# How to simplify expression

What are some ways to simplify the following expression?

(A <= B && B <= C) || (A <= D && D <= C) || (B <= A && A <= D && D <= C) || (D >= C && A <= B && B <= C ) || (B <= A && D >= C)

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to try to answer, I tried the code below (which must surely not be correct) but the kernel never stops !

log = SymbolicLogic()
s = log.statement("((A <= B) | (B <= C)) | ((A <= D) & (D <= C)) | ((B <= A) & ((A <= D) & (D <= C)) | ((D >= C) & (A <= B)) & ((B <= C ) | (B <= A) & (D >= C))")


From the manual :

"Formulas consist of the following operators:

• & – and

• | – or

• ~ – not

• ^ – xor

• -> – if-then

• <-> – if and only if

Operators can be applied to variables that consist of a leading letter and trailing underscores and alphanumerics. Parentheses may be used to explicitly show order of operation."

Your code uses symbols not in the above list ; no wonder that it can't compile...

More important : "A>=B" is a predicative assertion, not a proposition. First-order predicate logic is a horse of another color...

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A couple of answer elements :

• Sage per se has not (yet) symbolic logical operators (but see Trac#31911).

• Your (Mathematica ?) syntax is not Python : there is no such operators as && or ||...

With these limitations in mind, you can do :

sage: var("A, B, C, D")
(A, B, C, D)
sage: import sympy
sage: sympy.sympify('((A <= B) & (B <= C)) | ((A <= D) & (D <= C)) | ((B <= A) & (A <= D) & (D <= C)) | ((D >= C) & (A <= B) & (B <= C) ) | ((B <= A) & (D >= C))').simplify()
((A >= B) & (C <= D)) | ((C >= D) & (A <= D)) | ((A <= B) & (B <= C))


but this result cannot (yet) be translated in Sage.

There is some support for propositional logic in Sage, but these tools use string representation of the formulæ ; the translation to Sage symbolic expressions would require the addition of some support for symbolic logical functions (see the Trac cicket...).

FWIW :

sage: mathematica("FullSimplify[(A <= B && B <= C) || (A <= D && D <= C) || (B <= A && A <= D && D <= C) || (D >= C && A <= B && B <= C ) || (B <= A && D >= C)]")
(D >= C && B <= A) || (A <= B && B <= C) || (A <= D && D <= C)

sage: maxima_console()
Maxima 5.45.0 https://maxima.sourceforge.io
using Lisp ECL 21.2.1
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) simplify((A <= B and B <= C) or (A <= D and D <= C) or (B <= A and A <= D and D <= C) or (D >= C and A <= B and B <= C ) or (B <= A and D >= C));
(%o1) simplify(((A <= B) and (B <= C)) or ((A <= D) and (D <= C))
or ((B <= A) and (A <= D) and (D <= C))
or ((D >= C) and (A <= B) and (B <= C)) or ((B <= A) and (D >= C)))


HTH,

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If we assume that all $A,B,C,D$ are real, we can use cylindrical algebraic decomposition (CAD) to simplify such an expression (in general, CAD can handle logical formulas involving polynomial (in)equalities). Sage has an interface to QEPCAD (QEPCAD itself, however, I believe is not shipped with sage, so has to be installed) which is one of the most popular implementations of CAD. Here is how QEPCAD simplifies your expression:

sage: from sage.interfaces.qepcad import qepcad_formula, qepcad, qformula
sage: var("A,B,C,D")
sage: expr = qf.or_([qf.and_([A<=B, B<=C]), qf.and_([A<=D, D<=C]), qf.and_([B<=A, A<=D, D<=C]), qf.and_([D>=C, A<=B, B<=C]), qf.and_([B<=A, D>=C])])
[ B - A >= 0 /\ C - B >= 0 ] \/ [ D - A >= 0 /\ D - C <= 0 ] \/ [ B - A <= 0 /\ D - C >= 0 ]

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