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12(3GentzenInt)

12(3GentzenInt) - Gentzen Sequent Calculus for...

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Gentzen Sequent Calculus for Intuitionistic Logic PART 3: Proof Search Heuristics Before we deﬁne a heuristic method of search- ing for proof in LI let’s make some obser- vations. Observation 1 : the logical rules of LI are similar to those in Gentzen type classical formalizations we examined in previous chap- ters in a sense that each of them introduces a logical connective. Observation 2 : The process of searching for a proof is hence a decomposition process in which we use the inverse of logical and structural rules as decomposition rules. 1

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For example the implication rule: ( →⇒ ) A, Γ -→ B Γ -→ ( A B ) becomes an implication decomposition rule (we use the same name ( →⇒ ) in both cases) ( →⇒ ) Γ -→ ( A B ) A, Γ -→ B . Observation 3 : we write our proofs in a form of trees, instead of sequences of expres- sions, so the proof search process is a pro- cess of building a decomposition tree. 2
To facilitate the process we write the decom- position rules in a ”tree ” form. For exam- ple the the above implication decomposi- tion rule is written as follows Γ -→ ( A B ) | ( →⇒ ) A, Γ -→ B. The two premisses implication rule ( ⇒→ ) writ- ten as the tree decomposition rule becomes ( A B ) , Γ -→ ^ ( ⇒→ ) Γ -→ A B, Γ -→ 3

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12(3GentzenInt) - Gentzen Sequent Calculus for...

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