subproof

A subproof is a self-contained logical argument nested within a larger proof. It is used to establish intermediate results that can then be used in the main proof. Subproofs are common in formal logic, mathematics, and computer science, where they help to break down complex reasoning into manageable steps.

The structure of a subproof typically involves stating a new assumption or hypothesis, deriving conclusions from that assumption using logical rules, and then concluding the subproof. Once the subproof is completed, its conclusion can be used as if it were a given premise in the main proof. This allows for a more organized and streamlined presentation of arguments.

A key feature of subproofs is that assumptions made within a subproof are local to that subproof. They do not affect the truth or validity of statements outside of it. This isolation is crucial for maintaining the integrity of the overall proof. For example, in a proof by contradiction, one might assume the negation of the statement to be proven within a subproof. If this assumption leads to a contradiction, it proves the original statement without negating the assumption permanently.

Subproofs can be nested within other subproofs, creating a hierarchical structure of reasoning. This allows for even more intricate logical arguments to be constructed systematically. The use of subproofs is a fundamental technique for constructing sound and comprehensible proofs in various formal systems.