To add two numbers, each column needs the carry from the column below — so a naive adder waits, bit by bit, for the carry to ripple all the way up. A carry-lookahead adder cheats: it computes every carry AT ONCE from ‘generate’ and ‘propagate’ signals, turning a slow chain into a fast fan-out. Slide the inputs and watch the carries appear together.
For each bit i, define generate gᵢ = aᵢ·bᵢ (this column makes a carry no matter what) and propagate pᵢ = aᵢ⊕bᵢ (it passes a carry through). Then every carry is cᵢ₊₁ = gᵢ + pᵢ·cᵢ, which unrolls so ALL carries can be computed in parallel from the g’s and p’s — O(log n) depth instead of the ripple’s O(n). A fail-loud self-check throws unless the lookahead sum equals a+b for every pair of inputs, matching the slow ripple exactly — same answer, far fewer gate-delays.
The real speed-up needs a tree of lookahead blocks (a full CLA is hierarchical); shown here as the flat carry equations. The g/p logic and the equivalence to ordinary addition are exact.