proof_by_exhaustion

Automate case analysis for finite domains in Lean 4 by partitioning goals into cases using fin_cases and discharging each branch with decide or omega. This works for small finite types like Fin n and bounded Nat ranges.

The best way to split a goal into Fin n cases in Lean 4 is applying the fin_cases tactic, followed by using decide or omega to independently verify each resulting branch.

Proof by exhaustion works with structured disjunctions in mathlib by using the rcases tactic to handle the disjunction and omega to verify each independent case automatically.

You need Lean 4 with mathlib to use interval_cases and fin_cases for bounded Nat ranges, as these tactics provide the necessary environment to partition and verify finite domains.

fin_cases fails to discharge branches after splitting Fin n goals when the individual cases require additional arithmetic reasoning, requiring you to manually apply decide or omega to close each branch.

You can use omega to verify each case after applying rcases in Lean 4, especially when handling structured disjunctions where each branch can be closed by linear arithmetic decision procedures.