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with orwith without
Combined from primary sources listed below.

In Control flow§

See primary documentation in context for with orwith without

The with statement is like if, but tests for definedness rather than truth, and it topicalizes on the condition, much like given:

with "abc".index("a") { .say }      # prints 0

Similarly to elsif, orwith may be used to chain definedness tests:

# The below code says "Found 'a' at 0"
my $s = "abc";
with   $s.index("a") { say "Found 'a' at $_" }
orwith $s.index("b") { say "Found 'b' at $_" }
orwith $s.index("c") { say "Found 'c' at $_" }
else                 { say "Didn't find 'a', 'b' or 'c'" }

You may intermix if-based and with-based clauses.

# This says "Yes"
if 0 { say "No" } orwith Nil { say "No" } orwith 0 { say "Yes" };

As with unless, you may use without to check for undefinedness, but you may not add an else clause:

my $answer = Any;
without $answer { warn "Got: {$_.raku}" }

There are also with and without statement modifiers:

my $answer = (Any, True).roll;
say 42 with $answer;
warn "undefined answer" without $answer;

As with the other chainable constructs, an else completing a with/if..orwith/elsif chain will itself topicalize to the value of the prior (failed) condition's topic (either the topic of with or the final orwith or elsif).

In the case of an else following a with or orwith, topicalizing a value guaranteed to be undefined may seem useless. But it makes for a useful idiom when used in conjunction with operations that may fail, because Failure values are always undefined:

sub may_fail( --> Numeric:D ) {
  my $value = (^10).pick || fail "Zero is unacceptable";
  fail "Odd is also not okay" if $value % 2;
  return $value;
}

with may_fail() -> $value { # defined, so didn't fail
  say "I know $value isn't zero or odd."
} else { # undefined, so failed, and the Failure is the topic
  say "Uh-oh: {.exception.message}."
}

Note that while topicalizing a Failure marks it handled—so you can use the with/else to proceed safely with execution—it doesn't make the Failure value itself safe. Even within the else clause, if you try to use the value directly, it will result in your else clause itself failing (or, in Rakudo, "promoting" the Failure into a thrown exception).

But as seen above, you can use the methods of a handled Failure object the else topicalizes, such as exception, if you wish to provide diagnostics or interrogate the underlying Exception.