A cut-free sequent calculus for the bi-intuitionistic logic 2Int
aa r X i v : . [ c s . L O ] S e p A cut-free sequent calculus for the bi-intuitionisticlogic
Sara AyhanRuhr University [email protected] 2020
The purpose of this paper is to introduce a bi-intuitionistic sequent calculus andto give proofs of admissibility for its structural rules. Since I will ponder overthe philosophical problems and implications of this calculus in a different paper,I only want to make some brief comments on these matters here. The calculusI will present, called
SC2Int , is a sequent calculus for the bi-intuitionistic logic , which Wansing presents in [2016a]. There he also gives a natural deductionsystem for this logic,
N2Int , to which
SC2Int is equivalent in terms of what isderivable. I will spell out below what this amounts to exactly. What is importantis that these calculi represent a kind of bilateralist reasoning, since they do notonly internalize processes of verification or provability but also the dual processes interms of falsification or what is called dual provability . In [Wansing, 2017] a normalform theorem for
N2Int is stated, here, I want to prove a cut-elimination theoremfor
SC2Int , i.e. if successful, this would extend the results existing so far.
SC2Int
The language L Int of , as given by Wansing, is defined in Backus-Naur formas follows: A ::= p | ⊥ | ⊤ | ( A ∧ A ) | ( A ∨ A ) | ( A → A ) | ( A (cid:6) A ).1s can be seen, we have a non-standard connective in this language, namely the op-erator of co-implication (cid:6) , which acts as a dual to implication, just like conjunctionand disjunction can be seen as dual connectives. With that, we are in the realmsof so-called bi-intuitionistic logic, which is a conservative extension of intuitionisticlogic with co-implication. We read A (cid:6) B as ‘B co-implies A’.The general design of SC2Int resembles the intuitionistic sequent calculus
G3ip . Thedistinguishing features of this calculus consist in the shared contexts for all the log-ical rules, the axiom (in our calculus the reflexivity rules) being restricted to atomicformulas and the admissibility of all structural rules (cf. [Negri and von Plato, 2001,p. 28-30] for more information about the origins of this calculus). Another distin-guishing feature is the repetition of A → B in the left premise of the left introductionrule for implication, which is necessary for the proof of admissibility of contraction.Here, this happens in → L a as well as with A (cid:6) B in (cid:6) L c .We will use p, q, r, ... for atomic formulas, A, B, C, ... for arbitrary formulas, andΓ , ∆ , Γ ′ , ... for multisets of formulas. Sequents are of the form (Γ; ∆) ⊢ ∗ C (with Γand ∆ being finite, possibly empty multisets), which are read as “From the verifi-cation of all formulas in Γ and the falsification of all formulas in ∆ one can derivethe verification (resp. falsification) of C for ∗ = + (resp. ∗ = − )”. Thus, we havea calculus in which a duality of derivability relations is considered, not only theone of verification but also the one of falsification. The formulas in Γ can thenbe understood as assumptions , while the formulas in ∆ can be understood as coun-terassumptions . SC2Int is equivalent to
N2Int in that we have a proof in
N2Int of A from the pair (Γ; ∆) of assumptions Γ and counterassumptions ∆, iff the sequent(Γ; ∆) ⊢ + A is derivable in SC2Int and we have a dual proof of A from the pair(Γ; ∆) of assumptions Γ and counterassumptions ∆, iff the sequent (Γ; ∆) ⊢ − A isderivable in SC2Int .In contrast to
G3ip , there will be no distinction between axioms and logical rulesbut within the logical rules the zero-premise rules, which comprise Rf + , Rf − , ⊥ L a , ⊤ L c , ⊥ R − , and ⊤ R + , are distinguished from the non-zero-premise rules due to thespecial role of the former for the admissibility proofs below. Each of the logical rules Note that there is also a use of bi-intuitionistic logic in the literature to refer to a specific system,namely
BiInt , also called
Heyting-Brouwer logic (e.g. [Rauszer, 1974, Gor´e, 2000, Postniece, 2010,Kowalski and Ono, 2017]). Co-implication is there to be understood to internalize the preservationof non-truth from the conclusion to the premises in a valid inference. The system 2Int, which istreated here, uses the same language as
BiInt , but the meaning of co-implication differs (cf.[Wansing, 2016a,b, 2017, p. 30f.]). In N2Int this is indicated by using single lines for verification and double lines for falsification. context designated by Γ and ∆, active formulas designated by A and B and a principal formula , which is the one introduced on the left or right side of ⊢ ∗ . Withinthe right introduction rules we need to distinguish whether the derivability relationexpresses verification or falsification by using the superscripts + and -. Within theleft rules this is not necessary, but what is needed here is distinguishing an intro-duction of the principal formula into the assumptions from an introduction into the counterassumptions . The former are indexed by superscript a , while the latter areindexed by superscript c . The set of R + and L a rules are the proof rules ; the set of R − and L c rules are the dual proof rules . SC2Int
For ∗ ∈ { +, - } :(Γ , p ; ∆) ⊢ + p Rf + (Γ; ∆ , p ) ⊢ − p Rf − (Γ , ⊥ ; ∆) ⊢ ∗ C ⊥ L a (Γ; ∆ , ⊤ ) ⊢ ∗ C ⊤ L c (Γ; ∆) ⊢ − ⊥ ⊥ R − (Γ; ∆) ⊢ + ⊤ ⊤ R + (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A ∧ B ∧ R + (Γ , A, B ; ∆) ⊢ ∗ C (Γ , A ∧ B ; ∆) ⊢ ∗ C ∧ L a (Γ; ∆) ⊢ − A (Γ; ∆) ⊢ − A ∧ B ∧ R − (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A ∧ B ∧ R − (Γ; ∆ , A ) ⊢ ∗ C (Γ; ∆ , B ) ⊢ ∗ C (Γ; ∆ , A ∧ B ) ⊢ ∗ C ∧ L c (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ + A ∨ B ∨ R +1 (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A ∨ B ∨ R +2 (Γ , A ; ∆) ⊢ ∗ C (Γ , B ; ∆) ⊢ ∗ C (Γ , A ∨ B ; ∆) ⊢ ∗ C ∨ L a (Γ; ∆) ⊢ − A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A ∨ B ∨ R − (Γ; ∆ , A, B ) ⊢ ∗ C (Γ; ∆ , A ∨ B ) ⊢ ∗ C ∨ L c (Γ , A ; ∆) ⊢ + B (Γ; ∆) ⊢ + A → B → R + (Γ , A → B ; ∆) ⊢ + A (Γ , B ; ∆) ⊢ ∗ C (Γ , A → B ; ∆) ⊢ ∗ C → L a ⊢ + A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A → B → R − (Γ , A ; ∆ , B ) ⊢ ∗ C (Γ; ∆ , A → B ) ⊢ ∗ C → L c (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ + A (cid:6) B (cid:6) R + (Γ , A ; ∆ , B ) ⊢ ∗ C (Γ , A (cid:6) B ; ∆) ⊢ ∗ C (cid:6) L a (Γ; ∆ , B ) ⊢ − A (Γ; ∆) ⊢ − A (cid:6) B (cid:6) R − (Γ; ∆ , A (cid:6) B ) ⊢ − B (Γ; ∆ , A ) ⊢ ∗ C (Γ; ∆ , A (cid:6) B ) ⊢ ∗ C (cid:6) L c Note that the rules for ∧ L a , ∨ L c , → L c and (cid:6) L a could also be given in the formof two rules, each with only one active formula A or B , as it is for example donein Gentzen’s original calculus for the left conjunction rule. We need this singlerule formulation, however, in order to get the invertibility of these rules (cf. Lemma3.3.1 below), which is important for the proof of admissibility of contraction. As saidabove, the structural rules do not have to be taken as primitive in the calculus butcan be shown to be admissible. We want to consider rules for weakening, contractionand cut. Due to the dual nature of the calculus, we need two rules for each of theserules:(Γ; ∆) ⊢ ∗ C (Γ , A ; ∆) ⊢ ∗ C W a (Γ; ∆) ⊢ ∗ C (Γ; ∆ , A ) ⊢ ∗ C W c (Γ , A, A ; ∆) ⊢ ∗ C (Γ , A ; ∆) ⊢ ∗ C C a (Γ; ∆ , A, A ) ⊢ ∗ C (Γ; ∆ , A ) ⊢ ∗ C C c (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c The proofs of admissibility of the structural rules and especially of cut-eliminationare conducted analogously to the respective proofs of Negri and von Plato [2001, p.30-40] for
G3ip . The proofs will use induction on weight of formulas and heightof derivations . Definition 3.1.1.
The weight w(A) of a formula A is defined inductively by w ( ⊥ ) = w ( ⊤ ) = 0 , ( p ) = 1 for atoms p , w ( A B ) = w ( A ) + w ( B ) + 1 for ∈ {∧ , ∨ , → , (cid:6) } . Definition 3.1.2.
A derivation in
SC2Int is either an instance of a zero-premiserule, or an application of a logical rule to derivations concluding its premises. The height of a derivation is the greatest number of successive applications of rules in it,where zero-premise rules have height 0.
First, I will show that the reflexivity rules can be generalized to instances witharbitrary formulas, not only atomic formulas.
Lemma 3.1.3.
The sequents (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derivable foran arbitrary formula C and arbitrary context (Γ; ∆) .Proof. The proof is by induction on weight of C . If w ( C ) ≤
1, we have the 19 caseslisted below. Note that for some of the derivations there is more than one possibilityto derive the desired sequent and also some of the conclusions of zero-premise rulesare conclusions of more than one of those rules. I will just show one exemplaryderivation for each case, since this is enough for the proof. C = ⊥ . Then (Γ , C ; ∆) ⊢ + C is an instance of ⊥ L a and (Γ; ∆ , C ) ⊢ − C is aninstance of ⊥ R − . C = ⊤ . Then (Γ , C ; ∆) ⊢ + C is an instance of ⊤ R + and (Γ; ∆ , C ) ⊢ − C is aninstance of ⊤ L c . C = p for some atom p . Then (Γ , C ; ∆) ⊢ + C is an instance of Rf + and (Γ; ∆ , C ) ⊢ − C is an instance of Rf − . C = ⊥ ∧ ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ , ⊥ ; ∆) ⊢ + ⊥ ∧ ⊥ ⊥ L a (Γ , ⊥ ∧ ⊥ ; ∆) ⊢ + ⊥ ∧ ⊥ ∧ R + and (Γ; ∆ , ⊥ ∧ ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊥ ∧ ⊥ ) ⊢ − ⊥ ∧ ⊥ ∧ R − C = ⊥ ∨ ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by (Γ , ⊥ ; ∆) ⊢ + ⊥ ∨ ⊥ ⊥ L a (Γ , ⊥ ; ∆) ⊢ + ⊥ ∨ ⊥ ⊥ L a (Γ , ⊥ ∨ ⊥ ; ∆) ⊢ + ⊥ ∨ ⊥ ∨ L a and (Γ; ∆ , ⊥ ∨ ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊥ ∨ ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊥ ∨ ⊥ ) ⊢ − ⊥ ∨ ⊥ ∨ R − C = ⊥ → ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ → ⊥ , ⊥ ; ∆) ⊢ + ⊥ ⊥ L a (Γ , ⊥ → ⊥ ; ∆) ⊢ + ⊥ → ⊥ → R + and (Γ , ⊥ ; ∆ , ⊥ ) ⊢ − ⊥ → ⊥ ⊥ L a (Γ; ∆ , ⊥ → ⊥ ) ⊢ − ⊥ → ⊥ → L c C = ⊥ (cid:6) ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by5Γ , ⊥ ; ∆ , ⊥ ) ⊢ + ⊥ (cid:6) ⊥ ⊥ L a (Γ , ⊥ (cid:6) ⊥ ; ∆) ⊢ + ⊥ (cid:6) ⊥ (cid:6) L a and (Γ; ∆ , ⊥ (cid:6) ⊥ , ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊥ (cid:6) ⊥ ) ⊢ − ⊥ (cid:6) ⊥ (cid:6) R − C = ⊥ ∧ ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ , ⊤ ; ∆) ⊢ + ⊥ ∧ ⊤ ⊥ L a (Γ , ⊥ ∧ ⊤ ; ∆) ⊢ + ⊥ ∧ ⊤ ∧ L a and (Γ; ∆ , ⊥ ∧ ⊤ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊥ ∧ ⊤ ) ⊢ − ⊥ ∧ ⊤ ∧ R − C = ⊥ ∨ ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ ∨ ⊤ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊥ ∨ ⊤ ; ∆) ⊢ + ⊥ ∨ ⊤ ∨ R +2 and (Γ; ∆ , ⊥ , ⊤ ) ⊢ − ⊥ ∨ ⊤ ⊤ L c (Γ; ∆ , ⊥ ∨ ⊤ ) ⊢ − ⊥ ∨ ⊤ ∨ L c C = ⊥ → ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ → ⊤ , ⊥ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊥ → ⊤ ; ∆) ⊢ + ⊥ → ⊤ → R + and (Γ , ⊥ ; ∆ , ⊤ ) ⊢ − ⊥ → ⊤ ⊤ L c (Γ; ∆ , ⊥ → ⊤ ) ⊢ − ⊥ → ⊤ → L c C = ⊥ (cid:6) ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊥ ; ∆ , ⊤ ) ⊢ + ⊥ (cid:6) ⊤ ⊥ L a (Γ , ⊥ (cid:6) ⊤ ; ∆) ⊢ + ⊥ (cid:6) ⊤ (cid:6) L a and (Γ; ∆ , ⊥ (cid:6) ⊤ , ⊤ ) ⊢ − ⊥ ⊤ L c (Γ; ∆ , ⊥ (cid:6) ⊤ ) ⊢ − ⊥ (cid:6) ⊤ (cid:6) R − C = ⊤ ∧ ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊤ , ⊥ ; ∆) ⊢ + ⊤ ∧ ⊥ ⊥ L a (Γ , ⊤ ∧ ⊥ ; ∆) ⊢ + ⊤ ∧ ⊥ ∧ L a and (Γ; ∆ , ⊤ ∧ ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊤ ∧ ⊥ ) ⊢ − ⊤ ∧ ⊥ ∧ R − C = ⊤ ∨ ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊤ ∨ ⊥ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ ∨ ⊥ ; ∆) ⊢ + ⊤ ∨ ⊥ ∨ R +1 and (Γ; ∆ , ⊤ , ⊥ ) ⊢ − ⊤ ∨ ⊥ ⊤ L c (Γ; ∆ , ⊤ ∨ ⊥ ) ⊢ − ⊤ ∨ ⊥ ∨ L c C = ⊤ → ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by (Γ , ⊤ → ⊥ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊥ ; ∆) ⊢ + ⊤ → ⊥ ⊥ L a (Γ , ⊤ → ⊥ ; ∆) ⊢ + ⊤ → ⊥ → L a and (Γ; ∆ , ⊤ → ⊥ ) ⊢ + ⊤ ⊤ R + (Γ; ∆ , ⊤ → ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊤ → ⊥ ) ⊢ − ⊤ → ⊥ → R − C = ⊤ (cid:6) ⊥ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by (Γ , ⊤ (cid:6) ⊥ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ (cid:6) ⊥ ; ∆) ⊢ − ⊥ ⊥ L a (Γ , ⊤ (cid:6) ⊥ ; ∆) ⊢ + ⊤ (cid:6) ⊥ (cid:6) R + and (Γ; ∆ , ⊤ (cid:6) ⊥ ) ⊢ − ⊥ ⊥ R − (Γ; ∆ , ⊤ ) ⊢ − ⊤ (cid:6) ⊥ ⊤ L c (Γ; ∆ , ⊤ (cid:6) ⊥ ) ⊢ − ⊤ (cid:6) ⊥ (cid:6) L c C = ⊤ ∧ ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by (Γ , ⊤ ∧ ⊤ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ ∧ ⊤ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ ∧ ⊤ ; ∆) ⊢ + ⊤ ∧ ⊤ ∧ R + and (Γ; ∆ , ⊤ ) ⊢ − ⊤ ∧ ⊤ ⊤ L c (Γ; ∆ , ⊤ ) ⊢ − ⊤ ∧ ⊤ ⊤ L c (Γ; ∆ , ⊤ ∧ ⊤ ) ⊢ − ⊤ ∧ ⊤ ∧ L c C = ⊤ ∨ ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊤ ∨ ⊤ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ ∨ ⊤ ; ∆) ⊢ + ⊤ ∨ ⊤ ∨ R + and (Γ; ∆ , ⊤ , ⊤ ) ⊢ − ⊤ ∨ ⊤ ⊤ L c (Γ; ∆ , ⊤ ∨ ⊤ ) ⊢ − ⊤ ∨ ⊤ ∨ L c C = ⊤ → ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by6Γ , ⊤ → ⊤ , ⊤ ; ∆) ⊢ + ⊤ ⊤ R + (Γ , ⊤ → ⊤ ; ∆) ⊢ + ⊤ → ⊤ → R + and (Γ , ⊤ ; ∆ , ⊤ ) ⊢ − ⊤ → ⊤ ⊤ L c (Γ; ∆ , ⊤ → ⊤ ) ⊢ − ⊤ → ⊤ → L c C = ⊤ (cid:6) ⊤ . Then (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derived by(Γ , ⊤ ; ∆ , ⊤ ) ⊢ + ⊤ (cid:6) ⊤ ⊤ L c (Γ , ⊤ (cid:6) ⊤ ; ∆) ⊢ + ⊤ (cid:6) ⊤ (cid:6) L a and (Γ; ∆ , ⊤ (cid:6) ⊤ , ⊤ ) ⊢ − ⊤ ⊤ L c (Γ; ∆ , ⊤ (cid:6) ⊤ ) ⊢ − ⊤ (cid:6) ⊤ (cid:6) R − The inductive hypothesis is that (Γ , C ; ∆) ⊢ + C and (Γ; ∆ , C ) ⊢ − C are derivablefor all formulas C with w ( C ) ≤ n , and we have to show that (Γ , D ; ∆) ⊢ + D and(Γ; ∆ , D ) ⊢ − D are derivable for formulas D of weight ≤ n + 1. There are four cases: D = A ∧ B . By the definition of weight and our inductive hypothesis, w ( A ) ≤ n and w ( B ) ≤ n .We can derive (Γ , A ∧ B ; ∆) ⊢ + A ∧ B by(Γ , A, B ; ∆) ⊢ + A (Γ , A ∧ B ; ∆) ⊢ + A ∧ L a (Γ , A, B ; ∆) ⊢ + B (Γ , A ∧ B ; ∆) ⊢ + B ∧ L a (Γ , A ∧ B ; ∆) ⊢ + A ∧ B ∧ R + and (Γ; ∆ , A ∧ B ) ⊢ − A ∧ B by(Γ; ∆ , A ) ⊢ − A (Γ; ∆ , A ) ⊢ − A ∧ B ∧ R − (Γ; ∆ , B ) ⊢ − B (Γ; ∆ , B ) ⊢ − A ∧ B ∧ R − (Γ; ∆ , A ∧ B ) ⊢ − A ∧ B ∧ L c (Γ; ∆ , A ) ⊢ − A and (Γ; ∆ , B ) ⊢ − B are derivable by the inductive hypothesis andsince the context is arbitrary, so are (Γ ′ , A ; ∆) ⊢ + A and (Γ ′′ , B ; ∆) ⊢ + B , forΓ ′ = Γ , B and Γ ′′ = Γ , A . D = A ∨ B . As before, w ( A ) ≤ n and w ( B ) ≤ n .We can derive (Γ , A ∨ B ; ∆) ⊢ + A ∨ B by(Γ , A ; ∆) ⊢ + A (Γ , A ; ∆) ⊢ + A ∨ B ∨ R +1 (Γ , B ; ∆) ⊢ + B (Γ , B ; ∆) ⊢ + A ∨ B ∨ R +2 (Γ , A ∨ B ; ∆) ⊢ + A ∨ B ∨ L a and (Γ; ∆ , A ∨ B ) ⊢ − A ∨ B by(Γ; ∆ , A, B ) ⊢ − A (Γ; ∆ , A ∨ B ) ⊢ − A ∨ L c (Γ; ∆ , A, B ) ⊢ − B (Γ; ∆ , A ∨ B ) ⊢ − B ∨ L c (Γ; ∆ , A ∨ B ) ⊢ − A ∨ B ∨ R − Again, by inductive hypothesis we get the derivability of (Γ , A ; ∆) ⊢ + A and(Γ , B ; ∆) ⊢ + B and since the context is arbitrary, (Γ; ∆ ′ , A ) ⊢ − A and (Γ; ∆ ′′ , B ) ⊢ − B are derivable, for ∆ ′ = ∆ , B and ∆ ′′ = ∆ , A . D = A → B . As before, w ( A ) ≤ n and w ( B ) ≤ n .7e can derive (Γ , A → B ; ∆) ⊢ + A → B by(Γ , A, A → B ; ∆) ⊢ + A (Γ , A, B ; ∆) ⊢ + B (Γ , A, A → B ; ∆) ⊢ + B → L a (Γ , A → B ; ∆) ⊢ + A → B → R + and (Γ; ∆ , A → B ) ⊢ − A → B by(Γ , A ; ∆ , B ) ⊢ + A (Γ , A ; ∆ , B ) ⊢ − B (Γ , A ; ∆ , B ) ⊢ − A → B → R − (Γ; ∆ , A → B ) ⊢ − A → B → L c The case of (Γ , A, B ; ∆) ⊢ + B was already mentioned in the case of conjunction andwith the same reasoning (Γ ′ , A ; ∆) ⊢ + A for Γ ′ = Γ , A → B , (Γ , A ; ∆ ′ ) ⊢ + A for∆ ′ = ∆ , B as well as (Γ ′ ; ∆ , B ) ⊢ − B for Γ ′ = Γ , A are derivable. D = A (cid:6) B . As before, w ( A ) ≤ n and w ( B ) ≤ n .We can derive (Γ , A (cid:6) B ; ∆) ⊢ + A (cid:6) B by(Γ , A ; ∆ , B ) ⊢ + A (Γ , A ; ∆ , B ) ⊢ − B (Γ , A ; ∆ , B ) ⊢ + A (cid:6) B (cid:6) R + (Γ , A (cid:6) B ; ∆) ⊢ + A (cid:6) B (cid:6) L a and (Γ; ∆ , A (cid:6) B ) ⊢ − A (cid:6) B by(Γ; ∆ , B, A (cid:6) B ) ⊢ − B (Γ; ∆ , A, B ) ⊢ − A (Γ; ∆ , B, A (cid:6) B ) ⊢ − A (cid:6) L c (Γ; ∆ , A (cid:6) B ) ⊢ − A (cid:6) B (cid:6) R − With the same reasoning as above (Γ; ∆ ′ , B ) ⊢ − B is derivable for ∆ ′ = ∆ , A (cid:6) B and all other cases are already dealt with above. I will now start with the proof of admissibility of weakening by induction on heightof derivation. The general procedure when proving admissibility of a rule with thisis to prove it for applications of this rule to conclusions of zero-premise rules andthen generalize by induction on the number of applications of the rule to arbitraryderivations. Thus, we can assume that there is only one instance - as the last stepin the derivation - of the rule in question.
Theorem 3.2.1 (Height-preserving weakening) . If (Γ; ∆) ⊢ ∗ C is derivable with aheight of derivation at most n, then (Γ , D ; ∆) ⊢ ∗ C and (Γ; ∆ , D ) ⊢ ∗ C are derivablewith a height of derivation at most n for arbitrary D.Proof. If n = 0, then (Γ; ∆) ⊢ ∗ C is a zero-premise rule, which means that one of8he following six cases holds. C is an atom and 1) a formula in Γ with ∗ = + or2) a formula in ∆ with ∗ = − . Otherwise it can be the case that 3) C is ⊤ with ∗ = + or 4) C is ⊥ with ∗ = − . Lastly, it could be that 5) ⊥ is a formula in Γor 6) ⊤ a formula in ∆. In either case, (Γ , D ; ∆) ⊢ ∗ C and (Γ; ∆ , D ) ⊢ ∗ C areconclusions of the respective zero-premise rules. Our inductive hypothesis is nowthat height-preserving weakening is admissible up to derivations of height ≤ n . Let(Γ; ∆) ⊢ ∗ C be derivable with a height of derivation at most n + 1.If the last rule applied is ∧ L a , then Γ = Γ ′ , A ∧ B and the last step is(Γ ′ , A, B ; ∆) ⊢ ∗ C (Γ ′ , A ∧ B ; ∆) ⊢ ∗ C ∧ L a So (Γ ′ , A, B ; ∆) ⊢ ∗ C is derivable in ≤ n steps. By inductive hypothesis, also(Γ ′ , A, B, D ; ∆) ⊢ ∗ C and (Γ ′ , A, B ; ∆ , D ) ⊢ ∗ C are derivable in ≤ n steps. Thus,the application of ∧ L a gives a derivation of (Γ ′ , A ∧ B, D ; ∆) ⊢ ∗ C and (Γ ′ , A ∧ B ; ∆ , D ) ⊢ ∗ C in ≤ n + 1 steps.If the last rule applied is ∧ L c , then ∆ = ∆ ′ , A ∧ B and the last step is(Γ; ∆ ′ , A ) ⊢ ∗ C (Γ; ∆ ′ , B ) ⊢ ∗ C (Γ; ∆ ′ , A ∧ B ) ⊢ ∗ C ∧ L c So (Γ; ∆ ′ , A ) ⊢ ∗ C and (Γ; ∆ ′ , B ) ⊢ ∗ C are derivable in ≤ n steps. By inductivehypothesis, also (Γ , D ; ∆ ′ , A ) ⊢ ∗ C , (Γ; ∆ ′ , A, D ) ⊢ ∗ C , (Γ , D ; ∆ ′ , B ) ⊢ ∗ C and(Γ; ∆ ′ , B, D ) ⊢ ∗ C are derivable in ≤ n steps. Thus, the application of ∧ L c tothe first and the third premise and to the second and the fourth premise gives aderivation of (Γ , D ; ∆ ′ , A ∧ B ) ⊢ ∗ C and (Γ; ∆ ′ , A ∧ B, D ) ⊢ ∗ C , respectively, in ≤ n + 1 steps.If the last rule applied is ∧ R + , then C = A ∧ B and the last step is(Γ; ∆) ⊢ + A (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A ∧ B ∧ R + So (Γ; ∆) ⊢ + A and (Γ; ∆) ⊢ + B are derivable in ≤ n steps. By inductive hypothesis,also (Γ , D ; ∆) ⊢ + A , (Γ; ∆ , D ) ⊢ + A , (Γ , D ; ∆) ⊢ + B and (Γ; ∆ , D ) ⊢ + B arederivable in ≤ n steps. Thus, the application of ∧ R + to the first and the thirdpremise and to the second and the fourth premise gives a derivation of (Γ , D ; ∆) ⊢ + A ∧ B and (Γ; ∆ , D ) ⊢ + A ∧ B , respectively, in ≤ n + 1 steps.If the last rule applied is ∧ R − , then C = A ∧ B and the last step is(Γ; ∆) ⊢ − A (Γ; ∆) ⊢ − A ∧ B ∧ R − So (Γ; ∆) ⊢ − A is derivable in ≤ n steps. By inductive hypothesis, also (Γ , D ; ∆) ⊢ − and (Γ; ∆ , D ) ⊢ − A are derivable in ≤ n steps. Thus, the application of ∧ R − gives a derivation of (Γ , D ; ∆) ⊢ − A ∧ B and (Γ; ∆ , D ) ⊢ − A ∧ B in ≤ n + 1 steps.For the other logical rules the same can be shown with similar steps.Now I want to show one other thing related to weakening because we will need thisresult later in our proof for the admissibility of the cut rules, namely that for thespecial case that the weakening formula is ⊤ for W a and respectively ⊥ for W c , theweakening rules are invertible, i.e.:(Γ , ⊤ ; ∆) ⊢ ∗ C (Γ; ∆) ⊢ ∗ C W ⊤ inv (Γ; ∆ , ⊥ ) ⊢ ∗ C (Γ; ∆) ⊢ ∗ C W ⊥ inv Lemma 3.2.2 (Special case of inverted weakening) . If (Γ , ⊤ ; ∆) ⊢ ∗ C or (Γ; ∆ , ⊥ ) ⊢ ∗ C are derivable with a height of derivation at most n, then so is (Γ; ∆) ⊢ ∗ C .Proof. If n = 0, then exactly the same reasoning as for Theorem 3.2.1 can beapplied here.Now we assume height-preserving invertibility for these two special cases of weak-ening up to height n , and let (Γ , ⊤ ; ∆) ⊢ ∗ C and (Γ; ∆ , ⊥ ) ⊢ ∗ C be derivable witha height of derivation ≤ n + 1. The proof works correspondingly to the proof ofheight-preserving weakening above, I will show it for the case of the → L c -rule thistime, just to choose one that is not familiar in ‘usual’ calculi, but it works similarfor all logical connectives and their rules.If the last rule applied is → L c , then we have ∆ = ∆ ′ , A → B and the last step is(Γ , A, ⊤ ; ∆ ′ , B ) ⊢ ∗ C (Γ , ⊤ ; ∆ ′ , A → B ) ⊢ ∗ C → L c or respectively (Γ , A ; ∆ ′ , B, ⊥ ) ⊢ ∗ C (Γ; ∆ ′ , A → B, ⊥ ) ⊢ ∗ C → L c So, (Γ , A, ⊤ ; ∆ ′ , B ) ⊢ ∗ C and (Γ , A ; ∆ ′ , B, ⊥ ) ⊢ ∗ C are derivable in ≤ n steps. Thenby inductive hypothesis, (Γ , A ; ∆ ′ , B ) ⊢ ∗ C is derivable in ≤ n steps. If we apply → L c to this, this gives us (Γ; ∆ ′ , A → B ) ⊢ ∗ C in ≤ n + 1 steps. Before we can prove the admissibility of the contraction rules, we need to provethe following lemma about the invertibility of premises and conclusions of the log-ical rules for the left introduction of formulas. Note that for → L a and (cid:6) L c theinvertibility only holds for the right premises. Negri and von Plato [2001, p. 33] give a counterexample for the implication rule. The analo-gous counterexamples for
SC2Int would be the derivability of the sequents ( ⊥ → ⊥ ; ∅ ) ⊢ + ⊥ → ⊥ emma 3.3.1 (Inversion) . ( i ) If (Γ , A ∧ B ; ∆) ⊢ ∗ C is derivable with a heightof derivation at most n, then (Γ , A, B ; ∆) ⊢ ∗ C is derivable with a height ofderivation at most n. ( i ) If (Γ; ∆ , A ∧ B ) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ; ∆ , A ) ⊢ ∗ C and (Γ; ∆ , B ) ⊢ ∗ C are derivable with a height of derivation atmost n. ( ii ) If (Γ , A ∨ B ; ∆) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ , A ; ∆) ⊢ ∗ C and (Γ , B ; ∆) ⊢ ∗ C are derivable with a height of derivation atmost n. ( ii ) If (Γ; ∆ , A ∨ B ) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ; ∆ , A, B ) ⊢ ∗ C is derivable with a height of derivation at most n. ( iii ) If (Γ , A → B ; ∆) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ , B ; ∆) ⊢ ∗ C is derivable with a height of derivation at most n. ( iii ) If (Γ; ∆ , A → B ) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ , A ; ∆ , B ) ⊢ ∗ C is derivable with a height of derivation at most n. ( iv ) If (Γ , A (cid:6) B ; ∆) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ , A ; ∆ , B ) ⊢ ∗ C is derivable with a height of derivation at most n. ( iv ) If (Γ; ∆ , A (cid:6) B ) ⊢ ∗ C is derivable with a height of derivation at most n, then (Γ; ∆ , A ) ⊢ ∗ C is derivable with a height of derivation at most n.Proof. The proof is by induction on n .1.) If (Γ , A B ; ∆) ⊢ ∗ C with ∈ {∧ , ∨ , → , (cid:6) } is the conclusion of a zero-premiserule, then so are (Γ , A, B ; ∆) ⊢ ∗ C , (Γ , A ; ∆) ⊢ ∗ C , (Γ , B ; ∆) ⊢ ∗ C , (Γ; ∆ , B ) ⊢ ∗ C since A B is neither atomic nor ⊥ nor ⊤ .Now we assume height-preserving inversion up to height n , and let (Γ , A B ; ∆) ⊢ ∗ C be derivable with a height of derivation ≤ n + 1.( i ) Either A ∧ B is principal in the last rule or not. If A ∧ B is the principalformula, the premise (Γ , A, B ; ∆) ⊢ ∗ C has a derivation of height n . If A ∧ B is not principal in the last rule, then there must be one or two premises(Γ ′ , A ∧ B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , A ∧ B ; ∆ ′′ ) ⊢ ∗ C ′′ with a height of derivation ≤ n . and ( ∅ ; ⊤ (cid:6) ⊤ ) ⊢ − ⊤ (cid:6) ⊤ . ′ , A, B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , A, B ; ∆ ′′ ) ⊢ ∗ C ′′ are derivable with a height of derivation ≤ n . Now the last rule can be appliedto these premises to conclude (Γ , A, B ; ∆) ⊢ ∗ C in at most n + 1 steps.( ii ) Either A ∨ B is principal in the last rule or not. If A ∨ B is the principalformula, the premises (Γ , A ; ∆) ⊢ ∗ C and (Γ , B ; ∆) ⊢ ∗ C have a derivationof height ≤ n . If A ∨ B is not principal in the last rule, then there mustbe one or two premises (Γ ′ , A ∨ B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , A ∨ B ; ∆ ′′ ) ⊢ ∗ C ′′ with aheight of derivation ≤ n . Then, by inductive hypothesis, also (Γ ′ , A ; ∆ ′ ) ⊢ ∗ C ′ ,(Γ ′ , B ; ∆ ′ ) ⊢ ∗ C ′ and (Γ ′′ , A ; ∆ ′ ) ⊢ ∗ C ′′ , (Γ ′′ , B ; ∆ ′′ ) ⊢ ∗ C ′′ are derivable witha height of derivation ≤ n . Now the last rule can be applied to the first andthird premise to conclude (Γ , A ; ∆) ⊢ ∗ C and to the second and fourth premiseto conclude (Γ , B ; ∆) ⊢ ∗ C in at most n + 1 steps.( iii ) Either A → B is principal in the last rule or not. If A → B is the principalformula, the premise (Γ , B ; ∆) ⊢ ∗ C has a derivation of height ≤ n . If A → B is not principal in the last rule, then there must be one or two premises(Γ ′ , A → B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , A → B ; ∆ ′′ ) ⊢ ∗ C ′′ with a height of derivation ≤ n . Then, by inductive hypothesis, also (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , B ; ∆ ′′ ) ⊢ ∗ C ′′ are derivable with a height of derivation ≤ n . Now the last rule can be appliedto these premises to conclude (Γ , B ; ∆) ⊢ ∗ C in at most n + 1 steps.( iv ) Either A (cid:6) B is principal in the last rule or not. If A (cid:6) B is the principalformula, then the premise (Γ , A ; ∆ , B ) ⊢ ∗ C has a derivation of height n . If A (cid:6) B is not principal in the last rule, then there must be one or two premises(Γ ′ , A (cid:6) B ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , A (cid:6) B ; ∆ ′′ ) ⊢ ∗ C ′′ with a height of derivation ≤ n .Then, by inductive hypothesis, also (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C ′ , (Γ ′′ , A ; ∆ ′′ , B ) ⊢ ∗ C ′′ are derivable with a height of derivation ≤ n . Now the last rule can be appliedto these premises to conclude (Γ , A ; ∆ , B ) ⊢ ∗ C in at most n + 1 steps.2.) If (Γ; ∆ , A B ) ⊢ ∗ C with ∈ {∧ , ∨ , → , (cid:6) } is the conclusion of a zero-premiserule, then so are (Γ; ∆ , A ) ⊢ ∗ C , (Γ; ∆ , B ) ⊢ ∗ C , (Γ; ∆ , A, B ) ⊢ ∗ C , (Γ , A ; ∆) ⊢ ∗ C since A B is neither atomic nor ⊥ nor ⊤ .Now we assume height-preserving inversion up to height n , and let (Γ; ∆ , A B ) ⊢ ∗ C be derivable with a height of derivation ≤ n + 1.( i ) Either A ∧ B is principal in the last rule or not. If A ∧ B is the principalformula, the premises (Γ; ∆ , A ) ⊢ ∗ C and (Γ; ∆ , B ) ⊢ ∗ C have a derivation12f height ≤ n . If A ∧ B is not principal in the last rule, then there mustbe one or two premises (Γ ′ ; ∆ ′ , A ∧ B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A ∧ B ) ⊢ ∗ C ′′ with aheight of derivation ≤ n . Then, by inductive hypothesis, also (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C ′ ,(Γ ′ ; ∆ ′ , B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A ) ⊢ ∗ C ′′ , (Γ ′′ ; ∆ ′′ , B ) ⊢ ∗ C ′′ are derivable with aheight of derivation ≤ n . Now the last rule can be applied to the first andthird premise to conclude (Γ; ∆ , A ) ⊢ ∗ C and to the second and fourth premiseto conclude (Γ; ∆ , B ) ⊢ ∗ C in at most n + 1 steps.( ii ) Either A ∨ B is principal in the last rule or not. If A ∨ B is the principalformula, the premise (Γ; ∆ , A, B ) ⊢ ∗ C has a derivation of height n . If A ∨ B is not principal in the last rule, then there must be one or two premises(Γ ′ ; ∆ ′ , A ∨ B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A ∨ B ) ⊢ ∗ C ′′ with a height of derivation ≤ n .Then, by inductive hypothesis, also (Γ ′ ; ∆ ′ , A, B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A, B ) ⊢ ∗ C ′′ are derivable with a height of derivation ≤ n . Now the last rule can be appliedto these premises to conclude (Γ; ∆ , A, B ) ⊢ ∗ C in at most n + 1 steps.( iii ) Either A → B is principal in the last rule or not. If A → B is the principalformula, the premise (Γ , A ; ∆ , B ) ⊢ ∗ C has a derivation of height n . If A → B is not principal in the last rule, then there must be one or two premises(Γ ′ ; ∆ ′ , A → B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A → B ) ⊢ ∗ C ′′ with a height of derivation ≤ n .Then, by inductive hypothesis, also (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C ′ , (Γ ′′ , A ; ∆ ′′ , B ) ⊢ ∗ C ′′ are derivable with a height of derivation ≤ n . Now the last rule can be appliedto these premises to conclude (Γ , A ; ∆ , B ) ⊢ ∗ C in at most n + 1 steps.( iv ) Either A (cid:6) B is principal in the last rule or not. If A (cid:6) B is the principalformula, the premise (Γ; ∆ , A ) ⊢ ∗ C has a derivation of height ≤ n . If A (cid:6) B is not principal in the last rule, then there must be one or two premises(Γ ′ ; ∆ ′ , A (cid:6) B ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A (cid:6) B ) ⊢ ∗ C ′′ with a height of derivation ≤ n .Then, by inductive hypothesis, also (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C ′ , (Γ ′′ ; ∆ ′′ , A ) ⊢ ∗ C ′′ arederivable with a height of derivation ≤ n . Now the last rule can be applied tothese premises to conclude (Γ; ∆ , A ) ⊢ ∗ C in at most n + 1 steps.Next, I will prove the admissibility of the contraction rules in SC2Int . Theorem 3.3.2 (Height-preserving contraction) . If (Γ , D, D ; ∆) ⊢ ∗ C is derivablewith a height of derivation at most n, then (Γ , D ; ∆) ⊢ ∗ C is derivable with a height of erivation at most n and if (Γ; ∆ , D, D ) ⊢ ∗ C is derivable with a height of derivationat most n, then (Γ; ∆ , D ) ⊢ ∗ C is derivable with a height of derivation at most n .Proof. The proof is again by induction on the height of derivation n .If (Γ , D, D ; ∆) ⊢ ∗ C (resp. (Γ; ∆ , D, D ) ⊢ ∗ C ) is the conclusion of a zero-premiserule, then either C is an atom and contained in the antecedent, in the assumptionsfor ⊢ + or in the counterassumptions for ⊢ − , or ⊥ is part of the assumptions, or ⊤ is part of the counterassumptions, or C = ⊤ for ⊢ + , or C = ⊥ for ⊢ − . In eithercase, also (Γ , D ; ∆) ⊢ ∗ C (resp. (Γ; ∆ , D ) ⊢ ∗ C ) is a conclusion of the respectivezero-premise rule.Let contraction be admissible up to derivation height n and let (Γ , D, D ; ∆) ⊢ ∗ C (resp. (Γ; ∆ , D, D ) ⊢ ∗ C ) be derivable in at most n + 1 steps. Either the contractionformula is not principal in the last inference step or it is principal.If D is not principal in the last rule concluding the premise of contraction (Γ , D, D ; ∆) ⊢ ∗ C , there must be one or two premises (Γ ′ , D, D ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , D, D ; ∆ ′′ ) ⊢ ∗ C ′ with a height of derivation ≤ n . So by inductive hypothesis, we can derive(Γ ′ , D ; ∆ ′ ) ⊢ ∗ C ′ , (Γ ′′ , D ; ∆ ′′ ) ⊢ ∗ C ′ with a height of derivation ≤ n . Now the lastrule can be applied to these premises to conclude (Γ , D ; ∆) ⊢ ∗ C in at most n + 1steps. For the case of (Γ; ∆ , D, D ) ⊢ ∗ C being the premise of contraction, the sameargument applies respectively.If D is principal in the last rule, we have to consider four cases for each contractionrule according to the form of D . I will show the cases for C c this time; for C a thesame arguments apply respectively. D = A ∧ B . Then the last rule applied must be ∧ L c and we have as premises(Γ; ∆ , A ∧ B, A ) ⊢ ∗ C and (Γ; ∆ , A ∧ B, B ) ⊢ ∗ C with a derivation height ≤ n . Bythe inversion lemma this means that (Γ; ∆ , A, A ) ⊢ ∗ C and (Γ; ∆ , B, B ) ⊢ ∗ C arealso derivable with a derivation height ≤ n . Then by inductive hypothesis, we get(Γ; ∆ , A ) ⊢ ∗ C and (Γ; ∆ , B ) ⊢ ∗ C with a height of derivation ≤ n and by applying ∧ L c we can derive (Γ; ∆ , A ∧ B ) ⊢ ∗ C in at most n + 1 steps. D = A ∨ B . Then the last rule applied must be ∨ L c and (Γ; ∆ , A ∨ B, A, B ) ⊢ ∗ C is derivable with a height of derivation ≤ n . By the inversion lemma, also(Γ; ∆ , A, B, A, B ) ⊢ ∗ C is derivable with a derivation height ≤ n . Then by inductivehypothesis (applied twice), we get (Γ; ∆ , A, B ) ⊢ ∗ C with a height of derivation ≤ n and by applying ∨ L c we can derive (Γ; ∆ , A ∨ B ) ⊢ ∗ C in at most n + 1 steps.14 = A → B . Then the last rule applied must be → L c and accordingly (Γ , A ; ∆ , B,A → B ) ⊢ ∗ C is derivable with a height of derivation ≤ n . By the inversionlemma, then also (Γ , A, A ; ∆ , B, B ) ⊢ ∗ C is derivable with a derivation height ≤ n .By inductive hypothesis (applied twice), we get (Γ , A ; ∆ , B ) ⊢ ∗ C with a height ofderivation ≤ n and by applying → L c we can derive (Γ; ∆ , A → B ) ⊢ ∗ C in at most n + 1 steps. D = A (cid:6) B . Then the last rule applied must be (cid:6) L c and we have as premises(Γ; ∆ , A (cid:6) B, A (cid:6) B ) ⊢ − B and (Γ; ∆ , A (cid:6) B, A ) ⊢ ∗ C with a derivation height ≤ n . The inductive hypothesis applied to the first, gives us (Γ; ∆ , A (cid:6) B ) ⊢ − B with a derivation height ≤ n and the inversion lemma applied to the second, also(Γ; ∆ , A, A ) ⊢ ∗ C and again by inductive hypothesis (Γ; ∆ , A ) ⊢ ∗ C with a derivationheight ≤ n . By applying (cid:6) L c we can now derive (Γ; ∆ , A (cid:6) B ) ⊢ ∗ C in at most n + 1steps. Now, I will come to the main result, the proof of cut-elimination. The proof showsthat cuts can be permuted upward in a derivation until they reach one of the zero-premise rules the derivation started with. When cut has reached zero-premise rules,the derivation can be transformed into one beginning with the conclusion of the cut,which can be shown by the following reasoning.When both premises of cut are conclusions of a zero-premise rule, then the conclusionof cut is also a conclusion of one of these rules: If the left premise is (Γ , ⊥ ; ∆) ⊢ ∗ D ,then the conclusion also has ⊥ in the assumptions of the antecedent. If the leftpremise is (Γ; ∆ , ⊤ ) ⊢ ∗ D , then the conclusion also has ⊤ in the counterassumptionsof the antecedent. If the left premise of Cut a is (Γ; ∆) ⊢ + ⊤ or the left premise of Cut c is (Γ; ∆) ⊢ − ⊥ , then the right premise is (Γ ′ , ⊤ ; ∆ ′ ) ⊢ ∗ C or (Γ ′ ; ∆ ′ , ⊥ ) ⊢ ∗ C respectively. These are conclusions of zero-premise rules only in one of the followingcases: • C is an atom in Γ ′ for ∗ = + or C is an atom in ∆ ′ for ∗ = - • C = ⊤ for ∗ = + or C = ⊥ for ∗ = - • ⊥ is in Γ ′ or ⊤ is in ∆ ′ In each case the conclusion of cut (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of the15ame zero-premise rule as the right premise. The last two possibilities are that theleft premise is (Γ , p ; ∆) ⊢ + p for Cut a or (Γ; ∆ , p ) ⊢ − p for Cut c respectively. Forthe former case this means that the right premise is (Γ ′ , p ; ∆ ′ ) ⊢ ∗ C . This is theconclusion of a zero-premise rule only in one of the following cases: • For ∗ = +: C = p , or C is an atom in Γ ′ , or C = ⊤• For ∗ = -: C is an atom in ∆ ′ , or C = ⊥• ⊥ is in Γ ′ , or ⊤ is in ∆ ′ In each case the conclusion of cut (Γ , p, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of thesame zero-premise rule as the right premise. For the latter case this means that theright premise is (Γ ′ ; ∆ ′ , p ) ⊢ ∗ C . This is the conclusion of a zero-premise rule onlyin one of the following cases: • For ∗ = +: C is an atom in Γ ′ , or C = ⊤• For ∗ = -: C = p , or C is an atom in ∆ ′ , or C = ⊥• ⊥ is in Γ ′ , or ⊤ is in ∆ ′ In each case the conclusion of cut (Γ , Γ ′ ; ∆ , p, ∆ ′ ) ⊢ ∗ C is also a conclusion of thesame zero-premise rule as the right premise. So, when cut has reached zero-premiserules as premises, the derivation can be transformed into one beginning with theconclusion of the cut by deleting the premises.The proof is - as before - conducted in a manner corresponding to the proof ofcut-elimination for G3ip by Negri and von Plato [2001], which means that it is byinduction on the weight of the cut formula and a subinduction on the cut-height,the sum of heights of derivations of the two premises of cut.
Definition 3.4.1.
The cut-height of an application of one of the rules of cut in aderivation is the sum of heights of derivation of the two premises of the rule.
In the proof permutations are given that always reduce the weight of the cut formulaor the cut-height of instances of the rules. When the cut formula is not principalin at least one (or both) of the premises of cut, cut-height is reduced. In the othercases, i.e. in which the cut formula is principal in both premises, it is shown that cut-height and/or the weight of the cut formula can be reduced. This process terminatessince atoms cannot be principal formulas.16he difference between the height of a derivation and cut-height needs to be em-phasized here, because it is essential to understand that if there are two instances ofcut, one occurring below the other in the derivation, this does not necessarily meanthat the lower instance has a greater cut-height than the upper. Let us suppose theupper instance of cut occurs in the derivation of the left premise of the lower cut.The upper instance can have a cut-height which is greater than the height of eitherits premises because the sum of the premises is what matters. However, the lowerinstance can have as a right premise one with a much shorter derivation height thaneither of the premises of the upper cut, making the sum of the derivation heightsof those two premises lesser than the one from the upper cut. So, what follows isthat it is not enough to show that occurrences of cut can be permuted upward ina derivation in order to show that cut-height decreases, but we need to calculateexactly the cut-height of each derivation in our proof. As before, it can be assumedthat in a given derivation the last instance is the one and only occurrence of cut.
Theorem 3.4.2.
The cut rules (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a and (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c are admissible in SC2Int .Proof.
The proof is then organized as follows. First, I consider the case that at leastone premise in a cut is a conclusion of one of the zero-premise rules and show howcut can be eliminated in these cases. Otherwise three cases can be distinguished:1.) The cut formula is not principal in either premise of cut, 2.) the cut formulais principal in just one premise of cut, and 3.) the cut formula is principal in bothpremises of cut.
Cut with a conclusion of a zero-premise rule aspremise
Cut with a conclusion of Rf + , Rf − , ⊥ L a , ⊤ L c , ⊥ R − , or ⊤ R + as premise: If at least one of the premises of cut is a conclusion of one of the zero-premise rules,we distinguish three cases for both cut rules:17
1- Cut a -1.1- The left premise (Γ; ∆) ⊢ + D is a conclusion of a zero-premise-rule. There arefour subcases:(a) The cut formula D is an atom in Γ. Then the conclusion (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is derived from (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C by W a and W c .(b) ⊥ is a formula in Γ. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of ⊥ L a .(c) ⊤ is a formula in ∆. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of ⊤ L c .(d) ⊤ = D. Then the right premise is (Γ ′ , ⊤ ; ∆ ′ ) ⊢ ∗ C and (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C follows by W ⊤ inv as well as W a and W c .-1.2- The right premise (Γ ′ , D ; ∆ ′ ) ⊢ + C is a conclusion of a zero-premise rule.There are six subcases:(a) C is an atom in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of Rf + .(b) C = D . Then the left premise is (Γ; ∆) ⊢ + C and (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C follows by W a and W c .(c) ⊥ is in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊥ L a .(d) ⊥ = D. Then the left premise is (Γ; ∆) ⊢ + ⊥ and is either a conclusion of ⊥ L a or ⊤ L c (in which case cf. 1.1 (b) or 1.1 (c)) or it has been derivedby a left rule. There are eight cases according to the rule used which canbe transformed into derivations with lesser cut-height. I will not showthis here, since this is only a special case of the cases 3.1-3.8 below.(e) ⊤ is in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊤ L c .(f) ⊤ = C. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊤ R + .-1.3- The right premise (Γ ′ , D ; ∆ ′ ) ⊢ − C is a conclusion of a zero-premise rule.There are five subcases:(a) C is an atom in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of Rf − .(b) ⊥ is in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊥ L a .(c) ⊥ = D. Then the left premise is (Γ; ∆) ⊢ + ⊥ and the same as mentionedin 1.2 (d) holds.(d) ⊤ is in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊤ L c .(e) ⊥ = C. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊥ R − .18
2- Cut c -2.1- The left premise (Γ; ∆) ⊢ − D is a conclusion of a zero-premise rule. There arefour subcases:(a) The cut formula D is an atom in ∆. Then the conclusion (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is derived from (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C by W a and W c .(b) ⊥ is in Γ. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of ⊥ L a .(c) ⊤ is in ∆. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C is also a conclusion of ⊤ L c .(d) ⊥ = D. Then the right premise is (Γ ′ ; ∆ ′ , ⊥ ) ⊢ ∗ C and (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C follows by W ⊥ inv as well as W a and W c .-2.2- The right premise (Γ ′ ; ∆ ′ , D ) ⊢ + C is a conclusion of a zero-premise rule.There are five subcases:(a) C is an atom in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of Rf + .(b) ⊥ is in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊥ L a .(c) ⊤ is in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊤ L c .(d) ⊤ = D. Then the left premise is (Γ; ∆) ⊢ − ⊤ and the same as mentionedin 1.2 (d) holds.(e) ⊤ = C. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + C is also a conclusion of ⊤ R + .-2.3- The right premise (Γ ′ ; ∆ ′ , D ) ⊢ − C is a conclusion of a zero-premise rule.There are six subcases:(a) C is an atom in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of Rf − .(b) C = D . Then the left premise is (Γ; ∆) ⊢ − C and (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C follows by W a and W c .(c) ⊥ is in Γ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊥ L a .(d) ⊤ is in ∆ ′ . Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊤ L c .(e) ⊤ = D. Then the left premise is (Γ; ∆) ⊢ − ⊤ and the same as mentionedin 1.2 (d) holds.(f) ⊥ = C. Then (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − C is also a conclusion of ⊥ R − .19 ut with neither premise a conclusion of a zero-premise rule We distinguish the cases that a left rule is used to derive the left premise (cf. 3), aright rule is used to derive the left premise (cf. 5), a right or a left rule is used toderive the right premise with the cut formula not being principal there (cf. 4), andthat a left rule is used to derive the right premise with the cut formula being principal(cf. 5). These cases can be subsumed in a more compact form as categorized below.We assume, like Negri and von Plato [2001], that in the derivations the topsequents,from left to right, have derivation heights n , m , k ,... -3- Cut not principal in the left premise If the cut formula D is not principal in the left premise, this means that this premiseis derived by a left introduction rule. By permuting the order of the rules for thelogical connectives with the cut rules, cut-height can be reduced in each of thefollowing eight cases:-3.1- ∧ L a is the last rule used to derive the left premise with Γ = Γ ′′ , A ∧ B . Thederivations for Cut a and Cut c with cuts of cut-height n + 1 + m are (Γ ′′ , A, B ; ∆) ⊢ + D (Γ ′′ , A ∧ B ; ∆) ⊢ + D ∧ L a (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A ∧ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A, B ; ∆) ⊢ − D (Γ ′′ , A ∧ B ; ∆) ⊢ − D ∧ L a (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A ∧ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ ′′ , A, B ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A, B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A ∧ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C ∧ L a (Γ ′′ , A, B ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A, B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ ′′ , A ∧ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C ∧ L a -3.2- ∧ L c is the last rule used to derive the left premise with ∆ = ∆ ′′ , A ∧ B . Thederivations with cuts of cut-height max ( n, m ) + 1 + k are (Γ; ∆ ′′ , A ) ⊢ + D (Γ; ∆ ′′ , B ) ⊢ + D (Γ; ∆ ′′ , A ∧ B ) ⊢ + D ∧ Lc (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A ∧ B, ∆ ′ ) ⊢ ∗ C Cuta (Γ; ∆ ′′ , A ) ⊢ − D (Γ; ∆ ′′ , B ) ⊢ − D (Γ; ∆ ′′ , A ∧ B ) ⊢ − D ∧ Lc (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A ∧ B, ∆ ′ ) ⊢ ∗ C Cutc
These can be transformed into derivations each with two cuts of cut-height n + k and m + k , respectively: (Γ; ∆ ′′ , A ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆ ′′ , B ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , B, ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ ′′ , A ∧ B, ∆ ′ ) ⊢ ∗ C ∧ L c Γ; ∆ ′′ , A ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, ∆ ′ ) ⊢ ∗ C Cut c (Γ; ∆ ′′ , B ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , B, ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ ′′ , A ∧ B, ∆ ′ ) ⊢ ∗ C ∧ L c -3.3- ∨ L a is the last rule used to derive the left premise with Γ = Γ ′′ , A ∨ B . Thederivations with cuts of cut-height max ( n, m ) + 1 + k are (Γ ′′ , A ; ∆) ⊢ + D (Γ ′′ , B ; ∆) ⊢ + D (Γ ′′ , A ∨ B ; ∆) ⊢ + D ∨ L a (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A ∨ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A ; ∆) ⊢ − D (Γ ′′ , B ; ∆) ⊢ − D (Γ ′′ , A ∨ B ; ∆) ⊢ − D ∨ L a (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A ∨ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations each with two cuts of cut-height n + k and m + k , respectively: (Γ ′′ , A ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , B ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A ∨ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C ∨ L a (Γ ′′ , A ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ ′′ , B ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ ′′ , A ∨ B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C ∨ L a -3.4- ∨ L c is the last rule used to derive the left premise with ∆ = ∆ ′′ , A ∨ B . Thederivations with cuts of cut-height n + 1 + m are (Γ; ∆ ′′ , A, B ) ⊢ + D (Γ; ∆ ′′ , A ∨ B ) ⊢ + D ∨ L c (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A ∨ B, ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆ ′′ , A, B ) ⊢ − D (Γ; ∆ ′′ , A ∨ B ) ⊢ − D ∨ L c (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A ∨ B, ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆ ′′ , A, B ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, B, ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ ′′ , A ∨ B, ∆ ′ ) ⊢ ∗ C ∨ L c (Γ; ∆ ′′ , A, B ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, B, ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ ′′ , A ∨ B, ∆ ′ ) ⊢ ∗ C ∨ L c -3.5- → L a is the last rule used to derive the left premise with Γ = Γ ′′ , A → B . Thederivations with cuts of cut-height max ( n, m ) + 1 + k are (Γ ′′ , A → B ; ∆) ⊢ + A (Γ ′′ , B ; ∆) ⊢ + D (Γ ′′ , A → B ; ∆) ⊢ + D → L a (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A → B ; ∆) ⊢ + A (Γ ′′ , B ; ∆) ⊢ − D (Γ ′′ , A → B ; ∆) ⊢ − D → L a (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c m + k : (Γ ′′ , A → B ; ∆) ⊢ + A (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ + A W a/c (Γ ′′ , B ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C → L a (Γ ′′ , A → B ; ∆) ⊢ + A (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ + A W a/c (Γ ′′ , B ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ ′′ , A → B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C → L a -3.6- → L c is the last rule used to derive the left premise with ∆ = ∆ ′′ , A → B .The derivations with cuts of cut-height n + 1 + m are (Γ , A ; ∆ ′′ , B ) ⊢ + D (Γ; ∆ ′′ , A → B ) ⊢ + D → L c (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A → B, ∆ ′ ) ⊢ ∗ C Cut a (Γ , A ; ∆ ′′ , B ) ⊢ − D (Γ; ∆ ′′ , A → B ) ⊢ − D → L c (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A → B, ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ , A ; ∆ ′′ , B ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , A, Γ ′ ; ∆ ′′ , B, ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ ′′ , A → B, ∆ ′ ) ⊢ ∗ C → L c (Γ , A ; ∆ ′′ , B ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , A, Γ ′ ; ∆ ′′ , B, ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ ′′ , A → B, ∆ ′ ) ⊢ ∗ C → L c -3.7- (cid:6) L a is the last rule used to derive the left premise with Γ = Γ ′′ , A (cid:6) B . Thederivations with cuts of cut-height n + 1 + m are (Γ ′′ , A ; ∆ , B ) ⊢ + D (Γ ′′ , A (cid:6) B ; ∆) ⊢ + D (cid:6) L a (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A (cid:6) B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A ; ∆ , B ) ⊢ − D (Γ ′′ , A (cid:6) B ; ∆) ⊢ − D (cid:6) L a (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A (cid:6) B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ ′′ , A ; ∆ , B ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A, Γ ′ ; ∆ , B, ∆ ′ ) ⊢ ∗ C Cut a (Γ ′′ , A (cid:6) B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C (cid:6) L a (Γ ′′ , A ; ∆ , B ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A, Γ ′ ; ∆ , B, ∆ ′ ) ⊢ ∗ C Cut c (Γ ′′ , A (cid:6) B, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C (cid:6) L a -3.8- (cid:6) L c is the last rule used to derive the left premise with ∆ = ∆ ′′ , A (cid:6) B . Thederivations with cuts of cut-height max ( n, m ) + 1 + k are (Γ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ; ∆ ′′ , A ) ⊢ + D (Γ; ∆ ′′ , A (cid:6) B ) ⊢ + D (cid:6) L c (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ; ∆ ′′ , A ) ⊢ − D (Γ; ∆ ′′ , A (cid:6) B ) ⊢ − D (cid:6) L c (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height m + k :22 Γ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ − B W a/c (Γ; ∆ ′′ , A ) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ ∗ C (cid:6) L c (Γ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ − B W a/c (Γ; ∆ ′′ A ) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ ′′ , A, ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ ′′ , A (cid:6) B, ∆ ′ ) ⊢ ∗ C (cid:6) L c As said above, cut-height is reduced in all cases. -4- Cut formula D principal in the left premise only The cases distinguished here concern the way the right premise is derived. We candistinguish 16 cases and show for each case that the derivation of the right premisecan be transformed into one containing only occurrences of cut with a reduced cut-height.-4.1- ∧ L a is the last rule used to derive the right premise with Γ ′ = Γ ′′ , A ∧ B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′′ , A, B, D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A ∧ B, D ; ∆ ′ ) ⊢ ∗ C ∧ L a (Γ , Γ ′′ , A ∧ B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′′ , A, B ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A ∧ B ; ∆ ′ , D ) ⊢ ∗ C ∧ L a (Γ , Γ ′′ , A ∧ B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′′ , A, B, D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′′ , A, B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′′ , A ∧ B ; ∆ , ∆ ′ ) ⊢ ∗ C ∧ L a (Γ; ∆) ⊢ − D (Γ ′′ , A, B ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′′ , A, B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′′ , A ∧ B ; ∆ , ∆ ′ ) ⊢ ∗ C ∧ L a -4.2- ∧ L c is the last rule used to derive the right premise with ∆ ′ = ∆ ′′ , A ∧ B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A ) ⊢ ∗ C (Γ ′ , D ; ∆ ′′ , B ) ⊢ ∗ C (Γ ′ , D ; ∆ ′′ , A ∧ B ) ⊢ ∗ C ∧ L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∧ B ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A, D ) ⊢ ∗ C (Γ ′ ; ∆ ′′ , B, D ) ⊢ ∗ C (Γ ′ ; ∆ ′′ , A ∧ B, D ) ⊢ ∗ C ∧ L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∧ B ) ⊢ ∗ C Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , B ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∧ B ) ⊢ ∗ C ∧ L c Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A, D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A ) ⊢ ∗ C Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , B, D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , B ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∧ B ) ⊢ ∗ C ∧ L c -4.3- ∨ L a is the last rule used to derive the right premise with Γ ′ = Γ ′′ , A ∨ B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′′ , A, D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , B, D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A ∨ B, D ; ∆ ′ ) ⊢ ∗ C ∨ L a (Γ , Γ ′′ , A ∨ B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′′ , A ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , B ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A ∨ B ; ∆ ′ , D ) ⊢ ∗ C ∨ L a (Γ , Γ ′′ , A ∨ B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′′ , A, D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′′ , A ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ + D (Γ ′′ , B, D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′′ , B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′′ , A ∨ B ; ∆ , ∆ ′ ) ⊢ ∗ C ∨ L a (Γ; ∆) ⊢ − D (Γ ′′ , A ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′′ , A ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ; ∆) ⊢ − D (Γ ′′ , B ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′′ , B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′′ , A ∨ B ; ∆ , ∆ ′ ) ⊢ ∗ C ∨ L a -4.4- ∨ L c is the last rule used to derive the right premise with ∆ ′ = ∆ ′′ , A ∨ B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A, B ) ⊢ ∗ C (Γ ′ , D ; ∆ ′′ , A ∨ B ) ⊢ ∗ C ∨ L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∨ B ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A, B, D ) ⊢ ∗ C (Γ ′ ; ∆ ′′ , A ∨ B, D ) ⊢ ∗ C ∨ L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∨ B ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A, B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A, B ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′′ A ∨ B ) ⊢ ∗ C ∨ L c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A, B, D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A, B ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′′ , A ∨ B ) ⊢ ∗ C ∨ L c -4.5- → L a is the last rule used to derive the right premise with Γ ′ = Γ ′′ , A → B .The derivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′′ , A → B, D ; ∆ ′ ) ⊢ + A (Γ ′′ , B, D ; ∆ ′ ) ⊢ ∗ C (Γ ′′ , A → B, D ; ∆ ′ ) ⊢ ∗ C → L a (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a Γ; ∆) ⊢ − D (Γ ′′ , A → B ; ∆ ′ , D ) ⊢ + A (Γ ′′ , B ; ∆ ′ , D ) ⊢ ∗ C (Γ ′′ , A → B ; ∆ ′ , D ) ⊢ ∗ C → L a (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′′ , A → B, D ; ∆ ′ ) ⊢ + A (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ; ∆) ⊢ + D (Γ ′′ , B, D ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′′ , B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ ∗ C → L a (Γ; ∆) ⊢ − D (Γ ′′ , A → B ; ∆ ′ , D ) ⊢ + A (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ + A Cut c (Γ; ∆) ⊢ − D (Γ ′′ , B ; ∆ ′ , D ) ⊢ ∗ C (Γ , Γ ′′ , B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′′ , A → B ; ∆ , ∆ ′ ) ⊢ ∗ C → L a -4.6- → L c is the last rule used to derive the right premise with ∆ ′ = ∆ ′′ , A → B .The derivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , A, D ; ∆ ′′ , B ) ⊢ ∗ C (Γ ′ , D ; ∆ ′′ , A → B ) ⊢ ∗ C → L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A → B ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′ , A ; ∆ ′′ , B, D ) ⊢ ∗ C (Γ ′ ; ∆ ′′ , A → B, D ) ⊢ ∗ C → L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A → B ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , A, D ; ∆ ′′ , B ) ⊢ ∗ C (Γ , Γ ′ , A ; ∆ , ∆ ′′ , B ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′′ , A → B ) ⊢ ∗ C → L c (Γ; ∆) ⊢ − D (Γ ′ , A ; ∆ ′′ , B, D ) ⊢ ∗ C (Γ , Γ ′ , A ; ∆ , ∆ ′′ , B ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′′ , A → B ) ⊢ ∗ C → L c -4.7- (cid:6) L a is the last rule used to derive the right premise with Γ ′ = Γ ′′ , A (cid:6) B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′′ , A, D ; ∆ ′ , B ) ⊢ ∗ C (Γ ′′ , A (cid:6) B, D ; ∆ ′ ) ⊢ ∗ C (cid:6) L a (Γ , Γ ′′ , A (cid:6) B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ − D (Γ ′′ , A ; ∆ ′ , B, D ) ⊢ ∗ C (Γ ′′ , A (cid:6) B ; ∆ ′ , D ) ⊢ ∗ C (cid:6) L a (Γ , Γ ′′ , A (cid:6) B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′′ , A, D ; ∆ ′ , B ) ⊢ ∗ C (Γ , Γ ′′ , A ; ∆ , ∆ ′ , B ) ⊢ ∗ C Cut a (Γ , Γ ′′ , A (cid:6) B ; ∆ , ∆ ′ ) ⊢ ∗ C (cid:6) L a (Γ; ∆) ⊢ − D (Γ ′′ , A ; ∆ ′ , B, D ) ⊢ ∗ C (Γ , Γ ′′ , A ; ∆ , ∆ ′ , B ) ⊢ ∗ C Cut c (Γ , Γ ′′ , A (cid:6) B ; ∆ , ∆ ′ ) ⊢ ∗ C (cid:6) L a -4.8- (cid:6) L c is the last rule used to derive the right premise with ∆ ′ = ∆ ′′ , A (cid:6) B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ ′ , D ; ∆ ′′ , A ) ⊢ ∗ C (Γ ′ , D ; ∆ ′′ , A (cid:6) B ) ⊢ ∗ C (cid:6) L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ ∗ C Cut a Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A (cid:6) B, D ) ⊢ − B (Γ ′ ; ∆ ′′ , A, D ) ⊢ ∗ C (Γ ′ ; ∆ ′′ , A (cid:6) B, D ) ⊢ ∗ C (cid:6) L c (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ ∗ C Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A (cid:6) B ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ − B Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′′ , A ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ ∗ C (cid:6) L c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A (cid:6) B, D ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ − B Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′′ , A, D ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′′ , A ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′′ , A (cid:6) B ) ⊢ ∗ C (cid:6) L c -4.9- ∧ R + is the last rule used to derive the right premise with C = A ∧ B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ ′ , D ; ∆ ′ ) ⊢ + B (Γ ′ , D ; ∆ ′ ) ⊢ + A ∧ B ∧ R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∧ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ ′ ; ∆ ′ , D ) ⊢ + B (Γ ′ ; ∆ ′ , D ) ⊢ + A ∧ B ∧ R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∧ B Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∧ B ∧ R + (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∧ B ∧ R + -4.10.1- ∧ R − is the last rule used to derive the right premise with C = A ∧ B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − A (Γ ′ , D ; ∆ ′ ) ⊢ − A ∧ B ∧ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − A (Γ ′ ; ∆ ′ , D ) ⊢ − A ∧ B ∧ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B ∧ R − (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B ∧ R − -4.10.2- ∧ R − is the last rule used to derive the right premise with C = A ∧ B . Thederivations with cuts of cut-height n + m + 1 are26 Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ ′ , D ; ∆ ′ ) ⊢ − A ∧ B ∧ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ ′ ; ∆ ′ , D ) ⊢ − A ∧ B ∧ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B ∧ R − (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∧ B ∧ R − -4.11.1- ∨ R +1 is the last rule used to derive the right premise with C = A ∨ B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ ′ , D ; ∆ ′ ) ⊢ + A ∨ B ∨ R +1 (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ ′ ; ∆ ′ , D ) ⊢ + A ∨ B ∨ R +1 (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B ∨ R +1 (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B ∨ R +1 -4.11.2- ∨ R +2 is the last rule used to derive the right premise with C = A ∨ B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + B (Γ ′ , D ; ∆ ′ ) ⊢ + A ∨ B ∨ R +2 (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + B (Γ ′ ; ∆ ′ , D ) ⊢ + A ∨ B ∨ R +2 (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B ∨ R +2 (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A ∨ B ∨ R +2 -4.12- ∨ R − is the last rule used to derive the right premise with C = A ∨ B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − A (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ ′ , D ; ∆ ′ ) ⊢ − A ∨ B ∨ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∨ B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − A (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ ′ ; ∆ ′ , D ) ⊢ − A ∨ B ∨ R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∨ B Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∨ B ∨ R − Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A ∨ B ∨ R − -4.13- → R + is the last rule used to derive the right premise with C = A → B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , A, D ; ∆ ′ ) ⊢ + B (Γ ′ , D ; ∆ ′ ) ⊢ + A → B → R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A → B Cut a (Γ; ∆) ⊢ − D (Γ ′ , A ; ∆ ′ , D ) ⊢ + B (Γ ′ ; ∆ ′ , D ) ⊢ + A → B → R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A → B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , A, D ; ∆ ′ ) ⊢ + B (Γ , Γ ′ , A ; ∆ , ∆ ′ ) ⊢ + B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A → B → R + (Γ; ∆) ⊢ − D (Γ ′ , A ; ∆ ′ , D ) ⊢ + B (Γ , Γ ′ , A ; ∆ , ∆ ′ ) ⊢ + B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A → B → R + -4.14- → R − is the last rule used to derive the right premise with C = A → B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ ′ , D ; ∆ ′ ) ⊢ − A → B → R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A → B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ ′ ; ∆ ′ , D ) ⊢ − A → B → R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A → B Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A → B → R − (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A → B → R − -4.15- (cid:6) R + is the last rule used to derive the right premise with C = A (cid:6) B . Thederivations with cuts of cut-height n + max ( m, k ) + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ ′ , D ; ∆ ′ ) ⊢ + A (cid:6) B (cid:6) R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A (cid:6) B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ ′ ; ∆ ′ , D ) ⊢ + A (cid:6) B (cid:6) R + (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A (cid:6) B Cut c These can be transformed into derivations each with two cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A (cid:6) B (cid:6) R + Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut c (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , D ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A (cid:6) B (cid:6) R + -4.16- (cid:6) R − is the last rule used to derive the right premise with C = A (cid:6) B . Thederivations with cuts of cut-height n + m + 1 are (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ , B ) ⊢ − A (Γ ′ , D ; ∆ ′ ) ⊢ − A (cid:6) B (cid:6) R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A (cid:6) B Cut a (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , B, D ) ⊢ − A (Γ ′ ; ∆ ′ , D ) ⊢ − A (cid:6) B (cid:6) R − (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A (cid:6) B Cut c These can be transformed into derivations with cuts of cut-height n + m : (Γ; ∆) ⊢ + D (Γ ′ , D ; ∆ ′ , B ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ , B ) ⊢ − A Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A (cid:6) B (cid:6) R − (Γ; ∆) ⊢ − D (Γ ′ ; ∆ ′ , B, D ) ⊢ − A (Γ , Γ ′ ; ∆ , ∆ ′ , B ) ⊢ − A Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − A (cid:6) B (cid:6) R − It is shown that cut-height is reduced in all cases. -5- Cut formula D principal in both premises For each cut rule four cases can be distinguished. Here, it can be shown for eachcase that the derivations can be transformed into ones in which the occurrences ofcut have a reduced cut-height or the cut formula has a lower weight (or both).-5.1- D = A ∧ B . The derivation for Cut a with a cut of cut-height max ( n, m ) + 1 + k + 1 is (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A ∧ B ∧ R + (Γ ′ , A, B ; ∆ ′ ) ⊢ ∗ C (Γ ′ , A ∧ B ; ∆ ′ ) ⊢ ∗ C ∧ L a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a and can be transformed into a derivation with two cuts of cut-height (fromtop to bottom) n + k and m + max ( n, k ) + 1: (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A (Γ ′ , A, B ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ , B ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ , Γ ′ ; ∆ , ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c Note that in both cases the weight of the cut formula is reduced. The uppercut is also reduced in height, while with the lower cut we have a case wherecut-height is not necessarily reduced.The possible derivations for
Cut c with a cut of cut-height n +1+ max ( m, k )+1are 29 Γ; ∆) ⊢ − A (Γ; ∆) ⊢ − A ∧ B ∧ R − (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C (Γ ′ ; ∆ ′ , B ) ⊢ ∗ C (Γ ′ ; ∆ ′ , A ∧ B ) ⊢ ∗ C ∧ L c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c or (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A ∧ B ∧ R − (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C (Γ ′ ; ∆ ′ , B ) ⊢ ∗ C (Γ ′ ; ∆ ′ , A ∧ B ) ⊢ ∗ C ∧ L c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c and those can be transformed into derivations with cuts of cut-height n + m or n + k , respectively: (Γ; ∆) ⊢ − A (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ; ∆) ⊢ − B (Γ ′ ; ∆ ′ , B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c Here, both cut-height and weight of the cut formulas are reduced.-5.2- D = A ∨ B . The possible derivations for Cut a with a cut of cut-height n +1 + max ( m, k ) + 1 are (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ + A ∨ B ∨ R +1 (Γ ′ , A ; ∆ ′ ) ⊢ ∗ C (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C (Γ ′ , A ∨ B ; ∆ ′ ) ⊢ ∗ C ∨ L a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a or (Γ; ∆) ⊢ + B (Γ; ∆) ⊢ + A ∨ B ∨ R +2 (Γ ′ , A ; ∆ ′ ) ⊢ ∗ C (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C (Γ ′ , A ∨ B ; ∆ ′ ) ⊢ ∗ C ∨ L a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a and those can be transformed into derivations with cuts of cut-height n + m and n + k , respectively: (Γ; ∆) ⊢ + A (Γ ′ , A ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ; ∆) ⊢ + B (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a Again, both cut-height and weight of the cut formulas are reduced.The derivation for
Cut c with a cut of cut-height max ( n, m ) + 1 + k + 1 is (Γ; ∆) ⊢ − A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A ∨ B ∨ R − (Γ ′ ; ∆ ′ , A, B ) ⊢ ∗ C (Γ ′ ; ∆ ′ , A ∨ B ) ⊢ ∗ C ∨ L c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c and can be transformed into a derivation with two cuts of cut-height n + k and m + max ( n, k ) + 1: (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A (Γ ′ ; ∆ ′ , A, B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ , B ) ⊢ ∗ C Cut c (Γ , Γ , Γ ′ ; ∆ , ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c D = A → B . The derivation for Cut a with a cut of cut-height n + 1 + max ( m, k ) + 1 is (Γ , A ; ∆) ⊢ + B (Γ; ∆) ⊢ + A → B → R + (Γ ′ , A → B ; ∆ ′ ) ⊢ + A (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C (Γ ′ , A → B ; ∆ ′ ) ⊢ ∗ C → L a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a and this can be transformed into a derivation with three cuts of cut-height(from left to right and from top to bottom) n + 1 + m , n + k , and max ( n +1 , m ) + 1 + max ( n, k ) + 1 respectively: (Γ , A ; ∆) ⊢ + B (Γ; ∆) ⊢ + A → B → R + (Γ ′ , A → B ; ∆ ′ ) ⊢ + A (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ + A Cut a (Γ , A ; ∆) ⊢ + B (Γ ′ , B ; ∆ ′ ) ⊢ ∗ C (Γ , A, Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ , Γ ′ , Γ ′ ; ∆ , ∆ , ∆ ′ , ∆ ′ ) ⊢ ∗ C Cut a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c In the first case cut-height is reduced, in the second case cut-height and weightof the cut formula is reduced and in the third case weight of the cut formulais reduced.The derivation for
Cut c with a cut of cut-height max ( n, m ) + 1 + k + 1 is (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ − A → B → R − (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C (Γ ′ ; ∆ ′ , A → B ) ⊢ ∗ C → L c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c This can be transformed into a derivation with two cuts of cut-height n + k and m + max ( n, k ) + 1: (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ + A (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ , B ) ⊢ ∗ C Cut a (Γ , Γ , Γ ′ ; ∆ , ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c In the first case cut-height and weight of the cut formula is reduced, while inthe second case the weight of the cut formula is reduced. Here we can observea result specific for this calculus due to the mixture of derivability relations ⊢ + and ⊢ − in → R − and the position of the active formulas in the assumptions and in the counterassumptions in → L c : Derivations containing instances of31 ut c are not necessarily transformed into derivations with a lesser cut-heightor a reduced weight of the cut formula of another instance of Cut c but it canalso happen that Cut c is replaced by Cut a .-5.4- D = A (cid:6) B . The derivation for Cut a with a cut of cut-height max ( n, m ) + 1 + k + 1 is (Γ; ∆) ⊢ + A (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ + A (cid:6) B (cid:6) R + (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C (Γ ′ , A (cid:6) B ; ∆ ′ ) ⊢ ∗ C (cid:6) L a (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut a This can be transformed into a derivation with two cuts of cut-height n + k and m + max ( n, k ) + 1: (Γ; ∆) ⊢ − B (Γ; ∆) ⊢ + A (Γ ′ , A ; ∆ ′ , B ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , ∆ ′ , B ) ⊢ ∗ C Cut a (Γ , Γ , Γ ′ ; ∆ , ∆ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c Again, due to the mixture of derivability relations ⊢ + and ⊢ − in (cid:6) R + and thepresence of the active formulas both in assumptions and counterassumptionsin (cid:6) L a , in this case Cut a can be replaced by instances of Cut c with a reducedweight of the cut formula. In the upper cut we have a reduction of bothcut-height and weight of the cut formula.The derivation for Cut c with a cut of cut-height n + 1 + max ( m, k ) + 1 is (Γ; ∆ , B ) ⊢ − A (Γ; ∆) ⊢ − A (cid:6) B (cid:6) R − (Γ ′ ; ∆ ′ , A (cid:6) B ) ⊢ − B (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C (Γ ′ ; ∆ ′ , A (cid:6) B ) ⊢ ∗ C (cid:6) L c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C Cut c and this can be transformed into a derivation with three cuts of cut-height(from left to right and from top to bottom) n + 1 + m , n + k , and max ( n +1 , m ) + 1 + max ( n, k ) + 1 respectively: (Γ; ∆ , B ) ⊢ − A (Γ; ∆) ⊢ − A (cid:6) B (cid:6) R − (Γ ′ ; ∆ ′ , A (cid:6) B ) ⊢ − B (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ − B Cut c (Γ; ∆ , B ) ⊢ − A (Γ ′ ; ∆ ′ , A ) ⊢ ∗ C (Γ , Γ ′ ; ∆ , B, ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ , Γ ′ , Γ ′ ; ∆ , ∆ , ∆ ′ , ∆ ′ ) ⊢ ∗ C Cut c (Γ , Γ ′ ; ∆ , ∆ ′ ) ⊢ ∗ C C a/c In the first case cut-height is reduced, in the second case cut-height and weightof the cut formula and in the third case weight of the cut formula.32
Conclusion
By applying the proof methods that Negri and von Plato [2001] use for their calculus
G3ip , we were able to show that
SC2Int is a cut-free sequent calculus for the bi-intuitionistic logic . A proof can be given for the admissibility of the structuralrules of weakening, contraction and cut in the system.
References
Rajeev Gor´e. Dual intuitionistic logic revisited. In R. Dyckhoff, editor,
AutomatedReasoning with Analytic Tableaux and Related Methods. TABLEAUX 2000 , pages252–267. Springer-Verlag, Berlin, 2000.Tomasz Kowalski and Hiroakira Ono. Analytic cut and interpolation for bi-intuitionistic logic.
The Review of Symbolic Logic , 10(2):259–283, 2017.Sara Negri and Jan von Plato.
Structural Proof Theory . Cambridge University Press,Cambridge/New York, 2001.Linda Postniece.
Proof Theory and Proof Search of Bi-Intuitionistic and TenseLogic . PhD thesis, The Australian National University, Canberra, 2010.Cecylia Rauszer. A formalization of the propositional calculus of h-b logic.
StudiaLogica , 33(1):23–34, 1974.Heinrich Wansing. Falsification, natural deduction and bi-intuitionistic logic.
Jour-nal of Logic and Computation , 26(1):425–450, 2016a.Heinrich Wansing. On split negation, strong negation, information, falsification, andverification. In Katalin Bimb´o, editor,
J. Michael Dunn on Information BasedLogics. Outstanding Contributions to Logic , volume 8, pages 161–189. Springer,2016b.Heinrich Wansing. A more general general proof theory.