function checkstricture0
n = 7;
global A
for t=1:1
    aa = randn(n,1);
    bb = randn(n,1);
    H = [];
    TH = 0:.01:1;
    for i=1:length(TH)
        th = TH(i);
        %h = sum(abs(aa*th+bb));  % GOOD
        h = max(abs(aa*th+bb));  % BAD
        H(end+1)=h;
    end
    [x,y] = max(H);
    if ((y>1)&(y<length(TH)))
        'bad'
        keyboard
    end
    fprintf('t=%d good \n',t);
end
    
plot(TH,H);
return


function checkstricture1
n = 7;
global A
for t=1:10000
    A = pnormdim(rand(n),1);
    H = [];
    TH = 0:.01:1;
    for i=1:length(TH)
        B = collapse(A,TH(i));
        H(end+1)=strict(B);
    end
    [x,y] = max(H);
    if ((y>1)&(y<length(TH)))
        'bad'
        keyboard
    end
    fprintf('t=%d good \n',t);
end
    
plot(TH,H);
return



function checkstricture0
n = 5;

for t=1:100000
    A = pnormdim(rand(n),1);
    h5 = strict(A);
    B = collapse(A);
    b=compallpairs(A,B);
    %h4 = strict(A);
    %A = collapse(A);
    %h3 = strict(A);
    %A = collapse(A);
    %h2 = strict(A);
    %%[h1 h2]
    %fprintf('h2=%1.9f h3=%1.9f h4=%1.9f h5=%1.9f\n',h2,h3,h4,h5)
    fprintf('t=%d good \n',t);
    %%if (h2>h1+1e-10)
    %[b,margin,x] = is_increasing([h2 h3 h4 h5]);
    %if (margin < -1e-10)
    if ~b
        keyboard
    end
end

return

function A = collapse(A,th)
%th = rand;  % mixing coeff
[n,n] = size(A);
% collapse states n-1 and n
A(n-1,:) = A(n-1,:) + A(n,:);
A(:,n-1) = th*A(:,n-1) + (1-th)*A(:,n);
A = A(1:n-1,1:n-1);
return


function h = strict(A)
[n,n] = size(A);
d = 0;
for i=1:n
  for j=i+1:n
    x = A(:,i);
    y = A(:,j);
    d = max(d,.5*sum(abs(x-y)));
    % this stricture has the VERY NICE property
    % of not increasing after state collapse
    %d = max(d,max(abs(x-y)));
    % this is the OLD, BAD stricture def.
    % that's the problem that Sinai was so quick to point out
    % in my terminology, this stricture might increase after collapse
  end
end
h=d;
return

function b=compallpairs(A,B)
global n
[n,n] = size(A);
b = 1;
for i=1:n
  for j=i+1:n
      for k=1:n
          a1 = A(k,i);
          a2 = A(k,j);
          da = abs(a1-a2);
          
          b1 = B(ci(k),ci(i));
          b2 = B(ci(k),ci(j));
          db = abs(b1-b2);

          b = b & (db <= da);
      end
  end
end
return

function i = ci(i)
global n
if i==n
    i=n-1;
end
return