A Simple and Practical Valuation Tree Calculus for First-Order Logic

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6 Ján Kľuka and Paul J. VodaThe “real” sequent rules are used only in the omitted upper branches where theindicated initial properties are derived.The tree in Fig. 3(a) is somewhat similar to the derivation of P in theSmullyan’s [Smu68] calculus of signed tableaux. The tableau is given in Fig. 3(c).The beauty of the tree Fig. 3(a) is that it can be also read positively. This isthe view taken by humans operating a proof assistant in the style of Fig. 1. Thetheorem to be proved is ((B → C) → B) → B. The box below it contains its proofwhere we first assume its antecedent (B → C) → B, and under this assumptionwe prove the “lemma” B inside its own box. The proof of the lemma consists ofone case analysis on B → C.2.12 Lemma (Proof weakening). If D ⊢p T ; Γ ⇒ p S; ∆, then also D ⊢p(T, T ′ ); (Γ, Γ ′ ) ⇒ p (S, S ′ ); (∆, ∆ ′ ).Proof. By induction on D.⊓⊔2.13 Free cut free valuation trees. A valuation tree E p-witnessing T, Γ ⇒ pS, ∆ contains a free cut if it contains a non-leaf subtree such thatD 1 D 2A⊢p T ; Γ ′ ⇒ p S; ∆ ′and the cut formula A is neither in T, S nor it is an immediate subformula of aformula from Γ ′ , ∆ ′ . If E contains no free cuts, it is free cut free.2.14 Lemma (Reduction of G3cp proofs to ⊢p trees). If there is a derivationD of a closed sequent Γ ⇒ ∆ in G3cp, then there is a valuation tree D ∗p-witnessing Γ ⇒ p ∆. If D is cut free, then D ∗ is free cut free.Proof. By induction on the derivation D of a sequent Γ ⇒ ∆ in the calculusG3cp. If D is an axiom of the sequent calculus, there is A ∈ Γ ∩ ∆, and ◦ ⊢pΓ ⇒ p ∆ by the initial property (Ax) from Fig. 2.If D derives Γ ⇒ ∆ by application of a rule, then for i = 1 or i = 1, 2 thederivation D has predecessors D i deriving Γ i ⇒ ∆ i . By induction hypotheses,there are valuation trees D ∗ i ⊢ Γ i ⇒ p ∆ i , and we construct a tree D ∗ ⊢p Γ ⇒ p ∆as indicated in Fig. 4. If, for instance, D is formed by the rule L→, then we haveΓ = A → B, Γ 1 ; Γ 2 = B, Γ 1 ; ∆ 1 = A, ∆; and ∆ 2 = ∆. We construct thederivation D ∗ as follows:D2 ∗ ⊢ B, A, A → B, Γ 1 ⇒ p ∆B◦ ⊢ A, A → B, Γ 1 ⇒ p B, ∆AD ∗ 1 ⊢ A → B, Γ 1 ⇒ p A, ∆⊢ A → B, Γ 1 ⇒ p ∆Note that by Lemma 2.12 both D1 ∗ and D2 ∗ witness also the indicated weakeningsof Γ i ⇒ p ∆ i , and that the leaf in the construction witnesses a weakening of (L→).If D is cut free, then there are no free cuts in D ∗ .⊓⊔
A Simple and Practical Valuation Tree Calculus for First-Order Logic 7DD ∗Ax A, Γ ⇒ A, ∆ ◦ ⊢ (Ax)L¬D 1Γ ⇒ A, ∆¬A, Γ ⇒ ∆◦ ⊢ (L¬) D ∗ 1AR¬D 1A, Γ ⇒ ∆Γ ⇒ ¬A, ∆D ∗ 1◦ ⊢ (R¬)AL→D 1D 2Γ ⇒ A, ∆ B, Γ ⇒ ∆A → B, Γ ⇒ ∆D ∗ 2◦ ⊢ (L→)B D ∗ 1AR→L∧D 1A, Γ ⇒ B, ∆Γ ⇒ A → B, ∆D 1A, B, Γ ⇒ ∆A ∧ B, Γ ⇒ ∆◦ ⊢ (R→ 2) D ∗ 1B ◦ ⊢ (R→ 1)AD ∗ 1 ◦ ⊢ (L∧ 2)B ◦ ⊢ (L∧ 1)AR∧D 1D 2Γ ⇒ A, ∆ Γ ⇒ B, ∆Γ ⇒ A ∧ B, ∆◦ ⊢ (R∧) D ∗ 1B D ∗ 2AL∨D 1D 2A, Γ ⇒ ∆ B, Γ ⇒ ∆A ∨ B, Γ ⇒ ∆D ∗ 1D ∗ 2A◦ ⊢ (L∨)BR∨D 1Γ ⇒ A, B, ∆Γ ⇒ A ∨ B, ∆◦ ⊢ (R∨ 1)A◦ ⊢ (R∨ 2) D ∗ 1BCutD 1 D 2A, Γ ⇒ ∆ Γ ⇒ A, ∆Γ ⇒ ∆D ∗ 1 D ∗ 2AFig. 4. Translation of G3cp proofs to valuation trees.
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