Quick cut-elimination for strictly positive cuts

Abstract

In this paper we show that the intuitionistic theory ID i <ω (SP ) for finitely many iterations of strictly positive operators is a conservative extension of the Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic. * The paper has been finished when I visited München. I would like to thank to Prof. W. Buchholz for his interests, the valuable comments and the hospitality in my visit.

Proof Theory of Weak Compactness

“…An inspection to Definition 4.13 shows that there exists a strictly positive formula H n such that the relation (H γ,n [Θ 0 ], Θ, κ, n) ⊢ a b Γ is a fixed point of H n as in (6).…”

Intuitionistic fixed point theories over set theories

“…Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope of negations ¬. Indeed as shown in [5] FiX i (HA) is a conservative extension of HA.…”

“…When the set theory T is sufficiently strong, e.g., when T comprises KripkePlatek set theory, we could prove Theorem 1.2 as in [5], i.e., first the finitary derivations of set-theoretic sentences ϕ in FiX i (T ) are embedded to infinitary derivations of a sequent θ ⇒ ϕ for a provable sentence θ in T , then partial cut-elimination is possible. This results in a ∆ 1 -definable infinitary derivation of the same sequent θ ⇒ ϕ in which there occur no fixed point formulae.…”

“…Then they conclude that ID i (strict) is conservative over the intuitionistic arithmetic HA with respect to negative formulas. Let us try to prove the full conservation result in [5] along the line in [18]. The intuitionistic version of Lemma 1.1.2 is easy to see, which says that ID i (acc) is a conservative extension of HA.…”

“…Exponentiation is used first in the rule (E), i.e., to measure an increase of ordinal depths in lowering cut rank, and second in the rule (chain). The assignment Ω Ω α (α k # · · · #α 1 ) in (chain) comes from Lemma 9 in [5], which in turn is inspired by the quick cut-elimination strategy in [4,17] along Kleene-Brouwer ordering of infinitary derivations. Lexicographic comparing, i.e., multiplication of Ω Ω α and α k # · · · #α 1 is used in Case 9, and a doubly exponential Ω Ω α is needed in Case 6 and Case 7, once multiplications are introduced.…”