sursumCorda

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These are questions asked by sursumCorda

Let a, b, c, d be real numbers. Define two geometric regions implicitly: abcd>0∧abacadbcbdcda2b2c2d2, and abcd>0∧abcabdacdbcd>0. Suppose that somebody wants to prove that the former region is a subset of the latter one. Can we implement such an implication simply using certain ready-made commands in basic Maple? In other words, it is hoped that 

restart;
RealDomain:-simplify(forall([a, b, c, d], a + b​​​​​​​ + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies a*b*c*d > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0));

Unfortunately, simplify just returns the statement unevaluated. So, how to address it in internal Maple? 

Note. The desired result (or return value) is true (see below).

  1. restart; # Do not with(Logic): here
    RegularChains:-SemiAlgebraicSetTools:-QuantifierElimination(&A [d, c, b, a], ((d + c + b + a > 0) &and (b*a + c*a + d*a + c*b + d*b + d*c > a^2 + b^2 + c^2 + d^2)) &implies ((d*c*b*a > 0) &and (d*c*b + d*c*a + d*b*a + c*b*a > 0)));
  2. restart;
    not SMTLIB:-Satisfiable(`not`(a + b + c + d > 0 and a*b + a*c + a*d + b*c + b*d + c*d > a^2 + b^2 + c^2 + d^2 implies d*c*b*a > 0 and a*b*c + a*b*d + a*c*d + b*c*d > 0)); # assuming real 

​​​​​​​​​​​​​​​​​​​​​Either ⒈ or ⒉ needs external calls. It seems that the Maple standard library functions in List of Commands still fail in simplifying expressions in the aforementioned form. Am I right?

GraphTheory/IsHamiltonian says that the method can be one of legacy, sat, or tsp. But why method = tsp brings about an error? 

GraphTheory:-IsHamiltonian(GraphTheory:-CompleteGraph(3, 3), method = tsp); # Maple 2022.2

Attachment: HamiltonianGraphQ.mws

Let x, y, z be nonnegative real numbers. Obviously, x3+y3+z3-4xyz is not always nonnegative, since its value is -1 when x=y=z=1. But why Maple's minimize gives wrong result? 

minimize(x^3 + y^3 + z^3 - 4*x*y*z, x = 0 .. infinity, y = 0 .. infinity, z = 0 .. infinity);

Attachment: MinValue.mws.

MMA's Minimize finds correct infimum: 
So, can Maple always find correct exact (i.e., not just numerical) global minima (or maxima) at least for such problems with a small number of variables as well?

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