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This overview explains how the D10 rule engine finds exact rewrites,

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and why fewer operators do not automatically mean faster execution.

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This matters because finding, checking, and applying a rewrite
adds work that must be recovered by the complete execution path.

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This matters because finding, checking, and applying a rewrite
adds work that must be recovered by the complete execution path.

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The proposed rewrite removes two repeated A terms from an exclusive-or chain.

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We will verify the answer separately from counting the operations, because a smaller

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program is not by itself a proof of equivalence.

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For A zero, zero exclusive-or zero is zero.

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For A one, one exclusive-or one is also zero.

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Exclusive-or with the remaining B therefore returns B for both B values.

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All four complete assignments match the replacement.

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Under a binary exclusive-or evaluation model, the original chain takes two combinations.

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The replacement returns B directly and takes none.

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This count shows a strict reduction in that operation measure, while

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the preceding proof established correctness.

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the preceding proof established correctness.

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An index narrows the candidates.

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Proved rules preserve the Boolean result.

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Each admitted rewrite reduces a nonnegative operator count.

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The engine records what changed so it can validate replay.

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They do not make its overhead disappear.

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For a conceptual XOR-cancellation rule,
A exclusive-or A exclusive-or B equals B.

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Check the four input pairs in true-first order:

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11, 10, 01, 00.

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The original expression and B agree in every row.

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In this binary teaching tree, replacing the
expression with B removes two XOR operators.

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An operator index can direct matching toward relevant XOR nodes.

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It still must inspect candidates and establish the permitted match.

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Then update identities and execute the result.

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The retained engine contains mux,

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comparator, carry and XOR-cancellation families,
yet whole-path profitability remained negative.

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Our tiny identity is a teaching proof, not a claim that this
exact displayed instance supplied the retained timing.

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Strict decrease supports termination of the stated rule
process, not correctness of an unproved replacement.

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Work a second complete case with A 1, B 1.

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Keep this assignment fixed throughout the calculation.

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Each symbol retains the meaning already introduced; a new value does not

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create a new logical rule.

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The two repeated A terms first cancel to zero.

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Combining zero with B one returns one.

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The replacement directly returns that same B value.

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Cancellation does not mean the entire expression must become false; the remaining terms

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still determine the answer.

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This case establishes a specific contrast.

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Only the repeated pair disappears.

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The remaining input keeps its meaning.

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Keep the input, intermediate values, and conclusion together when comparing

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it with the previous case.

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The calculated answer belongs to this complete assignment under the stated rule.

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The calculated answer belongs to this complete assignment under the stated rule.

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An index can help find candidate patterns, but the matcher must verify the actual operator

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and variable bindings before applying the proved rewrite.

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A retrieved pattern is not automatically an eligible match.

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If the two terms are distinct A and C, they do not cancel in general.

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With A one, C zero, and B zero, the original returns one while B returns zero.

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The exact binding check prevents this false rewrite.

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In a fictional request, saving two units of execution while spending three on matching and

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updating increases cost by one.

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A nonnegative integer measure that strictly decreases on every successful rewrite can

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prevent an infinite chain of those rewrites.

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That termination argument does not replace semantic proof or complete cost measurement.

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A useful rewrite must beat the unchanged exact control after index,

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A useful rewrite must beat the unchanged exact control after index,

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matching, proof, cache, compilation,
execution, and delivery costs are included.

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The retained D10 engine preserved exactness, but
its complete path still lost on the tested workload.