Singular Manual: ADGT

| #### D.2.4.22 ADGT Procedure from library `grobcov.lib` (see grobcov_lib). Usage: ADGT(ideal H, ideal T, ideal H1,ideal T1[,options]); H: ideal in Q[a][x] hypothesis T: ideal in Q[a][x] thesis H1: ideal in Q[a][x] negative hypothesis, only dependent on [a] T1: ideal in Q[a][x] negative thesis of the Proposition (H and not H1) => (T and not T1) Return: The list [[1,S1],[2,S2],..], S1, S2, .. represent the set of parameter values that must be verified as supplementary conditions for the Proposition to become a Theorem. They are given by default in Prep. If the proposition is generally true, (the proposition is already a theorem), then the generic segment of the internal grobcov called is also returned to provide information about the values of the variables determined for every value of the parameters. Options: An option is a pair of arguments: string, integer. To modify the default options, pairs of arguments -option name, value- of valid options must be added to the call. Option "rep",0-1: The default is ("rep",0) and then the segments are given in canonical P-representation. Option ("rep",1) represents the segments in canonical C-representation, Option "gseg",0-1: The default is "gseg",1 and then when the proposition is generally true, ADGT outputs a second element which is the "generic segment" to provide supplementary information. With option "gseg",0 this is avoided. Note: The basering R, must be of the form Q[a][x], (a=parameters, x=variables), and should be defined previously. The ideals must be defined on R. Example:
    LIB "grobcov.lib";
    // Determine the supplementary conditions
    // for the non-degenerate  triangle A(-1,0), B(1,0), C(x,y)
    // to have an ortic non-degenerate isosceles triangle
    if(defined(R)){kill R;}
    ring R=(0,x,y),(x2,y2,x1,y1),lp;
    // Hypothesis H
    ideal H=-y*x1+(x-1)*y1+y,
    (x-1)*(x1+1)+y*y1,
    -y*x2+(x+1)*y2-y,
    (x+1)*(x2-1)+y*y2;
    // Thesis T
    ideal T=(x1-x)^2+y1^2-(x2-x)^2-y2^2;
    // Negative Hypothesis H1
    ideal H1=y;
    // Negative Thesis T1
    ideal T1=x*(y1-y2)-y*(x1-x2)+x1*y2-x2*y1;
    // Complementary conditions for the
    // Proposition (H and not H1) => (T and not T1)
    // to be true
    ADGT(H,T,H1,T1);
    ==> [1]:
    ==>    [1]:
    ==>       1
    ==>    [2]:
    ==>       [1]:
    ==>          [1]:
    ==>             _[1]=(x^2-y^2-1)
    ==>          [2]:
    ==>             [1]:
    ==>                _[1]=(y)
    ==>                _[2]=(x-1)
    ==>             [2]:
    ==>                _[1]=(y)
    ==>                _[2]=(x+1)
    ==>             [3]:
    ==>                _[1]=(y^2+1)
    ==>                _[2]=(x)
    ==>       [2]:
    ==>          [1]:
    ==>             _[1]=(x)
    ==>          [2]:
    ==>             [1]:
    ==>                _[1]=(y-1)
    ==>                _[2]=(x)
    ==>             [2]:
    ==>                _[1]=(y)
    ==>                _[2]=(x)
    ==>             [3]:
    ==>                _[1]=(y^2+1)
    ==>                _[2]=(x)
    ==>             [4]:
    ==>                _[1]=(y+1)
    ==>                _[2]=(x)
    //********************************************
    // Example of automatic demonstrations of a theorem
    // The Nine Points Circle Theorem:
    if(defined(R1)){kill R1;}
    ring R1=(0,x,y),(x1,y1,x2,y2,yh,a,b,r),dp;
    short=0;
    //Hypothesis H
    ideal H=y*x1+y1-x*y1-y,
    (x-1)*(x1+1)+y*y1,
    -y*x2+y2+x*y2-y,
    (x+1)*(x2-1)+y*y2,
    (x1-a)^2+(y1-b)^2-r^2,
    (x2-a)^2+(y2-b)^2-r^2,
    (x-a)^2+b^2-r^2,
    -yh+x*y1-x1*yh+y1;
    // Thesis T
    ideal T=yh+x*y2-x2*yh-y2,
    (x-1-2*a)^2+(yh-2*b)^2-4*r^2,
    (x+1-2*a)^2+(yh-2*b)^2-4*r^2,
    4*(x-a)^2+(y+yh-2*b)^2-4*r^2,
    (x-1-2*a)^2+(y-2*b)^2-4*r^2,
    (x+1-2*a)^2+(y-2*b)^2-4*r^2,
    4*(x-a)^2+(yh+y-2*b)^2-4*r^2;
    ADGT(H,T,y,1);
    ==> [1]:
    ==>    [1]:
    ==>       1
    ==>    [2]:
    ==>       [1]:
    ==>          [1]:
    ==>             _[1]=0
    ==>          [2]:
    ==>             [1]:
    ==>                _[1]=(x^2-2*x+y^2+1)
    ==>             [2]:
    ==>                _[1]=(x^2+2*x+y^2+1)
    ==>             [3]:
    ==>                _[1]=(y)
    ==> [2]:
    ==>    [1]:
    ==>       Generic segment
    ==>    [2]:
    ==>       [1]:
    ==>          _[1]=b
    ==>          _[2]=a
    ==>          _[3]=yh
    ==>          _[4]=y2
    ==>          _[5]=x2
    ==>          _[6]=y1
    ==>          _[7]=x1
    ==>          _[8]=r^2
    ==>       [2]:
    ==>          _[1]=(4*y)*b+(x^2-y^2-1)
    ==>          _[2]=2*a+(-x)
    ==>          _[3]=(y)*yh+(x^2-1)
    ==>          _[4]=(x^2+2*x+y^2+1)*y2+(-2*x*y-2*y)
    ==>          _[5]=(x^2+2*x+y^2+1)*x2+(-x^2-2*x+y^2-1)
    ==>          _[6]=(x^2-2*x+y^2+1)*y1+(2*x*y-2*y)
    ==>          _[7]=(x^2-2*x+y^2+1)*x1+(x^2-2*x-y^2+1)
    ==>          _[8]=(16*y^2)*r^2+(-x^4-2*x^2*y^2+2*x^2-y^4-2*y^2-1)
    ==>       [3]:
    ==>          [1]:
    ==>             [1]:
    ==>                _[1]=0
    ==>             [2]:
    ==>                [1]:
    ==>                   _[1]=(x^2-2*x+y^2+1)
    ==>                [2]:
    ==>                   _[1]=(x^2+2*x+y^2+1)
    ==>                [3]:
    ==>                   _[1]=(y)
    // Thus the theorem is true.

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