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FreeQ

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

FreeQ[expr, form]
    yields True if no subexpression of expr matches form, False otherwise.
FreeQ[expr, form, levelspec]
    restricts the search to parts of expr at the levels specified by
    levelspec.
FreeQ[form]
    is the operator form: FreeQ[form][expr] == FreeQ[expr, form].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= FreeQ[{1, 2, 4, 1, 0}, 0]
Out[1]= False

In[2]:= FreeQ[{a, b, b, a, a, a}, _Integer]
Out[2]= True

In[3]:= {f[3 x, x], f[a x, x], f[(1 + x) x, x]}
Out[3]= {3 f[x, x], a f[x, x], f[x (1 + x), x]}

Implementation notes

builtin_freeq (src/funcprog.c) returns True iff no subexpression of the first argument matches the pattern form. freeq_at_level walks the tree depth-first, calling the pattern matcher match at each level permitted by the level spec (default {0, Infinity}), and short-circuits to False on the first match. Rational/Complex are treated as atomic; Heads -> False excludes function heads from the search.

  • Protected.
  • By default, explores levels {0, Infinity} and option Heads -> True is enabled.
  • form can be a structural pattern.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= FreeQ[x^2 + y^2, z]
Out[1]= True

In[1]:= FreeQ[x^2 + y^2, y]
Out[1]= False

In[1]:= FreeQ[D[Sin[x] Exp[x], x], Cos]
Out[1]= False

In[1]:= Select[Range[20], FreeQ[FactorInteger[#], {2, _}] &]
Out[1]= {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

Notes

FreeQ[expr, form] tests whether no subexpression of expr matches the pattern form, searching at every level. form may be a literal symbol or a full pattern: the third example confirms that differentiating Sin[x] Exp[x] introduces a Cos term. The last example is a structural sieve — keeping only integers whose FactorInteger output is free of any {2, _} pair recovers the odd numbers, with FreeQ matching against the {prime, exponent} factor structure rather than a numeric value.