Suppose you know that 0 <= x1 <= x2 <= x3 <= ... etc. How do you know whether one set of coefficients [a1, a2, ..., an] produces numbers that are always less than another set [b1, b2, ..., bn]? That is, how do you determine for all 0 <= x1 <= x2 <= x3 <= ... <= xn that a1x1 + ... + anxn <= b1x1 + ... + bnxn ? Primitive operations: If you increment a coefficient, you increase the result: [ ... +1 ... ] If you subtract 1 from a lower coeff and add it to a higher coeff, you increase the result: [ ... -1 .... +1 ... ] Question: Does the cumulative operator achieve the comparison? L = [[1 0 0 0] [1 1 0 0] [1 1 1 0] [1 1 1 1]] 1. It has the correct result on the zero vector, returning 0. 2. It's linear, so we just have to show that the primitive increments have the right effect. Increment a coefficient: L [... +1 ...] = [0 0 ... 0 1 1 1 ... 1] Decrement and increment: L [... -1 ... +1 ...] = [0 0 0 -1 -1 -1 ... -1 0 0 0 0] No go. Hm. Maybe we need it to go right to left? L = [[0 0 0 1] [0 0 1 1] [0 1 1 1] [1 1 1 1]] L [ ...+1...] = [1 ... 1 0 0 0 0 0] L [ ... -1 ... +1 ...] = [0 0 0 ... 1 1 1 1 0 0 0] Now we've got something that always increments! Excellent.