(* ::Package:: *)

oneTOone[f_,region_]:=Module[{tmp,n,sol,exp,list,y}, (* x should not be in the local var list*)


Off[Solve::ratnz];
sol=Solve[f==y &&region, x, Reals];
exp=x/.sol;
list=Last@Level[#,1]&/@exp;
n=Length@list;
R=ParametricRegion [{{y},list[[#]]},{y}]&/@Range[n];

(*Length of R says in general how many solutions there are for y *)
If[Length@R>1,False,True]

];


(*KLudge x and y ar free unbound *)
onTO[f_,regionX_, regionY_]:= Module[{R,list,n,m1,exp,sol},
Off[Solve::ratnz];
sol=Solve[y-f==0&&regionX&&regionY,{x},MaxExtraConditions->All];
exp=x/.sol;
list=Last@Level[#,1]&/@exp;
n=Length@list;
R=ParametricRegion [{{z},list[[#]]},{z}]&/@Range[n];

m1=RegionMember[RegionUnion[Sequence@R],{y}];

If[FindInstance[Not[m1]&&regionY,{y}]=={}, True, False]

];






makeFUNC[type_]:=Module[{syms,symbolsDOMAIN,symbolsRANGE,domain,range,range2,e1,e2,e1NOT,v1coor,v2coor,domainLABLE,rangeLABLE,list,list2,shift,n,index,same,drop,degen1,degen2,rand, image,preimage},

syms={CharacterRange["\[Alpha]","\[Omega]"],CharacterRange["a","z"],CharacterRange["\[CapitalAlpha]","\[CapitalOmega]"],CharacterRange["A","Z"]};
{symbolsDOMAIN,symbolsRANGE}=RandomSample[syms,2];

Which[type=="1to1 and not Onto"  ,
domain=RandomSample[Range@Length@symbolsDOMAIN, 10];
range=RandomSample[Range@Length@symbolsRANGE,  15];
e1=RandomSample[domain,n=RandomInteger[{1,8}]];
e2=RandomSample[range,n];,

type=="not 1to1 and not Onto" ,
domain=RandomSample[Range@Length@symbolsDOMAIN, 10];
range=RandomSample[Range@Length@symbolsRANGE,  15];
e1=RandomSample[domain,n=RandomInteger[{5,8}]];
e2=RandomSample[range,n];,


type=="not 1to1 and Onto" ,
n=RandomInteger[{3,10}];
domain=RandomSample[Range@Length@symbolsDOMAIN, 15];
range=RandomSample[Range@Length@symbolsRANGE,  n];
e1=RandomSample[domain,15];
e2=RandomSample[range,n];,

type=="not function"  ,
domain=RandomSample[Range@Length@symbolsDOMAIN, 10];
range=RandomSample[Range@Length@symbolsRANGE,  15];
e1=RandomSample[domain,n=RandomInteger[{4,8}]];
e2=RandomSample[range,n];,


True, (*1to1 and onto*)
n=RandomInteger[{3,15}];
domain=RandomSample[Range@Length@symbolsDOMAIN,n];
range=RandomSample[Range@Length@symbolsRANGE,  n];
e1=RandomSample[domain,n];
e2=RandomSample[range,n];

];

shift=RandomInteger[{0,2}];

(*KLUDGE: change the layout here by adjusting shift*)
v1coor=Table[{0,i+shift}, {i,1,Length@domain}];
v2coor=Table[{5,i}+{0,-2}, {i,1,Length@range}];

v1coor=RandomSample[v1coor,Length@v1coor];
v2coor=RandomSample[v2coor,Length@v2coor];

domainLABLE=Table[Text[symbolsDOMAIN[[i]],v1coor[[i]]+{-0.3,0}],{i,1,Length@e1}];
rangeLABLE=Table[Text[symbolsRANGE[[range[[i]]]],v2coor[[i]]+{0.3,0}],{i,1,Length@range}];

Which[

type=="1to1 and not Onto",
list=Arrow/@Table[{v1coor[[i]],v2coor[[i]]},{i,1,n}];
list=Table[Tooltip[list[[i]],"f("<>symbolsDOMAIN[[i]]<>")="<>symbolsRANGE[[range[[i]]]]], {i,1,n}];
range2=Table[symbolsRANGE[[range[[i]]]], {i,1,n}];
rand=RandomChoice@Drop[Subsets[Range@n],1];
image={symbolsDOMAIN[[rand]], symbolsRANGE[[range[[rand]]]]};
rand=RandomChoice@Drop[Subsets[Range@n],1];
preimage={symbolsDOMAIN[[rand]], symbolsRANGE[[range[[rand]]]]};

,

type=="1to1 and Onto",
list=Arrow/@Table[{v1coor[[i]],v2coor[[i]]},{i,1,n}];
list=Table[Tooltip[list[[i]],"f("<>symbolsDOMAIN[[i]]<>")="<>symbolsRANGE[[range[[i]]]]], {i,1,n}];
range2=Table[symbolsRANGE[[range[[i]]]], {i,1,n}];
rand=RandomChoice@Drop[Subsets[Range@n],1];
image={symbolsDOMAIN[[rand]], symbolsRANGE[[range[[rand]]]]};
rand=RandomChoice@Drop[Subsets[Range@n],1];
preimage={symbolsDOMAIN[[rand]], symbolsRANGE[[range[[rand]]]]};


,

type=="not 1to1 and not Onto" ,
same=RandomSample[Range[n],4];
index=Range[n];
index[[same[[1]]]]=index[[same[[2]]]];
index[[same[[3]]]]=index[[same[[4]]]];

list=Arrow/@Table[{v1coor[[i]],v2coor[[index[[i]]]]},{i,1,n}];
list=Table[Tooltip[list[[i]],"f("<>symbolsDOMAIN[[i]]<>")="<>symbolsRANGE[[range[[index[[i]]]]]]], {i,1,n}];
range2=Table[symbolsRANGE[[range[[index[[i]]]]]], {i,1,n}];

rand=RandomChoice@Drop[Subsets[Range@n],1];
image=DeleteDuplicates/@{symbolsDOMAIN[[rand]], symbolsRANGE[[range[[index[[rand]]]]]]};
rand=RandomChoice@Drop[Subsets[Range@n],1];
preimage=DeleteDuplicates/@{If[ContainsAny[symbolsRANGE[[range[[index[[rand]]]]]],{symbolsRANGE[[range[[index[[#]]]]]]}]==True,symbolsDOMAIN[[#]],Nothing]&/@Range[n] , symbolsRANGE[[range[[index[[rand]]]]]]};

,

type=="not 1to1 and Onto" ,
index=Table[If[i<=n, i, Mod[i,n]+1],{i,1,15}];

list=Arrow/@Table[{v1coor[[i]],v2coor[[index[[i]]]]},{i,1,15}];
list=Table[Tooltip[list[[i]],"f("<>symbolsDOMAIN[[i]]<>")="<>symbolsRANGE[[range[[index[[i]]]]]]], {i,1,15}];
range2=Table[symbolsRANGE[[range[[index[[i]]]]]], {i,1,15}];

rand=RandomChoice@Drop[Subsets[Range@5],1];
image=DeleteDuplicates/@{symbolsDOMAIN[[rand]], symbolsRANGE[[range[[index[[rand]]]]]]};
rand=RandomChoice@Drop[Subsets[Range@5],1];
preimage=DeleteDuplicates/@{If[ContainsAny[symbolsRANGE[[range[[index[[rand]]]]]],{symbolsRANGE[[range[[index[[#]]]]]]}]==True,symbolsDOMAIN[[#]],Nothing]&/@Range[15] , symbolsRANGE[[range[[index[[rand]]]]]]};

,

type=="not function"  ,

n=RandomChoice@{n,n-1};
list=Table[{v1coor[[i]],v2coor[[i]]},{i,1,n}];
{degen1,degen2}=RandomSample[Range[2,Length@list],2];
list=list~Join~{{list[[degen1]][[1]],list[[degen2]][[2]]}};

list=Arrow/@list;

list2=Table[Tooltip[list[[i]],"f("<>symbolsDOMAIN[[i]]<>")="<>symbolsRANGE[[range[[i]]]]], {i,1,Length@list-1}];
list=list2~Join~{Tooltip[list[[Length@list]],"f("<>symbolsDOMAIN[[degen1]]<>")="<>symbolsRANGE[[range[[degen2]]]]]};
(*list=Drop[list,{RandomInteger[{2,Length@list-2}]}];*)


];



Which[type=="types", {"not function" ,"not 1to1 and Onto","not 1to1 and not Onto" ,"1to1 and Onto","1to1 and not Onto"},

type=="not function",
<|
"plot"->Graphics[{Opacity@0,Point[v1coor],Point[v2coor],Opacity@1,domainLABLE,rangeLABLE,Blue,list},ImageSize->200],
"domain"->symbolsDOMAIN[[Range[Length@e1]]],
"range"->symbolsRANGE[[range]]


|>,

True,
<|
"plot"->Graphics[{Opacity@0,Point[v1coor],Point[v2coor],Opacity@1,domainLABLE,rangeLABLE,Blue,list},ImageSize->200],
"domain"->symbolsDOMAIN[[Range[Length@e1]]],
"range"->DeleteDuplicates@range2,
"rangeALL"->symbolsRANGE[[range]],
"image"->image,
"preimage"->preimage

|>]


];