By Hellmut Baumgartel

ISBN-10: 3764316640

ISBN-13: 9783764316648

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B ⇒ D). (Recall that all axioms are atomic). Clearly |di | ≤ |d| and ρ(di ) ≤ ρ(d), for i = 1, 2. The remaining cases are analogous. Remark 5. (⊃, l), (∀, l) and (∃, r) are not invertible. Concerning (∨, r), one has G ∪ {Γ ⇒ A ∨ B} can be inverted to G ∪ {Γ ⇒ A} ∪ {Γ ⇒ B} (slightly changing the above bounds). The Sch¨ utte-Tait style cut-elimination procedure presuppose that at least one of the two premises of the cut rule is invertible. As we shall see, we will use (i), (iii), (iv), (v) and (vi).

Remark 1. By the presence of (c, l) and (w, l) (resp. (EW) and (EC)), one can derive equivalent versions of the above rules with multiplicative internal (resp. , [17] for this terminology). In fact, HIF has been originally deﬁned in [8] using a diﬀerent version of the communication rule, namely G | Π1 , Γ 1 ⇒ A G | Π2 , Γ 2 ⇒ B (com ) G | Π1 , Π2 ⇒ A | Γ1 , Γ2 ⇒ B However, using (w, l) and (c, l), (com) and (com ) are interderivable (see [3]). 28 M. Baaz and A. Ciabattoni Deﬁnition 3. The complexity |A| of a formula A is inductively deﬁned as follows: – |A| = 0 if A is atomic – |A ∧ B| = |A ∨ B| = |A ⊃ B| = max(|A|, |B|) + 1 – |∀xA(x)| = |∃xA(x)| = |A| + 1 The right (left) rank of a cut is the number of consecutive hypersequents containing the cut formula, counting upward from the right (left) upper sequent of the cut.

K , Γ2 , . . , Γn ⇒ Bk . We now apply again the induction hypothesis, based on the reduced complexity of the cut formula, to γ and δ . The desired result is obtained by several applications of (c, l), (w, l) and (EC). 2. Suppose that δ ends as follows ·· δ · 1 G | Γ1 , A(a) ⇒ C | . . | Γn ⇒ C G | Γ1 , ∃xA(x) ⇒ C | . . | Γn ⇒ C (∃,l) Applying the induction hypothesis to the proof γ of the hypersequent H | Σ1 , C n1 ⇒ B1 | . . | Σk , C nk ⇒ Bk and to δ1 one gets a proof γ of (a) H | G | Σ1 , Γ, A(a) ⇒ B1 | .

### Analytic perturbation theory for matrices and operators by Hellmut Baumgartel

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