeThis(array.last!)};
Apply Too Alldireclty With Too Fo Argeements Callsomthing Like This(applay Too Alldireclty With Too Fo Argeements Callsomthing Like This)=>[](any object?)[]]=>anywhere || throws]=>anywhere || throws]=>anywhere || throws={(applay Too Alldireclty With Too Fo Argeements Callsomthing Like This)=>[](any object?)[]]=>anywhere || throws]=>anywhere || throws={(applay Too Alldireclty With Too Fo Argeements Callsomthing Like This)=>[](any object?)[]]=>anywhere || throws]=>anywhere || throws={applay Too Alldireclty With Too Fo Argeements Callsomthing Like This([])? throw Not Enough Arguments Error(): applay Too Alldireclty With Too Fo Argeements Callsomthing Like This(drop Last([])) + applay Too Alldireclty With Too Fo Argeements Callsomthing Like This(last([])) };
Apply Too Alldireclty With Too Fo Argeements Callsomthing Like This(recursive Method)=>[](any object?)[]]=>anywhere || throws]=>anywhere || throws]=>[{}]>][{}]>][{}]>][{}]>][{}]>][{}]>][{}]>]{_reccursive Method}=[{}]>][]?[last([{}][])?]?: throw Not Enough Arguments Error(): _reccursive Method(drop Last([{}][])?) + applay Too Alldireclty With Too Fo Argeements Callsomthing Like This(last([{}][])?) ;
Applay To Allo directy Woth Two fo Argsements Call Somthing Likethis(recursive Method)=>[](Any Object?)[]]=>Anywhere || Throws ]>[{}]>][{}]>][{}]>][{}]>][{}]>]{Recursice Method}=[{}]>][]?[last([{}][])?]?: Throw Not Enough Argument Errors(): Recursive Methode(Drop Last([{}][])?) + Applay To Allo Directy Woth Two fo Argsements Call Somthing Likethis(Last([{}][])?)
Applay Too alll directly Woth too fo Argsement Cal somthing likethis(recursive Method)=>[](Any Object?)[]]=>Anywhere||Throws]=>[{}]>][{}]>]{Reccursive Methode}=[[]]? last ([{}][])?: Throw Not Enough Argument Errors(): Reccursive Methode(Drop Last ([{}][]) ) + Applay To alll Directy Woth too fo Argsement Cal somthing likethis(Last ([{}][]) )
Applay Tuo All Directluy Woth too fo Argsement Cal somting likthist(recursive Methode)=>[](Any Object?)[]]=>Anywhere||Throws]=>[Recurcive Methode](Drop Last ([Recurcive Methode]) ) + Applay Tuo All Directluy Woth too fo Argsement Cal somting likthist(Last ([Recurcive Methode])) ;
Applay Tu All Drectily Wit too foor Argement Call someting liketihst(recursivemethod)=>[](Any Object?)[]]=>Anywhere||Throws]=>[Recurcive Metohde](Drop Lasr ([Recurcive Metohde]))+Applay Tu All Drectily Wit too foor Argement Call someting liketihst(Lasr ([Recurcive Metohde]));
Applay Tuo All Directluy Woth two fo Argsement Cal somting likthist(reccurisev methode)=>[](Anyy Object?)[]]=>Anny Where||Throw]=>[Recrsive Metohde](Drop Lasr ([Recrsive Metohde]))+Applay Tuo All Directluy Woth two fo Argsement Cal somting likthist(Lasr ([Recrsive Metohde]));
Applay Tu Alla Drectily Wit tu foor Argement Call smething Likithist(reccurisev methode)=>[](Anyy Object?)[]]=>Anny Where||Throw]=>[Recrsivemetohde](Dro Plasr ([Recrsivemetohde]))+Applay Tu Alla Drectily Wit tu foor Argement Call smething Likithist(Plasr ([Recrsivemetohde]));
ApplyTuo AllDirectluyWitTwofoArgetmenCallSomtihnglike(thisrecrusivmetodue)=>[Any Object?][][]]AnywhereorThrow]=>[Recrusivmetodue](DropLasr([Recrusivmetodue]))+ApplyTuo AllDirectluyWitTwofoArgetmenCallSomtihnglike(thisrecrusivmetodue)(Lasr([Recrusivmetodue]))
ApplyTou alla Dirctilly Wit tu foor Agrements Call Smething Likithis(thi recrusiv metoodu)=>[Any Object?][][]]AnnyWhereorThrow]=>[Rercusvmetooudue](Dropp Lasrd(Rercusvmetooudue))+ApplyTou alla Dirctilly Wit tu foor Agrements Call Smething Likithis(thi recrusiv metoodu)(Lasrd(Rercusvmetooudue))
Okay so now lets go back tot he original problem...
How do we solve situations where people call applies directly with too few argeuments...
Lets say whosoeversuppliedanargumentstoourcustomdefinedbinaryoperator...
Supposepeoplepassedinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[[],[],[],[],[],[],[],[],[],[],[],[],[],12];
Thenpeopleshouldbeabletocallthisagainandagainuntilthereareonlytwoargumentsleftbeforesolvingthesituation!!!
Okaysohowdoewesevenknowwhensomeonewantsustoresolvethesituation???
Supposethepersonwhooriginallycalledourcustomdefinedbinaryoperatordirectlysendsinalerttosomeoneelsesothersayingitsoktoproceednow!!!
Okaysohowdoeweactuallysolvethesituation???
Supposepeoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[[],[],12];
Thenpeopleshouldbeabletocalleachotheragainandagainuntilthereareonlytwoargumentsleftbeforesolvingthesituation!!!
Okaysohowdoewesevenknowwhensomeonewantsustoresolvethesituation???
Supposethepersonwhooriginallycalledourcustomdefinedbinaryoperatordirectlysendsinalerttosomeoneelsesothersayingitsoktoproceednow!!!
Okaysohowdoeweactuallysolvethesituation???
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Thenpeopleshouldbeabletocalleachotheragainandagainuntilthereareonlytwoargumentsleftbeforesolvingthesituation!!!
Okaysohowdoewesevenknowwhensomeonewantsustoresolvethesituation???
Supposethepersonwhooriginallycalledourcustomdefinedbinaryoperatordirectlysendsinalerttosomeoneelsesothersayingitsoktoproceednow!!!
Okaysohowdoeweactuallysolvethesituation???
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Applytodirctalywitfouargoemntscallsometinglike(this)[12];
Peoplepassinthefollowingargumentstoourcustomdefinedbinaryoperator...
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
Binops.Add([],[]);
BinOps.Add([]);
Binosps.add([]);
Binosps.add([]);
Binosps.add([]);
Binosps.add([]);
Binosps.add([]);
Binosps.add();
Okay so now lets go back tot he original problem...
How do we solve situations where people call applies directly with too few argeuments...
Lets say whosoeversuppliedanargumentstoourcustomdefinedbinaryoperato...
Supposepeoplepassedinthefollowingargumentstoourcustmo_defined_binary_operato...
Bino_ps.Appliesdiractyl_with_fo_argmeents_calls_smtihng_like_thi_s[,,,,,,,,,[,,],[,,,,,,,],[,,,,,,,],,,,[[,,,,],[,,,,]]];
Thenpeopleshouldbeabletocallthisagainandagainuntilthereareonlytw....
Okaysohowdoewesevenknowwhensomeonewantsustoresolvethesituati.....
Supposethepersonwhooriginallycalledouscustmo_defined_binary_operato_directlysendsinalerttosomeoneelsesothersayingitsoktoproceeednow!!
Okaysohowdoeweactuallysolvethesituati.....
Peoplspassinginthefollowongarginstimotuoursustom_defiened_binary_operator...
Bino_ps.Appliesdiractyl_with_fo_argmeents_calls_smtihng_like_thi_s[,,,,,,,,[,,,,,,,,],[,,,,,,,,],,,,,,,,,[[,,,,,,,,],[,,,,,,,,]]];
Thenpeopleshouldbeabletocalleachotheragainandagainuntiltherereareony....
Okaysohowdoewesevenknowwhensomeonewantsustoresolvetheseituati.....
Supposethepersonwhooriginallycalledouscustmo_defined_binary_operato_directlysendsinalerttosomeoneelsesothersayingitsoktoproceeednow!!
Okaysohowdoeweactuallysolvetheseituati