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replacement - Pattern does not match with Orderless head



Why does the following pattern matching not succeed?


ClearAll[f];
SetAttributes[f,Orderless];
HoldForm[f[a,d[],b,c]]/.HoldPattern[f[a,d[],b,z_]]:>{z}
(* f[a,d[],b,c] *)

I expected that because f is Orderless, the pattern matcher should try all possible orders of arguments and find a match.


If the expression is not held, I find:


ClearAll[f];
SetAttributes[f,Orderless];

f[a,d[],b,c]/.HoldPattern[f[a,d[],b,z_]]:>{z}
(* {c} *)

I.e. the desired result. I know that because f is Orderless and the expression is evaluated, the arguments are sorted into canonical order a, b, c, d[]. It would appear in this case that the pattern matcher really does try all possible orders of arguments to find a match.


Why doesn't it work in the first example?


EDIT:


Here are some further examples:


ClearAll[f];
SetAttributes[f, Orderless];


Hold[f[a, b, c, d[]]] /. HoldPattern[f[a, b, c, z_]] :> {"Matched", {z}}
(* Hold[{"Matched", {d[]}}] *)

Hold[f[a, b, c[], d]] /. HoldPattern[f[a, b, c[], z_]] :> {"Matched", {z}}
(* Hold[{"Matched", {d}}] *)

Hold[f[a, b[], c, d]] /. HoldPattern[f[a, b[], c, z_]] :> {"Matched", {z}}
(* Hold[f[a, b[], c, d]] *)

Hold[f[a[], b, c, d]] /. HoldPattern[f[a[], b, c, z_]] :> {"Matched", {z}}

(* Hold[f[a[], b, c, d]] *)

This seems very inconsistent to me; how can I understand what is happening here?


EDIT:


An illuminating example:


Hold[f[a, c, b[], d]] /. HoldPattern[f[a, b[], c, z_]] :> {"Matched", {z}}
(* Hold[{"Matched", {d}}] *)

The Trace shows no argument reordering, but it appears that the pattern matcher must actually be reordering the pattern to (the canonical order) f[a, c, b[], z_] because f is Orderless, even though HoldPattern should prevent "regular" evaluation.


What remains confusing is what exactly is meant by "In matching patterns with Orderless functions, all possible orders of arguments are tried." in the documentation. On face value, the four examples shown above contradict this.




Answer



I am betting that this is almost certainly an optimization short-cut. If Orderless had to try every ordering it would be extremely slow when there are a moderate number of arguments, but it is not.


Consider for example:


f @@@ Hold @@ {RandomSample[Range@12]}
Hold @@ {f @@ Range@12 /. {7 -> _}}
MatchQ[%%, %]


Hold[f[2, 6, 11, 7, 12, 10, 4, 1, 3, 5, 8, 9]]


Hold[f[1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, _]]

False

This test is surely not performing 479,001,600 comparisons, one for each permutation. Instead I suppose that pattern matching assumes that arguments of Orderless heads will first be placed in order and applies logic accordingly.


Since the first two of your "further examples" match I also suppose that this is because the sorted order of arguments on the RHS matches the verbatim order on the left; perhaps this case is tried first.


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