Associativity of "in" in Python?

I'm making a Python parser, and this is really confusing me:

>>> 1 in [] in 'a'
False

>>> (1 in []) in 'a'
TypeError: 'in <string>' requires string as left operand, not bool

>>> 1 in ([] in 'a')
TypeError: 'in <string>' requires string as left operand, not list

How exactly does in work in Python, with regards to associativity, etc.?

Why do no two of these expressions behave the same way?

1 in [] in 'a' is evaluated as (1 in []) and ([] in 'a').¹

Since the first condition (1 in []) is False, the whole condition is evaluated as False; ([] in 'a') is never actually evaluated, so no error is raised.

We can see how Python executes each statement using the dis module:

>>> from dis import dis
>>> dis("1 in [] in 'a'")
  1           0 LOAD_CONST               0 (1)
              2 BUILD_LIST               0
              4 DUP_TOP
              6 ROT_THREE
              8 CONTAINS_OP              0        # `in` is the contains operator
             10 JUMP_IF_FALSE_OR_POP    18        # skip to 18 if the first 
                                                  # comparison is false
             12 LOAD_CONST               1 ('a')  # 12-16 are never executed
             14 CONTAINS_OP              0        # so no error here (14)
             16 RETURN_VALUE
        >>   18 ROT_TWO
             20 POP_TOP
             22 RETURN_VALUE
>>> dis("(1 in []) in 'a'")
  1           0 LOAD_CONST               0 (1)
              2 LOAD_CONST               1 (())
              4 CONTAINS_OP              0        # perform 1 in []
              6 LOAD_CONST               2 ('a')  # now load 'a'
              8 CONTAINS_OP              0        # check if result of (1 in []) is in 'a'
                                                  # throws Error because (False in 'a')
                                                  # is a TypeError
             10 RETURN_VALUE
>>> dis("1 in ([] in 'a')")
  1           0 LOAD_CONST               0 (1)
              2 BUILD_LIST               0
              4 LOAD_CONST               1 ('a')
              6 CONTAINS_OP              0        # perform ([] in 'a'), which is 
                                                  # incorrect, so it throws a TypeError
              8 CONTAINS_OP              0        # if no Error then this would 
                                                  # check if 1 is in the result of ([] in 'a')
             10 RETURN_VALUE

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1. Except that [] is only evaluated once. This doesn't matter in this example but if you (for example) replaced [] with a function that returned a list, that function would only be called once (at most). The documentation explains also this.

Python does special things with chained comparisons.

The following are evaluated differently:

x > y > z   # in this case, if x > y evaluates to true, then
            # the value of y is used, again, and compared with z

(x > y) > z # the parenthesized form, on the other hand, will first
            # evaluate x > y. And, compare the evaluated result
            # with z, which can be "True > z" or "False > z"

In both cases though, if the first comparison is False, the rest of the statement won't be looked at.

For your particular case,

1 in [] in 'a'   # this is false because 1 is not in []

(1 in []) in a   # this gives an error because we are
                 # essentially doing this: False in 'a'

1 in ([] in 'a') # this fails because you cannot do
                 # [] in 'a'

Also to demonstrate the first rule above, these are statements that evaluate to True.

1 in [1,2] in [4,[1,2]] # But "1 in [4,[1,2]]" is False

2 < 4 > 1               # and note "2 < 1" is also not true

Precedence of Python operators: https://docs.python.org/3/reference/expressions.html#comparisons

From the documentation:

> Comparisons can be chained arbitrarily, e.g., x < y <= z is equivalent to x < y and y <= z, except that y is evaluated only once (but in both cases z is not evaluated at all when x < y is found to be false).

What this means is, that there no associativity in x in y in z!

The following are equivalent:

1 in  []  in 'a'
# <=>
middle = []
#            False          not evaluated
result = (1 in middle) and (middle in 'a')


(1 in  []) in 'a'
# <=>
lhs = (1 in []) # False
result = lhs in 'a' # False in 'a' - TypeError


1 in  ([] in 'a')
# <=>
rhs = ([] in 'a') # TypeError
result = 1 in rhs

The short answer, since the long one is already given several times here and in excellent ways, is that the boolean expression is short-circuited, this is has stopped evaluation when a change of true in false or vice versa cannot happen by further evaluation.

(see http://en.wikipedia.org/wiki/Short-circuit_evaluation)

It might be a little short (no pun intended) as an answer, but as mentioned, all other explanation is allready done quite well here, but I thought the term deserved to be mentioned.