 Sequential SATPlan correctness

A sequential SATPlan encoding has a complete initial assignment of planning fluents, final goal constraints, action precondition axioms, successor-state axioms and action exclusion axioms allowing at most one action per layer. Zero selected actions is an idle step: the successor-state axioms preserve all facts. This encodes an upper bound on plan length rather than forcing exactly $T$ physical actions.

For soundness, read selected ground actions in layer order. The initial assignment is the real initial state. By mathematical induction, each selected action has its preconditions in the current state and the successor-state axioms give precisely its add/delete update; an empty layer leaves that state unchanged. The final goal constraint therefore proves that the decoded plan succeeds. Conversely, take any plan of length at most $T$, assign its actual state trajectory to the fluent variables, select its actions in successive layers and pad with idle layers. All encoding components hold. Complete initial negations and persistence conditions are essential: omitting either can introduce unsupported facts and spurious plans.