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. If the propsition is false for every values of the parameters, then the emply list is returned. 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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