A Cointuitionistic Adjoint Logic
AA COINTUITIONISTIC ADJOINT LOGIC
HARLEY EADES III AND GIANLUIGI BELLIN e-mail address : [email protected] Science, Augusta University, Augusta, GA e-mail address : [email protected] di Informatica, Universit`a di Verona, Strada Le Grazie, 37134 Verona, ItalyA bstract . Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to includethe duals of each logical connective. One leading question with respect to BINT is, what does BINTlook like across the three arcs – logic, typed λ -calculi, and category theory – of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first beingintuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being thedual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesiancoclosed category. Crolard [11] showed that combining these two categories into the same categoryresults in it degenerating to a poset. However, this degeneration does not occur when both logicsare linear. We propose that IL and coIL need to be separated, and then mixed in a controlled wayusing the modalities from linear logic. This separation can be ultimately achieved by an adjointformalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: thefirst is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionisticlogic. They are then related via two adjunctions. The adjunction between IL and ILL is known asa Linear / Non-linear model (LNL model) of ILL, and is due to Benton [4]. However, the dual toLNL models which would amount to the adjunction between coILL and coIL has yet to appear inthe literature. In this paper we fill this gap by studying the dual to LNL models which we call dualLNL models. We conduct a similar analysis to that of Benton for dual LNL models by showing thatdual LNL models correspond to dual linear categories, the dual to Bierman’s [5] linear categoriesproposed by Bellin [3]. Following this we give the definition of bi-LNL models by combining ourdual LNL models with Benton’s LNL models to obtain a categorical model of bi-intuitionistic logic,but we leave its analysis and corresponding logic to a future paper. Finally, we give a correspondingsequent calculus, natural deduction, and term assignment for dual LNL models.
1. I ntroduction
Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to include the dualsof each logical connective. That is, BINT contains the usual intuitionistic logical connectives such astrue, conjunction, and implication, but also their duals false, disjunction, and coimplication. Oneleading question with respect to BINT is, what does BINT look like across the three arcs – logic,typed λ -calculi, and category theory – of the Curry-Howard-Lambek correspondence? A non-trivial(does not degenerate to a poset) categorical model of BINT is currently an open problem. This paperdirectly contributes to the solution of this open problem by giving a new categorical model based onadjunctions for cointuitionistic logic, and then proposing a new categorical model for BINT. LOGICAL METHODSIN COMPUTER SCIENCE DOI:10.2168/LMCS-??? c (cid:13)
Harley Eades III and Gianluigi BellinCreative Commons a r X i v : . [ c s . L O ] A ug HARLEY EADES III AND GIANLUIGI BELLIN
BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), whichis modeled categorically by a cartesian closed category (CCC), and the second being the dual tointuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosedcategory (coCCC). Crolard [11] showed that combining these two categories into the same categoryresults in it degenerating to a poset, i.e. there is at most one morphism between any two objects; wereview this result in Section 2.2. However, this degeneration does not occur when both logics arelinear.Notice that atoms are not dualized, at least in the main stream tradition of BINT started by C.Rauszer [24, 25]. For this reason T. Crolard [11] p. 160, describes the relation between IL and coILwithin BINT as “pseudo duality”. A duality on atoms could be added and this has been attemptedwith linguistic motivations [2] (see the section on Related Work). This avoids the collapse but yieldsa di ff erent framework. Here we are concerned mainly with the main stream tradition.We propose that IL and coIL need to be separated, and then mixed in a controlled way using themodalities from linear logic. This separation can be ultimately achieved by an adjoint formalizationof bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first isintuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionisticlogic. They are then related via two adjunctions as depicted by the following diagram: (cid:97) IL (cid:97) coILILL coILLBi-ILLThe adjunction between IL and ILL is known as a Linear / Non-linear model (LNL model) of ILL,and is due to Benton [4]. However, the dual to LNL models which would amount to the adjunctionbetween coILL and coIL has yet to appear in the literature.Suppose ( I , , × , → ) is a cartesian closed category, and ( L , (cid:62) , ⊗ , −◦ ) is a symmetric monoidalclosed category. Then relate these two categories with a symmetric monoidal adjunction I : F (cid:97) G : L (Definition 11), where F and G are symmetric monoidal functors. The later point implies that thereare natural transformations m X , Y : F X ⊗ F Y (cid:47) (cid:47) F ( X × Y ) and n A , B : G A × G B (cid:47) (cid:47) G ( A ⊗ B ), andmaps m (cid:62) : (cid:62) (cid:47) (cid:47) F n : 1 (cid:47) (cid:47) G (cid:62) subject to several coherence conditions; see Definition 7.Furthermore, the functor F is strong which means that m X , Y and m (cid:62) are isomorphisms. This setupturns out to be one of the most beautiful models of intuitionistic linear logic called a LNL model dueto Benton [4]. In fact, the linear modality of-course can be defined by ! A = F ( G ( A )) which definesa symmetric monoidal comonad using the adjunction; see Section 2.2 of [4]. This model is muchsimpler than other known models, and resulted in a logic called LNL logic which supports mixingintuitionistic logic with linear logic. The main contribution of this paper is the definition and studyof the dual to Benton’s LNL models as models of cointuitionistic logic.Taking the dual of the previous model results in what we call dual LNL models. They consist ofa cocartesian coclosed category, ( C , , + , − ) where − : C × C (cid:47) (cid:47) C is left adjoint to the coproduct, asymmetric monoidal coclosed category (Definition 4), ( L (cid:48) , ⊥ , ⊕ , (cid:21) ), where (cid:21) : L (cid:48) × L (cid:48) (cid:47) (cid:47) L (cid:48) isleft adjoint to cotensor (usually called par ), and a symmetric comonoidal adjunction (Definition 12) L (cid:48) : H (cid:97) J : C , where H and J are symmetric comonoidal functors. Dual to the above, this implies COINTUITIONISTIC ADJOINT LOGIC 3 that there are natural transformations m X , Y : J ( X + Y ) (cid:47) (cid:47) J X ⊕ J Y and n A , B : H ( A ⊕ B ) (cid:47) (cid:47) H A + H B ,and maps m : J (cid:47) (cid:47) ⊥ and n ⊥ : H ⊥ (cid:47) (cid:47) A = JH A , and hence, is the monad inducedby the adjunction.Bellin [3] was the first to propose the dual to Bierman’s [5] linear categories which he namesdual linear categories as a model of cointuitionistic linear logic. We conduct a similar analysis tothat of Benton for dual LNL models by showing that dual LNL models are dual linear categories(Section 2.3.2), and that from a dual linear category we may obtain a dual LNL model (Section 2.3.3).Following this we give the definition of bi-LNL models by combining our dual LNL models withBenton’s LNL models to obtain a categorical model of bi-intuitionistic logic (Section 2.4), but weleave its analysis and corresponding logic to a future paper.Benton [4] showed that, syntactically, LNL models have a corresponding logic by first definingintuitionistic logic, whose sequent is denoted, Θ (cid:96) C X , and then intuitionistic linear logic, Θ ; Γ (cid:96) L A ,but the key insight was that Θ contains non-linear assumptions while Γ contains linear assumptions,but one should view their separation as merely cosmetic; all assumptions can consistently be mixedwithin a single context. The two logics are then connected by syntactic versions of the functors F and G which allow formulas to move between both fragments.Following Benton’s lead the design of dual LNL logic is similar. We have a non-linear coin-tuitionistic fragment, T (cid:96) C Ψ , and a linear cointuitionistic fragment, A (cid:96) C ∆ ; Ψ , where ∆ containslinear conclusions and Ψ contains non-linear conclusions, but again the separation of contexts is onlycosmetic. The non-linear fragment has the following structural rules: S (cid:96) C Ψ S (cid:96) C T , Ψ C weak S (cid:96) C T , T , Ψ S (cid:96) C T , Ψ C contrThen we connect these two fragments together using the following rules for the functors H and J : A (cid:96) L · ; Ψ H A (cid:96) C Ψ H L A (cid:96) L ∆ , B ; Ψ A (cid:96) L ∆ ; H B , Ψ H R T (cid:96) C Ψ J T (cid:96) L · ; Ψ J L A (cid:96) L ∆ ; T , Ψ A (cid:96) L ∆ , J T ; Ψ J R These allow for linear and non-linear formulas to move from one fragment to the other. We will givea sequent calculus and natural deduction formalization (Section 3.1 and Section 3.2) as well as a termassignment (Section 3.3). The latter is particularly interesting, because of the fact that cointuitionisticlogic has multiple conclusions, but only a single hypothesis.2. T he A djoint M odel In this section we define dual LNL models (Definition 21) and then relate them to Bellin’s dual linearcategories (Definition 22), but first we introduce the basic categorical machinery needed for the latersections and summarize Crolard’s result showing that the combination of cartesian closed categorieswith cocartesian coclosed categories is degenerate. Following these we conclude this section byintroducing a categorical model for full BINT called a mixed bilinear / non-linear model that combinesLNL models with dual LNL models (Definition 2.4).2.1. Symmetric (co)Monoidal Categories.
We now introduce the necessary definitions relatedto symmetric monoidal categories that our model will depend on. Most of these definitions areequivalent to the ones given by Benton [4], but we give a lesser known definition of symmetriccomonoidal functors due to Bellin [3]. In this section we also introduce distributive categories, the
HARLEY EADES III AND GIANLUIGI BELLIN notion of coclosure, and finally, the definition of bilinear categories. The reader may wish to simplyskim this section, but refer back to it when they encounter a definition or result they do not know.
Definition 1. A symmetric monoidal category (SMC) is a category, M , with the following data: • An object (cid:62) of M , • A bi-functor ⊗ : M × M (cid:47) (cid:47) M , • The following natural isomorphisms: λ A : (cid:62) ⊗ A (cid:47) (cid:47) A ρ A : A ⊗ (cid:62) (cid:47) (cid:47) A α A , B , C : ( A ⊗ B ) ⊗ C (cid:47) (cid:47) A ⊗ ( B ⊗ C ) • A symmetry natural transformation: β A , B : A ⊗ B (cid:47) (cid:47) B ⊗ A • Subject to the following coherence diagrams: A ⊗ ( B ⊗ ( C ⊗ D )) A ⊗ (( B ⊗ C ) ⊗ D ) (cid:111) (cid:111) id A ⊗ α B , C , D ( A ⊗ B ) ⊗ ( C ⊗ D ) A ⊗ ( B ⊗ ( C ⊗ D )) α A , B , C ⊗ D (cid:15) (cid:15) ( A ⊗ B ) ⊗ ( C ⊗ D ) A ⊗ (( B ⊗ C ) ⊗ D )( A ⊗ B ) ⊗ ( C ⊗ D )(( A ⊗ B ) ⊗ C ) ⊗ D ( A ⊗ B ) ⊗ ( C ⊗ D ) α A ⊗ B , C , D (cid:15) (cid:15) (( A ⊗ B ) ⊗ C ) ⊗ D ( A ⊗ ( B ⊗ C )) ⊗ D α A , B , C ⊗ id D (cid:47) (cid:47) ( A ⊗ ( B ⊗ C )) ⊗ D ( A ⊗ ( B ⊗ C )) ⊗ DA ⊗ (( B ⊗ C ) ⊗ D ) α A , B ⊗ C , D (cid:15) (cid:15) ( B ⊗ A ) ⊗ C B ⊗ ( A ⊗ C ) α B , A , C (cid:47) (cid:47) ( A ⊗ B ) ⊗ C ( B ⊗ A ) ⊗ C β A , B ⊗ id C (cid:15) (cid:15) ( A ⊗ B ) ⊗ C A ⊗ ( B ⊗ C ) α A , B , C (cid:47) (cid:47) A ⊗ ( B ⊗ C ) B ⊗ ( A ⊗ C ) B ⊗ ( A ⊗ C ) B ⊗ ( C ⊗ A ) id B ⊗ β A , C (cid:47) (cid:47) A ⊗ ( B ⊗ C ) B ⊗ ( A ⊗ C ) A ⊗ ( B ⊗ C ) ( B ⊗ C ) ⊗ A β A , B ⊗ C (cid:47) (cid:47) ( B ⊗ C ) ⊗ AB ⊗ ( C ⊗ A ) α B , C , A (cid:15) (cid:15) ( A ⊗ (cid:62) ) ⊗ B A ⊗ B ρ A (cid:31) (cid:31) ( A ⊗ (cid:62) ) ⊗ B A ⊗ ( (cid:62) ⊗ B ) α A , (cid:62) , B (cid:47) (cid:47) A ⊗ ( (cid:62) ⊗ B ) A ⊗ B λ B (cid:127) (cid:127) B ⊗ A A ⊗ B β B , A (cid:47) (cid:47) A ⊗ BB ⊗ A β A , B (cid:15) (cid:15) A ⊗ B A ⊗ B id A ⊗ B (cid:31) (cid:31) (cid:62) ⊗ A A λ A (cid:31) (cid:31) (cid:62) ⊗ A A ⊗ (cid:62) β (cid:62) , A (cid:47) (cid:47) A ⊗ (cid:62) A ρ A (cid:127) (cid:127) Categorical modeling implication requires that the model be closed; which can be seen as aninternalization of the notion of a morphism.
COINTUITIONISTIC ADJOINT LOGIC 5
Definition 2. A symmetric monoidal closed category (SMCC) is a symmetric monoidal category,( M , (cid:62) , ⊗ ), such that, for any object B of M , the functor − ⊗ B : M (cid:47) (cid:47) M has a specified rightadjoint. Hence, for any objects A and C of M there is an object B −◦ C of M and a natural bijection: Hom M ( A ⊗ B , C ) (cid:27) Hom M ( A , B −◦ C )We call the functor −◦ : M × M (cid:47) (cid:47) M the internal hom of M .Symmetric monoidal closed categories can be seen as a model of intuitionistic linear logic witha tensor product and implication [5]. What happens when we take the dual? First, we have thefollowing result: Lemma 3 (Dual of Symmetric Monoidal Categories) . If ( M , (cid:62) , ⊗ ) is a symmetric monoidal category,then M op is also a symmetric monoidal category.The previous result follows from the fact that the structures making up symmetric monoidalcategories are isomorphisms, and so naturally taking their opposite will yield another symmetricmonoidal category. To emphasize when we are thinking about a symmetric monoidal category inthe opposite we use the notation ( M , ⊥ , ⊕ ) which gives the suggestion of ⊕ corresponding to adisjunctive tensor product which we call the cotensor of M . The next definition describes when asymmetric monoidal category is coclosed. Definition 4. A symmetric monoidal coclosed category (SMCCC) is a symmetric monoidalcategory, ( M , ⊥ , ⊕ ), such that, for any object B of M , the functor − ⊕ B : M (cid:47) (cid:47) M has a specifiedleft adjoint. Hence, for any objects A and C of M there is an object C (cid:21) B of M and a naturalbijection: Hom M ( C , A ⊕ B ) (cid:27) Hom M ( C (cid:21) B , A )We call the functor (cid:21) : M × M (cid:47) (cid:47) M the internal cohom of M .We combine a symmetric monoidal closed category with a symmetric monoidal coclosedcategory in a single category. First, we define the notion of a distributive category due to Cockett andSeely [10]. Definition 5.
We call a symmetric monoidal category, ( M , (cid:62) , ⊗ , ⊥ , ⊕ ) equipped with the structureof a cotensor ( M , ⊥ , ⊕ ), a distributive category if there are natural transformations: δ LA , B , C : A ⊗ ( B ⊕ C ) (cid:47) (cid:47) ( A ⊗ B ) ⊕ C δ RA , B , C : ( B ⊕ C ) ⊗ A (cid:47) (cid:47) B ⊕ ( C ⊗ A )subject to several coherence diagrams. Due to the large number of coherence diagrams we do not listthem here, but they all can be found in Cockett and Seely’s paper [10].Requiring that the tensor and cotensor products have the corresponding right and left adjoints resultsin the following definition. Definition 6. A bilinear category is a distributive category ( M , (cid:62) , ⊗ , ⊥ , ⊕ ) such that ( M , (cid:62) , ⊗ ) isclosed, and ( M , ⊥ , ⊕ ) is coclosed. We will denote bi-linear categories by ( M , (cid:62) , ⊗ , −◦ , ⊥ , ⊕ , (cid:21) ).Originally, Lambek defined bilinear categories to be similar to the previous definition, but thetensor and cotensor were non-commutative [9], however, the bilinear categories given here are. Weretain the name in homage to his original work. As we will see below bilinear categories form thecore of a categorical model for bi-intuitionism.A symmetric monoidal category is a category with additional structure subject to severalcoherence diagrams. Thus, an ordinary functor is not enough to capture this structure, and hence, theintroduction of symmetric monoidal functors. HARLEY EADES III AND GIANLUIGI BELLIN
Definition 7.
Suppose we are given two symmetric monoidal categories( M , (cid:62) , ⊗ , α , λ , ρ , β ) and ( M , (cid:62) , ⊗ , α , λ , ρ , β ). Then a symmetric monoidal functor is a functor F : M (cid:47) (cid:47) M , a map m (cid:62) : (cid:62) (cid:47) (cid:47) F (cid:62) and a natural transformation m A , B : FA ⊗ F B (cid:47) (cid:47) F ( A ⊗ B ) subject to the following coherence conditions: F (( A ⊗ B ) ⊗ C ) F ( A ⊗ ( B ⊗ C )) F α A , B , C (cid:47) (cid:47) F ( A ⊗ B ) ⊗ FCF (( A ⊗ B ) ⊗ C ) m A ⊗ B , C (cid:15) (cid:15) F ( A ⊗ B ) ⊗ FC FA ⊗ F ( B ⊗ C ) FA ⊗ F ( B ⊗ C ) F ( A ⊗ ( B ⊗ C )) m A , B ⊗ C (cid:15) (cid:15) F ( A ⊗ B ) ⊗ FC FA ⊗ F ( B ⊗ C )( FA ⊗ F B ) ⊗ FCF ( A ⊗ B ) ⊗ FC m A , B ⊗ id FC (cid:15) (cid:15) ( FA ⊗ F B ) ⊗ FC FA ⊗ ( F B ⊗ FC ) α FA , FB , FC (cid:47) (cid:47) FA ⊗ ( F B ⊗ FC ) FA ⊗ F ( B ⊗ C ) id FA ⊗ m B , C (cid:15) (cid:15) F (cid:62) ⊗ FA F ( (cid:62) ⊗ A ) m (cid:62) , A (cid:47) (cid:47) (cid:62) ⊗ FAF (cid:62) ⊗ FA m (cid:62) ⊗ id FA (cid:15) (cid:15) (cid:62) ⊗ FA FA λ FA (cid:47) (cid:47) FAF ( (cid:62) ⊗ A ) (cid:79) (cid:79) F λ A FA ⊗ F (cid:62) F ( A ⊗ (cid:62) ) m A , (cid:62) (cid:47) (cid:47) FA ⊗ (cid:62) FA ⊗ F (cid:62) id FA ⊗ m (cid:62) (cid:15) (cid:15) FA ⊗ (cid:62) FA ρ FA (cid:47) (cid:47) FAF ( A ⊗ (cid:62) ) (cid:79) (cid:79) F ρ A F ( A ⊗ B ) F ( B ⊗ A ) F β A , B (cid:47) (cid:47) FA ⊗ F BF ( A ⊗ B ) m A , B (cid:15) (cid:15) FA ⊗ F B F B ⊗ FA β FA , FB (cid:47) (cid:47) F B ⊗ FAF ( B ⊗ A ) m B , A (cid:15) (cid:15) The following is dual to the previous definition.
Definition 8.
Suppose we are given two symmetric monoidal categories( M , ⊥ , ⊕ , α , λ , ρ , β ) and ( M , ⊥ , ⊕ , α , λ , ρ , β ). Then a symmetric comonoidal functor is a functor F : M (cid:47) (cid:47) M , a map m ⊥ : F ⊥ (cid:47) (cid:47) ⊥ and a natural transformation m A , B : F ( A ⊕ B ) (cid:47) (cid:47) FA ⊕ F B subject to the following coherence conditions: FA ⊕ F ( B ⊕ C )) FA ⊕ ( F B ⊕ FC ) id FA ⊕ m B , C (cid:47) (cid:47) F ( A ⊕ ( B ⊕ C )) FA ⊕ F ( B ⊕ C )) m A , B ⊕ C (cid:15) (cid:15) F ( A ⊕ ( B ⊕ C )) ( FA ⊕ F B ) ⊕ FC ( FA ⊕ F B ) ⊕ FCFA ⊕ ( F B ⊕ FC ) α FA , FB , FC (cid:15) (cid:15) F ( A ⊕ ( B ⊕ C )) ( FA ⊕ F B ) ⊕ FCF (( A ⊕ B ) ⊕ C ) F ( A ⊕ ( B ⊕ C )) F α A , B , C (cid:15) (cid:15) F (( A ⊕ B ) ⊕ C ) F ( A ⊕ B ) ⊕ FC m A ⊕ B , C (cid:47) (cid:47) F ( A ⊕ B ) ⊕ FC ( FA ⊕ F B ) ⊕ FC m A , B ⊕ id FC (cid:15) (cid:15) COINTUITIONISTIC ADJOINT LOGIC 7 FA ⊥ ⊕ FA λ − FA (cid:47) (cid:47) F ( ⊥ ⊕ A ) FA F λ A (cid:15) (cid:15) F ( ⊥ ⊕ A ) F ⊥ ⊕ FA m ⊥ , A (cid:47) (cid:47) F ⊥ ⊕ FA ⊥ ⊕ FA m ⊥ ⊕ id FA (cid:15) (cid:15) FA FA ⊕ ⊥ ρ − FA (cid:47) (cid:47) F ( A ⊕ ⊥ ) FA F ρ A (cid:15) (cid:15) F ( A ⊕ ⊥ ) FA ⊕ F ⊥ m A , ⊥ (cid:47) (cid:47) FA ⊕ F ⊥ FA ⊕ ⊥ id FA ⊕ m ⊥ (cid:15) (cid:15) F ( B ⊕ A ) F B ⊕ FA m B , A (cid:47) (cid:47) F ( A ⊕ B ) F ( B ⊕ A ) F β A , B (cid:15) (cid:15) F ( A ⊕ B ) FA ⊕ F B m A , B (cid:47) (cid:47) FA ⊕ F BF B ⊕ FA β FA , FB (cid:15) (cid:15) Naturally, since functors are enhanced to handle the additional structure found in a symmetricmonoidal category we must also extend natural transformations, and adjunctions.
Definition 9.
Suppose ( M , (cid:62) , ⊗ ) and ( M , (cid:62) , ⊗ ) are SMCs, and ( F , m ) and ( G , n ) are a symmet-ric monoidal functors between M and M . Then a symmetric monoidal natural transformation is a natural transformation, f : F (cid:47) (cid:47) G , subject to the following coherence diagrams: GA ⊗ GB G ( A ⊗ B ) n A , B (cid:47) (cid:47) FA ⊗ F BGA ⊗ GB f A ⊗ f B (cid:15) (cid:15) FA ⊗ F B F ( A ⊗ B ) m A , B (cid:47) (cid:47) F ( A ⊗ B ) G ( A ⊗ B ) f A ⊗ B (cid:15) (cid:15) F (cid:62) (cid:62) (cid:95) (cid:95) m (cid:62) F (cid:62) G (cid:62) f (cid:62) (cid:47) (cid:47) G (cid:62) (cid:62) (cid:63) (cid:63) n (cid:62) Definition 10.
Suppose ( M , ⊥ , ⊕ ) and ( M , ⊥ , ⊕ ) are SMCs, and ( F , m ) and ( G , n ) are a sym-metric comonoidal functors between M and M . Then a symmetric comonoidal natural trans-formation is a natural transformation, f : F (cid:47) (cid:47) G , subject to the following coherence diagrams: G ( A ⊕ B ) GA ⊕ GB n A , B (cid:47) (cid:47) F ( A ⊕ B ) G ( A ⊕ B ) f A ⊕ B (cid:15) (cid:15) F ( A ⊕ B ) FA ⊕ F B m A , B (cid:47) (cid:47) FA ⊕ F BGA ⊕ GB f A ⊕ f B (cid:15) (cid:15) ⊥ F ⊥ (cid:95) (cid:95) m ⊥ ⊥ G ⊥ (cid:111) (cid:111) n ⊥ G ⊥ F ⊥ (cid:63) (cid:63) f ⊥ Definition 11.
Suppose ( M , (cid:62) , ⊗ ) and ( M , (cid:62) , ⊗ ) are SMCs, and ( F , m ) is a symmetric monoidalfunctor between M and M and ( G , n ) is a symmetric monoidal functor between M and M . Thena symmetric monoidal adjunction is an ordinary adjunction M : F (cid:97) G : M such that the unit, η A : A → GFA , and the counit, ε A : FGA → A , are symmetric monoidal natural transformations. HARLEY EADES III AND GIANLUIGI BELLIN
Thus, the following diagrams must commute: A ⊗ B FGA ⊗ FGB (cid:111) (cid:111) ε A ⊗ B FGA ⊗ FGBA ⊗ B ε A ⊗ ε B (cid:15) (cid:15) FGA ⊗ FGB F ( GA ⊗ GB ) m GA , GB (cid:47) (cid:47) F ( GA ⊗ GB ) FGA ⊗ FGB Fn A , B (cid:15) (cid:15) (cid:62) (cid:62) F (cid:62) (cid:62) (cid:79) (cid:79) m (cid:62) F (cid:62) FG (cid:62) Fn (cid:62) (cid:47) (cid:47) FG (cid:62) (cid:62) ε (cid:62) (cid:15) (cid:15) G ( FA ⊗ F B ) GF ( A ⊗ B ) m A , B (cid:47) (cid:47) GFA ⊗ GF BG ( FA ⊗ F B ) n FA , FB (cid:15) (cid:15) GFA ⊗ GF B A ⊗ B (cid:111) (cid:111) η A ⊗ η B A ⊗ BGF ( A ⊗ B ) η A ⊗ B (cid:15) (cid:15) (cid:62) (cid:62) G (cid:62) (cid:62) (cid:79) (cid:79) n (cid:62) G (cid:62) GF (cid:62) Gm (cid:62) (cid:47) (cid:47) GF (cid:62) (cid:62) (cid:79) (cid:79) η (cid:62) Definition 12.
Suppose ( M , ⊥ , ⊕ ) and ( M , ⊥ , ⊕ ) are SMCs, and ( F , m ) is a symmetric comonoidalfunctor between M and M and ( G , n ) is a symmetric comonoidal functor between M and M .Then a symmetric comonoidal adjunction is an ordinary adjunction M : F (cid:97) G : M suchthat the unit, η A : A → GFA , and the counit, ε A : FGA → A , are symmetric comonoidal naturaltransformations. Thus, the following diagrams must commute: GFA ⊕ GF B G ( FA ⊕ F B ) (cid:111) (cid:111) m FA , FB A ⊕ BGFA ⊕ GF B η A ⊕ η B (cid:15) (cid:15) A ⊕ B GF ( A ⊕ B ) η A ⊕ B (cid:47) (cid:47) GF ( A ⊕ B ) G ( FA ⊕ F B ) Gm A , B (cid:15) (cid:15) ⊥ ⊥ GF ⊥ ⊥ (cid:79) (cid:79) η ⊥ GF ⊥ G ⊥ Gm ⊥ (cid:47) (cid:47) G ⊥ ⊥ n ⊥ (cid:15) (cid:15) A ⊕ B FGA ⊕ FGB (cid:111) (cid:111) ε A ⊕ ε B FG ( A ⊕ B ) A ⊕ B ε A ⊕ B (cid:15) (cid:15) FG ( A ⊕ B ) F ( GA ⊕ GB ) Fn A , B (cid:47) (cid:47) F ( GA ⊕ GB ) FGA ⊕ FGB m GA , GB (cid:15) (cid:15) FG ⊥ F ⊥ Fn ⊥ (cid:47) (cid:47) FG ⊥ FG ⊥ FG ⊥ ⊥ ε ⊥ (cid:47) (cid:47) ⊥ F ⊥ (cid:79) (cid:79) m ⊥ We will be defining, and making use of the why-not exponentials from linear logic, but these cor-respond to a symmetric comonoidal monad. In addition, whenever we have a symmetric comonoidaladjunction, we immediately obtain a symmetric comonoidal comonad on the left, and a symmetriccomonoidal monad on the right.
Definition 13. A symmetric comonoidal monad on a symmetric monoidal category C is a triple( T , η, µ ), where ( T , n ) is a symmetric comonoidal endofunctor on C , η A : A (cid:47) (cid:47) T A and µ A : T A → T A are symmetric comonoidal natural transformations, which make the following diagrams commute: T A T A µ A (cid:47) (cid:47) T AT A T µ A (cid:15) (cid:15) T A T A µ TA (cid:47) (cid:47) T AT A µ A (cid:15) (cid:15) T A T A η TA (cid:47) (cid:47) T A T A (cid:111) (cid:111) T η A T AT A T AT A (cid:79) (cid:79) µ A T A T A
COINTUITIONISTIC ADJOINT LOGIC 9
The assumption that η and µ are symmetric comonoidal natural transformations amount to thefollowing diagrams commuting: A ⊕ B T A ⊕ T B η A ⊕ η B (cid:47) (cid:47) A ⊕ BT ( A ⊕ B ) η A (cid:15) (cid:15) T A ⊕ T BT ( A ⊕ B ) (cid:57) (cid:57) n A , B ⊥ ⊥⊥ T ⊥ η ⊥ (cid:47) (cid:47) T ⊥⊥ n ⊥ (cid:127) (cid:127) T ( A ⊕ B ) T ( A ⊕ B ) T ( A ⊕ B ) µ A ⊕ B (cid:15) (cid:15) T ( A ⊕ B ) T ( T A ⊕ T B ) T n A , B (cid:47) (cid:47) T ( T A ⊕ T B ) T A ⊕ T BT ( T A ⊕ T B ) T ( T A ⊕ T B ) T A ⊕ T B n TA , TB (cid:47) (cid:47) T A ⊕ T BT A ⊕ T B µ A ⊕ µ B (cid:15) (cid:15) T ( A ⊕ B ) T A ⊕ T B n A , B (cid:47) (cid:47) T ⊥ ⊥ n ⊥ (cid:47) (cid:47) T ⊥ T ⊥ µ ⊥ (cid:15) (cid:15) T ⊥ T ⊥ T n ⊥ (cid:47) (cid:47) T ⊥⊥ n ⊥ (cid:15) (cid:15) Finally, the dual concept of a symmetric comonoidal comonad.
Definition 14. A symmetric comonoidal comonad on a symmetric monoidal category C is atriple ( T , ε, δ ), where ( T , m ) is a symmetric comonoidal endofunctor on C , ε A : T A (cid:47) (cid:47) A and δ A : T A → T A are symmetric comonoidal natural transformations, which make the followingdiagrams commute: T A T A δ TA (cid:47) (cid:47) T AT A δ A (cid:15) (cid:15) T A T A δ A (cid:47) (cid:47) T AT A T δ A (cid:15) (cid:15) T A T A (cid:111) (cid:111) ε TA T A T A T ε A (cid:47) (cid:47) T AT A T AT A δ A (cid:15) (cid:15) T A T A
The assumption that ε and δ are symmetric monoidal natural transformations amount to the followingdiagrams commuting: T ( A ⊕ B ) T A ⊕ T B m A , B (cid:47) (cid:47) T ( A ⊕ B ) A ⊕ B ε A ⊕ B (cid:39) (cid:39) T A ⊕ T BA ⊕ B ε A ⊕ ε B (cid:15) (cid:15) T ⊥ T ⊥ T ⊥ ⊥ ε ⊥ (cid:47) (cid:47) ⊥ T ⊥ (cid:63) (cid:63) m ⊥ T ( A ⊕ B ) T ( T A ⊕ T B ) T m A , B (cid:47) (cid:47) T ( A ⊕ B ) T ( A ⊕ B ) δ A ⊕ B (cid:15) (cid:15) T ( A ⊕ B ) T ( T A ⊕ T B ) T ( T A ⊕ T B ) T A ⊕ T B m TA , TB (cid:47) (cid:47) T ( T A ⊕ T B ) T A ⊕ T BT A ⊕ T BT A ⊕ T B δ A ⊕ δ B (cid:15) (cid:15) T ( A ⊕ B ) T A ⊕ T B m A , B (cid:47) (cid:47) T ⊥ T ⊥ T m ⊥ (cid:47) (cid:47) T ⊥ T ⊥ δ ⊥ (cid:15) (cid:15) T ⊥ ⊥ m ⊥ (cid:47) (cid:47) ⊥ T ⊥ (cid:79) (cid:79) m ⊥ Cartesian Closed and Cocartesian Coclosed Categories.
The notion of a cartesian closedcategory is well-known, but for completeness we define them here. However, their dual is lesserknown, especially in computer science, and so we given their full definition. We also review someknow results concerning cocartesian coclosed categories and categories that are both cartesian closedand cocartesian coclosed.
Definition 15. A cartesian category is a category, ( C , , × ), with an object, 1, and a bi-functor, × : C × C (cid:47) (cid:47) C , such that for any object A there is exactly one morphism (cid:5) : A →
1, and for anymorphisms f : C (cid:47) (cid:47) A and g : C (cid:47) (cid:47) B there is a morphism (cid:104) f , g (cid:105) : C → A × B subject to thefollowing diagram: A A × B (cid:111) (cid:111) π A × B B π (cid:47) (cid:47) CA f (cid:127) (cid:127) CA × B (cid:104) f , g (cid:105) (cid:15) (cid:15) C B g (cid:31) (cid:31) A cartesian category models conjunction by the product functor, × : C × C (cid:47) (cid:47) C , and the unitof conjunction by the terminal object. As we mention above modeling implication requires closure,and since it is well-known that any cartesian category is also a symmetric monoidal category thedefinition of closure for a cartesian category is the same as the definition of closure for a symmetricmonoidal category (Definition 2). We denote the internal hom for cartesian closed categories by A → B .The dual of a cartesian category is a cocartesian category. They are a model of intuitionisticlogic with disjunction and its unit. Definition 16. A cocartesian category is a category, ( C , , + ), with an object, 0, and a bi-functor, + : C × C (cid:47) (cid:47) C , such that for any object A there is exactly one morphism (cid:3) : 0 → A , and for anymorphisms f : A (cid:47) (cid:47) C and g : B (cid:47) (cid:47) C there is a morphism [ f , g ] : A + B (cid:47) (cid:47) C subject to thefollowing diagram: A A + B ι (cid:47) (cid:47) A + B B (cid:111) (cid:111) ι CA (cid:63) (cid:63) f CA + B (cid:79) (cid:79) [ f , g ] C B (cid:95) (cid:95) g Coclosure, just like closure for cartesian categories, is defined in the same way that coclosureis defined for symmetric monoidal categories, because cocartesian categories are also symmetricmonoidal categories. Thus, a cocartesian category is coclosed if there is a specified left-adjoint,which we denote S − T , to the coproduct.There are many examples of cocartesian coclosed categories. Basically, any interesting cartesiancategory has an interesting dual, and hence, induces an interesting cocartesian coclosed category. Theopposite of the category of sets and functions between them is isomorphic to the category of completeatomic boolean algebras, and both of which, are examples of cocartesian coclosed categories. Aswe mentioned above bi-linear categories [9] are models of bi-linear logic where the left adjoint tothe cotensor models coimplication. Similarly, cocartesian coclosed categories model cointuitionisticlogic with disjunction and intuitionistic coimplicationWe might now ask if a category can be both cartesian closed and cocartesian coclosed just asbi-linear categories, but this turns out to be where the matter meets antimatter in such away that the COINTUITIONISTIC ADJOINT LOGIC 11 category degenerates to a preorder. That is, every homspace contains at most one morphism. Werecall this proof here, which is due to Crolard [11]. We need a couple basic facts about cartesianclosed categories with initial objects.
Lemma 17.
In any cartesian category C , if 0 is an initial object in C and Hom C ( A ,
0) is non-empty,then A (cid:27) A × Proof.
This follows easily from the universial mapping property for products.
Lemma 18.
In any cartesian closed category C , if 0 is an initial object in C , then so is 0 × A for anyobject A of C . Proof.
We know that the universal morphism for the initial object is unique, and hence, the homspace
Hom C (0 , A ⇒ B ) for any object B of C contains exactly one morphism. Then using the right adjointto the product functor we know that Hom C (0 , A ⇒ B ) (cid:27) Hom C (0 × A , B ), and hence, there is onlyone arrow between 0 × A and B .The following lemma is due to Joyal [18], and is key to the next theorem. Lemma 19 (Joyal) . In any cartesian closed category C , if 0 is an initial object in C and Hom C ( A , A is an initial object in C . Proof.
Suppose C is a cartesian closed category, such that, 0 is an initial object in C , and A is anarbitrary object in C . Furthermore, suppose Hom C ( A ,
0) is non-empty. By the first basic lemmaabove we know that A (cid:27) A ×
0, and by the second A × A is initial.Finally, the following theorem shows that any category that is both cartesian closed and cocarte-sian coclosed is a preorder. Theorem 20 ((co)Cartesian (co)Closed Categories are Preorders (Crolard[11])) . If C is both cartesianclosed and cocartesian coclosed, then for any two objects A and B of C , Hom C ( A , B ) has at most oneelement. Proof.
Suppose C is both cartesian closed and cocartesian coclosed, and A and B are objects of C .Then by using the basic fact that the initial object is the unit to the coproduct, and the coproducts leftadjoint we know the following: Hom C ( A , B ) (cid:27) Hom C ( A , + B ) (cid:27) Hom C ( B − A , Hom C ( A , B ) has at most one element.Notice that the previous result hinges on the fact that there are initial and terminal objects, and thus,this result does not hold for bi-linear categories, because the units to the tensor and cotensor are notinitial nor terminal.The repercussions of this result are that if we do not want to work with preorders, but do want towork with all of the structure, then we must separate the two worlds. Thus, this result can be seen asthe motivation for the current work. We enforce the separation using linear logic, but through thepower of linear logic this separation is not over a large distance. A Mixed Linear / Non-Linear Model for Co-Intuitionistic Logic.
Benton [4] showed thatfrom a LNL model it is possible to construct a linear category, and vice versa. Bellin [3] showed thatthe dual to linear categories are su ffi cient to model co-intuitionistic linear logic. We show that fromthe dual to a LNL model we can construct the dual to a linear category, and vice versa, thus, carryingout the same program for co-intuitionistic linear logic as Benton did for intuitionistic linear logic.Combining a symmetric monoidal coclosed category with a cocartesian coclosed category via asymmetric comonoidal adjunction defines a dual LNL model. Definition 21. A mixed linear / non-linear model for co-intuitionistic logic (dual LNL model) , L : H (cid:97) J : C , consists of the following:i. a symmetric monoidal coclosed category ( L , ⊥ , ⊕ , (cid:21) ),ii. a cocartesian coclosed category ( C , , + , − ), andiv. a symmetric comonoidal adjunction L : H (cid:97) J : C , where η A : A (cid:47) (cid:47) JH A and ε R : HJ R (cid:47) (cid:47) R arethe unit and counit of the adjunction respectively.It is well-known that an adjunction L : H (cid:97) J : C induces a monad H ; J : L (cid:47) (cid:47) L , but whenthe adjunction is symmetric comonoidal we obtain a symmetric comonoidal monad, in fact, H ; J defines the linear exponential why-not denoted ? A = JH A . By the definition of dual LNL models weknow that both H and J are symmetric comonoidal functors, and hence, are equipped with naturaltransformations h A , B : H ( A ⊕ B ) (cid:47) (cid:47) H A + H B and j R , S : J ( R + S ) (cid:47) (cid:47) J R ⊕ J S , and maps h ⊥ : H ⊥ (cid:47) (cid:47) j : J (cid:47) (cid:47) ⊥ . We will make heavy use of these maps throughout the sequel.Compare this definition with that of Bellin’s dual linear category from [3], and we can easily seethat the definition of dual LNL models – much like LNL models – is more succinct. Definition 22. A dual linear category , L , consists of the following data:i. A symmetric monoidal coclosed category ( L , ⊕ , ⊥ , (cid:21) ) withii. a symmetric co-monoidal monad (? , η, µ ) on L such thata. each free ?-algebra carries naturally the structure of a commutative ⊕ -monoid. This impliesthat there are distinguished symmetric monoidal natural transformations w A : ⊥ (cid:47) (cid:47) ? A and c A : ? A ⊕ ? A (cid:47) (cid:47) ? A which form a commutative monoid and are ?-algebra morphisms.b. whenever f : (? A , µ A ) (cid:47) (cid:47) (? B , µ B ) is a morphism of free ?-algebras, then it is also a monoidmorphism.2.3.1. A Useful Isomorphism.
One useful property of Benton’s LNL model is that the maps asso-ciated with the symmetric monoidal left adjoint in the model are isomorphisms. Since dual LNLmodels are dual we obtain similar isomorphisms with respect to the right adjoint.
Lemma 23 (Symmetric Comonoidal Isomorphisms) . Given any dual LNL model L : H (cid:97) J : C , thenthere are the following isomorphisms: J ( R + S ) (cid:27) J R ⊕ J S and J (cid:27) ⊥ Furthermore, the former is natural in R and S . Proof.
Suppose L : H (cid:97) J : C is a dual LNL model. Then we can define the following family ofmaps: j − R , S : = J R ⊕ J S η (cid:47) (cid:47) JH ( J R ⊕ J S ) Jh A , B (cid:47) (cid:47) J ( HJ R + HJ S ) J ( ε R + ε S ) (cid:47) (cid:47) J ( R + S ) j − : = ⊥ η (cid:47) (cid:47) JH ⊥ Jh ⊥ (cid:47) (cid:47) J COINTUITIONISTIC ADJOINT LOGIC 13
It is easy to see that j − R , S is natural, because it is defined in terms of a composition of naturaltransformations. All that is left to be shown is that j − R , s and j − are mutual inverses with j R , S and j ;for the details see Appendix B.1.Just as Benton we also do not have similar isomorphisms with respect to the functor H . One fact thatwe can point out, that Benton did not make explicit – because he did not use the notion of symmetriccomonoidal functor – is that j − makes J also a symmetric monoidal functor. Corollary 24.
Given any dual LNL model L : H (cid:97) J : C , the functor ( J , j − ) is symmetric monoidal. Proof.
This holds by straightforwardly reducing the diagrams defining a symmetric monoidal functor,Definition 7, to the diagrams defining a symmetric comonoidal functor, Definition 8, using the factthat j − is an isomorphism.2.3.2. Dual LNL Model Implies Dual Linear Category.
The next result shows that any dual LNLmodel induces a symmetric comonoidal monad.
Lemma 25 (Symmetric Comonoidal Monad) . Given a dual LNL model L : H (cid:97) J : C , the functor,? = H ; J , defines a symmetric comonoidal monad. Proof.
Suppose ( H , h ) and ( J , j ) are two symmetric comonoidal functors, such that, L : H (cid:97) J : C is a dual LNL model. We can easily show that ? A = JH A is symmetric monoidal by defining thefollowing maps: r ⊥ : = ? ⊥ JH ⊥ Jh ⊥ (cid:47) (cid:47) J j ⊥ (cid:47) (cid:47) ⊥ r A , B : = ?( A ⊕ B ) JH ( A ⊕ B ) Jh A , B (cid:47) (cid:47) J ( H A + H B ) j H A , H B (cid:47) (cid:47) JH A ⊕ JH B ? A ⊕ ? B The fact that these maps satisfy the appropriate symmetric comonoidal functor diagrams fromDefinition 8 is obvious, because symmetric comonoidal functors are closed under composition.We have a dual LNL model, and hence, we have the symmetric comonoidal natural transforma-tions η A : A (cid:47) (cid:47) JH A and ε R : HJ R (cid:47) (cid:47) R which correspond to the unit and counit of the adjunctionrespectively. Define µ A : = J ε H A : JHJH A (cid:47) (cid:47) JH A . This implies that we have maps η A : A (cid:47) (cid:47) ? A and µ A : ? ? A (cid:47) (cid:47) ? A , and thus, we can show that (? , η, µ ) is a symmetric comonoidal monad. Allthe diagrams defining a symmetric comonoidal monad hold by the structure given by the adjunction.For the complete proof see Appendix B.2.The monad from the previous result must be equipped with the additional structure to model theright weakening and contraction structural rules. Lemma 26 (Right Weakening and Contraction) . Given a dual LNL model L : H (cid:97) J : C , then forany ? A there are distinguished symmetric comonoidal natural transformations w A : ⊥ (cid:47) (cid:47) ? A and c A : ? A ⊕ ? A (cid:47) (cid:47) ? A that form a commutative monoid, and are ? -algebra morphisms with respectto the canonical definitions of the algebras ? A , ⊥ , ? A ⊕ ? A . Proof.
Suppose ( H , h ) and ( J , j ) are two symmetric comonoidal functors, such that, L : H (cid:97) J : C isa dual LNL model. Again, we know ? A = H ; J : L (cid:47) (cid:47) L is a symmetric comonoidal monad byLemma 25.We define the following morphisms: w A : = ⊥ j − ⊥ (cid:47) (cid:47) J J (cid:5) H A (cid:47) (cid:47) JH A ? A c A : = ? A ⊕ ? A JH A ⊕ JH A j − H A , H A (cid:47) (cid:47) J ( H A + H A ) J (cid:96) H A (cid:47) (cid:47) JHA ? A The remainder of the proof is by carefully checking all of the required diagrams. Please seeAppendix B.3 for the complete proof.
Lemma 27 (?-Monoid Morphisms) . Suppose L : H (cid:97) J : C is a dual LNL model. Then if f : (? A , µ A ) (cid:47) (cid:47) (? B , µ B ) is a morphism of free ?-algebras, then it is a monoid morphism. Proof.
Suppose L : H (cid:97) J : C is a dual LNL model. Then we know ? A = JH A is a symmetriccomonoidal monad by Lemma 25. Bellin [3] remarks that by Maietti, Maneggia de Paiva and Ritter’sProposition 25 [19], it su ffi ces to show that µ A : ? ? A (cid:47) (cid:47) ? A is a monoid morphism. For the detailssee the complete proof in Appendix B.4.Finally, we may now conclude the following corollary. Corollary 28.
Every dual LNL model is a dual linear category.2.3.3.
Dual Linear Category implies Dual LNL Model.
This section shows essentially the inverse tothe result from the previous section. That is, from any dual linear category we may construct a dualLNL model. By exploiting the duality between LNL models and dual LNL models this result followsstraightforwardly from Benton’s result. The proof of this result must first find a symmetric monoidcoclosed category, a cocartesian coclosed category, and finally, a symmetric comonoidal adjunctionbetween them. Take the symmetric monoid coclosed category to be an arbitrary dual linear category L . Then we may define the following categories. • The Eilenberg-Moore category, L ? , has as objects all ?-algebras, ( A , h A : ? A (cid:47) (cid:47) A ), and asmorphisms all ?-algebra morphisms. • The Kleisli category, L ? , is the full subcategory of L ? of all free ?-algebras (? A , µ A : ? ? A (cid:47) (cid:47) ? A ).The previous three categories are related by a pair of adjunctions: L L ? F (cid:41) (cid:41) L L ? (cid:104) (cid:104) U L L ? F (cid:41) (cid:41) L L ? (cid:104) (cid:104) U L L ? LLL L ? L ? L ? i (cid:79) (cid:79) The functor F ( A ) = (? A , µ A ) is the free functor, and the functor U ( A , h A ) = A is the forgetfulfunctor. Note that we, just as Benton did, are overloading the symbols F and U . Lastly, the functor i : L ? (cid:47) (cid:47) L ? is the injection of the subcategory of free ?-algebras into its parent category.We are now going to show that both L ? and L ? are induce two cocartesian coclosed categories.Then we could take either of those when constructing a dual LNL model from a dual linear category.First, we show C ? is cocartesian. Lemma 29. If L is a dual linear category, then L ? has finite coproducts. Proof.
We give a proof sketch of this result, because the proof is essentially by duality of Benton’scorresponding proof for LNL models (see Lemma 9, [4]). Suppose L is a dual linear category. Thenwe first need to identify the initial object which is defined by the ?-algebra ( ⊥ , r ⊥ : ? ⊥ (cid:47) (cid:47) ⊥ ).The unique map between the initial map and any other ?-algebra ( A , h A : ? A (cid:47) (cid:47) A ) is defined by ⊥ w A (cid:47) (cid:47) ? A h A (cid:47) (cid:47) A . The coproduct of the ?-algebras ( A , h A : ? A (cid:47) (cid:47) A ) and ( B , h B : ? B (cid:47) (cid:47) B ) is( A ⊕ B , r A , B ; ( h A ⊕ h B )). Injections and the codiagonal map are defined as follows: COINTUITIONISTIC ADJOINT LOGIC 15 • Injections: ι : = A ρ A (cid:47) (cid:47) A ⊕ ⊥ id A ⊕ w B (cid:47) (cid:47) A ⊕ ? B id ⊕ h b (cid:47) (cid:47) A ⊕ B ι : = B λ A (cid:47) (cid:47) ⊥ ⊕ B w A ⊕ id B (cid:47) (cid:47) ? A ⊕ B h A ⊕ id B (cid:47) (cid:47) A ⊕ B • Codiagonal map: (cid:104) : = A ⊕ A η A ⊕ η A (cid:47) (cid:47) ? A ⊕ ? A c A (cid:47) (cid:47) ? A h A (cid:47) (cid:47) A Showing that these respect the appropriate diagrams is straightforward.Notice as a direct consequence of the previous result we know the following.
Corollary 30.
The Kleisli category, L ? , has finite coproducts.Thus, both L ? and L ? are cocartesian, but we need a cocartesian coclosed category, and ingeneral these are not coclosed, and so we follow Benton’s lead and show that there are actually twosubcategories of L ? that are coclosed. Definition 31.
We call an object, A , of a category, L , subtractable if for any object B of L , theinternal cohom A (cid:21) B exists.We now have the following results: Lemma 32. In L ? , all the free ?-algebras are subtractable, and the internal cohom is a free ?-algebra. Proof.
The internal cohom is defined as follows:(? A , δ A ) (cid:21) ( B , h B ) : = (?( A (cid:21) B ) , δ A (cid:21) B )We can capitalize on the adjunctions involving F and U from above to lift the internal cohom of L into L ? : Hom L ? ((?( A (cid:21) B ) , δ A (cid:21) B ) , ( C , h C )) = Hom L ? ( F ( A (cid:21) B ) , ( C , h C )) (cid:27) Hom L ( A (cid:21) B , U ( C , h C )) = Hom L ( A (cid:21) B , C ) (cid:27) Hom L ( A , C ⊕ B ) = Hom L ( A , U ( C ⊕ B , h C ⊕ B )) (cid:27) Hom L ? ( F A , ( C ⊕ B , h C ⊕ B )) = Hom L ? ((? A , δ A ) , ( C ⊕ B , h C ⊕ B ))The previous equation holds for any h C ⊕ B making C ⊕ B a ?-algebra, in particular, the co-productin L ? (Lemma 29), and hence, we may instantiate the final line of the previous equation with thefollowing: Hom L ? ((? A , δ A ) , ( C , h c ) + ( B , δ A ))Thus, we obtain our result. Lemma 33.
We have the following cocartesian coclosed categories:i. The full subcategory,
Sub ( L ? ), of L ? consisting of objects the subtractable ?-algebras iscocartesian coclosed, and contains the Kleisli category.ii. The full subcategory, L ∗ ? , of Sub ( L ? ) consisting of finite coproducts of free ?-algebras iscocartesian coclosed.Let C be either of the previous two categories. Then we must exhibit a adjunction between C and L ,but this is easily done. Lemma 34.
The adjunction L : F (cid:96) U : C , with the free functor, F , and the forgetful functor, U , issymmetric comonoidal. Proof.
Showing that F and U are symmetric comonoidal follows similar reasoning to Benton’s result,but in the opposite; see Lemma 13 and Lemma 14 of [4]. Lastly, showing that the unit and the counitof the adjunction are comonoidal natural transformations is straightforward, and we leave it to thereader. The reasoning is similar to Benton’s, but in the opposite; see Lemma 15 and Lemma 16 of[4]. Corollary 35.
Any dual linear category gives rise to a dual LNL model.2.4.
A Mixed Bilinear / Non-Linear Model.
The main goal of our research program is to give anon-trivial categorical model of bi-intuitionistic logic. In this section we give a introduction of themodel we have in mind, but leave the details and the study of the logical and programmatic sides tofuture work.The naive approach would be to try and define a LNL-style model of bi-intuitionistic logicas an adjunction between a bilinear category and a bi-cartesian bi-closed category, but this resultsin a few problems. First, should the adjunction be monoidal or comonoidal? Furthermore, weknow bi-cartesian bi-closed categories are trivial (Theorem 20), and hence, this model is not veryinteresting nor correct. We must separate the two worlds using two dual adjunctions, and hence, wearrive at the following definition.
Definition 36. A mixed bilinear / non-linear model consists of the following:i. a bilinear category ( L , (cid:62) , ⊗ , −◦ , ⊥ , ⊕ , (cid:21) ),ii. a cartesian closed category ( I , , × , → ),iii. a cocartesian coclosed category ( C , , + , − ),iv. a LNL model I : F (cid:97) G : L , andv. a dual LNL model L : H (cid:97) J : C .Since L is a bilinear category then it is also a linear category, and a dual linear category. Thus,the LNL model intuitively corresponds to an adjunction between I and the linear subcategory of L ,and the dual LNL model corresponds to an adjunction between the dual linear subcategory of L and C . In addition, both intuitionistic logic and cointuitionistic logic can be embedded into L via thelinear modalities of-course, ! A , and why-not, ? A , using the well-known Girard embeddings. Thisimplies that we have a very controlled way of mixing I and C within L , and hence, linear logic isthe key. COINTUITIONISTIC ADJOINT LOGIC 17 S (cid:96) C S C id S (cid:96) C Ψ S (cid:96) C T , Ψ C weak S (cid:96) C T , T , Ψ S (cid:96) C T , Ψ C contr R (cid:96) C Ψ , S , T , Ψ R (cid:96) C Ψ , T , S , Ψ C ex0 (cid:96) C Ψ C 0 T (cid:96) C Ψ T (cid:96) C Ψ T + T (cid:96) C Ψ , Ψ C + L R (cid:96) C Ψ , T R (cid:96) C Ψ , T + T C + R R (cid:96) C Ψ , T R (cid:96) C Ψ , T + T C + R T (cid:96) C T , Ψ T − T (cid:96) C Ψ C − L S (cid:96) C Ψ , T T (cid:96) C Ψ S (cid:96) C Ψ , Ψ , T − T C − R S (cid:96) C Ψ , T T (cid:96) C Ψ S (cid:96) C Ψ , Ψ C cut A (cid:96) L · ; Ψ H A (cid:96) C Ψ H L Figure 1: Non-linear fragment of the DLNL logic3. D ual
LNL L ogic
We now turn to developing the syntactic side of dual LNL models called dual LNL logic (DLNL).First, we give a sequent calculus formalization which we will simply refer to as DLNL logic,then a natural deduction formalization called DND logic, and finally a term assignment to thenatural deduction version. Each of these systems will consistently use the same syntax and namingconventions for formulas, types, and contexts given by the following definition.
Definition 37.
The the syntax for formulas, types, and contexts are given as follows:(non-linear formulas / types) R , S , T :: = | S + T | S − T | H A (linear formulas / types) A , B , C :: = ⊥| A ⊕ B | A (cid:21) B | J S (non-linear contexts) Ψ , Θ :: = · | T | Ψ , Θ (linear contexts) Γ , ∆ :: = · | A | Γ , ∆ The term assignment will index contexts by terms, but we will maintain the same naming conventionthroughout.3.1.
The Sequent Calculus for Dual LNL Logic.
In this section we take the dual of Benton’s [4]sequent calculus for LNL logic to obtain the sequent calculus for dual LNL logic. The inferencerules for the non-linear fragment can be found in Figure 1 and the linear fragment in Figure 2. Theremainder of this section is devoted to proving cut-elimination. However, the proof is simply adualization of Benton’s [4] proof of cut-elimination for LNL logic.Just as Benton we use n -ary cuts: S (cid:96) C Ψ , S n S (cid:96) C Ψ (cid:48) S (cid:96) C Ψ , Ψ (cid:48) C cut n A (cid:96) L ∆ ; Ψ , S n S (cid:96) C Ψ (cid:48) A (cid:96) L ∆ ; Ψ , Ψ (cid:48) LC cut n where S n = S , . . . , S n -times. We call DLNL + the system DLNL with n -cuts replacing ordinary1-cuts. Such cuts are admissible in DLNL and cut-elimination for DLNL + implies cut-eliminationfor DLNL.We begin with a few standard definitions. The rank of a formula, denoted by | A | or | S | , is thenumber of the logical symbols in the given formula. The cut-rank of a derivation Π , denoted by c ( Π ), is the maximum of the ranks of the cut formulas in Π plus one; if Π is cut-free its cut rank is A (cid:96) L A ; · LL id A (cid:96) L ∆ ; Ψ A (cid:96) L ∆ ; T , Ψ LC weak A (cid:96) L ∆ ; T , T , Ψ A (cid:96) L ∆ ; T , Ψ LC contr A (cid:96) L ∆ , A , B , ∆ ; Ψ A (cid:96) L ∆ , B , A , ∆ ; Ψ LL ex A (cid:96) L ∆ ; Ψ , S , T , Ψ A (cid:96) L ∆ ; Ψ , T , S , Ψ LC ex A (cid:96) L ∆ , B ; Ψ B (cid:96) L ∆ ; Ψ A (cid:96) L ∆ , ∆ ; Ψ , Ψ LL cut A (cid:96) L ∆ ; Ψ , T T (cid:96) C Ψ A (cid:96) L ∆ ; Ψ , Ψ LC cut ⊥(cid:96) L · ; · LL ⊥ L A (cid:96) L ∆ ; Ψ A (cid:96) L ⊥ , ∆ ; Ψ LL ⊥ R A (cid:96) L ∆ ; Ψ , T A (cid:96) L ∆ ; Ψ , T + T LC + R A (cid:96) L ∆ ; Ψ , T A (cid:96) L ∆ ; Ψ , T + T LC + R B (cid:96) L ∆ ; Ψ B (cid:96) L ∆ ; Ψ B ⊕ B (cid:96) L ∆ , ∆ ; Ψ , Ψ LL ⊕ L A (cid:96) L ∆ , B , C ; Ψ A (cid:96) L ∆ , B ⊕ C ; Ψ LL ⊕ R B (cid:96) L B , ∆ ; Ψ B (cid:21) B (cid:96) L ∆ ; Ψ LL (cid:21) L A (cid:96) L B , ∆ ; Ψ B (cid:96) L ∆ ; Ψ A (cid:96) L B (cid:21) C , ∆ , ∆ ; Ψ , Ψ LL (cid:21) R A (cid:96) L ∆ ; Ψ , T T (cid:96) C Ψ A (cid:96) L ∆ ; Ψ , Ψ , T − T LL − R T (cid:96) C Ψ J T (cid:96) L · ; Ψ J L A (cid:96) L ∆ ; T , Ψ A (cid:96) L ∆ , J T ; Ψ J R A (cid:96) L ∆ , B ; Ψ A (cid:96) L ∆ ; H B , Ψ H R Figure 2: Linear fragment of the DLNL logic0. Finally, the depth of a derivation Π , denoted by d ( Π ), is the length of the longest path in Π . Thefollowing three results establish cut elimination. Lemma 38 (Cut Reduction) . The following defines the cut reduction procedure:(1) If Π is a derivation of T (cid:96) C Ψ , S n and Π is a derivation of S (cid:96) C Ψ (cid:48) with c ( Π ) , c ( Π ) ≤ | S | , thenthere exists a derivation Π of T (cid:96) C Ψ , Ψ (cid:48) with c ( Π ) ≤ | S | ;(2) If Π is a derivation of T (cid:96) L ∆ ; Ψ , S n and Π is a derivation of S (cid:96) C Ψ (cid:48) with c ( Π ) , c ( Π ) ≤ | S | ,then there exists a derivation Π of T (cid:96) L ∆ ; Ψ , Ψ (cid:48) with c ( Π ) ≤ | S | ;(3) If Π is a derivation of B (cid:96) L ∆ ; Ψ , A n and Π is a derivation of A (cid:96) L ∆ (cid:48) , Ψ (cid:48) with c ( Π ) , c ( Π ) ≤ | S | ,then there exists a derivation Π of B (cid:96) C ∆ , ∆ (cid:48) , Ψ , Ψ (cid:48) with c ( Π ) ≤ | A | . Proof.
By induction on d ( Π ) + d ( Π ). We give one case where the last inferences of Π and Π arelogical inferences; please see Appendix B.5 for the complete proof. •− right / •− left. We have COINTUITIONISTIC ADJOINT LOGIC 19 π A (cid:96) L ∆ ; Ψ , B π B (cid:96) L ∆ ; Ψ Π = LL (cid:21) R A (cid:96) L B •− B , ∆ , ∆ ; Ψ , Ψ π B (cid:96) L B , ∆ ; ΨΠ = LL (cid:21) L B •− B (cid:96) L ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , ∆ ; Ψ , Ψ , Ψ reduces to Π π A (cid:96) L ∆ , B ; Ψ π B (cid:96) L B , ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , B ; Ψ , Ψ π B (cid:96) L ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , ∆ ; Ψ , Ψ , Ψ The resulting derivation Π has cut rank c ( Π ) = max ( | B | + , c ( π ) , c ( π ) , | B | + , c ( π )) ≤ | B •− B | . Lemma 39 (Decrease in Cut-Rank) . Let Π be a DLNL + proof of a sequent S (cid:96) C Ψ or A (cid:96) L ∆ ; Ψ with c ( Π ) >
0. Then there exists a proof Π (cid:48) of the same sequent with c ( Π (cid:48) ) < c ( Π ). Proof.
By induction on d ( Π ). If the last inference is not a cut, then we apply the induction hypothesis.If the last inference is a cut on a formula A , but A is not of maximal rank among the cut formulas, sothat c ( Π ) > | A | +
1, then we apply the induction hypothesis. Finally, if the last inference is a cut on A and c ( Π ) = | A | + Π B (cid:96) L ∆ , A ; Ψ Π A (cid:96) L ∆ (cid:48) ; Ψ (cid:48) Π =
LL cut B (cid:96) L ∆ , ∆ (cid:48) ; Ψ , Ψ (cid:48) Now since c ( Π ) , c ( Π ) ≤ | A | + Π (cid:48) and Π (cid:48) with c ( Π (cid:48) ) ≤ | A | and c ( Π (cid:48) ) ≤ | A | . Thenby cut reduction we can construct a derivation Π (cid:48) proving B (cid:96) L ∆ , ∆ (cid:48) ; Ψ , Ψ (cid:48) with c ( Π (cid:48) ) ≤ | A | asrequired. Theorem 40 (Cut Elimination) . Let Π be a proof of a sequent S (cid:96) C Ψ or A (cid:96) L ∆ ; Ψ such that c ( Π ) > Proof.
By induction on c ( Π ) using the previous lemma.3.2. Sequent-style Natural Deduction.
The inference rules for the non-linear and linear fragmentsof the sequent-style natural deduction formalization of DLNL (DND) can be found in Figure 3 andFigure 4 respectively.
Remark 41.
In DLNL logic contexts are treated multiplicatively and so are in DND. Non-linearcontext could also be treated additively. In the case of the minor premises of non-linear disjunctionelimination (rule NLL + E of Figure 3 an additive interpretation is required, namely, both minorpremises must have the same right context, to match the categorical interpretation of disjunctionas coproduct. The same holds for the term assignment in the rule T C + E of Figure 5. Of courseadditive contexts can be simulated using weakening and contraction. This is what we do in the caseof disjunction elimination.We now recall a correspondence between DND and DLNL logic. First, we need the admissiblerule of cut, i.e., substitution. S (cid:96) C S NC id S (cid:96) C Ψ S (cid:96) C T , Ψ NC weak S (cid:96) C T , T , Ψ S (cid:96) C T , Ψ NC contr S (cid:96) C , Ψ S (cid:96) C Ψ , ... , S n (cid:96) C Ψ n S (cid:96) C Ψ , Ψ , ... , Ψ n NC 0 E S (cid:96) C Ψ , T S (cid:96) C Ψ , T + T NC + I S (cid:96) C Ψ , T S (cid:96) C Ψ , T + T NC + I S (cid:96) C Ψ , T + T T (cid:96) C Ψ T (cid:96) C Ψ S (cid:96) C Ψ , Ψ NC + E S (cid:96) C Ψ , T T (cid:96) C Ψ S (cid:96) C Ψ , Ψ , T − T NC − I S (cid:96) C Ψ , T − T T (cid:96) C T , Ψ S (cid:96) C Ψ , Ψ NC − E S (cid:96) C Ψ , H A A (cid:96) L · ; Ψ S (cid:96) C Ψ , Ψ NC H E Figure 3: Non-linear fragment of DND logic A (cid:96) L A ; · NLL id A (cid:96) L ∆ ; Ψ A (cid:96) L ∆ ; T , Ψ NLC weak A (cid:96) L ∆ ; T , T , Ψ A (cid:96) L ∆ ; T , Ψ NLC contr A (cid:96) L ∆ ; Ψ A (cid:96) L ∆ , ⊥ ; Ψ NLL ⊥ I A (cid:96) L ⊥ , ∆ ; · A (cid:96) L ∆ ; · NLL ⊥ E A (cid:96) L ∆ , B , B ; Ψ A (cid:96) L ∆ , B ⊕ B ; Ψ NLL ⊕ I A (cid:96) L ∆ , B ⊕ B ; Ψ B (cid:96) L ∆ ; Ψ B (cid:96) L ∆ ; Ψ A (cid:96) L ∆ , ∆ , ∆ ; Ψ , Ψ , Ψ NLL ⊕ E A (cid:96) L ∆ , B ; Ψ B (cid:96) L ∆ ; Ψ A (cid:96) L B (cid:21) B , ∆ , ∆ ; Ψ , Ψ NLL (cid:21) I A (cid:96) L ∆ , B (cid:21) B ; Ψ B (cid:96) L B , ∆ ; Ψ A (cid:96) L ∆ , ∆ ; Ψ , Ψ NLL (cid:21) E A (cid:96) L ∆ ; T , Ψ A (cid:96) L ∆ , J T ; Ψ NLL J I A (cid:96) L ∆ , J T ; Ψ T (cid:96) C Ψ A (cid:96) L ∆ ; Ψ , Ψ NLL J E A (cid:96) L ∆ , B ; Ψ A (cid:96) L ∆ ; H B , Ψ NLL H I A (cid:96) L ∆ ; Ψ , H A A (cid:96) L · ; Ψ A (cid:96) L ∆ ; Ψ , Ψ NLL H E Figure 4: Linear fragment of DND logic
COINTUITIONISTIC ADJOINT LOGIC 21
Lemma 42 (Admissible Rules in DND) . The following rules are admissible in DND: S (cid:96) C Ψ , T T (cid:96) C Ψ S (cid:96) C Ψ , Ψ NC cut A (cid:96) L ∆ ; Ψ , T T (cid:96) C Ψ A (cid:96) L ∆ ; Ψ , Ψ NLC cut A (cid:96) L ∆ , B ; Ψ B (cid:96) L ∆ ; Ψ A (cid:96) L ∆ , ∆ ; Ψ , Ψ NLL cutUsing these admissible rules we can construct a proof preserving translation between DND andDLNL logic.
Lemma 43 (Translations between DND and DLNL logic) . There are functions S : DND → DLNL and N : DLNL → DND from natural deduction to sequent calculus derivations.Notice that the right rules of the sequent calculus and the introductions of natural deduction have thesame form. Elimination rules are derivable from left rules with cut and left rules are derivable usingthe admissible cut rule in DND. For instance, the NC 0 E rule S (cid:96) C , Ψ S (cid:96) C Ψ , ... , S n (cid:96) C Ψ n S (cid:96) C Ψ , Ψ , ... , Ψ n NC 0 E is derivable in the sequent calculus as follows: S (cid:96) C , Ψ (cid:96) S , . . . S n C cut S (cid:96) C Ψ , S , . . . S n S (cid:96) C Ψ C cut S (cid:96) C Ψ , Ψ , S , . . . , S n ... S (cid:96) C Ψ , Ψ , . . . Ψ n − , S n S n (cid:96) Ψ n C cut S (cid:96) C Ψ , Ψ , . . . Ψ n − , Ψ n Term Assignment.
We now turn to giving a term assignment to DND logic called TND,which is greatly influenced by Crolard’s term assignment for subtractive logic in the paper
Aformulae-as-types interpretation of subtractive logic
JLC 2004. Crolard based his term assignmenton Parigot’s [20] λµ -calculus. He then shows that a type theory of coroutines can be given bysubtractive types and it is this result we pull inspiration from.TND pushes beyond Crolard’s work on subtractive logic. He restricts a classical calculus toprovide a constructive version of subtraction called safe coroutines . TND is based on the work ofthe second author where he used a variant of Crolard’s constructive calculus as a term assignmentto co-intuitionistic logic and to linear co-intuitionistic logic [3] without using the λµ -calculus. Inthis formulation, distinct terms are assigned to distinct formulas in the context and the reduction of aterm in context may impact other terms in the context.The syntax of TND terms is defined by the following definition. Definition 44.
The syntax for TND terms and typing judgments are given by the following grammar:(non-linear terms) s , t :: = x | ε | t · t | false t | x ( t ) | mkc ( t , x ) | inl t | inr t | case t of x . t , y . t | H e | let J x = e in t | postp ( x (cid:55)→ t , t ) | let H x = t in t (linear terms) e , u :: = x | connect ⊥ to e | postp ⊥ e | postp ( x (cid:55)→ e , e ) | mkc ( e , x ) | x ( e ) | e ⊕ e | casel e | caser e | J t (non-linear judgment) x : R (cid:96) C Ψ (linear judgment) x : A (cid:96) L ∆ ; Ψ Contexts, ∆ and Ψ , are the straightforward extension where each type is annotated with a term fromthe respective fragment.To aid the reader in understanding the variable structure, which variable annotations are bound,deployed throughout the TND term syntax we give the definitions of the free variable functions inthe following definition. Definition 45.
The free variable functions, FV ( t ) and FV ( e ), for linear and non-linear terms t and e are defined by mutual recursion as follows: linear terms: FV ( x ) = { x } FV ( connect ⊥ to e ) = FV ( e ) FV ( x ( e )) = FV ( e ) FV ( mkc ( e , y )) = FV ( e ) FV ( e ⊕ e ) = FV ( e ) ∪ FV ( e ) FV ( casel e ) = FV ( e ) FV ( caser e ) = FV ( e ) FV ( J t ) = FV ( t ) non-linear terms: FV ( x ) = { x } FV ( ε ) = ∅ FV ( t · t ) = FV ( t ) ∪ FV ( t ) FV ( false t ) = FV ( t ) FV ( x ( t )) = FV ( t ) FV ( mkc ( t , y )) = FV ( t ) FV ( inl t ) = FV ( inr t ) = FV ( t ) FV ( case t of x . t , y . t ) = FV ( t ) ∪ FV ( t ) (cid:114) { x } ∪ FV ( t ) (cid:114) { y } FV ( let J y = e in t ) = FV ( e ) ∪ FV ( t ) (cid:114) { y } FV ( let H y = t in t ) = FV ( t ) ∪ FV ( t ) (cid:114) { y } FV ( H e ) = FV ( e ) The free variables of a p -term are defined s follows: FV ( postp ⊥ e ) = FV ( e ) FV ( postp ( x (cid:55)→ e , e )) = FV ( e ) (cid:114) { x } ∪ FV ( e )and similarly for terms postp ( x (cid:55)→ t , t ).Terms are then typed by annotating the previous term structure over DND derivations, and thisis accomplished by annotating the DND inference rules. The typing rules for the non-linear fragmentof TND can be found in Figure 5, and the typing rules for the linear fragment of TND can be foundin Figure 6. Remark 46.
Let us call terms of the form postp ( x (cid:55)→ t , t ), postp ( x (cid:55)→ e , e ), and postp ⊥ ep - terms . Then say that a term t is p-normal if t does not contain any p -term as a proper subterm. Ina typed calculus, linear p -terms can be typed with ⊥ . Non-linear p terms can be typed with 0: inpresence of the TC 0 E rule this yieds instances of the ex falso rule. This is what happens in Crolard’scalculus, where the analogue of the postp ( x (cid:55)→ t , t ), namely, resume t with x (cid:55)→ t , always goeswith a weakening operation. The term ε is the identity of the contraction binary operator t · t . COINTUITIONISTIC ADJOINT LOGIC 23 x : S (cid:96) C x : S TC id x : S (cid:96) C Ψ x : S (cid:96) C Ψ , ε : T TC weak x : S (cid:96) C t : T , t : T , Ψ x : S (cid:96) C ( t · t ) : T , Ψ TC contr x : S (cid:96) C t : 0 , Ψ x : S (cid:96) C Ψ , ... , x n : S n (cid:96) C Ψ n x : S (cid:96) C Ψ , [ false t / x ] Ψ , ... , [ false t / x n ] Ψ n TC 0 E x : S (cid:96) C Ψ , t : T x : S (cid:96) C Ψ , inl t : T + T TC + I x : S (cid:96) C Ψ , t : T x : S (cid:96) C Ψ , inr t : T + T TC + I x : S (cid:96) C Ψ , t : T + T y : T (cid:96) C Ψ z : T (cid:96) C Ψ | Ψ | = | Ψ | x : S (cid:96) C Ψ , case t of y . Ψ , z . Ψ TC + E x : S (cid:96) C Ψ , t : T y : T (cid:96) C Ψ x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , [ y ( t ) / y ] Ψ TC − I x : S (cid:96) C Ψ , s : T − T y : T (cid:96) C t : T , Ψ x : S (cid:96) C Ψ , postp ( y (cid:55)→ t , s ) , [ y ( s ) / y ] Ψ TC − E x : S (cid:96) C Ψ , t : H A y : A (cid:96) L · ; Ψ x : S (cid:96) C Ψ , let H y = t in Ψ TC H E Figure 5: Non-linear fragment of the term assignment for TNDHowever when within a non- p -normal term an expression of the form postp ( x (cid:55)→ t , t ) is eliminatedas a β -redex, there is a choice of the syntax for the contextual reduction. In absence of a more detailedanalysis of the matter, we prefer to leave the typing of p terms implicit in the syntax, to enforcethe requirement of p -normality and to use the rule of weakening in place of the TC 0 E rule in thiscontext.The typing rules depend on the extension of let and case expressions to typing contexts. We usethe following notation for parallel composition of typing contexts: ∆ = e : A (cid:107) · · · (cid:107) e n : A n This operation should be regarded as associative, commutative and having the empty context as itsidentity. The extension of let expressions to contexts is given as follows: let p = t in · = · let p = t in ( t : A ) = let p = t in t : A let p = t in ( Ψ (cid:107) Ψ ) = ( let p = t in Ψ ) (cid:107) ( let p = t in Ψ )where p = H y or p = J y . Case expressions are handled similarly.Similarly to DND logic we have the following admissible rules. x : A (cid:96) L x : A ; · TLL id x : A (cid:96) L ∆ ; Ψ x : A (cid:96) L ∆ ; Ψ , ε : T TL weak x : A (cid:96) L ∆ ; t : T , t : T , Ψ x : A (cid:96) L ∆ ; t · t : T , Ψ TLC contr x : A (cid:96) L ∆ ; Ψ e : B ∈ ∆ x : A (cid:96) L ∆ , connect ⊥ to e : ⊥ ; Ψ TLL ⊥ I x : A (cid:96) L e : ⊥ , ∆ ; · x : A (cid:96) L postp ⊥ e , ∆ ; · TLL ⊥ E x : A (cid:96) L ∆ , e : B , e : B ; Ψ x : A (cid:96) L ∆ , e ⊕ e : B ⊕ B ; Ψ TLL ⊕ I x : A (cid:96) L ∆ , e : B ⊕ B ; Ψ y : B (cid:96) L ∆ ; Ψ z : B (cid:96) L ∆ ; Ψ x : A (cid:96) L ∆ , [ casel ( e ) / y ] ∆ , [ caser ( e ) / z ] ∆ ; Ψ , [ casel ( e ) / y ] Ψ , [ caser ( e ) / z ] Ψ TLL ⊕ E x : A (cid:96) L ∆ , e : B ; Ψ y : B (cid:96) L ∆ ; Ψ x : A (cid:96) L mkc ( e , y ) : B (cid:21) B , ∆ , [ y ( e ) / y ] ∆ ; Ψ , [ y ( e ) / y ] Ψ TLL (cid:21) I x : A (cid:96) L ∆ , e : B (cid:21) B ; Ψ y : B (cid:96) L e : B , ∆ ; Ψ x : A (cid:96) L ∆ , postp ( y (cid:55)→ e , e ) , ∆ ; Ψ , Ψ TLL (cid:21) E x : A (cid:96) L ∆ ; t : T , Ψ x : A (cid:96) L ∆ , J t : J T ; Ψ TLL J I x : A (cid:96) L ∆ , e : J T ; Ψ y : T (cid:96) C Ψ x : A (cid:96) L ∆ ; Ψ , let J y = e in Ψ TLL J E x : A (cid:96) L ∆ , e : B ; Ψ x : A (cid:96) L ∆ ; H e : H B , Ψ TLL H I x : A (cid:96) L ∆ ; Ψ , t : H A y : A (cid:96) L · ; Ψ x : A (cid:96) L ∆ ; Ψ , let H y = t in Ψ TLL H E Figure 6: Linear fragment of the term assignment for TND
Lemma 47 (Admissible Typing Rules) . The term assignment for the admissible rules of the calculusis as follows: x : S (cid:96) C Ψ , t : T y : T (cid:96) C Ψ x : S (cid:96) C Ψ , [ t / y ] Ψ TC cut x : A (cid:96) L ∆ ; Ψ , t : T y : T (cid:96) C Ψ x : A (cid:96) L ∆ ; Ψ , [ t / y ] Ψ NLC cut x : A (cid:96) L ∆ , e : B ; Ψ y : B (cid:96) L ∆ ; Ψ x : A (cid:96) L ∆ , [ e / y ] ∆ ; Ψ , [ e / y ] Ψ TLL cutWe generalize the rule of contraction on the non-linear side to contexts. Let m and m be multisetsof terms, then we denote by m · m the sum of multisets; if multisets are represented as lists, thenthe sum is representable as the appending of the lists. We denote singleton multisets, { t } , by the termthat inhabits it, e.g. t . We extend this to contexts, Ψ · Ψ , recursively as follows:( · ) · ( · ) = ( · )( t : S ) · ( t : S ) = t · t : S ( Ψ (cid:107) Ψ ) · ( Ψ (cid:107) Ψ ) = ( Ψ · Ψ ) (cid:107) ( Ψ · Ψ )where | Ψ | = | Ψ | and | Ψ | = | Ψ | . Whenever we write Ψ · Ψ we assume that | Ψ | = | Ψ | .At this point we are now ready to turn to computing in TND by specifying the reduction relation.This definition is perhaps the most interesting aspect of the theory, because reducing one term maya ff ect others. COINTUITIONISTIC ADJOINT LOGIC 25
Bottom: x : A (cid:96) L postp ⊥ ( connect ⊥ to e ) , ∆ ; Ψ (cid:32) x : A (cid:96) L ∆ ; Ψ H : x : B (cid:96) L ∆ ; Ψ · ( let H y = H e in Ψ ) (cid:32) x : B (cid:96) L ∆ ; Ψ · [ e / y ] Ψ J : x : A (cid:96) L ∆ ; Ψ · let J y = J t in Ψ (cid:32) x : A (cid:96) L ∆ ; Ψ · [ t / y ] Ψ Linear Subtraction: Ψ (cid:48) = [ y ( e ) / y ] Ψ , [ z ( mkc ( e , y )) / z ] Ψ Ψ (cid:48)(cid:48) = [[ e / z ] e / y ] Ψ , [ e / z ] Ψ x : B (cid:96) L ∆ , postp ( z (cid:55)→ e , mkc ( e , y )) , [ y ( e ) / y ] ∆ , [ z ( mkc ( e , y )) / z ] ∆ ; Ψ , Ψ (cid:48) (cid:32) x : B (cid:96) L ∆ , [[ e / z ] e / y ] ∆ , [ e / z ] ∆ ; Ψ , Ψ (cid:48)(cid:48) Par: Ψ (cid:48) = [ casel ( e ⊕ e ) / y ] Ψ , [ caser ( e ⊕ e ) / z ] Ψ Ψ (cid:48)(cid:48) = [ e / y ] Ψ , [ e / x ] Ψ x : A (cid:96) L ∆ , [ casel ( e ⊕ e ) / y ] ∆ , [ caser ( e ⊕ e ) / z ] ∆ ; Ψ , Ψ (cid:48) (cid:32) x : A (cid:96) L ∆ , [ e / y ] ∆ , [ e / x ] ∆ ; Ψ , Ψ (cid:48)(cid:48) Figure 7: Reductions for Linear Terms β -Reduction in TND. As we discussed above cointuitionistic logic can be interpreted as a theoryof coroutines that manipulate local context. Thus, reducing one term in a typing context could a ff ectother terms in the context. This implies that the definition of the reduction relation for TND mustaccount for more than a single term. We accomplish this by defining the reduction relation of termsin context, x : S (cid:96) C Ψ , t : T , Ψ and x : A (cid:96) L ∆ , e : B , ∆ ; Ψ , so that the manipulation of the contextis made explicit.The reduction rules for the linear and non-linear fragments can be found in Figure 7 andFigure 8 respectively. We denote the judgments for reduction by x : S (cid:96) C Ψ (cid:32) x : S (cid:96) C Ψ and x : A (cid:96) L ∆ ; Ψ (cid:32) x : A (cid:96) L ∆ ; Ψ . In the interest of readability we do not show full derivations, butit should be noted that it is assumed that every term mentioned in a reduction rule is typable with theexpected type given where it occurs in the judgment. Furthermore, the reduction relation depends ona few standard definitions and non-standard binding operations.The non-standard binding operations concern the variable y in mkc ( t , y ) and in postp ( y (cid:55)→ t , s )and the related expressions y ( t ) and y ( s ), respectively, occurring in the non-linear context; similaroperations occur in the linear case. Consider term assignment to the rule subtraction introductionTC − I in Figure 5. The variable y is the unique free variable occurring in the sequent y : T (cid:96) C Ψ ,the minor premise of the inference. In the conclusion x : S (cid:96) Ψ , mkc ( t , y ) : T − T , [ y ( t ) / y ] Ψ thevariable y is bound in mkc ( t , y ); moreover, the occurrences of the free variable y have been substitutedsimultaneously in the context Ψ by the expression y ( t ) which denotes a bound varianble, indexed Subtraction: x : S (cid:96) C Ψ , postp ( z (cid:55)→ t , mkc ( t , y )) , [ y ( t ) / y ] Ψ , [ mkc ( t , y ) / z ] Ψ (cid:32) x : S (cid:96) C Ψ , [[ t / z ] t / y ] Ψ , [ t / x ] Ψ Coproduct Left: x : S (cid:96) C Ψ , case ( inl t ) of y . Ψ , z . Ψ (cid:32) x : S (cid:96) C Ψ , [ t / y ] Ψ Coproduct Right: x : S (cid:96) C Ψ , case ( inr t ) of y . Ψ , z . Ψ (cid:32) x : S (cid:96) C Ψ , [ t / z ] Ψ H : x : H B (cid:96) C ( let H x = y in Ψ ) · let H z = ( let H x = y in H e ) in Ψ (cid:32) x : H B (cid:96) C ( let H x = y in Ψ ) · ( let H x = y in [ e / z ] Ψ ) Contraction with TC + E : x : S (cid:96) C Ψ , case ( t · t ) of y . Ψ , z . Ψ (cid:32) x : S (cid:96) C Ψ , ( case t of y . Ψ , z . Ψ ) · ( case t of y . Ψ , z . Ψ ) Contraction with TC + I : x : S (cid:96) C inl ( t · t ) : S + S , Ψ (cid:32) x : S (cid:96) C ( inl t ) · ( inl t ) : S + S , Ψ Contraction with TC + I : x : S (cid:96) C inr ( t · t ) : S + S , Ψ (cid:32) x : S (cid:96) C ( inr t ) · ( inr t ) : S + S , Ψ Contraction with TC − I : x : S (cid:96) C Ψ , mkc ( t · t , y ) : T − T , [ y ( t · t ) / y ] Ψ (cid:32) x : S (cid:96) C Ψ , ( mkc ( t , y ) · mkc ( t , y )) : T − T , ([ y ( t ) / y ] Ψ · [ y ( t ) / y ] Ψ ) Contraction with TC − E : x : S (cid:96) C Ψ , postp ( z (cid:55)→ s , t · t ) , [ y ( t · t ) / y ] Ψ (cid:32) x : S (cid:96) C ( Ψ , postp ( z (cid:55)→ s , t ) , [ y ( t ) / y ] Ψ ) · ( Ψ , postp ( z (cid:55)→ s , t ) , [ y ( t ) / y ] Ψ ) Contraction with TC H E : x : S (cid:96) C Ψ , let H y = t · t in Ψ (cid:32) x : S (cid:96) C Ψ , ( let H y = t in Ψ ) · ( let H y = t in Ψ )Figure 8: Reductions for Non-linear Termswith t . Similar explanations apply to the term assignment for subtraction elimination, and to thecorresponding linear rules in Figure 6.An analogue of the capture of a free variable by a binder in the λ -calculus, is an occurrence ofa bound variable y ( t ) whose binder is ambiguous, for instance in a context where there were two COINTUITIONISTIC ADJOINT LOGIC 27
Weakening with TC + E : x : S (cid:96) C Ψ , case ( ε ) of y . Ψ , z . Ψ (cid:32) x : S (cid:96) C Ψ , ε : S , ... , ε : S i where | Ψ | = | Ψ | and | Ψ | = S , ... , S i Weakening with TC + I : x : S (cid:96) C Ψ , inl ε : S + S (cid:32) x : S (cid:96) C Ψ , ε : S + S Weakening with TC + I : x : S (cid:96) C Ψ , inr ε : S + S (cid:32) x : S (cid:96) C Ψ , ε : S + S Weakening with TC − E : x : S (cid:96) C Ψ , postp ( z (cid:55)→ s , ε ) , [ y ( ε ) / y ] Ψ (cid:32) x : S (cid:96) C Ψ , [ ε/ y ] Ψ Weakening with TC − I : x : S (cid:96) C Ψ , mkc ( ε, y ) : T − T , [ y ( ε ) / y ] Ψ (cid:32) x : S (cid:96) C Ψ , ε : T − T , [ ε/ y ] Ψ Weakening with TC H E : x : S (cid:96) C Ψ , let H y = ε in Ψ (cid:32) x : S (cid:96) C Ψ , [ ε/ y ] Ψ Figure 9: Reductions for Non-linear Terms Continuedoccurrences of mkc ( t , y ), as a result of a contraction / cut reduction in a derivation. Such a context maybe the conclusion of the following derivation, if x = x , y = y ; here t = false x , t = false x : z : 0 (cid:96) C z : 0 x : S (cid:96) C x : S y : T (cid:96) C y : Tx : S (cid:96) C mkc ( x , y ) : S − T , y ( x ) : T x : A (cid:96) C x : A y : B (cid:96) C y : Bx : S (cid:96) C mkc ( x , y ) : S − T , y ( x ) : Tz : 0 (cid:96) C , mkc ( t , y ) : S − T , mkc ( t , y ) : S − T , y ( t ) : T , y ( t ) : T A formal notion of α conversion has been proposed for this notion of binding in untyped linearcontexts in [3]. Here (capture-avoiding) substitution, denoted by [ t / x ] t , [ e / x ] t , [ t / x ] e , and [ e / x ] e ,is defined in the usual way. We extend capture-avoiding substitution to multisets in the followingway: • [ t · . . . · t n / z ] s = [ t / z ] s · . . . · [ t n / z ] s • [ t · . . . · t n / z ] p = [ t / z ] p (cid:107) . . . (cid:107) [ t n / z ] p , where p is a p -termThe extension of the other flavors of substitution to multisets are similar. Standard extension ofsubstitution to contexts was also necessary.Finally, there are several commuting conversions that are required for reduction, for example,the following is one: y : T (cid:96) C Ψ , t : T + T x : S (cid:96) C Ψ , t : T + T z : T (cid:96) C Ψ , t : T + T x : S (cid:96) C Ψ , case t of y . t , z . t : T + T v : T (cid:96) C Ψ v : T (cid:96) C Ψ x : S (cid:96) C Ψ , case ( case t of y . t , z . t ) of v . Ψ , v . Ψ
58 HARLEY EADES III AND GIANLUIGI BELLIN commutes to x : S (cid:96) C Ψ , t : T + T Π Π x : S (cid:96) C Ψ , case t of y . ( Ψ , case t of v . Ψ , v . Ψ ) , y . ( Ψ , case t of v . Ψ , v . Ψ )where Π : y : T (cid:96) C Ψ , t : T + T v : T (cid:96) C Ψ v : T (cid:96) C Ψ y : T (cid:96) C Ψ , case t of v . Ψ , v . Ψ Π : y : T (cid:96) C Ψ , t : T + T v : T (cid:96) C Ψ v : T (cid:96) C Ψ y : T (cid:96) C Ψ , case t of v . Ψ , v . Ψ If t = inl s and t = inr s then after commutation y : T (cid:96) C Ψ , case ( inl s ) of v . Ψ , v . Ψ (cid:32) β y : T (cid:96) C Ψ , [ s / v ] Ψ and y : T (cid:96) C Ψ , case ( inr s ) of v . Ψ , v . Ψ (cid:32) β y : T (cid:96) C Ψ , [ s / v ] Ψ There are other commuting conversions as well, but as one can see, due to the complexities introducedin reduction arising from the fact that multiple terms in the context are a ff ected during reductionresults in the commuting conversions from being very compact. The remainder of the commutingconversions can be found in Appendix A. In the next section we give the interpretation of TND intothe categorical model.3.4. Categorical interpretation of rules.
We now turn to the interpretation of Dual LNL Logicinto our categorical model given in Section 2. We structure the proof similarly to Bierman [5], butthe proof itself follows similarly to Benton’s [4] proof for LNL Logic.Given a signature Sg , consisting of a collection of types σ i , where σ i = A or S , and a collectionof sorted function symbols f j : σ , . . . , σ n → τ and given a Symmetric Monoidal Category (SMC)( C , • , , α, λ, ρ, γ ), a structure M for Sg is an assignment of an object [[ σ ]] of L for each type σ and of a morphism [[ f ]] : [[ σ ]] • . . . • [[ σ n ]] → [[ τ ]] for each function f : σ , . . . , σ n → τ of Sg . The types of terms in context ∆ = [ e : A , . . . , e n : A n ] or ∆ = [ t : T , . . . , t n : T n ] areinterpreted into the SMC as [[ σ , σ , . . . , σ n ]] = ( . . . ([[ σ ]] • [[ σ ]]) . . . ) • [[ σ n ]]; left associativity isalso intended for concatenations of type sequences Γ , ∆ . Thus, we need the “book-keeping” functions Split ( Γ , ∆ ) : [[ Γ , ∆ ]] → [[ Γ ]] • [[ ∆ ]] and Join ( Γ , ∆ ) : [[ Γ ]] • [[ ∆ ]] → [[ Γ , ∆ ]] inductively defined usingthe associativity laws α and its inverse α − (cfr Bierman 1994, given also in Bellin 2015).The semantics of terms in context is then specified by induction on terms:[[ x : A (cid:96) L x : A ]] = d f id [[ A ] [[ x : A (cid:96) L f ( e , . . . , e n ) : B ]] = d f [[ x : A (cid:96) L e : A ]] • . . . • [[ x : σ (cid:96) L e n : A n ]]; [[ f ]]and similarly with non-linear types. Following this one then proves by induction on the typederivation that substitution in the term calculus corresponds to composition in the category ([5],Lemma 13). In this subsection only we use the symbol • and 1 for the monoidal binary operation and its unit in the categoricalstructure, distinguished from the ⊕ and ⊥ symbols in the formal language. We shall show that the interpretation of ⊕ isisomorphic to the operation • , so we shall be able to identify them ( and similarly for ⊥ and 1). COINTUITIONISTIC ADJOINT LOGIC 29
In the mixed sequents x : A (cid:96) L ∆ ; Ψ , t : T of TND non-linear terms are interpreted through thefunctor J : C → L . Thus, we have the following: x : A (cid:96) L ∆ ; Ψ , t : T = x : A (cid:96) L ∆ , J Ψ , J t : J T ; · Let M be a structure for a signature Sg in a SMC L . Equations in context will be denoted by x : A (cid:96) L Γ , e = e : B ; Ψ and x : S (cid:96) C Ψ , t = t : T , and are both defined to be the reflexive,symmetric, and transitive closure of the reduction relations defined by the rules in Figure 7 andFigure 8 respectively. Given such an equation: x : A (cid:96) L Γ , e = e : B ; Ψ we say that the structure satisfies the equation if it assigns the same morphisms to x : A (cid:96) L Γ , e : B ; Ψ .and to x : A (cid:96) L Γ , e : B ; Ψ . Similarly, M satisfies x : S (cid:96) C Ψ , t = t : T if it assigns thesame morphism to x : S (cid:96) C Ψ , t : T and to x : S (cid:96) C Ψ , t : T . Then given an algebraic theory Th = ( Sg , Ax ), a structure M for Sg is a model for Th if it satisfies all the axioms in Ax .We now go through some cases of the rules in TND to specify their categorical interpretation soas to satisfy the equations in context and to prove consistency of TND, and hence, DLNL logic in themodel. We do not give every case, but the ones we do not give are similar to the ones given here. Weanalyze the linear connectives, giving an argument for co-ILL that is analogue to Bierman’s for ILL.We conclude that as expected: • the cotensor par can be identified with the bifunctor • of the structure; • linear subtraction •− is the left adjoint to the bifunctor • ; • the unit ⊥ can be identified with 1.3.4.1. Linear Disjunction.
The introduction rule for Par is of the form x : A (cid:96) L ∆ , e : B , e : C ; Ψ x : A (cid:96) L ∆ , e ⊕ e : B ⊕ C ; Ψ TLL ⊕ I This suggests an operation on Hom-sets of the form: Φ A , ∆ J Ψ : L ( A , ∆ • ( B • C ) • J Ψ ) → L ( A , ∆ • B ⊕ C • J Ψ ) natural in ∆ , A and J Ψ . Given e : A → ∆ • ( B • C ) • J Ψ , a : A (cid:48) → A h : ∆ → ∆ (cid:48) , and p : J Ψ → J Ψ (cid:48) ,naturality yields: Φ A (cid:48) , ∆ (cid:48) , J Ψ (cid:48) ( a ; e ; h • ( id B • id C ) • p ) = a ; Φ A , ∆ , J Ψ ( e ); h • id B ⊕ C • p In particular, suppose we have d : A → ∆ • ( B • C ) • J Ψ , and let e = id ∆ • ( id B • id C ) • id J Ψ , h = id ∆ , and p = id J Ψ . Then we have Φ A , ∆ , J Ψ ( d ) = d ; Φ ( ∆ • ( B • C ) • J Ψ ) , ∆ , J Ψ ( id ∆ • ( id B • id C ) • id J Ψ ). By functoralityof • we have id B • id C = id B • C . Hence, writing (cid:76) for Φ ( ∆ • ( B • C ) • J Ψ ) , ∆ , J Ψ ( id ∆ • id B • C • id J Ψ ) we have Φ A , ∆ , J Ψ ( d ) = d ; (cid:76) . Finally, given the morphism ψ ∆ , B , C , P : (( ∆ • B ) • C ) • J Ψ → ∆ • ( B • C ) • J Ψ ,which is natural in all arguments and is definable using Split and
Join , we define: (cid:74) x : A (cid:96) L ∆ , e ⊕ e : B ⊕ C , J Ψ ; · (cid:75) = d f (cid:74) x : A (cid:96) L ∆ , e : B , e : C , J Ψ ; · (cid:75) ; ψ ; (cid:77) . The Par elimination rule has the form Notice that given a sequent x : A (cid:96) L ∆ ; Ψ where ∆ = e : A , . . . , e n : A n and Ψ = t : T , . . . , t m : T m we write L ( A , ∆ • J Ψ ) for the Hom-set L ([[ A ]] , [[ A ]] • . . . • [[ A n ]] • J [[ T ]] • . . . • J [[ T m ]]) . z : A (cid:96) L ∆ , e : B ⊕ C ; Ψ x : B (cid:96) L ∆ ; Ψ y : C (cid:96) L ∆ ; Ψ z : A (cid:96) L ∆ , [ casel ( e ) / x ] ∆ , [ caser ( e ) / y ] ∆ ; Ψ , [ casel ( e ) / x ] Ψ , [ caser ( e ) / y ] Ψ TLL ⊕ E This suggests an operation on Hom-sets of the form Ψ A , ∆ , J Ψ : L ( A , B ⊕ C • ∆ • J Ψ ) × L ( B , ∆ • J Ψ ) × L ( C , ∆ • J Ψ ) → L ( A , ∆ • J Ψ ) natural in A , ∆ , J Ψ where we write ∆ = ∆ • ∆ • ∆ and J Ψ = J Ψ • J Ψ • J Ψ . Given the followingmorphisms: g : A → B ⊕ C • ∆ • J Ψ e : B → ∆ • J Ψ f : C → ∆ • J Ψ a : A (cid:48) → A d : ∆ → ∆ (cid:48) d : ∆ → ∆ (cid:48) d : ∆ → ∆ (cid:48) p : J Ψ → J Ψ (cid:48) p : J Ψ → J Ψ (cid:48) p : J Ψ → J Ψ (cid:48) naturality yields: Ψ A (cid:48) , ∆ (cid:48) , Γ (cid:48) , J Ψ (cid:48) (cid:0) ( a ; g ; id B ⊕ C • d • p ) , ( e ; d • p ) , ( f ; d • p ) (cid:1) = a ; Ψ A , ∆ , J Ψ ( g , e , f ); d • d • d • p • p • p ; Join ( ∆ (cid:48) , J Ψ (cid:48) ) . In particular, set e = id B , f = id C , a = id A , d i = id ∆ i , and p i = id J Ψ i , and we get Ψ A , ∆ J Ψ ( g , e , f ) = Ψ A , ∆ , J Ψ ( g , id B , id C ); id ∆ • c • d ; Join ( ∆ , J Ψ )where the operation Join implements the required associativity. Writing ( x ) ∗ for Ψ D , ∆ ( x , id B , id C )we define[[ z : A (cid:96) L ∆ , [ casel e / x ] ∆ , [ caser e / y ] ∆ ; Ψ , [ casel e / x ] Ψ , [ caser e / y ] Ψ ]] = d f [[ z : A (cid:96) L ∆ , e : B ⊕ C ; Ψ ]] ∗ ; ( id ∆ • [[ x : B (cid:96) L ∆ ; Ψ ]] • [[ y : C (cid:96) L ∆ ; Ψ ]]); Join ( ∆ , J Ψ ).We now turn to the equations in context. Consider the following case: e ≡ casel ( e ⊕ e ) e (cid:48) ≡ caser ( e ⊕ e ) | ∆ | = | ∆ (cid:48) | | Ψ | = | Ψ (cid:48) | y : A (cid:96) L ∆ ; J Ψ = ∆ (cid:48) ; J Ψ (cid:48) | ∆ | = | ∆ (cid:48) | | Ψ | = | Ψ (cid:48) | x : A (cid:96) L ∆ ; J Ψ = ∆ (cid:48) ; J Ψ (cid:48) | ∆ | = | ∆ (cid:48) | | Ψ | = | Ψ (cid:48) | z : B (cid:96) L e : A , e : A , ∆ ; J Ψ = e (cid:48) : A , e (cid:48) : A , ∆ (cid:48) ; J Ψ (cid:48) z : B (cid:96) L ∆ , [ e / x ] ∆ , [ e (cid:48) / x ] ∆ ; Ψ , [ e / x ] Ψ , [ e (cid:48) / x ] Ψ = ∆ (cid:48) , [ e (cid:48) / x ] ∆ , [ e (cid:48) / x ] ∆ ; Ψ , [ e (cid:48) / x ] Ψ (cid:48) , [ e (cid:48) / x ] Ψ (cid:48) ⊕ - β Let q : B → A • A • ∆ • J Ψ , m : A → ∆ • J Ψ and n : A → ∆ • J Ψ . Then to satisfy the above equations in context we need that the following diagram commutes: B q (cid:47) (cid:47) ∆ • J Ψ • ( A • A ) ⊕ (cid:15) (cid:15) id ∆ • m • n (cid:47) (cid:47) ∆ • J Ψ • ∆ • J Ψ • ∆ • J Ψ ∆ • J Ψ • A ⊕ B ∗ (cid:47) (cid:47) ∆ • J Ψ • A • B id ∆ • m • n (cid:79) (cid:79) We make the assumption that the above decomposition is unique. Moreover, supposing ∆ to beempty and m = id A , n = id B , q = id A • id B = id A • B we obtain ( id A • id B ; (cid:76) ) ∗ = id A • id B andsimilarly ( id A ⊕ B ) ∗ ; (cid:76) = id A ⊕ B ; hence we may conclude that there is a natural isomorphism D → Γ • A • BD → Γ • A ⊕ B COINTUITIONISTIC ADJOINT LOGIC 31 so we can identify • and ⊕ . Finally we see that the following η equation in context is also satisfied: ⊕ − η rule | ∆ | = | ∆ (cid:48) | | Ψ | = | Ψ (cid:48) | z : B (cid:96) L ∆ ; Ψ = ∆ (cid:48) ; Ψ (cid:48) z : B (cid:96) L ( casel e ⊕ caser e ) : A ⊕ A , ∆ ; Ψ = e : A ⊕ A , ∆ (cid:48) ; Ψ (cid:48) (3.1)3.4.2. Linear subtraction.
Subtraction introduction.
The introduction rule for subtractionhas the form: x : A (cid:96) L ∆ , e : B ; Ψ y : C (cid:96) L ∆ ; Ψ | Ψ | = | Ψ | x : A (cid:96) L ∆ , mkc ( e , y ) : B (cid:21) C , [ y ( e ) / y ] ∆ ; Ψ , [ y ( e ) / y ] Ψ TLL (cid:21) I This suggests a natural transformation with components: Φ A , ∆ , J Ψ : L ( A , ∆ • B • J Ψ ) × L ( C , ∆ • J Ψ ) → L ( A , ∆ • ( B •− C ) • ∆ • J Ψ • J Ψ )natural in A , ∆ , ∆ , J Ψ , J Ψ . Taking morphisms e : A → ∆ • B • J Ψ , f : C → ∆ • J Ψ and also a : A (cid:48) → A , d : ∆ → ∆ (cid:48) , d : ∆ → ∆ (cid:48) , p : J Ψ → J Ψ (cid:48) , p : J Ψ → J Ψ (cid:48) , by naturalitywe have Φ A (cid:48) , ∆ (cid:48) , ∆ (cid:48) , J Ψ (cid:48) J Ψ (cid:48) (( a ; e ; d • id B • p ) , ( f ; d ; p )) == a ; Φ A , ∆ , J Ψ ( e , f ); d • d • id B •− C ; Join ( ∆ (cid:48) , ∆ (cid:48) , B •− C , J Ψ (cid:48) , J Ψ (cid:48) )In particular, taking a = id A , d = id ∆ , p = id J Ψ , p = id J Ψ but d : C → ∆ · J Ψ and f = id C we have: Φ A , ∆ , ∆ , J Ψ , J Ψ ( e , d ) = Φ A , ∆ ( e , id C ); id ∆ • d • id A •− B • id J Ψ • idJ Ψ ; Join ( ∆ , ∆ , A •− B , J Ψ , J Ψ )Writing MKC CA , ∆ , J Ψ ( e ) for Φ A , ∆ J Ψ ( e , id C ), Φ A , ∆ , J Ψ ( e , d ) can be expressed as the composition MKC CA , ∆ , J Ψ ( e ); id ∆ • d • id B •− C where MKC CA , ∆ , J Ψ is a natural transformation with components L ( A , ∆ • B • J Ψ ) × L ( C , C ) → L ( A , ∆ • C • C •− C )so we make the definition[[ x : A (cid:96) L ∆ , mkc ( e , y ) : B •− C , [ y ( e ) / y + ∆ ; Ψ · [ y ( e ) / y ] Ψ ]] = d f MKC CA , ∆ , J Ψ [[ x : A (cid:96) L ∆ , e : B ]]; id ∆ • [[ y : C (cid:96) L ∆ ; Ψ ]] • id B •− C ; Join ( ∆ , ∆ , B •− C , J Ψ , J Ψ )Notice that MKC CA , ∆ , J Ψ corresponds to the one-premise form of the subtraction introduction rule x : A (cid:96) L ∆ , e : B ; Ψ TLL (cid:21) I x : A (cid:96) L ∆ , mkc ( e , y ) : B •− C , y ( e ) : C ; Ψ which is equivalent in terms of provability to the more general form considered here [12].The subtraction elimination rule has the form: x : A (cid:96) L ∆ , e : B (cid:21) C ; Ψ y : B (cid:96) L e : C , ∆ ; Ψ | Ψ | = | Ψ | x : A (cid:96) L postp ( y (cid:55)→ e , e ) , ∆ , [ y ( e ) / y ] ∆ ; Ψ , [ y ( e ) / y ] Ψ TLL (cid:21) E This suggests a natural transformation with components Ψ A , ∆ , ∆ , J Ψ , J Ψ : L ( A , ∆ • ( B •− C ) • J Ψ ) × L ( B , C • ∆ • J Ψ ) → L ( A , ∆ • ∆ • J Ψ • J Ψ )natural in A , ∆ , ∆ , J Ψ , J Ψ . Here postp ( y (cid:55)→ e , e ) is given type 1 and an application of leftidentity λ , ∆ is assumed implicitly.Given e : A → ∆ • ( B •− C ) • J Ψ , f : B → C • ∆ • J Ψ and also a : A (cid:48) → A , d : ∆ → ∆ (cid:48) , d : ∆ → ∆ (cid:48) , p : J Ψ → J Ψ (cid:48) p : J Ψ → J Ψ (cid:48) naturalityyields Ψ A (cid:48) , ∆ (cid:48) , ∆ (cid:48) , J Ψ (cid:48) , J Ψ (cid:48) (( a ; e ; d • id B •− C • p ) , ( f ; id C • d • p )) = a ; Ψ A , ∆ , ∆ , J Ψ ( e , f ); λ , ∆ • d • d • p • p ; Join ( ∆ (cid:48) , ∆ (cid:48) , J Ψ , J Ψ )In particular, taking a : A → ∆ • ( B •− C ), e = id ∆ • ( B •− C ) , d = id ∆ , d : id ∆ , p = id J Ψ , p = idJ Ψ we obtain Ψ A , ∆ , ∆ ( a , f ) = a ; Ψ A , ∆ , ∆ ( id ∆ • ( C •− D ) • id J Ψ , f ); Join ( ∆ , ∆ , J Ψ , J Ψ )Writing POSTP ( f ) for Ψ A , ∆ , ∆ , J Ψ , J Ψ ( id ∆ • ( B •− C ) • J Ψ , f ) we define[[ x : A (cid:96) L ∆ , postp ( y (cid:55)→ e , e ) , [ y ( e ) / y ] ∆ ; Ψ , [ y ( e ) / y ] , Ψ ]] = d f [[ x : A (cid:96) L ∆ , e : B •− C ]]; id ∆ • POSTP [[ y : B (cid:96) L e : C , ∆ ; Ψ ]]; Join ( ∆ , ∆ , J Ψ , J Ψ )3.4.2.3. Equations in context.
We have equations in context of the form e z ≡ z ( mkc ( e , y )) | ∆ | = | ∆ (cid:48) | e p ≡ postp ( z (cid:55)→ e , mkc ( e , y )) | ∆ | = | ∆ (cid:48) | x : B (cid:96) L e : A , ∆ ; Ψ = e (cid:48) : A , ∆ (cid:48) ; Ψ (cid:48) | Ψ | = | Ψ (cid:48) | y : A (cid:96) L ∆ ; Ψ = ∆ (cid:48) ; Ψ (cid:48) | Ψ | = | Ψ (cid:48) | z : A (cid:96) L e : A , ∆ ; Ψ = e (cid:48) : A , ∆ (cid:48) ; Ψ (cid:48) x : B (cid:96) L ∆ , e p , [ y ( e ) / y ] ∆ , [ e z / z ] ∆ ; Ψ , [ y ( e ) / y ] Ψ , [ e z / z ] Ψ =∆ (cid:48) , [[ e (cid:48) / z ] e (cid:48) / y ] ∆ , [ e (cid:48) / z ] ∆ (cid:48) ; Ψ (cid:48) , [[ e (cid:48) / z ] e (cid:48) / y ] Ψ (cid:48) , [ e (cid:48) / z ] Ψ (cid:48) •− − β We repeat the derivations of the redex and of the reductum.
Redex: x : B (cid:96) L e : A , ∆ ; Ψ y : A (cid:96) L ∆ ; Ψ x : B (cid:96) L mkc ( e , y ) : A •− A , ∆ , [ y ( e ) / y ] ∆ ; Ψ , [ y ( e ) / y ] Ψ z : A (cid:96) L e : A , ∆ ; Ψ x : B (cid:96) L ∆ , e p (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) postp ( z (cid:55)→ e , mkc ( e , y )) , [ y ( e ) / y ] ∆ , [ e z (cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125)(cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123) z ( mkc ( e , y )) / z ] ∆ ; Ψ , [ y ( e ) / y ] Ψ , [ z ( mkc ( e , y ) / z )] : Ψ Reductum: x : B (cid:96) L e (cid:48) : A , ∆ (cid:48) ; Ψ (cid:48) z : A (cid:96) L e (cid:48) : A , ∆ (cid:48) ; Ψ (cid:48) x : B (cid:96) L ∆ (cid:48) , [ e (cid:48) / z ] ∆ (cid:48) , [ e (cid:48) / z ] e (cid:48) : A ; Ψ (cid:48) , [ e (cid:48) / z ] Ψ (cid:48) y : A (cid:96) L ∆ ; Ψ x : B (cid:96) L ∆ (cid:48) , [[ e (cid:48) / z ] e (cid:48) / y ] ∆ (cid:48) , [ e (cid:48) / z ] ∆ (cid:48) ; Ψ (cid:48) , [[ e (cid:48) / z ] e (cid:48) ] Ψ (cid:48) , [ e (cid:48) / z ] Ψ (cid:48) Given morphisms n : B → ∆ • A and m : A → ∆ • A , for these equations to be satisfied weneed the following diagram to commute (omitting non-linear terms): B MKC A ( n ) (cid:15) (cid:15) n (cid:47) (cid:47) ∆ • A id ∆ • m (cid:15) (cid:15) ∆ • ( A •− A ) • A POSTP ( m ) • id A (cid:47) (cid:47) ∆ • ∆ • A in particular, taking n = id A we have COINTUITIONISTIC ADJOINT LOGIC 33 A MKC A ( id A ) (cid:15) (cid:15) m (cid:47) (cid:47) ∆ • A ( A •− A ) • A POSTP ( m ) • id A (cid:56) (cid:56) Assuming the above decomposition to be unique, we can show that the η equation in context is alsosatisfied: | ∆ | = | ∆ (cid:48) | | Ψ | = | Ψ (cid:48) | z ; B (cid:96) L ∆ , Ψ = ∆ (cid:48) , Ψ (cid:48) z : B (cid:96) L postp ( x (cid:55)→ y , e ) , mkc ( x ( e ) , y ) : A •− A , ∆ , Ψ = e : A •− A , ∆ (cid:48) ; Ψ (cid:48) (3.2)and conclude that there is a natural isomorphism between the maps A → ∆ • BA •− B → ∆ i.e., that •− is the left adjoint to the bifunctor • .3.4.3. Functors.
Recall that a model of Linear-Non Linear co-intuitionistic logic consists of asymmetric comonoidal adjunction L : H (cid:97) J : C where L = ( L , ⊥ , ⊕ , −◦ ) is a symmetric monoidalcoclosed category and C = ( C , , + , − ) is a cocartesian coclosed category.We use the same symbols for the functors J : C → L and H : L → C in the models and for theoperators that represent them in the language.3.4.3.1 rules for J : C → L . T L J I x : A (cid:96) L , ∆ ; t : T , Ψ x : A (cid:96) L ∆ , Jt : JT ; Ψ (3.3)If ∆ = R : | ∆ | and Ψ = S : | Ψ | , then the categorical interpetation of the rule is an application of α − : A R • Jt • JS (cid:47) (cid:47) ∆ • JT • J Ψ A ( R • Jt ) • JS (cid:47) (cid:47) ( ∆ • JT ) • J Ψ T L J E eliminationx : A (cid:96) L ∆ , e : JT ; Ψ y : T (cid:96) C Ψ where | Ψ | = | Ψ | x : A (cid:96) L ∆ ; Ψ · let Jy = e in Ψ (3.4)If ∆ = R : | ∆ | , Ψ = R (cid:48) : | Ψ | , Ψ = S : | Ψ | , then the categorical interpretation of the rule is given byan operation of the form L ( A , ∆ • JT • J Ψ ) × C ( T , Ψ ) → L ( A , ∆ • J Ψ • J Ψ )given by the following compositions A R • e • JR (cid:48) (cid:47) (cid:47) ∆ • JT • J Ψ T S (cid:47) (cid:47) Ψ in C JT JS (cid:47) (cid:47) J Ψ in L A R • e • J ( R (cid:48) ) (cid:47) (cid:47) ∆ • J ( T ) • J ( Ψ ) id ∆ • J ( S ) • id J ( Ψ (cid:47) (cid:47) ∆ • J ( Ψ ) • J ( Ψ ) id ∆ • j − Ψ Ψ (cid:47) (cid:47) ∆ • J ( Ψ + Ψ )since | Ψ | = | Ψ | , id ∆ •∇ Ψ (cid:47) (cid:47) ∆ • J ( Ψ ) rules for H : L → C . T C H I x : A (cid:96) L R : ∆ , e : B ; Ψ x : A (cid:96) L R : ∆ ; He : HB , Ψ (3.5)Let ∆ = R : | ∆ | and Ψ = S : | Ψ | : then A R ⊕ e ⊕ J ( S ) (cid:47) (cid:47) ∆ • B • J ( Ψ ) using η B : B → JHBA R ⊕ JH ( e ) • J ( S ) (cid:47) (cid:47) ∆ • JH ( B ) • J ( Ψ ) T C H E x : B (cid:96) L ∆ ; t : HA , Ψ y : A (cid:96) L ; Ψ where | Ψ | = | Ψ | x : B (cid:96) L ∆ ; Ψ , let Hy = t in Ψ (3.6)The categorical interpretation of H elim is as follows: Let Ψ = R : | Ψ | and Ψ = S : | Ψ | .Then we have the following compositions: S t + R (cid:47) (cid:47) H ( A ) + Ψ A J ( S ) (cid:47) (cid:47) J ( Ψ ) in L HA HJ ( S ) (cid:47) (cid:47) H J ( Ψ ) (cid:15) Ψ (cid:47) (cid:47) Ψ in C S t + R (cid:47) (cid:47) H ( A ) + Ψ HJ ( S ) + id Ψ (cid:47) (cid:47) Ψ + Ψ ∇ Ψ (cid:47) (cid:47) Ψ
4. R elated and F uture W ork The most comprehensive treatment of ILL is in Gavin Bierman’s thesis [5]. There one finds theProof Theory (Chapter 2), i.e, the sequent calculus with cut-eliminaton, natural deduction andaxiomatic versions of ILL. Then (Chapter 3) a term assignment to the natural deduction and to thesequent calculus versions are presented with β -reductions and commutative conversions, and strongnormalization and confluence are proved for the resulting calculus. A painstaking analysis of therules of the labeled calculus leads to the construction of a categorical model of ILL, a linear category ,in particular of the exponential part, a main contribution of Bierman and of the Cambridge schoolof the 1990s with respect to previous models by Seely and Lafont. Bellin [3] presents a categoricalmodel of co-intuitionistic linear logic based on a dualization of Bierman [5] construction for ILL.Benton’s work [4] on LNL logic presents the categorical model for Linear-Non-Linear Intuition-istic logic LNL. Chapter 2 shows how to obtain a LNL model from a Linear Category and viceversa.Versions of the sequent calculus for LNL are considered and cut-elimination is proved for one suchversion. Then Natural Deduction is given with term assignment and the categorical interpretationof a fragment of the natural deduction system. Then β -reductions and commuting conversions arepresented. The present work follows Benton’s paper aiming at a (non-trivial) dualization of it.Bi-intuitionistic logic was introduced by C.Rauszer [24] with an algebraic and Kripke semantics[25] and a Gentzen style sequent calculus [23]. Co-intuitionistic logic requires a multiple conclusionsystem, because of the cotensor in the linear case and of contraction right in the non-linear one. Thisraises the problem of the relations between intuitionistic implication and disjunction, and, dually,between subtraction and conjunction. In the case of the logic FILL that extends ILL with the cotensor COINTUITIONISTIC ADJOINT LOGIC 35 ( par ) applying Maheara and Dragalin’s restriction that only one formula occurs in the succedentof the premise of an implication right, yields a calculus that does not satisfies cut-elimination, asnoticed by Schellinx [26]. Similarly, in the logic BILL ( Bi-Intuitionistic Linear Logic ) requiring thatonly one formula occurs in the antecedent of the premise of a subtraction left yields a system thatdoes not satisfy cut-elimination. Γ , A (cid:96) B −◦ R Γ (cid:96) A −◦ B A (cid:96) B , ∆ •− E A •− B (cid:96) ∆ As a simple counterexample, consider the sequent p ⇒ q , r → (( p − q ) ∧ r ) given by Pinto andUustalu around 2003 [21], which is provable with cut but not cut-free with Dragalin’s restrictions.Hyland and de Paiva introduced a sequent calculus for FILL labeled with terms y : Γ , x : A (cid:96) t : B , u : ∆ −◦ R y : Γ (cid:96) λ x : T A −◦ B , u : ∆ where x : A occurs in t : B if and only if there is an “essential dependency” of B from A . Therestriction on the −◦ I is that x does not occur in the terms u : ∆ . The original term assignment didnot guarantee cut-elimination, as noticed by Bierman [6]; the assignment to par left ( ⊕ L) had to befine tuned, as indicated by Bellin [1].A detailed presentation of the term calculus for FILL with a full proof of cut elimination byEades and de Paiva is in [17], where the correctness for a categorical semantics for FILL is alsoproved. Another correct formalization of FILL, a sequent calculus with a relational annotation, wasgiven by Bra¨uner and de Paiva [7], with a proof of cut-elimination. The second author [1] gave asystem of proof nets for FILL which sequentialize in the sequent calculus with term assignment; theessential fact here is that x : A occurs in t : B if and only if there is a “directed chain” between A andB in the proof structure.
Here cut elimination is proved by reduction to cut-elimination for proof nets.A system of two-sided proof nets (in the style of natural deduction) was given by Cockett andSeely [10]. For Bi-Intuitionistic Linear Logic, they gave also a system of proof nets, correspondingto a sequent calculus without annotations and restrictions that therefore collapses into classicalMLL. Recently, Clouston, Dawson, Gore and Tiu [8] gave an annotation-free formalization forBILL, alternative to sequent calculi, in the form of deep-inference and display calculi for BILL. Thiscalculus enjoys cut-elimination and is relevant to the categorical semantics bi-intuitionistic linearlogic. Annotation-free formalizations of Bi-Intuitionistic Logic use the display calculus [15], nestedsequents [16] and deep inference [22].Tristan Crolard [11, 12] made an in-depth study of Rauszer’s logic. In [11] he showed thatmodels of Rauszer logic (called “subtractive logic”) based on bi-cartesian closed categories (withco-exponents) collapse to preorders. He also studied models of subtractive logic and showed that itsfirst order theory is constant-domain logic, thus it is not a conservative extension of intuitionisticlogic.Crolard [12] develops the type theory for subtractive logic, extending a system of multipleconclusion classical natural deduction with a connective of subtraction and then decorating proofswith a system of annotations of dependencies that allows us to identify “constructive proofs”: theseare derivations where only the premise of an implication introduction depends on the dischargedassumption and only the premise of a subtraction elimination depends on the discharged conclusion.Therefore Crolard’s sequent calculus with annotations is not a ff ected by the counterexamples tocut-eliminations.The type theory is Parigot λµ -calculus extended with operators for sums, products and subtrac-tion, where the operators for subtraction introduction and elimination are understood as a calculus ofco-routines. A constructive system of co-routines is then obtained by imposing restrictions on termscorresponding to the restrictions on constructive proofs. In a series of papers the second author gave a “pragmatic” interpretation of bi-intuitionism,where intuitionistic and co-intuitionistic logic are interpreted as logics of the acts of assertion andmaking a hypothesis, respectively, the interactions between the two sides depending on negations, see[2]. Here the separation between intuitionistic and co-intuitionistic logic and their models is givena linguistic motivation. Writing (cid:96) p for the type of assertions that p is true and using intuitionisticconnectives with the BHK interpretation, one gives a “pragmatic interpretatiion” of ILL, where anexpression A is justified or unjustified [13]. Similarly, writing H p for the type of hypotheses that p is true, and using co-intuitionistic connectives, one builds a co-intuitionistic language, for whichan analogue “pragmatic interpretation” has been attempted. Both languages may be given a modalinterpretattion in S4, with ( (cid:96) p ) M = (cid:3) p and ( H p ) M = ♦ p . Notice that here there is a semanticduality between an assertion (cid:96) p and a hypothesis H ¬ p , as (cid:3) p and ♦¬ p are contradictory. Similarlythere is a semantic duality between H p and (cid:96) ¬ p , but not between (cid:96) p and the hypothesis H p .A useful direction of research in the proof theory of bi-intuitionism may be the investigation therelations between co-intuitionistic proofs and intuitionistic refutations.It is in this context that a term assignment for co-intuitionistic logic has been developed, startingfrom Crolard’s definition but independently of the λµ -framework. This calculus was used here as aterm assignment of Dual LNL logic.Tra ff ord [27] defines an interpretation of co-intuitionistic logic into a topos-theoretic model torepresent both proofs, in an elementary topoi, and refutations, in a complement topoi. He then showsthat classical logic can be simulated in his model. Earlier Estrada-Gonz´alez [14] gave a sequentcalculus for BINT based on complement topoi.Finally, to achieve the project outlined in the introduction of putting together intuitionistic andco-intuitionistic adjoint logic in the environment of BILL the definition of a suitable syntax for BILLwill play a key role. R eferences [1] Gianluigi Bellin. Subnets of proof-nets in multiplicative linear logic with MIX. Mathematical Structures inComputer Science , 7(6):663–699, 1997. URL: http://journals.cambridge.org/action/displayAbstract?aid=44699 .[2] Gianluigi Bellin.
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Journal of Logic and Computation , 1(4):537–559,1991.[27] James Tra ff ord. Structuring co-constructive logic for proofs and refutations. Logica Universalis , 10(1):67–97, 2016. A ppendix A. C ommuting C onversions Non linear rules. (1) disjunction intro TC + I and TC + I commute upwards with every inference and the terms obtainedare the same.(2) disjunction elim TC + E commutes upwrds with inferences in the derivation of the major premise,the terms assigned to the resulting subderivations are equated. For instance y : T (cid:96) C Ψ , t : T + T z : T (cid:96) C Ψ , t : T + T x : S (cid:96) C Ψ , t : T + T | Ψ | = | Ψ | x : S (cid:96) C Ψ , case t of y . Ψ , z . Ψ , case t of y . t , z . t : T + T v : T (cid:96) C Ψ v : T (cid:96) C Ψ | Ψ | = | Ψ | x : S (cid:96) C Ψ , case t of y . Ψ , z . Ψ , case ( case t of y . t , z . t ) of v . Ψ , v . Ψ commutes to x : S (cid:96) C Ψ , t : T + T | Ψ | = | Ψ | v : T (cid:96) C Ψ v : T (cid:96) C Ψ y : T (cid:96) C Ψ , t : T + T y : T (cid:96) C Ψ , case t of v . Ψ , v . Ψ | Ψ | = | Ψ | v : T (cid:96) C Ψ v : T (cid:96) C Ψ z : T (cid:96) C Ψ , t : T + T z : T (cid:96) C Ψ , case t of v . Ψ , v . Ψ x : S (cid:96) C Ψ , case t of y . ( Ψ , case t of v . Ψ , v . Ψ ) , z . ( Ψ , case t of v . Ψ , v . Ψ ) Remark 48. If t = inl s and u = inr s then after commutation y : T (cid:96) C Ψ , case inl s of v . Ψ , v . Ψ (cid:32) β y : T (cid:96) C Ψ , [ s / v ] Ψ y : T (cid:96) C Ψ , case inr s of v . Ψ , v . Ψ (cid:32) β y : T (cid:96) C Ψ , [ s / v ] Ψ (3) Subtraction introduction TC − I commutes upwards with inferences in both branches with anyinference I : x : S (cid:96) C t (cid:48) : T , Ψ (cid:48) I x : S (cid:96) C t : T , Ψ y : T (cid:96) C Ψ TC − I x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , [ y ( t ) / y ] Ψ x : S (cid:96) C t : T , Ψ y : T (cid:96) C Ψ (cid:48) I y : T (cid:96) C Ψ TC − I x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , [ y ( t ) / y ] Ψ commutes to commutes to x : S (cid:96) C t : T , Ψ (cid:48) y : T (cid:96) C Ψ TC − I x : S (cid:96) C Ψ (cid:48) , mkc ( t , y ) : T − T , [ y ( t (cid:48) ) / y ] Ψ I x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , [ y ( t ) / y ] Ψ x : S (cid:96) C t : T , Ψ y : T (cid:96) C Ψ (cid:48) TC − I x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , Ψ (cid:48) I x : S (cid:96) C Ψ , mkc ( t , y ) : T − T , Ψ (4) Subtraction elimination TC − E commutes upwards. For instance, x : S (cid:96) C w : S , Ψ z : S (cid:96) C Ψ , t : T − T x : S (cid:96) C Ψ , mkc ( w , z ) : S − S , [ z ( w ) / z ] Ψ , [ z ( w ) / z ] t : T − T y : T (cid:96) C t : T , Ψ x : S (cid:96) C Ψ , mkc ( w , z ) : S − S , [ z ( w ) / z ] Ψ , postp ( y (cid:55)→ t , [ z ( w ) / z ] t ) , [ y ([ z ( w ) / z ] t ) / y ] Ψ commutes to x : S (cid:96) C w : S , Ψ z : S (cid:96) C Ψ , t : T − T y : T (cid:96) C t : T , Ψ z : S (cid:96) C Ψ , postp ( y (cid:55)→ t , t ) , [ y ( t ) / y ] Ψ x : S (cid:96) C Ψ , mkc ( w , z ) : S − S , [ z ( w ) / z ] Ψ , postp ( y (cid:55)→ t , [ z ( w ) / z ] t ) , [ z ( w ) / z ][ y ( t ) / y ] Ψ where [ z ( w ) / z ][ y ( t ) / y ] Ψ = [ y ([ z ( w ) / z ] t ) / y ] Ψ (A.1)since z (cid:60) Ψ . Linear rules. (5) The ⊥ introduction rule TILL ⊥ I rule commutes with any inference, as connect ⊥ to e can be“rewired” to any term in the context.(6) The commutations of the rules for linear subtraction TILL •− I and TILL •− E are similar to thosefor non-linear subtraction.(7) Linear disjunction ( par ) introduction (TLL ⊕ I ) commutes with any inference. Linear disjunctionelimination (TLL ⊕ E ) also commutes upwards. For example (writing a proof without non-linearparts for simplicity) we have the following: COINTUITIONISTIC ADJOINT LOGIC 39 x : A (cid:96) L ∆ , e : ( B ⊕ B ) y : B (cid:96) L ∆ , e : C ⊕ C v : C (cid:96) L ∆ w : C (cid:96) L ∆ ⊕ E y : B (cid:96) L ∆ , [ casel e / v ] ∆ , [ caser e / w ] ∆ z : B (cid:96) L ∆ ⊕ E x : A (cid:96) L ∆ , [ casel e / y ] ∆ , [ casel e / y ][ casel e / v ] ∆ , [ casel e / y ][ caser e / w ] ∆ , [ caser e / z ] ∆ commutes to x : A (cid:96) L ∆ , e : ( B ⊕ B ) y : B (cid:96) L ∆ , e : C ⊕ C z : B (cid:96) L ∆ ⊕ E x : A (cid:96) L ∆ , [ casel e / y ] ∆ , [ casel e / y ] e : C ⊕ C , [ caser e / z ] ∆ v : C (cid:96) L ∆ w : C (cid:96) L ∆ ⊕ E x : A (cid:96) L ∆ , [ casel e / y ] ∆ , [ casel [ casel e / y ] e / v ] ∆ , [ caser [ casel e / y ] e / w ] ∆ , [ caser e / z ] ∆ Now [ casel e / y ][ casel e / v ] ∆ = [ casel [ casel e / y ] e / v ] ∆ (A.2)because y does not occur in ∆ , only in e and[ casel e / y ][ casel e / w ] ∆ = [ caser [ casel e / y ] e / w ] ∆ because y does not occur in ∆ , only in e .A ppendix B. P roofs
B.1.
Proof of Lemma 23.
We show that both of the maps: j − R , S : = J R ⊕ J S η (cid:47) (cid:47) JH ( J R ⊕ J S ) Jh A , B (cid:47) (cid:47) J ( HJ R + HJ S ) J ( ε R + ε S ) (cid:47) (cid:47) J ( R + S ) j − : = ⊥ η (cid:47) (cid:47) JH ⊥ Jh ⊥ (cid:47) (cid:47) J j R , S : J ( R + S ) (cid:47) (cid:47) J R ⊕ J S and j : ⊥ (cid:47) (cid:47) J j − R , S ; j R , S = id : JHJ R ⊕ JHJ S J ( HJ R + HJ S ) (cid:111) (cid:111) j J R ⊕ J S JHJ R ⊕ JHJ S η ⊕ η (cid:15) (cid:15) J R ⊕ J S JH ( J R ⊕ J S ) η (cid:47) (cid:47) JH ( J R ⊕ J S ) J ( HJ R + HJ S ) Jh (cid:15) (cid:15) J R ⊕ J S JHJ R ⊕ JHJ S (cid:111) (cid:111) J ε ⊕ J ε J R ⊕ J S J R ⊕ J S J R ⊕ J S JHJ R ⊕ JHJ S J R ⊕ J S J ( HJ R + HJ S ) J R ⊕ J S J ( R + S ) (cid:107) (cid:107) j J ( HJ R + HJ S ) J ( R + S ) J ( ε + ε ) (cid:15) (cid:15) The two top diagrams both commute because η and ε are the unit and counit of the adjunctionrespectively, and the bottom diagram commutes by naturality of j .Case. The following diagram implies that j R , S ; j − R , S = id : JHJ ( R + S ) JH ( J R ⊕ J S ) JHj (cid:47) (cid:47) J ( R + S ) JHJ ( R + S ) η (cid:15) (cid:15) J ( R + S ) J R ⊕ J S j (cid:47) (cid:47) J R ⊕ J S JH ( J R ⊕ J S ) η (cid:15) (cid:15) J ( R + S ) JHJ ( R + S ) (cid:111) (cid:111) J ε J ( R + S ) J ( R + S ) J ( R + S ) JHJ ( R + S ) J ( R + S ) JH ( J R ⊕ J S ) J ( R + S ) J ( HJ R + HJ S ) (cid:107) (cid:107) J ( ε + ε ) JH ( J R ⊕ J S ) J ( HJ R + HJ S ) Jh (cid:15) (cid:15) The top left and bottom diagrams both commute because η and ε are the unit and counit of theadjunction respectively, and the top right diagram commutes by naturality of η .Case. The following diagram implies that j − ; j = id : ⊥ J (cid:111) (cid:111) j ⊥⊥⊥ JH ⊥ η (cid:47) (cid:47) JH ⊥ J Jh ⊥ (cid:15) (cid:15) This diagram holds because η is the unit of the adjunction.Case. The following diagram implies that j ; j − = id : J JH ⊥ (cid:111) (cid:111) Jh ⊥ JHJ J J ε (cid:119) (cid:119) JHJ JH ⊥ JHj (cid:39) (cid:39) JHJ J J JHJ η (cid:39) (cid:39) J J J JH ⊥ J J J ⊥ j (cid:47) (cid:47) ⊥ JH ⊥ η (cid:15) (cid:15) The top-left and bottom diagrams commute because η and ε are the unit and counit of the adjunctionrespectively, and the top-right digram commutes by naturality of η .B.2. Proof of Lemma 25.
Since ? is the composition of two symmetric comonoidal functors weknow it is also symmetric comonoidal, and hence, the following diagrams all hold:? A ⊕ ?( B ⊕ C )) ? A ⊕ (? B ⊕ ? C ) id ? A ⊕ r B , C (cid:47) (cid:47) ?( A ⊕ ( B ⊕ C ))? A ⊕ ?( B ⊕ C )) r A , B ⊕ C (cid:15) (cid:15) ?( A ⊕ ( B ⊕ C )) (? A ⊕ ? B ) ⊕ ? C (? A ⊕ ? B ) ⊕ ? C ? A ⊕ (? B ⊕ ? C ) α ? A , ? B , ? C (cid:15) (cid:15) ?( A ⊕ ( B ⊕ C )) (? A ⊕ ? B ) ⊕ ? C ?(( A ⊕ B ) ⊕ C )?( A ⊕ ( B ⊕ C )) ? α A , B , C (cid:15) (cid:15) ?(( A ⊕ B ) ⊕ C ) ?( A ⊕ B ) ⊕ ? C r A ⊕ B , C (cid:47) (cid:47) ?( A ⊕ B ) ⊕ ? C (? A ⊕ ? B ) ⊕ ? C r A , B ⊕ id ? C (cid:15) (cid:15) COINTUITIONISTIC ADJOINT LOGIC 41 ? A ⊥ ⊕ ? A λ − A (cid:47) (cid:47) ?( ⊥ ⊕ A )? A ? λ A (cid:15) (cid:15) ?( ⊥ ⊕ A ) ? ⊥ ⊕ ? A r ⊥ , A (cid:47) (cid:47) ? ⊥ ⊕ ? A ⊥ ⊕ ? A r ⊥ ⊕ id ? A (cid:15) (cid:15) ? A ? A ⊕ ⊥ ρ − A (cid:47) (cid:47) ?( A ⊕ ⊥ )? A ? ρ A (cid:15) (cid:15) ?( A ⊕ ⊥ ) ? A ⊕ ? ⊥ r A , ⊥ (cid:47) (cid:47) ? A ⊕ ? ⊥ ? A ⊕ ⊥ id ? A ⊕ r ⊥ (cid:15) (cid:15) ?( B ⊕ A ) ? B ⊕ ? A r B , A (cid:47) (cid:47) ?( A ⊕ B )?( B ⊕ A ) ? β A , B (cid:15) (cid:15) ?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) ? A ⊕ ? B ? B ⊕ ? A β ? A , ? B (cid:15) (cid:15) Next we show that (? , η, µ ) defines a monad where η A : A (cid:47) (cid:47) ? A is the unit of the adjunction, and µ A = J ε H A : ? ? A (cid:47) (cid:47) ? A . It su ffi ces to show that every diagram of Definition 13 holds.Case. ? A ? A µ A (cid:47) (cid:47) ? A ? A ? µ A (cid:15) (cid:15) ? A ? A µ ? A (cid:47) (cid:47) ? A ? A µ A (cid:15) (cid:15) It su ffi ces to show that the following diagram commutes: J ( H ? A ) J ( H A ) J ε H A (cid:47) (cid:47) J ( H (? A )) J ( H ? A ) J ( H µ A ) (cid:15) (cid:15) J ( H (? A )) J ( H ? A ) J ε H ? A (cid:47) (cid:47) J ( H ? A ) J ( H A ) J ε H A (cid:15) (cid:15) But this diagram is equivalent to the following:
H JH A H A ε H A (cid:47) (cid:47) HJHJH A H JH A H J ε H A (cid:15) (cid:15) HJHJH A H JH A ε HJH A (cid:47) (cid:47) H JH A H A ε H A (cid:15) (cid:15) The previous diagram commutes by naturality of ε .Case. ? A ? A η ? A (cid:47) (cid:47) ? A ? A (cid:111) (cid:111) ? η A ? A ? A ? A ? A (cid:79) (cid:79) µ A ? A ? A It su ffi ces to show that the following diagrams commutes: JHA JH JHA η JHA (cid:47) (cid:47)
JH JHA JHA (cid:111) (cid:111) JH η A JHAJHA JHAJH JHA (cid:79) (cid:79) J ε HA JHA JHA
Both of these diagrams commute because η and ε are the unit and counit of an adjunction.It remains to be shown that η and µ are both symmetric comonoidal natural transformations, butthis easily follows from the fact that we know η is by assumption, and that µ is because it is definedin terms of ε which is a symmetric comonoidal natural transformation. Thus, all of the followingdiagrams commute: A ⊕ B ? A ⊕ ? B η A ⊕ η B (cid:47) (cid:47) A ⊕ B ?( A ⊕ B ) η A (cid:15) (cid:15) ? A ⊕ ? B ?( A ⊕ B ) (cid:57) (cid:57) r A , B ⊥ ⊥⊥ ? ⊥ η ⊥ (cid:47) (cid:47) ? ⊥⊥ r ⊥ (cid:127) (cid:127) ?( A ⊕ B )? ( A ⊕ B )?( A ⊕ B ) µ A ⊕ B (cid:15) (cid:15) ? ( A ⊕ B ) ?(? A ⊕ ? B ) ? r A , B (cid:47) (cid:47) ?(? A ⊕ ? B ) ? A ⊕ ? B ?(? A ⊕ ? B )?(? A ⊕ ? B ) ? A ⊕ ? B r ? A , ? B (cid:47) (cid:47) ? A ⊕ ? B ? A ⊕ ? B µ A ⊕ µ B (cid:15) (cid:15) ?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) ? ⊥ ⊥ r ⊥ (cid:47) (cid:47) ? ⊥ ? ⊥ µ ⊥ (cid:15) (cid:15) ? ⊥ ? ⊥ ? r ⊥ (cid:47) (cid:47) ? ⊥⊥ r ⊥ (cid:15) (cid:15) B.3.
Proof of Lemma 26.
Suppose ( H , h ) and ( J , j ) are two symmetric comonoidal functors, suchthat, L : H (cid:97) J : C is a dual LNL model. Again, we know ? A = H ; J : L (cid:47) (cid:47) L is a symmetriccomonoidal monad by Lemma 25.We define the following morphisms: w A : = ⊥ j − (cid:47) (cid:47) J J (cid:5) H A (cid:47) (cid:47) JH A ? A c A : = ? A ⊕ ? A JH A ⊕ JH A j − H A , H A (cid:47) (cid:47) J ( H A + H A ) J (cid:96) H A (cid:47) (cid:47) JHA ? A Next we show that both of these are symmetric comonoidal natural transformations, but forwhich functors? Define W ( A ) = ⊥ and C ( A ) = ? A ⊕ ? A on objects of L , and W ( f : A (cid:47) (cid:47) B ) = id ⊥ and C ( f : A (cid:47) (cid:47) B ) = ? f ⊕ ? f on morphisms. So we must show that w : W (cid:47) (cid:47) ? and c : C (cid:47) (cid:47) ?are symmetric comonoidal natural transformations. We first show that w is and then we show that c is. Throughout the proof we drop subscripts on natural transformations for readability. COINTUITIONISTIC ADJOINT LOGIC 43
Case. To show w is a natural transformation we must show the following diagram commutes for anymorphism f : A (cid:47) (cid:47) B : W ( B ) ? B w B (cid:47) (cid:47) W ( A ) W ( B ) W ( f ) (cid:15) (cid:15) W ( A ) ? A w A (cid:47) (cid:47) ? A ? B ? f (cid:15) (cid:15) This diagram is equivalent to the following: ⊥ ? B w B (cid:47) (cid:47) ⊥⊥ id ⊥ (cid:15) (cid:15) ⊥ ? A w A (cid:47) (cid:47) ? A ? B ? f (cid:15) (cid:15) It further expands to the following: ⊥ J j − (cid:47) (cid:47) ⊥⊥ id ⊥ (cid:15) (cid:15) ⊥ J j − (cid:47) (cid:47) J J J JH B J ( (cid:5) H B ) (cid:47) (cid:47) J J J JH A J ( (cid:5) H A ) (cid:47) (cid:47) JH A JH B JH f (cid:15) (cid:15) This diagram commutes, because J ( (cid:5) H A ); J f = J ( (cid:5) H A ; f ) = J ( (cid:5) H B ), by the uniqueness of the initialmap.Case. The functor W is comonoidal itself. To see this we must exhibit a map s ⊥ : = id ⊥ : W ⊥ (cid:47) (cid:47) ⊥ and a natural transformation s A , B : = ρ − ⊥ : W ( A ⊕ B ) (cid:47) (cid:47) W A ⊕ W B subject to the coherence conditions in Definition 8. Clearly, the second map is a natural transfor-mation, but we leave showing they respect the coherence conditions to the reader. Now we canshow that w is indeed symmetric comonoidal.Case. ?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) W ( A ⊕ B )?( A ⊕ B ) w A ⊕ B (cid:15) (cid:15) W ( A ⊕ B ) W A ⊕ W B s A , B (cid:47) (cid:47) W A ⊕ W B ? A ⊕ ? B w A ⊕ w B (cid:15) (cid:15) Expanding the objects of the previous diagram results in the following: ?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) ⊥ ?( A ⊕ B ) w A ⊕ B (cid:15) (cid:15) ⊥ ⊥ ⊕ ⊥ s A , B (cid:47) (cid:47) ⊥ ⊕ ⊥ ? A ⊕ ? B w A ⊕ w B (cid:15) (cid:15) This diagram commutes, because the following fully expanded diagram commutes: JH ( A ⊕ B ) J ( H A + H B ) Jh (cid:47) (cid:47) J JH ( A ⊕ B ) J (cid:5) (cid:15) (cid:15) J J (0 + (cid:111) (cid:111) J ρ J (0 + J ( H A + H B ) J ( (cid:5) + (cid:5) ) (cid:15) (cid:15) J ( H A + H B ) JH A ⊕ JH B j (cid:47) (cid:47) J (0 + J ( H A + H B ) J (0 + J + J j (cid:47) (cid:47) J + J JH A ⊕ JH B J (cid:5)⊕ J (cid:5) (cid:15) (cid:15) J ⊥ J j − (cid:15) (cid:15) ⊥ ⊥ ⊕ ⊥ ρ − (cid:47) (cid:47) ⊥ ⊕ ⊥ J ⊕ ⊥ J ⊕ J id ⊕ j − (cid:47) (cid:47) ⊥ ⊕ ⊥ J ⊕ ⊥ j − ⊕ id (cid:123) (cid:123) ⊥ ⊕ ⊥ J ⊕ J j − ⊕ j − (cid:15) (cid:15) J ⊕ ⊥ J ⊕ J J ⊕ ⊥ J ⊕ ⊥ J ⊕ J J ⊕ ⊥ id ⊕ j (cid:119) (cid:119) J ⊕ J J ⊕ J J J ⊕ ⊥ ρ − (cid:60) (cid:60) (1) (2)(3) (4) (5)(6)Diagram 1 commutes because 0 is the initial object, diagram 2 commutes by naturality of j , diagram 3 commutes because J is a symmetric comonoidal functor, diagram 4 commutesbecause j is an isomorphism (Lemma 23), diagram 5 commutes by functorality of J , anddiagram 6 commutes by naturality of ρ .Case. ⊥ W ⊥ (cid:95) (cid:95) s ⊥ ⊥ ? ⊥ (cid:111) (cid:111) r ⊥ ? ⊥ W ⊥ (cid:63) (cid:63) w ⊥ Expanding the objects in the previous diagram results in the following: ⊥ ⊥⊥ ? ⊥ (cid:111) (cid:111) r ⊥ ? ⊥⊥ (cid:63) (cid:63) w ⊥ This diagram commutes because the following one does:
COINTUITIONISTIC ADJOINT LOGIC 45 J JH ⊥ J (cid:5) (cid:47) (cid:47) J J J JH ⊥ (cid:79) (cid:79) Jh ⊥ ⊥ J j − (cid:47) (cid:47) ⊥⊥⊥ J ⊥ J (cid:111) (cid:111) j The diagram on the left commutes because j is an isomorphism (Lemma 23), and the diagramon the right commutes because 0 is the initial object.Case. Now we show that c A : ? A ⊕ ? A (cid:47) (cid:47) ? A is a natural transformation. This requires the followingdiagram to commute (for any f : A (cid:47) (cid:47) B ): C B ? B c B (cid:47) (cid:47) C A C B C f (cid:15) (cid:15) C A ? A c A (cid:47) (cid:47) ? A ? B ? f (cid:15) (cid:15) This expands to the following diagram:? B ⊕ ? B ? B c B (cid:47) (cid:47) ? A ⊕ ? A ? B ⊕ ? B ? f ⊕ ? f (cid:15) (cid:15) ? A ⊕ ? A ? A c A (cid:47) (cid:47) ? A ? B ? f (cid:15) (cid:15) This diagram commutes because the following diagram does: JH B ⊕ JH B J ( H B + H B ) j − H B , H B (cid:47) (cid:47) JH A ⊕ JH A JH B ⊕ JH B JH f ⊕ JH f (cid:15) (cid:15) JH A ⊕ JH A J ( H A + H A ) j − H A , H A (cid:47) (cid:47) J ( H A + H A ) J ( H B + H B ) J ( H f + H f ) (cid:15) (cid:15) J ( H B + H B ) JH B J (cid:96) H B (cid:47) (cid:47) J ( H A + H A ) J ( H B + H B ) J ( H A + H A ) JH A J (cid:96) H A (cid:47) (cid:47) JH A JH B JH f (cid:15) (cid:15) The left square commutes by naturality of j − , and the right square commutes by naturality of thecodiagonal (cid:96) A : A + A (cid:47) (cid:47) A .Case. The functor C : L (cid:47) (cid:47) L is indeed symmetric comonoidal where the required maps are defined asfollows: t ⊥ : = ? ⊥ ⊕ ? ⊥ JH ⊥ ⊕ JH ⊥ j − (cid:47) (cid:47) J ( H ⊥ + H ⊥ ) J (cid:96) (cid:47) (cid:47) JH ⊥ Jh ⊥ (cid:47) (cid:47) J j (cid:47) (cid:47) ⊥ t A , B : = ?( A ⊕ B ) ⊕ ?( A ⊕ B ) r A , B ⊕ r A , B (cid:47) (cid:47) (? A ⊕ ? B ) ⊕ (? A ⊕ ? B ) iso (cid:47) (cid:47) (? A ⊕ ? A ) ⊕ (? B ⊕ ? B ) where iso is a natural isomorphism that can easily be defined using the symmetric monoidalstructure of L . Clearly, t is indeed a natural transformation, but we leave checking that the requireddiagrams in Definition 8 commute to the reader. We can now show that c A : ? A ⊕ ? A (cid:47) (cid:47) ? A issymmetric comonoidal. The following diagrams from Definition 10 must commute:Case. ?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) C ( A ⊕ B )?( A ⊕ B ) c A ⊕ B (cid:15) (cid:15) C ( A ⊕ B ) C A ⊕ C B t A , B (cid:47) (cid:47) C A ⊕ C B ? A ⊕ ? B c A ⊕ c B (cid:15) (cid:15) Expanding the objects in the previous diagram results in the following:?( A ⊕ B ) ? A ⊕ ? B r A , B (cid:47) (cid:47) ?( A ⊕ B ) ⊕ ?( A ⊕ B )?( A ⊕ B ) c A ⊕ B (cid:15) (cid:15) ?( A ⊕ B ) ⊕ ?( A ⊕ B ) (? A ⊕ ? A ) ⊕ (? B ⊕ ? B ) t A , B (cid:47) (cid:47) (? A ⊕ ? A ) ⊕ (? B ⊕ ? B )? A ⊕ ? B c A ⊕ c B (cid:15) (cid:15) This diagram commutes, because the following fully expanded one does:
COINTUITIONISTIC ADJOINT LOGIC 47 J H ( A ⊕ B ) J ( H A + H B ) J h (cid:47) (cid:47) J ( H ( A ⊕ B ) + H ( A ⊕ B )) J H ( A ⊕ B ) J (cid:96) (cid:15) (cid:15) J ( H ( A ⊕ B ) + H ( A ⊕ B )) J (( H A + H B ) + ( H A + H B )) J ( h + h ) (cid:47) (cid:47) J (( H A + H B ) + ( H A + H B )) J ( H A + H B ) J (cid:96) (cid:15) (cid:15) J ( H ( A ⊕ B ) + H ( A ⊕ B )) J (( H A + H B ) + ( H A + H B )) J ( h + h ) (cid:47) (cid:47) J H ( A ⊕ B ) ⊕ J H ( A ⊕ B ) J ( H ( A ⊕ B ) + H ( A ⊕ B )) j − (cid:15) (cid:15) J H ( A ⊕ B ) ⊕ J H ( A ⊕ B ) J ( H A + H B ) ⊕ J ( H A + H B ) J h ⊕ J h (cid:47) (cid:47) J ( H A + H B ) ⊕ J ( H A + H B ) J (( H A + H B ) + ( H A + H B )) j − (cid:15) (cid:15) J ( H A + H B ) J ( H A + H B ) J (( H A + H B ) + ( H A + H B )) J ( H A + H B ) J (cid:96) (cid:15) (cid:15) J (( H A + H B ) + ( H A + H B )) J (( H A + H A ) + ( H B + H B )) J i s o (cid:47) (cid:47) J (( H A + H A ) + ( H B + H B )) J ( H A + H B ) J ( (cid:96) + (cid:96) ) (cid:15) (cid:15) J ( H A + H B ) J H A ⊕ J H B j (cid:47) (cid:47) J (( H A + H A ) + ( H B + H B )) J ( H A + H B ) J ( (cid:96) + (cid:96) ) (cid:15) (cid:15) J (( H A + H A ) + ( H B + H B )) J ( H A + H A ) ⊕ J ( H B + H B ) j (cid:47) (cid:47) J ( H A + H A ) ⊕ J ( H B + H B ) J H A ⊕ J H B J (cid:96) ⊕ J (cid:96) (cid:15) (cid:15) J (( H A + H A ) + ( H B + H B )) J ( H A + H A ) ⊕ J ( H B + H B ) j (cid:47) (cid:47) ( J H A ⊕ J H A ) ⊕ ( J H B ⊕ J H B ) J (( H A + H A ) + ( H B + H B )) (cid:52) (cid:52) j ; ( j ⊕ j ) ( J H A ⊕ J H A ) ⊕ ( J H B ⊕ J H B ) J ( H A + H A ) ⊕ J ( H B + H B ) j − ⊕ j − (cid:15) (cid:15) J ( H A + H B ) ⊕ J ( H A + H B )( J H A ⊕ J H B ) ⊕ ( J H A ⊕ J H B ) j ⊕ j (cid:47) (cid:47) ( J H A ⊕ J H B ) ⊕ ( J H A ⊕ J H B )( J H A ⊕ J H A ) ⊕ ( J H B ⊕ J H B ) i s o (cid:47) (cid:47) ( )( ) ( )( ) ( )( ) Diagram 1 commutes by naturality of (cid:96) , diagram 2 commutes by naturality of j − , diagram 3commutes by straightforward reasoning on coproducts, diagram 4 commutes by straightforwardreasoning on the symmetric monoidal structure of J after expanding the definition of thetwo isomorphisms – here Jiso is the corresponding isomorphisms on coproducts – diagram 5commutes by naturality of j , and diagram 6 commutes because j is an isomorphism (Lemma 23).Case. ⊥ C ⊥ (cid:95) (cid:95) t ⊥ ⊥ ? ⊥ (cid:111) (cid:111) r ⊥ ? ⊥ C ⊥ (cid:63) (cid:63) c ⊥ Expanding the objects of this diagram results in the following:? ⊥ ⊕ ? ⊥ ? ⊥ ⊕ ? ⊥⊥ ? ⊥ ⊕ ? ⊥ (cid:79) (cid:79) t ⊥ ⊥ ? ⊥ (cid:111) (cid:111) r ⊥ ? ⊥ ? ⊥ ⊕ ? ⊥ (cid:79) (cid:79) c ⊥ Simply unfolding the morphisms in the previous diagram reveals the following: JH ⊥ ⊕ JH ⊥ JH ⊥ ⊕ JH ⊥ J ( H ⊥ + H ⊥ ) JH ⊥ ⊕ JH ⊥ (cid:79) (cid:79) j − J ( H ⊥ + H ⊥ ) J ( H ⊥ + H ⊥ ) J ( H ⊥ + H ⊥ ) JH ⊥ ⊕ JH ⊥ (cid:79) (cid:79) j − J ( H ⊥ + H ⊥ ) J ( H ⊥ + H ⊥ ) JH ⊥ J ( H ⊥ + H ⊥ ) (cid:79) (cid:79) J (cid:96) JH ⊥ JH ⊥ JH ⊥ J ( H ⊥ + H ⊥ ) (cid:79) (cid:79) J (cid:96) JH ⊥ JH ⊥ J JH ⊥ (cid:79) (cid:79) Jh ⊥ J J J JH ⊥ (cid:79) (cid:79) Jh ⊥ J J ⊥ J (cid:79) (cid:79) j ⊥ ⊥⊥ J (cid:79) (cid:79) j Clearly, this diagram commutes.At this point we have shown that w A : ⊥ (cid:47) (cid:47) ? A and c A : ? A ⊕ ? A (cid:47) (cid:47) ? A are symmetric comonoidalnaturality transformations. Now we show that for any ? A the triple (? A , w A , c A ) forms a commutativemonoid. This means that the following diagrams must commute:Case. COINTUITIONISTIC ADJOINT LOGIC 49 (? A ⊕ ? A ) ⊕ ? A (? A ⊕ ? A ) ⊕ ? A ? A ⊕ (? A ⊕ ? A ) α ? A , ? A , ? A (cid:47) (cid:47) ? A ⊕ (? A ⊕ ? A ) ? A ? A ⊕ (? A ⊕ ? A )? A ⊕ (? A ⊕ ? A ) ? A ⊕ ? A id ? A ⊕ c A (cid:47) (cid:47) ? A ⊕ ? A ? A c A (cid:15) (cid:15) ? A ⊕ ? A ? A c A (cid:47) (cid:47) (? A ⊕ ? A ) ⊕ ? A ? A ⊕ ? A c A ⊕ id A (cid:15) (cid:15) (? A ⊕ ? A ) ⊕ ? A ? A The previous diagram commutes, because the following one does (we omit subscripts for readabil-ity): J ( H A + H A ) ⊕ JH A J (( H A + H A ) + H A ) j − (cid:47) (cid:47) ( JH A ⊕ JH A ) ⊕ JH A J ( H A + H A ) ⊕ JH A j − ⊕ id (cid:15) (cid:15) ( JH A ⊕ JH A ) ⊕ JH A JH A ⊕ ( JH A ⊕ JH A ) α (cid:47) (cid:47) JH A ⊕ ( JH A ⊕ JH A ) J (( H A + H A ) + H A ) J (( H A + H A ) + H A ) J ( H A + ( H A + H A )) J α (cid:47) (cid:47) JH A ⊕ ( JH A ⊕ JH A ) J (( H A + H A ) + H A ) JH A ⊕ ( JH A ⊕ JH A ) JH A ⊕ J ( H A + H A ) id ⊕ j − (cid:47) (cid:47) JH A ⊕ J ( H A + H A ) J ( H A + ( H A + H A )) j − (cid:15) (cid:15) J ( H A + ( H A + H A )) J ( H A + H A ) J ( id + (cid:96) ) (cid:47) (cid:47) JH A ⊕ J ( H A + H A ) J ( H A + ( H A + H A )) JH A ⊕ J ( H A + H A ) JH A ⊕ JH A id ⊕ J (cid:96) (cid:47) (cid:47) JH A ⊕ JH A J ( H A + H A ) j − (cid:15) (cid:15) JH A ⊕ JH A J ( H A + H A ) j − (cid:47) (cid:47) J ( H A + H A ) ⊕ JH A JH A ⊕ JH A J (cid:96) ⊕ id (cid:15) (cid:15) J ( H A + H A ) ⊕ JH A J (( H A + H A ) + H A ) J (( H A + H A ) + H A ) J ( H A + H A ) J ( (cid:96) + id ) (cid:15) (cid:15) J ( H A + H A ) JH A J (cid:96) (cid:47) (cid:47) J ( H A + H A ) J ( H A + H A ) J ( H A + H A ) JH A J (cid:96) (cid:15) (cid:15) (1) (2)(3) (4) Diagram 1 commutes because J is a symmetric monoidal functor (Corollary 24), diagrams 2 and3 commute by naturality of j − , and diagram 4 commutes because ( H A , (cid:5) , (cid:96) ) is a commutativemonoid in C , but we leave the proof of this to the reader.Case. ? A ⊕ ? A ? A c A (cid:47) (cid:47) ? A ⊕ ⊥ ? A ⊕ ? A id ? A ⊕ w A (cid:15) (cid:15) ? A ⊕ ⊥ ? A ρ ? A (cid:37) (cid:37) The previous diagram commutes, because the following one does: JH A ⊕ JH A J ( H A + H A ) j − (cid:47) (cid:47) JH A ⊕ J JH A ⊕ JH A id ⊕ J (cid:5) (cid:15) (cid:15) JH A ⊕ J J ( H A + j − (cid:47) (cid:47) J ( H A + J ( H A + H A ) J ( id ⊕(cid:5) ) (cid:15) (cid:15) J ( H A + H A ) JH A J (cid:96) (cid:47) (cid:47) J ( H A + J ( H A + H A ) J ( H A + JH A J ρ (cid:47) (cid:47) JH A JH A JH A ⊕ J JH A JH A ⊕ ⊥ JH A ⊕ J id ⊕ j − (cid:15) (cid:15) JH A ⊕ ⊥ JH A ρ (cid:47) (cid:47) JH A JH A (1)(2) (3)Diagram 1 commutes because J is a symmetric monoidal functor (Corollary 24), diagram 2commutes by naturality of j − , and diagram 3 commutes because ( H A , (cid:5) , (cid:96) ) is a commutativemonoid in C , but we leave the proof of this to the reader.Case. ? A ⊕ ? A ? A c A (cid:47) (cid:47) ? A ⊕ ? A ? A ⊕ ? A β ? A , ? A (cid:15) (cid:15) ? A ⊕ ? A ? A c A (cid:37) (cid:37) This diagram commutes, because the following one does: JH A ⊕ JH A J ( H A + H A ) j − (cid:47) (cid:47) JH A ⊕ JH A JH A ⊕ JH A β (cid:15) (cid:15) JH A ⊕ JH A J ( H A + H A ) j − (cid:47) (cid:47) J ( H A + H A ) J ( H A + H A ) J β (cid:15) (cid:15) J ( H A + H A ) JH A J (cid:96) (cid:47) (cid:47) J ( H A + H A ) J ( H A + H A ) J ( H A + H A ) JH A J (cid:96) (cid:47) (cid:47) JH A JH A The left diagram commutes by naturality of j − , and the right diagram commutes because ( H A , (cid:5) , (cid:96) )is a commutative monoid in C , but we leave the proof of this to the reader.Finally, we must show that w A : ⊥ (cid:47) (cid:47) ? A and c A : ? A ⊕ ? A (cid:47) (cid:47) ? A are ? -algebra morphisms.The algebras in play here are (? A , µ : ? ? A (cid:47) (cid:47) ? A ), ( ⊥ , r ⊥ : ? ⊥ (cid:47) (cid:47) ⊥ ), and (? A ⊕ ? A , u A :?(? A ⊕ ? A ) (cid:47) (cid:47) ? A ⊕ ? A ), where u A : = ?(? A ⊕ ? A ) r ? A , ? A (cid:47) (cid:47) ? A ⊕ ? A µ A ⊕ µ A (cid:47) (cid:47) ? A ⊕ ? A . It su ffi cesto show that the following diagrams commute:Case. ? ? A ? A µ (cid:47) (cid:47) ? ⊥ ? ? A ? w (cid:15) (cid:15) ? ⊥ ⊥ r ⊥ (cid:47) (cid:47) ⊥ ? A w (cid:15) (cid:15) This diagram commutes, because the following fully expanded one does:
JHJH A JH A J ε (cid:47) (cid:47) JHJ JHJH A JHJ (cid:5) (cid:15) (cid:15)
JHJ J J ε (cid:47) (cid:47) J JH A J (cid:5) (cid:15) (cid:15) JHJ JH ⊥ JHJ JHj − (cid:15) (cid:15) JH ⊥ J Jh ⊥ (cid:47) (cid:47) J J J J ⊥ j (cid:47) (cid:47) ⊥ J j − (cid:15) (cid:15) JHJ JHJ JH ⊥ JHJ JHj − (cid:41) (cid:41) JH ⊥ JHJ JHJ JH ⊥ JHj (cid:47) (cid:47) JH ⊥ J Jh ⊥ (cid:32) (cid:32) (1) (2)(3) (4) COINTUITIONISTIC ADJOINT LOGIC 51
Diagram 1 commutes by naturality of ε , diagram 2 commutes because ε is the counit of thesymmetric comonoidal adjunction, diagram 3 clearly commutes, and diagram 4 commutes because j is an isomorphism (Lemma 23).Case. ? ? A ? A µ (cid:47) (cid:47) ?(? A ⊕ ? A )? ? A ? c (cid:15) (cid:15) ?(? A ⊕ ? A ) ? A ⊕ ? A u (cid:47) (cid:47) ? A ⊕ ? A ? A c (cid:15) (cid:15) This diagram commutes because the following fully expanded one does: J H J H A J H J ( H A + H A ) J H J H A J H J (cid:96) (cid:15) (cid:15) J H J ( H A + H A ) J H ( J H A ⊕ J H A ) J H j (cid:47) (cid:47) J H ( J H A ⊕ J H A ) J H J H A J H ( J H A ⊕ J H A ) J H ( J H A ⊕ J H A ) J ( H J H A + H J H A ) J h (cid:47) (cid:47) J ( H J H A + H J H A ) J H J H A J (cid:96) (cid:15) (cid:15) J H J H A J H J H A J H J H A J H J H A J H A J H J H A (cid:53) (cid:53) J ε J H A J H J H A (cid:105) (cid:105) J ε J H J ( H A + H A ) J ( H A + H A ) J ε (cid:41) (cid:41) J H J ( H A + H A ) J ( H J H A + H J H A ) J ( H J H A + H J H A ) J ( H A + H A ) J ( ε + ε ) (cid:117) (cid:117) J ( H A + H A ) J H A J (cid:96) (cid:15) (cid:15) J H J ( H A + H A ) J H ( J H A ⊕ J H A ) J H J ( H A + H A ) J H j − (cid:15) (cid:15) J H ( J H A ⊕ J H A ) J ( H J H A + H J H A ) J h (cid:47) (cid:47) J ( H J H A + H J H A ) J H ( J H A ⊕ J H A ) J ( H J H A + H J H A ) J ( H J H A + H J H A ) J H ( J H A ⊕ J H A ) J ( H J H A + H J H A ) J H J H A ⊕ J H J H A j (cid:47) (cid:47) J H J H A ⊕ J H J H A J ( H J H A + H J H A ) j − (cid:15) (cid:15) J H J H A J H A J ε (cid:47) (cid:47) J ( H J H A + H J H A ) J H J H A J (cid:96) (cid:15) (cid:15) J ( H J H A + H J H A ) J ( H A + H A ) J ( ε + ε ) (cid:47) (cid:47) J ( H A + H A ) J H A J (cid:96) (cid:15) (cid:15) J ( H J H A + H J H A ) J ( H A + H A ) J ( ε + ε ) (cid:47) (cid:47) J H J H A ⊕ J H J H A J ( H J H A + H J H A ) j − (cid:15) (cid:15) J H J H A ⊕ J H J H A J H A ⊕ J H A J ε ⊕ J ε (cid:47) (cid:47) J H A ⊕ J H A J ( H A + H A ) j − (cid:15) (cid:15) ( )( )( )( )( )( ) ( ) Diagram 1 clearly commutes, diagram 2 commutes by naturality of ε , diagram 3 commutes bynaturality of (cid:96) , diagram 4 commutes because ε is the counit of the symmetric comonoidal adjunction,diagram 5 commutes because j is an isomorphism (Lemma 23), diagram 6 commutes by naturality of j − , and diagram 7 is the same diagram as 3, but this diagram is redundant for readability.B.4. Proof of Lemma 27.
Suppose L : H (cid:97) J : C is a dual LNL model. Then we know ? A = JH A is a symmetric comonoidal monad by Lemma 25. Bellin [3] remarks that by Maietti, Maneggiade Paiva and Ritter’s Proposition 25 [19], it su ffi ces to show that µ A : ? ? A (cid:47) (cid:47) ? A is a monoidmorphism. Thus, the following diagrams must commute:Case. ? A ⊕ ? A ? A c A (cid:47) (cid:47) ? ? A ⊕ ? ? A ? A ⊕ ? A µ A ⊕ µ A (cid:15) (cid:15) ? ? A ⊕ ? ? A ? ? A c ? A (cid:47) (cid:47) ? ? A ? A µ A (cid:15) (cid:15) This diagram commutes because the following fully expanded one does: JH A ⊕ JH A J ( H A + H A ) j − (cid:47) (cid:47) JHJH A ⊕ JHJH A JH A ⊕ JH A J ε ⊕ J ε (cid:15) (cid:15) JHJH A ⊕ JHJH A J ( HJH A + HJH A ) j − (cid:47) (cid:47) J ( HJH A + HJH A ) J ( H A + H A ) J ( ε + ε ) (cid:15) (cid:15) J ( H A + H A ) JH A J (cid:96) (cid:47) (cid:47) J ( HJH A + HJH A ) J ( H A + H A ) J ( ε + ε ) (cid:15) (cid:15) J ( HJH A + HJH A ) JHJH A J (cid:96) (cid:47) (cid:47) JHJH A JH A J ε (cid:15) (cid:15) The left square commutes by naturality of j − and the right square commutes by naturality of thecodiagonal.Case. ? ? A ? A µ A (cid:47) (cid:47) ⊥ ? ? A w ? A (cid:127) (cid:127) ⊥ ? A w A (cid:31) (cid:31) This diagram commutes because the following fully expanded one does: J J ⊥ J j − (cid:15) (cid:15) ⊥ ⊥⊥ J j − (cid:15) (cid:15) JHJH A JH A J ε (cid:47) (cid:47) J JHJH A J (cid:5) (cid:15) (cid:15) J J J JH A J (cid:5) (cid:15) (cid:15) The top square trivially commutes, and the bottom square commutes by uniqueness of the initialmap.
COINTUITIONISTIC ADJOINT LOGIC 53
B.5.
Proof of Cut Reduction (Lemma 38).
By induction on d ( Π ) + d ( Π ). We consider only thecase where the last inferences of Π and Π are logical inferences. The other cases are handledmainly by permutation of inferences and use of the inductive hypothesis; we refer to Benton’s textfor them. Throughout the proof we will add an asterisk to the name of an inference rule to indicatethat the rule may be applied zero or more times. J right / J left . We have π A (cid:96) L ∆ , JT n ; T , ΨΠ = J R A (cid:96) L ∆ , JT n + ; Ψ π T (cid:96) C Ψ (cid:48) Π = J L JT (cid:96) L ; ∆ , JT n + , Ψ (cid:48) By the inductive hypothesis appled to Π and π there exists a proof Π (cid:48) of A (cid:96) L ∆ ; T , Ψ , Ψ (cid:48) with c ( Π (cid:48) ) ≤ | JT | = | T | +
1. Then the following derivation Π (cid:48) A (cid:96) L ∆ ; T , Ψ , Ψ (cid:48) π T (cid:96) C Ψ (cid:48) Π = LC cut A (cid:96) L ∆ , Ψ , Ψ (cid:48) , Ψ (cid:48) C contr ∗ A (cid:96) L ∆ , Ψ , Ψ (cid:48) has cut rank max ( | T | + , c ( Π (cid:48) ) , c ( π )) = | T | + = | JT | . H right / H left . We have π B (cid:96) L ∆ , A ; HA n ; ΨΠ = H R B (cid:96) L ∆ ; HA n + , Ψ π A (cid:96) L ; Ψ (cid:48) Π = H L HA (cid:96) C ; Ψ (cid:48) By the inductive hypothesis applied to Π and π there exists a proof Π (cid:48) of B (cid:96) L ∆ ; A , Ψ , Ψ (cid:48) with c ( Π (cid:48) ) ≤ | HA | = | A | +
1. Then the following derivation Π (cid:48) B (cid:96) L ∆ ; A , Ψ , Ψ (cid:48) π A (cid:96) L ; Ψ (cid:48) Π = LL cut B (cid:96) L ∆ , Ψ , Ψ (cid:48) , Ψ (cid:48) C contr ∗ B (cid:96) L ∆ , Ψ , Ψ (cid:48) has cut rank max ( | A | + , c ( Π (cid:48) ) , c ( π )) = | A | + = | HA | . + right / + left . We have π S (cid:96) C T , ( T + T ) n , ΨΠ = C + R S (cid:96) C ( T + T ) n + , Ψ π T (cid:96) C Ψ π T (cid:96) C Ψ Π = C + L T + T (cid:96) C ; Ψ , Ψ If n =
0, then the reduction is as follows: π S (cid:96) C T , ΨΠ = C + R S (cid:96) C T + T , Ψ π T (cid:96) C Ψ π T (cid:96) C Ψ Π = C + L T + T (cid:96) C Ψ , Ψ C cut S (cid:96) C Ψ , Ψ , Ψ reduces to π S (cid:96) C T , Ψ π T (cid:96) C Ψ C cut S (cid:96) C Ψ , Ψ Π C weak ∗ S (cid:96) C Ψ , Ψ , Ψ Here c ( Π ) = max ( | T + | , c ( π ) , c ( π )) ≤ | T + T | . If n >
0, then by the inductive hypothesis applied to Π and π there exists a proof Π (cid:48) of S (cid:96) C T , Ψ , Ψ , Ψ with c ( Π (cid:48) ) ≤ | T + T | = | T | + | T | +
1. Then the following derivation Π (cid:48) S (cid:96) C T , Ψ , Ψ , Ψ π T (cid:96) C Ψ Π =
C cut S (cid:96) C Ψ , Ψ , Ψ , Ψ C contr ∗ S (cid:96) C Ψ , Ψ , Ψ has cut rank max ( | T | + , c ( Π (cid:48) ) , c ( π )) ≤ | T + T | . •− right / •− left. We have π A (cid:96) L ∆ ; Ψ , B π B (cid:96) L ∆ ; Ψ Π = LL (cid:21) R A (cid:96) L B •− B , ∆ , ∆ ; Ψ , Ψ π B (cid:96) L B , ∆ ; ΨΠ = LL (cid:21) L B •− B (cid:96) L ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , ∆ ; Ψ , Ψ , Ψ reduces to Π π A (cid:96) L ∆ , B ; Ψ π B (cid:96) L B , ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , B ; Ψ , Ψ π B (cid:96) L ∆ ; Ψ LL cut A (cid:96) L ∆ , ∆ , ∆ ; Ψ , Ψ , Ψ The resulting derivation Π has cut rank c ( Π ) = max ( | B | + , c ( π ) , c ( π ) , | B | + , c ( π )) ≤ | B •− B | . This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copyof this license, visit http://creativecommons.org/licenses/by-nd/2.0/http://creativecommons.org/licenses/by-nd/2.0/