W={1,1,1,0,1,1,0,2,1,1,1,0,1,1}; -- Cerulli Irelli-Lanini convention
B={3,3,3,3,3,3,3,3,3,3,3,3,3,3};
S=QQ[p1,p2,p3,p4,p12,p13,p14,p23,p24,p34,p123,p124,p134,p234, Weights=>B-W];

--Ideal flag4
I4=ideal(
p34*p123-p23*p134+p13*p234,
p24*p123-p23*p124+p12*p234,
p14*p123-p13*p124+p12*p134,
p34*p124-p24*p134+p14*p234,
p4*p123-p3*p124+p2*p134-p1*p234,
p14*p23-p13*p24+p12*p34,
p3*p12-p2*p13+p1*p23,
p4*p23-p3*p24+p2*p34,
p4*p13-p3*p14+p1*p34,
p4*p12-p2*p14+p1*p24);

-- initial ideal 
J4=ideal(leadTerm(1,I4));
mingens J4:
p34*p123+p13*p234,
p24*p123-p23*p124+p12*p234,
p14*p123-p13*p124,
p34*p124+p14*p234,
p4*p123-p3*p124,
p14*p23-p13*p24+p12*p34,
p3*p12-p2*p13,
p4*p23-p3*p24+p2*p34,
p4*p13-p3*p14,
p4*p12-p2*p14);


*************************************
*************************************

--s1: vanishing of p3,p4,p13,p23,p14,p24,p34,p124,p134,p234
V=ideal(p3,p4,p13,p23,p14,p24,p34,p124,p134,p234);
IV=I4+V;
mingens IV
p23, p234, p134, p34, p24, p13, p3, p124, p14, p4

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p134, p34, p24, p13, p3, p124, p14, p4

JV==J4+V
true
*************************************

s2: vanishing of p2,p3,p4,p23,p14,p24,p34,p124,p134,p234
V=ideal(p2,p3,p4,p23,p14,p24,p34,p124,p134,p234);
IV=I4+V;
mingens IV
p34 p234 p134 p124 p24 p23 p14 p4 p3 p2

JV=ideal(leadTerm(1,IV));
mingens JV
p34 p234 p134 p124 p24 p23 p14 p4 p3 p2

JV==J4+V
true

*************************************

s3: vanishing of p2,p3,p4,p13,p23,p14,p24,p34,p134,p234
V=ideal( p2,p3,p4,p13,p23,p14,p24,p34,p134,p234);
IV=I4+V;
mingens IV
p34 p234 p134 p24 p23 p14 p13 p4 p3 p2

JV=ideal(leadTerm(1,IV));
mingens JV
p34 p234 p134 p24 p23 p14 p13 p4 p3 p2

JV==J4+V
true

**************************************
**************************************

s2s1: vanishing of p4,p23,p14,p24,p34,p124,p134,p234
V=ideal(p4,p23,p14,p24,p34,p124,p134,p234);
IV=I4+V;
mingens IV
p23, p234, p134, p34, p24, p124, p14, p4, p3*p12-p2*p13

JV=ideal(leadTerm(1,IV));
p23, p234, p134, p34, p24, p124, p14, p4, p3*p12-p2*p13

JV==J4+V
true


***************************************

s1s2: vanishing of p3,p4,p14,p24,p34,p124,p134,p234
V=ideal(p3,p4,p14,p24,p34,p124,p134,p234);
IV=I4+V;
mingens IV
p234, p134, p34, p24, p3, p124, p14, p4, p2*p13-p1*p23

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p134, p34, p24, p3, p124, p14, p4, p2*p13

JV==J4+V
true

***************************************

s1s3: vanishing of p3,p4,p13,p14,p23,p24,p34,p134,p234
V=ideal(p3,p4,p13,p14,p23,p24,p34,p134,p234);
IV=I4+V;
mingens IV
p23, p234, p134, p34, p24, p13, p3, p14, p4

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p134, p34, p24, p13, p3, p14, p4

JV==J4+V
true

******************************

s3s2: vanishing of p2,p3,p4,p23,p24,p34,p134,p234
V=ideal(p2,p3,p4,p23,p24,p34,p134,p234);
IV=I4+V;
mingens IV
p23, p234, p134, p34, p24, p3, p2, p4, p14*p123-p13*p124

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p134, p34, p24, p3, p2, p4, p14*p123-p13*p124

JV==J4+V
true

*******************************

s2s3 vanishing of p2,p3,p4,p14,p23,p24,p34,p234
V=ideal(p2,p3,p4,p14,p23,p24,p34,p234);
IV=I4+V;
mingens IV
p23, p234, p34, p24, p3, p2, p14, p4, p13*p124-p12*p134

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p34, p24, p3, p2, p14, p4, p13*p124

JV==J4+V
true

*********************************
*********************************

s1s2s1 vanishing of p4,p14,p24,p34,p124,p134,p234
V=ideal(p4,p14,p24,p34,p124,p134,p234);
IV=I4+V;
mingens IV
p234, p134, p34, p24, p124, p14, p4, p3*p12-p2*p13+p1*p23

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p134, p34, p24, p124, p14, p4, p3*p12-p2*p13

JV==J4+V
true

*********************************

s2s3s2 vanishing of p2,p3,p4,p23,p24,p34,p234
V=ideal(p2,p3,p4,p23,p24,p34,p234);
IV=I4+V;
mingens IV
p23, p234, p34, p24, p3, p2, p4, p14*p123-p13*p124+p12*p134

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p34, p24, p3, p2, p4, p14*p123-p13*p124

JV==J4+V
true

*********************************

s1s2s3 vanishing of p3,p4,p14,p24,p34
V=ideal(p3,p4,p14,p24,p34);
IV=I4+V;
mingens IV
p34, p24, p3, p14, p4, p13*p234-p23*p134, p2*p134-p1*p234,
      p23*p124-p12*p234, p2*p13-p1*p23, p13*p124-p12*p134

JV=ideal(leadTerm(1,IV));
mingens JV
p34, p24, p3, p14, p4, p13*p234, p2*p134-p1*p234,
       p23*p124-p12*p234, p2*p13, p13*p124

JV==J4+V
false

IV and JV are radical

*********************************

s3s2s1 vanishing of p23,p24,p34,p134,p234
V=ideal(p23,p24,p34,p134,p234);
IV=I4+V;
mingens IV
p23, p234, p134, p34, p24, p3*p12-p2*p13, p14*p123-p13*p124,
       p4*p123-p3*p124, p4*p13-p3*p14, p4*p12-p2*p14

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p134, p34, p24, p3*p12-p2*p13, p14*p123-p13*p124,
       p4*p123-p3*p124, p4*p13-p3*p14, p4*p12-p2*p14

JV==J4+V
true

********************************

s2s1s3 vanishing of p4,p14,p23,p24,p34,p234
V=ideal(p4,p14,p23,p24,p34,p234);
IV=I4+V;
mingens IV
p23, p234, p34, p24, p14, p4, p3*p12-p2*p13, p13*p124-p12*p134,
       p3*p124-p2*p134

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p34, p24, p14, p4, p3*p12-p2*p13, p13*p124, p3*p124

JV==J4+V
true

********************************

s1s3s2 vanishing of p3,p4,p34,p134,p234
V=ideal(p3,p4,p34,p134,p234);
IV=I4+V;
mingens IV
p234, p134, p34, p3, p4, p24*p123-p23*p124, p14*p23-p13*p24,
       p2*p13-p1*p23, p14*p123-p13*p124, p2*p14-p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p134, p34, p3, p4, p24*p123-p23*p124, p14*p23-p13*p24,
       p2*p13, p14*p123-p13*p124, p2*p14

JV==J4+V
true

*********************************
*********************************

s2s1s3s2 vanishing of p4,p234
V=ideal(p4,p234);
IV=I4+V;
mingens IV
p234, p4, p34*p123-p23*p134, p24*p123-p23*p124, p3*p24-p2*p34,
       p14*p23-p13*p24+p12*p34, p3*p12-p2*p13+p1*p23, p34*p124-p24*p134,
       p3*p124-p2*p134, p14*p123-p13*p124+p12*p134, p3*p14-p1*p34,
       p2*p14-p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p4, p34*p123, p24*p123-p23*p124, p3*p24-p2*p34,
       p14*p23-p13*p24+p12*p34, p3*p12-p2*p13, p34*p124, p3*p124,
       p14*p123-p13*p124, p3*p14, p2*p14

JV==J4+V
true

*********************************

s3s1s2s1 vanishing of p34,p134,p234
V=ideal(p34,p134,p234);
IV=I4+V;
mingens IV
p234, p134, p34, p24*p123-p23*p124, p14*p23-p13*p24,
       p4*p23-p3*p24, p3*p12-p2*p13+p1*p23, p14*p123-p13*p124, p4*p123-p3*p124,
       p4*p13-p3*p14, p4*p12-p2*p14+p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p134, p34, p24*p123-p23*p124, p14*p23-p13*p24,
       p4*p23-p3*p24, p3*p12-p2*p13, p14*p123-p13*p124, p4*p123-p3*p124,
       p4*p13-p3*p14, p4*p12-p2*p14

JV==J4+V
true

*********************************

s1s2s1s3 vanishing of p4,p14,p24,p34
V=ideal(p4,p14,p24,p34);
IV=I4+V;
mingens IV
p34, p24, p14, p4, p13*p234-p23*p134, p23*p124-p12*p234,
       p3*p12-p2*p13+p1*p23, p13*p124-p12*p134, p3*p124-p2*p134+p1*p234

JV=ideal(leadTerm(1,IV));
mingens JV
p34, p24, p14, p4, p13*p234, p23*p124-p12*p234, p3*p12-p2*p13,
       p13*p124, p3*p124

JV==J4+V
true

********************************

s1s2s3s2 vanishing of p3,p4,p34
V=ideal(p3,p4,p34);
IV=I4+V;
mingens IV
p34, p3, p4, p13*p234-p23*p134, p2*p134-p1*p234,
       p24*p123-p23*p124+p12*p234, p14*p23-p13*p24, p2*p13-p1*p23,
       p14*p234-p24*p134, p14*p123-p13*p124+p12*p134, p2*p14-p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p34, p3, p4, p13*p234, p2*p134-p1*p234,
       p24*p123-p23*p124+p12*p234, p14*p23-p13*p24, p2*p13, p14*p234,
       p14*p123-p13*p124, p2*p14

JV==J4+V
false

IV and JV are radical

*********************************

s2s3s2s1 vanishing of p23,p24,p34,p234
V=ideal(p23,p24,p34,p234);
IV=I4+V;
mingens IV
p23, p234, p34, p24, p3*p12-p2*p13, p14*p123-p13*p124+p12*p134,
       p4*p123-p3*p124+p2*p134, p4*p13-p3*p14, p4*p12-p2*p14

JV=ideal(leadTerm(1,IV));
mingens JV
p23, p234, p34, p24, p3*p12-p2*p13, p14*p123-p13*p124,
       p4*p123-p3*p124, p4*p13-p3*p14, p4*p12-p2*p14

JV==J4+V
true

**********************************
**********************************

s2s3s1s2s3 vanishing of p4

V=ideal(p4);
IV=I4+V;
mingens IV
p4, p34*p123+p13*p234-p23*p134, p24*p123-p23*p124+p12*p234,
       p3*p24-p2*p34, p14*p23-p13*p24+p12*p34, p3*p12-p2*p13+p1*p23,
       p34*p124+p14*p234-p24*p134, p3*p124-p2*p134+p1*p234,
       p14*p123-p13*p124+p12*p134, p3*p14-p1*p34, p2*p14-p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p4, p34*p123+p13*p234, p24*p123-p23*p124+p12*p234,
       p3*p24-p2*p34, p14*p23-p13*p24+p12*p34, p3*p12-p2*p13, p34*p124+p14*p234,
       p3*p124, p14*p123-p13*p124, p3*p14, p2*p14

JV==J4+V
true

**********************************

s1s3s2s1s3 vanishing of p34

V=ideal(p34);
IV=I4+V;
mingens IV
p34, p13*p234-p23*p134, p24*p123-p23*p124+p12*p234,
       p14*p23-p13*p24, p4*p23-p3*p24, p3*p12-p2*p13+p1*p23, p14*p234-p24*p134,
       p14*p123-p13*p124+p12*p134, p4*p123-p3*p124+p2*p134-p1*p234,
       p4*p13-p3*p14, p4*p12-p2*p14+p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p34, p13*p234, p24*p123-p23*p124+p12*p234, p14*p23-p13*p24,
       p4*p23-p3*p24, p3*p12-p2*p13, p14*p234, p14*p123-p13*p124,
       p4*p123-p3*p124, p4*p13-p3*p14, p4*p12-p2*p14

JV==J4+V
true

**********************************

s2s3s1s2s1 vanishing of p234

V=ideal(p234);
IV=I4+V;
mingens IV
p234, p34*p123-p23*p134, p24*p123-p23*p124,
       p14*p23-p13*p24+p12*p34, p4*p23-p3*p24+p2*p34, p3*p12-p2*p13+p1*p23,
       p34*p124-p24*p134, p14*p123-p13*p124+p12*p134, p4*p123-p3*p124+p2*p134,
       p4*p13-p3*p14+p1*p34, p4*p12-p2*p14+p1*p24

JV=ideal(leadTerm(1,IV));
mingens JV
p234, p34*p123, p24*p123-p23*p124, p14*p23-p13*p24+p12*p34,
       p4*p23-p3*p24+p2*p34, p3*p12-p2*p13, p34*p124, p14*p123-p13*p124,
       p4*p123-p3*p124, p4*p13-p3*p14, p4*p12-p2*p14

JV==J4+V
true

**********************************