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Transcript
Problem Set 6 Due Date: Thurs, March 6 Reading skim Chapter IV, p. 43-51 for Tuesday, March 4 Problems 1. A proof rule is reversible if its main goal is equivalent to the conjunction of all its subgoals. (a) Show that all rules of Gentzen’s multi-conclusioned sequent calculus are reversible. (b) Show that the refinement logic rule impL is irreversible 2. For the completeness proof of refinement logic we have introduced the following rules. H ` A ∨ (B ∨ C) by orAssocL H ` (A⊃B) ∨ G∗ by impR∗ H ` (A ∨ B) ∨ C H, A ` B ∨ G∗ Prove that both rules can be derived in refinement logic, i.e. give a proof fragment that simultates the rules. 3. Prove that the magic rule is equivalent to the following rule H ` G by contradiction H, ∼G ` f Show how to derive contradiction in refinement logic and how to derive magic in a refinement logic with contradiction (and without magic). Note that using the cut rule is not necessary but simplifies the derivation. 1