A methodology for producing formal, machine-checked proofs by combining human guidance with proof-assistant software and tactic languages to construct proof scripts that a verifier accepts.

Principle

Human–machine collaboration: the human supplies high-level strategy, invariants and lemmas while the assistant enforces formal rules and performs routine inference and checking; proofs are built compositionally by tactics following the syntax of the object logic.

Demonstration

Developing a correctness proof for a recursive sorting routine by (a) stating the specification formally in the system's logic, (b) using induction and user-provided lemmas to break goals into subgoals, and (c) applying automated tactics to discharge side conditions until the proof script is accepted by the checker.

Misapplication

Relying purely on opaque automation to 'finish' proofs without understanding the produced proof script, producing brittle tactics that fail under small specification changes, or treating the assistant's output as informal documentation rather than a formally checked artifact.

Consequence

When applied correctly, it yields machine-checked, reproducible proofs that increase trust, enable mechanized reuse of lemmas and extracted artifacts (e.g., verified code), and make subtle logical obligations explicit.

Reversal

Fully automated theorem proving replaces human-guided tactic orchestration with blind search; informal pen-and-paper proofs replace machine checking and thus sacrifice mechanical certainty.

Boundary

Applies only to formal proofs in a proof assistant or environment that produces machine-checkable derivations; it excludes informal proofs, empirical testing, and verification methods that do not produce a checkable proof object.

Semantic Tension

Competes with 'automated theorem proving' (emphasis on search and heuristics) and with 'proof engineering' (emphasis on maintainability and software-like practices); tension lies between trustworthiness from checking and productivity from automation.

Synthesis

Interactive theorem proving is the practice of constructing formally checkable proofs by iterating between human strategic choices and machine-enforced formal steps, producing reusable, high-assurance artifacts under the constraints of a chosen formal language and proof assistant.