# How can I get the progressive application of list of rules?

I have a very long list of rules, and using this list of rules I want to see how some elements evolve as I successively apply the next rules on the previous rule.

Here is the simpler version of my problem, but I don't have any clue how to proceed. What would be an approach to this problem?

ClearAll[Evaluate[StringJoin[Context[], "*"]]]
Needs["UtilitiesCleanSlate"];
CleanSlate[];
ClearInOut[];

myRules = {a5 -> a4/a3, a4 -> a3 + a2,
a3 -> a2^2 + a1, a2 -> a1 - 1, a1 -> b};


I wanted to get this list

{
myRules[] //. myRules[],

myRules[] //. myRules[] //.
myRules[],

myRules[] //. myRules[] //.
myRules[] //. myRules[],

myRules[] //. myRules[] //.
myRules[] //. myRules[] //.
myRules[]
}

• Does Rest@FoldList[#1 /. #2 &, myRules] do the trick? Is that what you wanted? Something that just automates what you wrote down? Aug 6 '20 at 21:36
• @march, If my question seems to be silly , i'm extremely sorry. Your code is super (just in one line). Truly thanks a lot. Aug 6 '20 at 21:45
• The question's not silly. I just wasn't sure what you wanted, so I asked to clarify in order to see if I should write an answer or not. I have written an answer below. Aug 6 '20 at 21:47

Here is a quick one-liner.

myRules = {a5 -> a4/a3, a4 -> a3 + a2, a3 -> a2^2 + a1, a2 -> a1 - 1, a1 -> b};
Rest@FoldList[#1 /. #2 &, myRules]
(* {a5 -> (a2 + a3)/a3,
a5 -> (a1 + a2 + a2^2)/(a1 + a2^2),
a5 -> (-1 + (-1 + a1)^2 + 2 a1)/((-1 + a1)^2 + a1),
a5 -> (-1 + (-1 + b)^2 + 2 b)/((-1 + b)^2 + b)} *)


Which is the same output as we get from the OP's code.

Alternatively, do

Rest@FoldList[ReplaceAll, myRules]


(Thanks to Lukas-Lang.)

• You should be able to write ReplaceAll instead of #1 /. #2& to improve readability (depending on who you ask of course) Aug 7 '20 at 6:15