Techniques and transformations applied to reduce the size, length, or structural complexity of a proof while preserving its correctness and verifiability.

Principle

Compression removes redundant subproofs, merges identical derivations into shared substructures (DAGification), introduces lemmas or abstractions, and exploits normalization or cut-introduction/elimination strategies to preserve validity while shrinking representation.

Demonstration

A long sequent-style proof with repeated derivations for the same intermediate lemma is transformed into a directed acyclic graph that shares the common subproof once, reducing total nodes and enabling faster rechecking by proof assistants.

Misapplication

Over-aggressive compression that removes traceable structure or replaces steps by implicit assumptions can make the proof unverifiable by standard checkers or reduce human interpretability, and introducing unproven lemmas destroys correctness.

Consequence

Reduces storage and transmission costs, accelerates automated proof checking and replay, and often exposes higher-level structure (lemmas and modular reasoning) useful for maintenance and understanding.

Reversal

Proof expansion: fully unfolding all macro-steps and inlining all lemmas into raw primitive steps, which increases size and can obscure structure despite maximizing explicitness.

Boundary

Applies to formal proofs in deductive systems where transformations preserve logical validity and checkability; does not include lossy summarization that sacrifices verifiability or informal sketching that omits derivations entirely.

Semantic Tension

Tension between maximal compression for machine efficiency and preservation of human-readable, inspectable proof structure; tension also with proof-certificates that prioritize checkability over minimal size.

Synthesis

Proof compression encompasses sound transformations—sharing subderivations, lemma extraction, normalization—that reduce a proof's representational size while maintaining correctness and verifiability, balancing machine efficiency with traceability.