Recognition D10

This overview explains how the D10 rule engine finds exact rewrites, and why fewer operators do not automatically mean faster execution. This matters because finding, checking, and applying a rewrite adds work that must be recovered by the complete execution path. The proposed rewrite removes two repeated A terms from an exclusive-or chain. We will verify the answer separately from counting the operations, because a smaller program is not by itself a proof of equivalence. For A zero, zero exclusive-or zero is zero. For A one, one exclusive-or one is also zero. Exclusive-or with the remaining B therefore returns B for both B values. All four complete assignments match the replacement. Under a binary exclusive-or evaluation model, the original chain takes two combinations. The replacement returns B directly and takes none. This count shows a strict reduction in that operation measure, while the preceding proof established correctness. An index narrows the candidates. Proved rules preserve the Boolean result. Each admitted rewrite reduces a nonnegative operator count. The engine records what changed so it can validate replay. They do not make its overhead disappear. For a conceptual XOR-cancellation rule, A exclusive-or A exclusive-or B equals B. Check the four input pairs in true-first order: 11, 10, 01, 00. The original expression and B agree in every row. In this binary teaching tree, replacing the expression with B removes two XOR operators. An operator index can direct matching toward relevant XOR nodes. It still must inspect candidates and establish the permitted match. Then update identities and execute the result. The retained engine contains mux, comparator, carry and XOR-cancellation families, yet whole-path profitability remained negative. Our tiny identity is a teaching proof, not a claim that this exact displayed instance supplied the retained timing. Strict decrease supports termination of the stated rule process, not correctness of an unproved replacement. Work a second complete case with A 1, B 1. Keep this assignment fixed throughout the calculation. Each symbol retains the meaning already introduced; a new value does not create a new logical rule. The two repeated A terms first cancel to zero. Combining zero with B one returns one. The replacement directly returns that same B value. Cancellation does not mean the entire expression must become false; the remaining terms still determine the answer. This case establishes a specific contrast. Only the repeated pair disappears. The remaining input keeps its meaning. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. An index can help find candidate patterns, but the matcher must verify the actual operator and variable bindings before applying the proved rewrite. A retrieved pattern is not automatically an eligible match. If the two terms are distinct A and C, they do not cancel in general. With A one, C zero, and B zero, the original returns one while B returns zero. The exact binding check prevents this false rewrite. In a fictional request, saving two units of execution while spending three on matching and updating increases cost by one. A nonnegative integer measure that strictly decreases on every successful rewrite can prevent an infinite chain of those rewrites. That termination argument does not replace semantic proof or complete cost measurement. A useful rewrite must beat the unchanged exact control after index, matching, proof, cache, compilation, execution, and delivery costs are included. The retained D10 engine preserved exactness, but its complete path still lost on the tested workload.