"""Proposition 22.5, step 4: from the Siegel-Whittaker coefficient (all positive roots of D_m) to the
group of positive degree for D_SW, by removals and exchanges; every hypothesis is checked."""
import sys; import os; sys.path.insert(0,os.path.dirname(os.path.abspath(__file__)))
from exchange_checker import State
def run(m):
    n=m//2
    def v(*terms):
        x=[0]*m
        for s,i in terms: x[i-1]+=s
        return tuple(x)
    def is_root(x):
        nz=[t for t in x if t]
        return len(nz)==2 and all(abs(t)==1 for t in nz)
    S=set(); chi=set()
    for i in range(1,m+1):
        for j in range(i+1,m+1):
            S.add(v((1,i),(-1,j))); S.add(v((1,i),(1,j)))
    for i in range(1,m): chi.add(v((1,i),(-1,i+1)))
    st=State(S,chi,is_root)
    for i in range(m-2,1,-1):
        if i>=n: st.remove(v((1,i+1),(1,i+2)),f'remove at i={i}')
        for j in range(m,max(i+2,m+2-i)-1,-1):
            st.exchange(v((1,i),(1,j)),v((-1,i+1),(-1,j)),f'({i},{j})')
    deg=lambda k:m+2-2*k
    exp=set()
    for i in range(1,m+1):
        for j in range(i+1,m+1):
            exp.add(v((1,i),(-1,j)))
            if deg(i)+deg(j)>0: exp.add(v((1,i),(1,j)))
            if deg(i)+deg(j)<0: exp.add(v((-1,i),(-1,j)))
    print(f'm={m}: problems={len(st.log)} final = positive degree for D_SW: {st.S==exp}',st.log[:4],
          'missing',sorted(exp-st.S)[:3],'extra',sorted(st.S-exp)[:3])
for m in (4,6,8,10,12): run(m)