Proof search is the systematic exploration of the space of possible derivations to locate a valid proof of a specified assertion, using algorithms, strategies, and resource limits.

Principle

Organize the exploration with search strategies (depth-first, breadth-first, best-first, goal-directed backward chaining, forward chaining), heuristics, and pruning while balancing completeness, soundness, and resource constraints.

Demonstration

Run a depth-bounded backward-chaining search in a sequent calculus to find a derivation of a goal, or perform resolution-based search with unification and clause selection heuristics to refute the negation of the conjecture.

Misapplication

Blindly applying brute-force search without pruning or heuristics leads to combinatorial explosion; using heuristics that bias the search can sacrifice completeness and miss existing proofs.

Consequence

Effective proof search locates proofs or certifiable refutations, informs complexity estimates, and provides the basis for automated theorem provers and tactic selection in interactive environments.

Reversal

The reverse is targeted model or counterexample search (refutation via countermodels) or relying solely on human insight to construct a proof without systematic exploration.

Boundary

Proof search refers to syntactic exploration of derivations and includes both complete and incomplete procedures; it excludes purely semantic methods that do not traverse syntactic derivation spaces unless those methods are framed as search.

Semantic Tension

There is tension between goal-directed backward search (which focuses on the target) and forward-chaining saturation (which derives consequences broadly); each has trade-offs in applicability and performance.

Synthesis

Proof search is the algorithmic, strategy-driven traversal of derivation space that applies inference rules, heuristics, and resource management to discover a valid derivation or refutation for a target assertion.