Coherence in Linear Predicate Logic

Coherence in Linear Predicate Logic Ko st a Do ˇ sen a nd Zo ran Petri ´ c Mathematical Institute, SANU Knez Mihailo v a 35, p.f. 367, 11001 Belgrade, Serbia email: { ko sta, zp etric } @mi.san u.ac.yu Abstract Coherence with resp ect to K elly-Mac Lane graphs is prov ed for categories that correspond to t he multiplicativ e fragmen t without constan t prop o- sitions of classical linear first-order predicate logic without or with mix. T o obt ain this result, coherence is fi rst established for categories that correspond to the m ultiplicativ e conjunction-disjunction fragmen t with first-order q uan tifiers of classical linear log ic, a fragment lacking nega- tion. These results exten d results of [7] and [8], where coherence w as established for categories of the corre sp onding fra gments of prop ositional classical linear logic, which are related to pro of nets, and which could b e described as star-autonomous categories without unit ob jects. Mathematics Subje ct Classific ation ( 2000 ): 03F52, 03F05, 03F07, 18D10, 18D15, 18C05, 18A15 Keywor ds : classical linear logic, first-order predicate logic, pro of-net category , coher- ence, Kelly-Mac Lane graphs, pro of, criteria of identit y , cut elimination, mix 0 In tro duction The goal of this pap e r is to pro ve coherence for categor ies that corresp ond to the m ultiplicative fragment without pro positiona l co nstan ts (n ullary connectives) of classical linear first-order predicate logic without or with mix. (In this fra gmen t the mo dal oper ators ! and ? are left out.) The pr opo s itional log ic cor respo nding to this fr a gmen t is the frag men t o f linear logic caught b y pro of nets. Coher- ence for categor ies that corresp ond to this pro positio na l logic, called pr o of-net c ate gories , was proved in [8], where it is a lso demo nstrated that the notion of pro of-net category is the righ t notio n of star-autonomous category without unit ob jects (and where references to rela ted w ork may be found; se e [1] for a general categoria l in tro duction to propo sitional linea r logic). The notion of pro of-net category is here extended with assumptions concerning first-or der quantifiers, and this yields the notion of c a tegory that corresp onds to the fra gmen t of linear 1 predicate logic men tioned in the first sentence. W e prove coherence for these categorie s. Co herence in [8] and her e is understo o d as in Kelly’s and Mac La ne’s coherence result of [14] for symmetric monoidal clo sed categ ories. It is coher ence with r espect to the same kind of gr aphs. Such coherence results are very useful, beca us e they enable us to decide eas ily equality of canonical arrows. The coherence r esults of this pap er are interesting also for general pro of theor y . They dea l w ith a plausible notion of iden tit y of pro ofs in log ic (see [7], Chapter 1). The addition o f first-order quantification to categories treated in [8] do es not bring anything new with resp ect to the graphs. Before they in v olved prop osi- tional letters, and no w they inv olve predicate letters. Individual v ariables do not bring an ything to these graphs. All the new arrows for quantifiers hav e iden- tit y graphs (see Section 1.5). Although it s eems we have made a n iness en tial addition (bas ed on a tr ivial adjunction; see Section 1.4), we don’t know how to reduce simply the coherence r esult prov ed in this pap er to the coherence result of [8]. The pro ofs in this pa per extend and m o dify those of [8], but they require considerable additional effort. W e omit the multiplicativ e pro positio nal constants from our treatment b e- cause they present sp ecial pro blems for co herence (though their additio n to o may be ba s ed on a trivial adjunction; see [7], Section 7 .9). These pro ble ms are comparable to tho s e that the unit o b ject o f symmetric monoidal closed cate- gories makes for co herence of these categories (see [14]). W e b eliev e that befor e attacking such problems one s hould first settle more tractable matters. It ma y seem t hat in the absence of the m ultiplicative propo sitional constan t ⊤ we will not be able express that a formula is a theorem with sequents of the form A ⊢ B , for A and B formulae. These are the sequents of categ o rial pr o of theory , and o f this pap er. F or a theorem B we should have a deriv a ble sequent ⊤ ⊢ B , but in the a bsence of ⊤ w e shall ha ve instead the sequen ts A ⊢ A ∧ B deriv able for every formula A . W e omit also the additive (lattice) connectives from our treatmen t. They would lead to the s ame kind of pro blem for co herence that arises for classica l or intuitionistic conjunctive-disjunctive logic with quantifiers added. This is a challenging ma tter , with which we in tend to deal on another o ccasion. The present pap er should lay the ground for this future work. The categories corr espo nding t o the fra gmen t of linear predica te logic that we cov er are her e pr esen ted equationally , in an axio matic, regular and surveyable manner. These axiomatic equatio ns should corresp ond to the combin atoria l building blo c ks of identit y of pro ofs in this fra gmen t of logic, as in kno t theory the Reidemeister moves are the combinatorial building blo c ks of iden tit y of knots and links (se e [3]). In this pap er, our appr oach to categories cor resp onding to first-order pre di- cate logic is quite sy n tactical. W e deal mainly with fr eely generated catego ries, which are a c a tegorial pr esen tation o f syntax. Ob jects are for m ulae, and ar rows are pro ofs, or deductions, i.e. equiv a lence classe s of deriv a tions. At the level of 2 ob jects, our first-or der la nguage is quite standa rd. After these freely generated categorie s ar e intro duced, other co ncrete c a tegories b elonging to the clas s es in which o ur categ ories are free may b e ta k en as mo dels—of pr o ofs, ra ther than formulae. T he only mo dels of this kind that we co nsider in this pap er ar e ca t- egories who se arrows a re gra phs. Our coherence results may b e under stoo d a s completeness r esults with resp ect to these mo dels. What is shown complete is the axio matization of equality betw een a rrows in the freely gener a ted category . Previous treatments of first-o r der quantifiers in categorial logic, which started with the work of Lawv ere (see Section 1.4, and refer ences given there), ar e less syntactical in spir it than our s. As we said ab ov e, the pro ofs in this pap er are ba sed on pro ofs to b e found in [8], which are themselves bas ed on pro ofs in [7]. W e will eschew rep eating this previously published mater ial with all its details, and so our pap er will not be self-c on tained. T o make it s elf-con tained would y ield a rather siza ble b oo k, ov e rlapping e xcessively with [8] and [7]. W e s uppose the reader is acquainted up to a p oint with [7] (at least Sections 3.2-3 , 7.6-8 a nd 8.4) and with [8] (at least Cha pters 2 and 6). Although to avoid unnecessary lengthy rep etitions we sometimes presupp ose the reader k no ws the prev ious materia l, and w e mak e only remarks co ncerning additions and changes, we hav e in general strived to make our text as self-contained as p ossible, so the r eader ca n get an idea of what we do from this text only . In the first part of the pa per we deal with categor ie s that corre s pond to the multiplicativ e conjunction-disjunction fragment with first-or der quantifiers of classical linea r log ic. Coherence pr o ved for these categorie s ex tends r esults prov ed fo r the corr esponding prop ositiona l fra gmen t in [7] (Chapter 7). When we add the m ultiplicative prop ositional constants to the corresp onding proposi- tional categ ories w e may obtain the linearly distributive catego r ies of [4] (see also [1], Section 2), or ca tegories with mor e eq ua tions pres upp osed, which coherence requires (see [7], Section 7 .9). In this first part w e intro duce in detail the categor ia l no tions brought by quantifiers. The gr eatest no velt y here ma y b e the treatmen t accor ded to renam- ing o f free individual v a riables, which is not us ually consider ed as a primitive rule of inference (see Sections 1.2, 1.8 and 2.2 ). T a king this r e na ming as primi- tive enables us to hav e ca tegorial ax io ms that ar e regula r, surveyable and ea sy to handle. W e prov e in the first part a catego rial cut-eliminatio n result, which says that every ar r o w is e qu al to a cut-free one. The pr o of of this result requir es a prepar ation involving change of individual v aria bles. This prepara tion gives a ca tegorial for m to ideas o f Gentzen and Kleene. In the second part of the pap er we a dd negation (the only connective missing from the firs t part), and pro ceed to prove co herence following the direction of [8] (Chapter 2). In the third part, we add the mix principle, and indicate what adjustment s should be made in the pro ofs of the previo usly obtained results in order to o btain coherence also in the pres ence o f mix. The exp osition in the first part, where we introduce new matter s co ncerning q ua n tifiers, is in g eneral more 3 detailed than the e xpositio n in the second a nd third part, where we rely ev en more heavily o n pr e viously published results, and where we suppos e that the reader has already a cquired some dexterity . A brief concluding section p oin ts to future work. 1 Coherence of QDS 1.1 The language L Let P be a set whose elemen ts we call pr e dic ate letters , for which we use P , R, . . . , sometimes with indices. T o every member of P we assign a natural nu mber n ≥ 0, called its arity . F or every n ≥ 0, w e assume tha t w e hav e infinitely many pr e dicate letters in P of arity n . T o build the first-o rder lang uage L gener ated by P , w e assume that we hav e infinitely man y individual variables , whic h we call simply variables , a nd for which we use x , y , z , u , v , . . . , sometimes with indices. Let x n stand for a sequence of v ariables of length n ≥ 0. The atomic formulae of L are all of the form P x n for P a member of P of a rit y n . W e a ssume througho ut this pap er that ξ ∈ { ∧ , ∨} and Q ∈ {∀ , ∃} . The symbols ∧ and ∨ are used here for the mult iplicative conjunction and disjunction co nnectiv es (for which ⊗ a nd the inv er ted amp ersand are used in [10]). The formulae o f L are defined inductively by the following cla uses: every a tomic formula is a for m ula; if A and B are formulae, then ( A ξ B ) is a formula; if A is a formula and x is a v ar iable, then Q x A is a for m ula. As usual, we will omit the outermost parentheses o f formulae a nd take them for granted. W e call Q x a quantifier pr efi x . (The adv antage of the qua n tifier prefix ∀ x ov e r the more usual ∀ x is that in ∀ x x = x we do not need pa ren theses, or a dot b efore x = x , for which a nee d is felt in ∀ xx = x ; in this pap er, where we use the schematic letter Q for q uan tifiers, we wan t the q uan tifier pr efix Q x to be cle a rly distinguishe d fro m a formula P x .) F or fo rm ulae we use A , B , C , . . . , sometimes with indices . The no tions o f free and bo und o ccurr ences of v aria ble s in a fo rm ula ar e understo o d as usual, and, a s usual, w e say that x is fr e e in A when there is at least one free o ccurrence o f x in A . W e say that x is b ound in A when Q x o ccurs in A (though the quantifier pr efix need not bind any o ccurrence of x , a s in ∀ x P y ). The v aria ble y is said to b e fr ee for subs titution for x in A when no free o ccurrence of x in A is in a subformula of A o f the form Q y B . W e wr ite A x y for the result of uniformly substituting y for the free o ccurrences of x in A , provided that, as usua l, y is free for substitution for x in A (this notation may b e found in [23]). If this proviso is not satisfied, then A x y is not defined. 4 1.2 The category QDS The categ ory QDS , which we introduce in this section, cor resp o nds to the m ultiplicative conjunction-disjunction fra gmen t with first- o rder quant ifiers of classical linear logic. This category e x tends with quantifi ers (this is where Q comes from) the prop ositiona l catego ry DS of [7] (Section 7.6). The o b jects of the category Q DS are the formulae of L . T o define the arrows of QDS , we define first inductively a set of expr essions c alled the arr ow terms o f QDS . Every arrow term will hav e a t yp e , which is a n or dered pair o f formulae o f L . W e write f : A ⊢ B when the arrow ter m f is of t yp e ( A, B ). Here A is the sourc e , and B the tar get of f . F or arr o w ter ms we use f , g , h, . . . , sometimes with indices. In tuitiv ely , the arr o w term f is the co de of a der iv ation of the conclusio n B from the premise A (which ex pla ins why we write ⊢ instead of → ). F or a ll formulae A , B and C of L , for every v ar ia ble x , and for all fo rm ulae D of L in which x is no t free, the following primitive arr ow terms : 1 A : A ⊢ A , ˆ b → A,B ,C : A ∧ ( B ∧ C ) ⊢ ( A ∧ B ) ∧ C , ˇ b → A,B ,C : A ∨ ( B ∨ C ) ⊢ ( A ∨ B ) ∨ C , ˆ b ← A,B ,C : ( A ∧ B ) ∧ C ⊢ A ∧ ( B ∧ C ), ˇ b ← A,B ,C : ( A ∨ B ) ∨ C ⊢ A ∨ ( B ∨ C ), ˆ c A,B : A ∧ B ⊢ B ∧ A , ˇ c A,B : B ∨ A ⊢ A ∨ B , d A,B ,C : A ∧ ( B ∨ C ) ⊢ ( A ∧ B ) ∨ C , ι ∀ x A : ∀ x A ⊢ A , ι ∃ x A : A ⊢ ∃ x A , γ ∀ x D : D ⊢ ∀ x D , γ ∃ x D : ∃ x D ⊢ D , ˇ θ ∀ x → A,D : ∀ x ( A ∨ D ) ⊢ ∀ x A ∨ D , ˆ θ ∃ x ← A,D : ∃ x A ∧ D ⊢ ∃ x ( A ∧ D ) are a rrow terms. (In tuitively , these a r e the axioms of our lo gic with the co des of their triv ial deriv a tions.) Next w e have the following inductive clauses: if f : A ⊢ B a nd g : B ⊢ C a re arrow terms, then ( g ◦ f ) : A ⊢ C is an arrow ter m; if f 1 : A 1 ⊢ B 1 and f 2 : A 2 ⊢ B 2 are arrow terms , then ( f 1 ξ f 2 ) : A 1 ξ A 2 ⊢ B 1 ξ B 2 is an a rrow term; if f : A ⊢ B is an arrow term, then Q x f : Q x A ⊢ Q x B a nd [ f ] x y : A x y ⊢ B x y are ar row terms, provided A x y and B x y are defined. (In tuitiv ely , the op erations on arrow terms ◦ , ξ , Q x and [ ] x y are c o des of the r ule s of infer e nc e o f our log ic.) This defines the arr o w terms of QDS . As we do with formulae, we will omit the o utermost parentheses o f ar row ter ms . 5 The types of the arr o w terms ι ∀ x A and γ ∀ x D are related to the lo gical pr inciples of universal inst antiation and universal gener alization resp ectiv ely (this is where ι a nd γ come fro m). The log ical pr inciple r elated to the type o f ι ∃ x A , and not of γ ∃ x D , is sometimes called existent ial gener alization , but fo r the sake of duality we use ι and γ with the exis ten tial quantifier as with the universal quantifier. The logica l principles of the types of ˇ θ ∀ x → A,D and ˆ θ ∃ x ← A,D are distributivity princi- ples. The firs t, which is the int uitionistically spurious c onstant domain pr inciple, is the conv erse of distribution of disjunction ov er universal quantification, and the seco nd is distribution o f conjunction ov er existential qua n tification. W e de- fine b elo w arr o w ter ms with the converse types, which are bo th in tuitionistically v alid (cf. also the end of the section). With Q x and [ ] x y we are given infinite families of op erations, indexed b y v ariable s . W e ca ll [ ] x y r enaming of fr e e variables , or for sho rt just r enaming . The op eratio ns Q x and ξ are total, but co mposition ◦ and r enaming are not total op erations on ar row ter ms . The result [ f ] x y of applying rena ming [ ] x y to f : A ⊢ B is defined iff A x y and B x y are defined. Note that renaming is not s ubs titution. The arrow terms [ 1 A ] x y : A x y ⊢ A x y and 1 A x y : A x y ⊢ A x y are different ar row terms. The renaming op eratio n is in the ob ject language of arr o w ter ms , while the substitution op eration x y of A x y is no t in the o b ject la nguage of formulae L , but o nly in the metalang ua ge. Note that [ g ◦ f ] x y may b e defined though [ f ] x y and [ g ] x y are not defined (for e x - ample, with f b eing ι ∃ y Rxy : R xy ⊢ ∃ y Rxy and g b eing ι ∃ x ∃ y Rxy : ∃ y Rxy ⊢ ∃ x ∃ y Rxy , where ( ∃ y Rxy ) x y is not defined). Note also that [ f ] x y and [ g ] x y may b e de- fined and comp osable without [ g ◦ f ] x y being defined (for exa mple, with f being ι ∀ y P y : ∀ y P y ⊢ P y and g b eing ι ∃ x P x : P x ⊢ ∃ x P x , where g ◦ f is not defined). Renaming [ ] x y is usually implicitly considered in pro of theor y a s a deriv a ble rule when it is applied to f : A ⊢ B with x not free e ither in A or in B . F or example, fo r x not free in D , we hav e D ⊢ B D ⊢ ∀ x B D ⊢ B x y It is also implicit in a deriv able rule, whic h we hav e in the prese nce o f implication: A ⊢ B ⊢ A → B ⊢ ∀ x ( A → B ) ⊢ A x y → B x y A x y ⊢ B x y 6 W e assume rena ming here in the abs ence of implication. Renaming corres ponds to a s tr uctural rule of logic, in Gentzen’s terminolo gy . Next we define inductively the set of e qu ations of QDS , which a re expres- sions of the form f = g , wher e f and g are arrow terms of QDS of the same t yp e. These equations ho ld whene ver b oth sides are defined. F o r example, in the equa tion ( ∀ γ nat ) below we a ssume that b oth sides are defined, which in- tro duces the proviso that x is free neither in A nor B . An analogo us proviso is in tro duced alrea dy by ( c at 2) b elow, where we assume that f and g , as well as g and h , ar e comp osable. W e will always ass ume these provisos, but we will usually not mention them explicitly . These tacit provisos are carried by the con- ven tions of the notation for arrow terms a nd conditions co ncerning s ubstitution of v ariable s in formulae. Int uitively , these equa tions should c a tc h a pla usible notion o f identit y of pro ofs, pr oofs b eing understo o d as e q uiv a lence c lasses of deriv a tions. Coherence results sho uld justify our ca lling this notion of equa lit y plausible. A justification may also be provided by purely syn tactical results, like cut elimination, a nd other similar nor mal-form results. The t w o justificatio ns may , but need not, coincide (see [7 ], Chapter 1 ). In this pap er, we concentrate o n cohe r ence res ults for the justification, but a s a to ol for demo ns trating this cohere nc e we establish cut-elimination and normal-form results. The latter r esults also provide a partial justification: they show the sufficiency of the equa tions a ssumed. W e do not consider her e (like in [6]) the question whether all these equations ar e also necessary fo r these or related syntactical r esults. (Suc h a question should first be pr ecisely phra sed.) In the lo ng list of a xiomatic eq ua tions b elo w, only the q ua n tificational equa- tions and the renaming equatio ns at the end are new. The preceding pro posi- tional DS equatio ns are taken from [7 ] a nd [8] (Section 2.1), while the first tw o categoria l eq ua tions a re, of course, omnipres en t. W e stipulate firs t that all the instances of f = f and of the following equa tions are equations of QDS : c ate gorial e quations : ( c at 1) f ◦ 1 A = 1 B ◦ f = f : A ⊢ B , ( c at 2) h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f , DS e quations : ( ξ 1) 1 A ξ 1 B = 1 A ξ B , ( ξ 2) ( g 1 ◦ f 1 ) ξ ( g 2 ◦ f 2 ) = ( g 1 ξ g 2 ) ◦ ( f 1 ξ f 2 ), for f : A ⊢ D , g : B ⊢ E and h : C ⊢ F , ( ξ b → nat ) (( f ξ g ) ξ h ) ◦ ξ b → A,B ,C = ξ b → D,E ,F ◦ ( f ξ ( g ξ h )), 7 (ˆ c nat ) ( g ∧ f ) ◦ ˆ c A,B = ˆ c D,E ◦ ( f ∧ g ), (ˇ c nat ) ( g ∨ f ) ◦ ˇ c B ,A = ˇ c E ,D ◦ ( f ∨ g ), ( d n at ) (( f ∧ g ) ∨ h ) ◦ d A,B ,C = d D,E ,F ◦ ( f ∧ ( g ∨ h )), ( ξ b ξ b ) ξ b → A,B ,C ◦ ξ b ← A,B ,C = 1 ( A ξ B ) ξ C , ξ b ← A,B ,C ◦ ξ b → A,B ,C = 1 A ξ ( B ξ C ) , ( ξ b 5) ξ b ← A,B ,C ξ D ◦ ξ b ← A ξ B ,C,D = ( 1 A ξ ξ b ← B ,C,D ) ◦ ξ b ← A,B ξ C,D ◦ ( ξ b ← A,B ,C ξ 1 D ), (ˆ c ˆ c ) ˆ c B ,A ◦ ˆ c A,B = 1 A ∧ B , (ˇ c ˇ c ) ˇ c A,B ◦ ˇ c B ,A = 1 A ∨ B , ( ˆ b ˆ c ) ( 1 B ∧ ˆ c C,A ) ◦ ˆ b ← B ,C,A ◦ ˆ c A,B ∧ C ◦ ˆ b ← A,B ,C ◦ (ˆ c B ,A ∧ 1 C ) = ˆ b ← B ,A,C , ( ˇ b ˇ c ) ( 1 B ∨ ˇ c A,C ) ◦ ˇ b ← B ,C,A ◦ ˇ c B ∨ C,A ◦ ˇ b ← A,B ,C ◦ (ˇ c A,B ∨ 1 C ) = ˇ b ← B ,A,C , ( d ∧ ) ( ˆ b ← A,B ,C ∨ 1 D ) ◦ d A ∧ B ,C,D = d A,B ∧ C,D ◦ ( 1 A ∧ d B ,C,D ) ◦ ˆ b ← A,B ,C ∨ D , ( d ∨ ) d D,C ,B ∨ A ◦ ( 1 D ∧ ˇ b ← C,B ,A ) = ˇ b ← D ∧ C ,B ,A ◦ ( d D,C ,B ∨ 1 A ) ◦ d D,C ∨ B ,A , for d R C,B ,A = d f ˇ c C,B ∧ A ◦ (ˆ c A,B ∨ 1 C ) ◦ d A,B ,C ◦ ( 1 A ∧ ˇ c B ,C ) ◦ ˆ c C ∨ B ,A : ( C ∨ B ) ∧ A ⊢ C ∨ ( B ∧ A ), ( d ˆ b ) d R A ∧ B ,C,D ◦ ( d A,B ,C ∧ 1 D ) = d A,B ,C ∧ D ◦ ( 1 A ∧ d R B ,C,D ) ◦ ˆ b ← A,B ∨ C,D , ( d ˇ b ) ( 1 D ∨ d C,B ,A ) ◦ d R D,C ,B ∨ A = ˇ b ← D,C ∧ B ,A ◦ ( d R D,C ,B ∨ 1 A ) ◦ d D ∨ C ,B ,A , quantific ational e quations : ( Q 1) Q x 1 A = 1 Q x A , ( Q 2) Q x ( g ◦ f ) = Q x g ◦ Q x f , for f : A ⊢ B , ( ∀ ι nat ) f ◦ ι ∀ x A = ι ∀ x B ◦ ∀ x f , ( ∃ ι nat ) ∃ x f ◦ ι ∃ x A = ι ∃ x B ◦ f , ( ∀ γ nat ) ∀ x f ◦ γ ∀ x A = γ ∀ x B ◦ f , ( ∃ γ nat ) f ◦ γ ∃ x A = γ ∃ x B ◦ ∃ x f , ( ∀ β ) ι ∀ x A ◦ γ ∀ x A = 1 A , ( ∃ β ) γ ∃ x A ◦ ι ∃ x A = 1 A , ( ∀ η ) ∀ x ι ∀ x A ◦ γ ∀ x ∀ x A = 1 ∀ x A , ( ∃ η ) γ ∃ x ∃ x A ◦ ∃ x ι ∃ x A = 1 ∃ x A , for ˇ θ ∀ x ← A,D = d f ∀ x ( ι ∀ x A ∨ 1 D ) ◦ γ ∀ x ∀ x A ∨ D : ∀ x A ∨ D ⊢ ∀ x ( A ∨ D ), ˆ θ ∃ x → A,D = d f γ ∃ x ∃ x A ∧ D ◦ ∃ x ( ι ∃ x A ∧ 1 D ) : ∃ x ( A ∧ D ) ⊢ ∃ x A ∧ D and ( ξ , Q ) ∈ { ( ∨ , ∀ ) , ( ∧ , ∃ ) } , ( Q ξ θ ξ θ ) ξ θ Q x ← A,D ◦ ξ θ Q x → A,D = 1 Q x ( A ξ D ) , ξ θ Q x → A,D ◦ ξ θ Q x ← A,D = 1 Q x A ξ D , 8 r enaming e quations : for x , y , z and v mutually different v ar iables and α A 1 ,...,A n a pr imitiv e arr o w term e x cept ι Q x A , ( r en α ) [ α A 1 ,...,A n ] x y = α ( A 1 ) x y ,..., ( A n ) x y , ( r en ◦ ) [ g ◦ f ] x y = [ g ] x y ◦ [ f ] x y , ( r en ξ ) [ f 1 ξ f 2 ] x y = [ f 1 ] x y ξ [ f 2 ] x y , ( r en Q ) [ Q z f ] x y = Q z [ f ] x y , ( r en 1) [ f ] x x = f , ( r en 2) [ f ] x y = f , if x is free neither in the source nor in the tar g et of f , ( r en 3) [[ f ] z v ] x y = [[ f ] x y ] z v , ( r en 4) [[ f ] z y ] x y = [[ f ] x y ] z y , ( r en 5) [[ f ] z x ] x y = [[ f ] z y ] x y , ( r en 6) [[ f ] y x ] x y = [ f ] x y . This concludes the list of axioma tic equations stipulated for QDS . T o define all the equations o f QDS it remains only to say that the s e t of these equatio ns is closed under sy mmetr y and transitivity of equality and under the r ules ( ◦ c ong ) f = f ′ g = g ′ g ◦ f = g ′ ◦ f ′ ( ξ c ong ) f 1 = f ′ 1 f 2 = f ′ 2 f 1 ξ f 2 = f ′ 1 ξ f ′ 2 ( Q c ong ) f = f ′ Q x f = Q x f ′ ( r en c ong ) f = f ′ [ f ] x y = [ f ′ ] x y On the a rrow terms of QDS we impo se the equations of QDS . This mea ns that an ar row of QDS is a n e quiv a lence clas s of a rrow terms o f Q DS defined with resp e ct to the smallest equiv alence relation such that the equations of QDS are satisfied (see [7], Section 2.3, for details). The equa tions ( ξ 1), ( ξ 2), ( Q 1) and ( Q 2) are called functorial equations . They say that ξ is a biendofunctor and Q x an endo functor of QDS (i.e. 2 - endo- functor and 1-endo functor res pectively , in the termino logy of [7], Section 2.4). The equations with “ nat ” in their names are called natu ra lity equations. The naturality equations above say that ξ b → , ξ c , d and ι Q x are natural tra nsformations 9 ( ξ b ← is a na tural tr a nsformation to o, due to ( ξ b ξ b )). The naturality equation ( Qγ n at ) says that γ Q x has some proper ties o f a natural transfor ma tion, but one side o f ( Qγ nat ) is not alwa ys defined when the other is. W e will see la ter (in Section 1.4 ) where γ Q x gives rise to a natural transforma tion. As γ Q x , so ˇ θ ∀ x → and ˆ θ ∃ x ← hav e so me prop erties of na tural transfor ma tions due to the equatio ns ( Q ξ θ ξ θ ). The equa tions ( ξ b ξ b ), ( ξ c ξ c ) and ( Q ξ θ ξ θ ) are equations o f isomo rphisms. In spite o f the equa tions ( r en 1 ) and ( r en ◦ ), which are also functorial equa- tions, r enaming combined with substitution a pplied to formulae do es not g iv e an e ndo functor of QDS , becaus e we do not hav e totally defined functions. If the renaming op erations were not assumed as primitive opera tions on arrow terms for defining QDS (we will see in Sectio ns 1.8 and 2.2 that [ ι Q x A ] x y could be assumed instead), then we would hav e pro ble ms in formulating assumptions that give the following equatio n of QDS : [ f ] x y ◦ [ ι ∀ x A ] x y = [ ι ∀ x B ] x y ◦ ∀ x f , which follows fro m ( ∀ ι n at ), ( r en c ong ) and ( r en ◦ ); we would not know what to write for [ f ] x y . The following eq uation: ( ∀ γ ι ) γ ∀ x A ◦ ι ∀ x A = 1 ∀ x A holds in Q DS . (By our tacit presupp osition, introduced be fo re pres en ting the equations of QDS , the v aria ble x is her e not free in A .) This equatio n is derived as fo llo ws: γ ∀ x A ◦ ι ∀ x A = ∀ x ι ∀ x A ◦ γ ∀ x ∀ x A , by ( ∀ γ nat ), = 1 ∀ x A , by ( ∀ η ). In an ana logous manner, we derive in QDS the equatio n ( ∃ γ ι ) ι ∃ x A ◦ γ ∃ x A = 1 ∃ x A . With the help of the equations ( Qγ ι ) w e derive eas ily the following equations analogo us to ( Q 1): ( Qι ) Q x ι Q x A = ι Q x Q x A , ( Qγ ) Q x γ Q x A = γ Q x Q x A . Note that ( Qι ) can replace ( Qη ) for axiomatizing the equa tio ns of QDS , but ( Qγ ) ca nnot do so , be c ause for it we presupp ose that x is no t free in A , which we do no t presupp ose for ( Qη ). Note that ( ∀ η ) and ( ∃ η ) could b e replaced resp ectively by the equa tions ( ∀ ex t ) ∀ x ( ι ∀ x A ◦ f ) ◦ γ ∀ x B = f : B ⊢ ∀ x A , ( ∃ ex t ) γ ∃ x B ◦ ∃ x ( g ◦ ι ∃ x A ) = g : ∃ x A ⊢ B , 10 which are ea sily derived with ( Qγ nat ) and ( Qη ). (In b oth of these e q uations we tacitly pr esuppos e that x is no t free in B .) By rely ing on ( Qβ ) and ( Qγ ι ) we c a n easily derive the following equations if x is free neither in A no r in D : ˇ θ ∀ x → A,D = ( γ ∀ x A ∨ 1 D ) ◦ ι ∀ x A ∨ D , ˆ θ ∃ x ← A,D = ι ∃ x A ∧ D ◦ ( γ ∃ x A ∧ 1 D ). Note that if x is free in A and not free in D , then in QDS we do not hav e arrows of the types conv erse to the type s of the following distributivity arrows: ˆ θ ∀ x ← A,D = d f ∀ x ( ι ∀ x A ∧ 1 D ) ◦ γ ∀ x ∀ x A ∧ D : ∀ x A ∧ D ⊢ ∀ x ( A ∧ D ), ˇ θ ∃ x → A,D = d f γ ∃ x ∃ x A ∨ D ◦ ∃ x ( ι ∃ x A ∨ 1 D ) : ∃ x ( A ∨ D ) ⊢ ∃ x A ∨ D , which are analog ous to the arrows ˇ θ ∀ x ← A,D and ˆ θ ∃ x → A,D resp ectively . (That these arrows do not exis t in Q DS is shown via cut elimination in GQDS ; see Sections 1.5-9.) So w e ca nnot hav e a prenex normal for m for formulae, i.e. ob jects. 1.3 Change of b ound v ariables W e call cha nge of bound v a r iables what could as well b e called r enaming of bo und v ariables, b ecause we do not wan t to confuse this rena ming with r enaming of free v ariables. W e define in QDS the following arrows, which for malize change of bo und v a riables ( τ might come from “ transcrib e”): τ ∀ x A,u,v = d f ∀ v [ ι ∀ u A x u ] u v ◦ γ ∀ v ∀ u A x u : ∀ u A x u ⊢ ∀ v A x v , τ ∃ x A,v ,u = d f γ ∃ v ∃ u A x u ◦ ∃ v [ ι ∃ u A x u ] u v : ∃ v A x v ⊢ ∃ u A x u , provided u and v a re not free in A . Note tha t τ Q x A,u,v is the sa me arr o w term as τ Q y A x y ,u,v for y not free in A , a nd a fortior i for y neither free nor b ound in A . The v ariable x in τ Q x A,u,v is just a place ho lder, which can alwa ys b e replaced by a n arbitrar y new v ar iable. W e can de r iv e the following equations of Q DS : ( Qτ r en ) [ τ Q x A,u,v ] y z = τ Q x A y z ,u,v , if x is not y or z (if y is free in A and z is u o r v , then the right-hand side of ( Qτ r en ) is undefined), ( Qτ nat ) Q v [ f ] x v ◦ τ Q x A,u,v = τ Q x B ,u,v ◦ Q u [ f ] x u , ( Qτ r ef ) τ Q x A,u,u = 1 Q u A x u , ( Qτ sym ) τ Q x A,v ,u ◦ τ Q x A,u,v = 1 Q u A x u , ( Qτ tr ans ) τ Q x A,v ,w ◦ τ Q x A,u,v = τ Q x A,u,w . 11 F rom the equation ( Qτ sym ) we see that τ Q x A,u,v and τ Q x A,v ,u are inv erse to each other. W e can also der iv e the following equations of QDS : ( ∀ τ ι ) ι ∀ v A x v ◦ τ ∀ x A,u,v = [ ι ∀ u A x u ] u v , ( ∃ τ ι ) τ ∃ x A,v ,u ◦ ι ∃ v A x v = [ ι ∃ u A x u ] u v , ( ∀ τ γ ) τ ∀ x A,u,v ◦ γ ∀ u A = γ ∀ v A , ( ∃ τ γ ) γ ∃ u A ◦ τ ∃ x A,v ,u = γ ∃ v A , ( ∀ τ ˇ θ ) ( τ ∀ x A,u,v ∨ 1 D ) ◦ ˇ θ ∀ u → A x u ,D = ˇ θ ∀ v → A x v ,D ◦ τ ∀ x A ∨ D ,u,v , ( ∃ τ ˆ θ ) τ ∃ x A ∧ D ,u,v ◦ ˆ θ ∃ u ← A x u ,D = ˆ θ ∃ v ← A x v ,D ◦ ( τ ∃ x A,u,v ∧ 1 D ). T o derive ( ∀ τ ˇ θ ) we derive ˇ θ ∀ v ← A x v ,D ◦ ( τ ∀ x A,u,v ∨ 1 D ) = τ ∀ x A ∨ D ,u,v ◦ ˇ θ ∀ u ← A x u ,D with the help of ( ∀ τ ι ) and ( ∀ τ γ ). W e pro ceed ana logously for ( ∃ τ ˆ θ ). Note that τ ∀ x and ˇ θ ∀ x ← , as well as τ ∃ x and ˆ θ ∃ x → , hav e analogous definitions. The following tw o equa tions of QDS are analogo us to the eq uations ( Qτ ι ): ι ∀ x A ∨ D ◦ ˇ θ ∀ x ← A,D = ι ∀ x A ∨ 1 D , ˆ θ ∃ x → A,D ◦ ι ∃ x A ∧ D = ι ∃ x A ∧ 1 D . As a co nsequence of these tw o equations w e hav e ( ∀ ˇ θ ι ) ( ι ∀ x A ∨ 1 D ) ◦ ˇ θ ∀ x → A,D = ι ∀ x A ∨ D , ( ∃ ˆ θ ι ) ˆ θ ∃ x ← A,D ◦ ( ι ∃ x A ∧ 1 D ) = ι ∃ x A ∧ D . 1.4 Quan tifiers and adjunction Lawv ere’s pr esen tation of predica te log ic in catego rial terms (see [1 8], [1 9] and [20], App endix A.1), and presentations that follow him more or less closely (see, for instance, [25], [5], [22] and [1 3], Chapter 4), are less syntactical than ours . They do no t pay close attention to syntax. If this syntax were to be supplied precisely , then a language without v ar iables, like Quine’s v a r iable-free lang ua ge for predicate logic (see [24], and references therein), called predic a te functor logic, w ould b e mor e appropriate. Our first- o rder language is on the co ntrary quite sta ndard. It should b e mentioned also that Lawvere’s approach is more general, whe r eas we concentrate on first-order logic. Lawv ere character iz ed q ua n tifiers in int uitionistic logic through a n a djoin t situation. In La wvere’s c haracter ization of quant ifiers, functors from which the univ ersal a nd existential qua n tifiers ar ise a re resp ectively the right and left adjoints o f a functor that is an insta nc e , inv olving pro duct types and pro jections , of a functor Lawv ere ca lls subst itution . An a pproach in this st yle to linear predicate logic w as firs t made in [26] (Section 2.5, Remar k 3). W e will now present tw o kinds o f a djoin t s ituations that inv olve the quanti- fiers of QDS . These a djunctions ar e r elated to Lawvere’s ideas , but, as we sa id 12 ab o ve, o ur appro ac h is mo re syntactical. In this syntactical approa c h s ubstitu- tion is not mentioned. (What we call r en aming plays no role in it.) Let QDS − x be the full sub categor y o f QDS whose ob jects are a ll for m ulae of L in which x is no t free. F rom QDS − x to QDS there is a n obvious inclusion functor, which we ca ll E . (It b ehav es like identit y on ob jects and on ar rows.) The functor E is full and faithful. B y restricting the co domain of the functors Q x from Q DS to QDS we obtain the functors Q x from Q DS to QDS − x . Then the functor ∀ x is rig ht a djoin t to E , and ∃ x is left adjoint to E . Consider first the adjunction inv olving ∀ x and E . In this adjunction, the arr o ws ι ∀ x make the counit and the arrows γ ∀ x the unit natural transfor mation, w hile ( ∀ β ) a nd ( ∀ η ) are the triangular equa tio ns o f this adjunction. In the adjunction inv olving ∃ x and E , the ar rows ι ∃ x make the unit and the arr o ws γ ∃ x the counit natural transformatio n, while ( ∃ β ) and ( ∃ η ) are the triang ula r eq ua tions. In other words, the full subc ategory QDS − x of Q DS is b oth co reflective and re flectiv e in QDS . The equations ( Q γ ι ) o f Section 1 .2 follow from Theorem 1 and its dual in [21] (Section IV.3). F ro m these theor e ms we also obtain that Q x A and A a re isomo rphic for e very ob ject A of QDS − x . Note that these tw o a djunctions, due to the presence of the equations ( Qγ ι ), or ( Qι ), or ( Qγ ), of Section 1.2, ar e trivial adjunctions in the following sense . If f and g o f the same type are arr o w terms of QDS made only of 1 , ◦ , Q x , ι Q x and γ Q x , then f = g in Q DS (see [6], Sections 4.6 .2 a nd 4.1 1). Whenever a full sub category of a ca tegory C is cor e fle c tiv e or reflective in C , w e hav e a trivial adjunction in the s ame s ense. (The notion o f trivial adjunction is clo sely related to Lambek ’s notion of idemp oten t mo nad o f [17], Section 1.) Note that the adjunctions inv o lving E and Q x do not deliver the distr ibu- tivit y arrows ˇ θ ∀ x → and ˆ θ ∃ x ← . Lawv ere was able to define ˆ θ ∃ x ← in the pr esence of intuitionistic implication, while the constant doma in arrows ˇ θ ∀ x → are not present, and not desired in intu itionistic logic. In an analog ous w ay , the a djunc- tions of pro duct a nd copro duct with the diagonal functor do not deliver dis tribu- tivit y is omorphisms of c o pro duct over pro duct and of pro duct over copro duct in bicar tesian categor ies, i.e. c ategories that ar e cartesian and co cartesian. In bicartesian closed ca tegories, where w e ha ve the exp onential functor, fro m which int uitionistic implication arises, we obtain distributivity iso mo rphisms of pr od- uct ov er co pro duct, but distributivity isomor phisms of co product over pro duct may b e missing. So Lawv er e’s thesis that log ical constants are characterized completely by adjoint situatio ns sho uld be taken with a grain of salt. Conjunction, which corr e- sp onds to pro duct, is characterized by rig h t-adjointn ess to the diago nal functor when it is alone, or when it is accompanied by intuitionistic implica tion. When conjunction and disjunction, which corresp onds to co pro duct, are alo ne, then the tw o adjunctions with the diagonal functor do not s uffice . So me distribu- tivit y arrows, which we would have in the prese nc e of implication, ar e missing. The situation is analogous with quantifiers and the distributivit y arr ows ˇ θ ∀ x → and ˆ θ ∃ x ← in in tuitionistic logic. 13 The situation is different in c la ssical logic, where duality reig ns. Both o f the distributivity arrows ˇ θ ∀ x → and ˆ θ ∃ x ← are defina ble in the presenc e of nega tion (see Section 2.7; nega tion yields implication and “coimplication” ). Both, when defined, happ en to b e is omorphisms in this pap er, a nd should be suc h in clas- sical lo gic, but neither the distribution of disjunction ov er conjunction no r the distribution o f conjunction ov er disjunction should b e iso morphisms in class ical logic, as we ar gued in [7]. The following remark is not ab out our immediate concerns here, but it is per haps worth mak ing once we hav e ra ised the issue of the iso morphism of dis- tribution o f conjunction o ver disjunction, i.e. o f pro duct over copro duct. It is not clea r that a mo dal transla tion ba sed on S 4 will turn this distribution, which should not b e an isomor phis m in classical logic, int o an isomorphism, as it should be in intuitionistic logic. So it is not cle a r that in the pro of theory of S 4 based on classica l lo gic we will be a ble to represent co rrectly the pro of theory of intu- itionistic logic, if the latter is based on bica r tesian clos ed catego r ies. Equa tions betw een proo fs need not b e the same. A similar phenomeno n, pointed o ut in [26] (to whic h the referee brought our attention), is that the mo dal trans la tion of intuitionistic logic in to linear logic ba sed o n Gira rd’s mo dal op erato r ! need not b e pro of-theoretically corr ect when disjunction and the exis ten tial quanti- fier are taken in to account. The Kleisli category of the c o monad of a Girard category in the sense o f [26], where o ne expects to find the mo dal tra nslation of intuitionistic lo gic, need not hav e copro ducts and a left adjoint to Lawv ere’s substitution functor based on pro jection. This ca tegory need not b e bica rtesian closed. 1.5 The category GQDS In this section we enlar ge the results of Section 7.7 o f [7], on which our expo sition will heavily rely . W e in tro duce a ca teg ory called GQDS , which extends with quantifiers the ca tegory GDS of [7]. In GQDS we will be able to per form in a manageable manner the Gentzenization of QDS (this is where G comes from). Let a for mula o f L b e called diversifie d when every predica te letter o ccurs in it at most o nce. A t yp e A ⊢ B is called diversifie d when A and B a re div ersified, and a n a rrow term is diversifie d when its type is diversified. It is eas y to verify that for every arr ow f : A ⊢ B of QDS there is a diver- sified a rrow term f ′ : A ′ ⊢ B ′ of QDS suc h that f is obtained by substituting uniformly pr edicate letters fo r s ome predicate letters in f ′ : A ′ ⊢ B ′ . Namely , f is a letter -for-letter substitution instance of f ′ (cf. [7], Sections 3 .3 a nd 7.6 ). Our aim is to sho w that QDS is a dive rsifie d pr e or der , whic h means that if f 1 , f 2 : A ⊢ B ar e div ersified arrow terms, then f 1 = f 2 in QDS . F o r that purp ose we introduce an aux ilia ry catego ry GQDS where the ξ b → arrows, ξ b ← arrows a nd ξ c a rrows a re identit y arr o ws. W e will prove that GQ DS is a pr e or der , which means that for all ar row ter ms f 1 and f 2 of the same type we have f 1 = f 2 14 in GQDS . That GQDS is a pr eorder will imply that QDS is a diversified preorder . F rom the fact that QDS is a diversified preo rder one can infer that there is a faithful functor G from QDS to the catego r y R el , whose ob jects a re finite ordinals and who se ar r o ws are r elations b et ween these ordinals (see [7], Sections 2.9 and 7 .6). This functor G is defined as for DS in [7] with the understanding that predicate le tters now stand for pro p ositiona l letters; we hav e moreov er that GQ x A = GA (so that GA is the num ber of o ccurre nce s of pr edicate letters in the formula A ), the ar r o w Gα for α b eing ι Q x A , γ Q x A , ˇ θ ∀ x → A,D and ˆ θ ∃ x ← A,D is an identit y arrow, while GQ x f = G [ f ] x y = Gf . The theorem that G is a faithful functor is called Q DS Coher enc e . Let L div be the set of diversified for mulae o f L . Consider the smalles t equiv- alence relation ≡ on L div that satisfies A ξ ( B ξ C ) ≡ ( A ξ B ) ξ C , A ξ B ≡ B ξ A , if A 1 ≡ B 1 and A 2 ≡ B 2 , then A 1 ξ A 2 ≡ B 1 ξ B 2 , if A ≡ B , then Q x A ≡ Q x B , and let [ A ] b e the equiv a lence class o f a diversified formula A with resp ect to this equiv alence rela tion. W e call [ A ] a form set (which follows the terminolo gy of [7], Sectio n 7.7 ). W e us e X , Y , Z , . . . , sometimes with indices, for form sets. It is clear that the form se t [ A ] can b e named by an y of the members of the equiv alence class [ A ]. In these names we may delete parentheses tied to ξ in the immediate scop e of ξ . A su bform set of a form set X is a form set [ A ] fo r A a subformula of a formula in X . Let the ob jects of the catego ry GQDS b e the form s ets we hav e just intro- duced. The a r row terms of GQDS are defined as the arrow terms o f QDS sav e that their indices are form s ets instead of formulae. The equations of GQDS are defined as thos e of Q DS save that we add the equations ξ b → X,Y ,Z = ξ b ← X,Y ,Z = 1 X ξ Y ξ Z , ξ c X,Y = 1 X ξ Y . This defines the category GQDS . F rom the fact that GQDS is a preorder we infer that QDS is a div ersified pr e order as in [7 ] (Sections 3 .3, 7.6 , beginning of 7.7 and end of 7.8). W e define by induction a s e t of terms for the arrows of GQDS , w hich we call Gentzen terms . The y are defined as in [7] (Section 7.7), sav e that to the Gentzen op er ations cut X , ∧ X 1 ,X 2 and ∨ X 1 ,X 2 we add the following Gentzen o peratio ns, where = dn is read “deno tes”: 15 f : X x y ∧ Z ⊢ U ∀ L x,X f = dn f ◦ ([ ι ∀ x X ] x y ∧ 1 Z ) : ∀ x X ∧ Z ⊢ U f : X x y ⊢ U ∀ L x,X f = dn f ◦ [ ι ∀ x X ] x y : ∀ x X ⊢ U f : U ⊢ X x u ∨ Z ∀ R x,X f = dn ( τ ∀ v X x v ,u,x ∨ 1 Z ) ◦ ˇ θ ∀ u → X x u ,Z ◦ ∀ u f ◦ γ ∀ u U : U ⊢ ∀ x X ∨ Z f : U ⊢ X x u ∀ R x,X f = dn τ ∀ v X x v ,u,x ◦ ∀ u f ◦ γ ∀ u U : U ⊢ ∀ x X f : U ⊢ X x y ∨ Z ∃ R x,X f = dn ([ ι ∃ x X ] x y ∨ 1 Z ) ◦ f : U ⊢ ∃ x X ∨ Z f : U ⊢ X x y ∃ R x,X f = dn [ ι ∃ x X ] x y ◦ f : U ⊢ ∃ x X f : X x u ∧ Z ⊢ U ∃ L x,X f = dn γ ∃ u U ◦ ∃ u f ◦ ˆ θ ∃ u ← X x u ,Z ◦ ( τ ∃ v X x v ,x,u ∧ 1 Z ) : ∃ x X ∧ Z ⊢ U f : X x u ⊢ U ∃ L x,X f = dn γ ∃ u U ◦ ∃ u f ◦ τ ∃ v X x v ,x,u : ∃ x X ⊢ U f : X ⊢ Y [ f ] x y : X x y ⊢ Y x y The usual pr oviso for the eigenvariable in connection with ∀ R and ∃ L is impos ed by the tacit pr o visos concer ning γ Q u U , ˇ θ ∀ u → X x u ,Z , ˆ θ ∃ u ← X x u ,Z , τ ∀ v X x v ,u,x and τ ∃ v X x v ,x,u . This proviso says that u , which is called the eigenvariable , is not free in the t yp e s o f ∀ R x,X f and ∃ L x,X f ; i.e., u is free neither in the sources nor in the targets. The types of all the subterms o f a Gentzen ter m make a der iv a tion tr ee usual in Gen tzen systems. (An exa mple may b e fo und in Section 1.10 .) It is easy to show that every ar row of GQDS is deno ted by a Gentzen term. F or that we rely on the Gen tzenization Lemma of Sec tio n 7 .7 of [7], together with the following equations of GQDS : 16 ι ∀ x X = ∀ L x,X 1 X , γ ∀ x U = ∀ R x,U 1 U , ∀ x f = ∀ R x,Y ∀ L x,X f , for f : X ⊢ Y , and the dual equa tions inv olving ∃ instead of ∀ . With the help of the equations ( ∀ τ ι ) (together with renaming), ( ∀ ˇ θ ι ), ( ∀ ι nat ) and ( ∀ β ) (see Sections 1.2-3 ) we derive the following equations of GQDS : ([ ι ∀ x X ] x u ∨ 1 Z ) ◦ ∀ R x,X f = f : U ⊢ X x u ∨ Z , ( ∀ β r e d ) [ ι ∀ x X ] x u ◦ ∀ R x,X f = f : U ⊢ X x u . With the help of the naturality of ˇ θ ∀ u → , ( ∀ τ nat ), ( r en 1), ( ∀ ι ), ( ∀ τ ι ), ( ∀ ˇ θ ι ), ( ∀ ι nat ) and ( ∀ β ) (see Sections 1.2 -3) we derive the following equations of GQDS : ∀ R x,X (([ ι ∀ x X ] x u ∨ 1 Z ) ◦ g ) = g : U ⊢ ∀ x X ∨ Z , ( ∀ η r e d ) ∀ R x,X ([ ι ∀ x X ] x u ◦ g ) = g : U ⊢ ∀ x X . W e der iv e analog o usly the dual equation of GQDS involving ∃ instead of ∀ , which are ca lled ( ∃ β r e d ) and ( ∃ η r e d ). 1.6 V ariable-purification F or pr o ving the results of the following sections w e need to replace arbitra ry Gent zen terms b y Gentzen terms in whose type no v a riable is b oth free a nd bo und. This is the same kind of condition that K leene had to satis fy in [15] (Section 78 ) in order to prove cut elimination in the predicate ca lc ulus . The condition is implicit in Gent zen’s [9], bec a use he did not use the same letters for fr ee and b ound v ariables. A v a riable x is free in the t yp e of f : X ⊢ Y when x is fre e either in X or in Y . W e say that x p articip ates fr e e in f when x is free in the t ype of some subterm of f . W e hav e a na logous definitions with “ free” re placed by “b ound”. W e say that x p articip ates in f when x pa rticipates either free or bo und in f . A Gentzen term of GQDS is variable-pur e when no v ariable pa rticipates in it bo th free and b ound. By changing only b ound v a riables one can transfor m an arbitra r y Gentzen term that is not v ar iable-pure in to a v ariable-pure Gentzen term. (W e could as w ell talk of r en aming o f b ound v ariables, but, as we said at the b eginning of Section 1 .3 , we do no t want to confuse this renaming with the r e naming of free v ariables.) The initial term a nd the r e sulting term need not b e of the same t yp e, a nd hence need not b e eq ual, but they will b e equal up to a n iso morphism, as we sha ll see b elow. Kleene’s purification was done for a s equen t where there w as no v ariable bo th free and bound, and his aim was to obtain a deriv a tion for it in whic h 17 no v ariable is b oth fre e and bound. F or that he could not just c hange b ound v ariable s , but he needed also to change free v ariables . Our aim is different, and we ca n change only bo und v a riables. W e hav e the following equations in GQDS : Q L x,X f = Q L y ,X x y f ◦ ( τ Q v X x v ,x,y ∧ 1 Z ), ( Q L τ ) Q L x,X f = Q L y ,X x y f ◦ τ Q v X x v ,x,y , Q R x,X f = ( τ Q v X x v ,y ,x ∨ 1 Z ) ◦ Q R y ,X x y f , ( Q R τ ) Q R x,X f = τ Q v X x v ,y ,x ◦ Q R y ,X x y f . T o prov e these equations we use e ssen tially the equations ( Qτ ι ) and ( Qτ tr ans ) of Section 1 .3 . W e define τ - t erm s inductively with the following cla us es: τ Q x X,u,v is a τ -ter m; if f is a τ -ter m and S is a qua n tifier prefix, then f ξ 1 Y and S f a re τ -terms. The unique subterm τ Q x X,u,v of a τ -term is ca lled its he ad . Then for every τ -term h there is a τ -term h ′ such that the following eq uations hold in GQDS : ( ξ h ) e quations : ξ X 1 ,X 2 ( f ◦ h, g ) = ξ X ′ 1 ,X 2 ( f , g ) ◦ h ′ , ξ X 1 ,X 2 ( h ◦ f , g ) = h ′ ◦ ξ X ′ 1 ,X 2 ( f , g ), ( cut h ) e quations : cut X ( f ◦ h, g ) = cut X ( f , g ) ◦ h ′ , cut X ( f , g ◦ h ) = cu t X ( f , g ) ◦ h ′ , cut X ( h ◦ f , g ) = h ′ ◦ cut X ( f , g ), cut X ( f , h ◦ g ) = h ′ ◦ cut X ( f , g ), cut X ( f ◦ h 1 , h 2 ◦ g ) = cu t X ′ ( f , g ), ( Qh ) e quations : for S ∈ { L, R } , Q S x,X ( f ◦ h ) = Q S x,X ′ f ◦ h ′ , Q S x,X ( h ◦ f ) = h ′ ◦ Q S x,X ′ f , ( r en h ) e quations : [ f ◦ h ] x y = [ f ] x y ◦ h ′ , [ h ◦ f ] x y = h ′ ◦ [ f ] x y . In these equations X ′ is either X or a different for m set. The τ -terms h 1 and h 2 in the last ( cut h ) equation differ in their heads, whic h a re inv er se to each other (see Section 1.3 ). 18 T o de r iv e the ( ξ h ) equations and the first four ( cut h ) eq uations we use essentially functorial and natur a lit y equations (see [7]). F or the la st ( cut h ) equation we also use ( Q τ s im ), and for the ( Qh ) and ( r en h ) eq uations w e use essentially ( Qτ r en ), ( Qι nat ) and ( Qτ nat ) (see Sectio ns 1.2-3 ). By applying the equations o f GQDS men tio ned in this section, we can establish the following. V ariable-Purifica tion Lemma. F or every Gentzen term f : X ⊢ Y ther e is a variable-pur e Gentzen term f ′ : X ′ ⊢ Y ′ such that in GQDS f = h 2 ◦ f ′ ◦ h 1 wher e h 1 and h 2 ar e c omp ositions of τ -t erms or 1 X or 1 Y . Let us expla in up to a po in t how we achiev e that. Let x b e new for f when x do es not participate in f (see the b eginning of the section) a nd do es not o ccur as an index in the Gentzen o pera tions of renaming that o ccur in f . Let x 1 , . . . , x n be all the v ar iables that pa r ticipate b ound in f . Then ta ke the v ar iables x ′ 1 , . . . , x ′ n all new for f , and apply firs t the equations ( Q L τ ) and ( Q R τ ) w ith x b eing x i and y being x ′ i . In Gentzen ter ms, 1 X o ccurs only with X atomic, and so for every v ariable that participates b ound in f there is a Gentzen op eration by which it was introduced. It rema ins then to apply the equations ( ξ h ), ( cut h ), ( Qh ) and ( r en h ). Note that h 1 and h 2 depe nd only on the t ype o f f and on the choice of the v ar iables x ′ 1 , . . . , x ′ n . 1.7 Renaming of eigenv ariables In this s ection we prov e the equations of GQDS o f the following form: ( Q S r en ) Q S x,X f = Q S x,X [ f ] u v for Q S ∈ {∀ R , ∃ L } , with u the eigenv ar ia ble and v a v aria ble no t free in the t yp e of f (see the b eginning of the prece ding sec tion). That v is not fr ee in the t yp e of f is satisfied a for tio ri w he n v is new for the left-hand side (see the end of the preceding section). The e q uations ( Q S r en ) say that GQDS cov ers the renaming of eigenv ar iables by new v ariables, which is a technique derived fr om [9] (Sectio n I I I.3.10). W e need the equations ( Q S r en ) to prov e the results of Sections 1 .9-10. W e derive now the equation ( ∀ R r en ) for f : U ⊢ X x u ∨ Z : ∀ R x,X f = (( τ ∀ w X x w ,v ,x ◦ τ ∀ w X x w ,u,v ) ∨ 1 Z ) ◦ ˇ θ ∀ u → X x u ,Z ◦ ∀ u [[ f ] u w ] w u ◦ γ ∀ u U , b y ( r en 6), ( r en 2) and ( ∀ τ tr ans ), = ( τ ∀ w X x w ,v ,x ∨ 1 Z ) ◦ ˇ θ ∀ v → X x v ,Z ◦ ∀ w [[ f ] u w ] w v ◦ τ ∀ w U,u,v ◦ γ ∀ u U , b y ( ∀ τ ˇ θ ) and ( ∀ τ nat ), = ∀ R x,X [ f ] u v , b y ( r en 5), ( r en 2) and ( ∀ τ γ ) (see Sections 1.2 -3). 19 The equation ( ∃ L r en ) is derived analog ously . W e can pr o ve also the equations ( Q T r en ) Q T x,X f = Q T x,X [ f ] y z for Q T ∈ {∀ L , ∃ R } , w ith f either of the type X x y ∧ Z ⊢ U or X x y ⊢ U or o f the t yp e U ⊢ X x y ∨ Z or U ⊢ X x y , provided y is not free in the type of Q T x,X f . So , though y is not here an eigenv ar iable, it co uld hav e b een o ne. T o derive the equation ( ∀ L r en ) for f : X x y ∧ Z ⊢ U we hav e ∀ L x,X f = [ f ◦ ([ ι ∀ x X ] x y ∧ 1 Z )] y z , b y ( r en 2 ), = [ f ] y z ◦ ([ ι ∀ x X ] x z ∧ 1 Z ) = ∀ L x,X [ f ] y z , by using, together with other renaming equations , ( r en 5) and ( r en 2 ) if y is different from x , since y is then not free in the type of ι ∀ x X , and by using ( r en 1) if y is x . The equa tion ( ∃ R r en ) is derived ana logously . 1.8 Elimination of renaming W e can e stablish the following prop osition for GQDS . Renaming Elimina tion. F or every varia ble-pur e and cut- fr e e Gentzen term t ther e is a variable-pur e, cu t -fr e e and r enaming-fr e e Gentzen t erm t ′ such that t = t ′ . Here cut-fr e e means o f cour se that no instance of the Gentzen op eration cut X o ccurs in t and t ′ , and r enaming-fr e e means that none o f the Gen tzen op erations [ ] x y o ccurs in t ′ . The pr o of o f Renaming Elimination is based o n the follo wing equatio ns of GQDS : [ ξ X 1 ,X 2 ( f , g )] x y = ξ X ′ 1 ,X ′ 2 ([ f ] x y , [ g ] x y ), [ Q S z ,X f ] x y = Q S z ,X ′ [ f ] x y , if z is neither x nor y . T o eliminate all o ccurrences of renaming w e eliminate one by one innermost o ccurrences of renaming, i.e. o ccurrence s of renaming within the scop e of which there is no r enaming. V a riable-purity ensures that the pro viso of the second equation is not an obstacle. W e will use Renaming Elimination for the pro o f o f the Cut-Elimination Theorem for GQDS in the next section. F or that we need a str engthene d version of R enaming Elimination , in which it is sp ecified that the Gentzen term t ′ is ex actly analogo us to t : only indices of its identit y a rrows and of its Gentzen op erations may change. F or f : X ⊢ Y and g : Y ⊢ X such that x is not free in X in GQDS we hav e 20 [ f ] x y = cut ∀ x Y ( ∀ R x,Y f , ∀ L x,Y 1 Y x y ), [ g ] x y = cut ∃ x Y ( ∃ R x,Y 1 Y x y , ∃ L x,X g ). So particular instances of r e naming (and Gentzen and Kleene did not envisage implicitly mor e tha n that) ca n be e asily eliminated provided we want to toler ate cut. (In the pres ence o f implication we could eliminate a ll instances of r enaming in the presence of cut, as we mentioned in Section 1.2.) O ur aim how ever is to eliminate bo th cut and renaming. If we delete “v ariable-pure” from Renaming Elimination, then this prop o- sition canno t b e prov ed. A co unterexample, a nalogous to a counterexample in [15] (Section 78, Exa mple 4), is the following: [ ∀ L x, ∀ y Rxy ∀ L y ,Ruy 1 Ruz ] u y : ∀ x ∀ y Rxy ⊢ Ry z . F rom this Gentzen term we can eliminate r enaming only by intro ducing cut, as ab o ve. Kleene in [15] also needed v ariable-purity to eliminate cut. But his coun- terexample, mentioned ab ov e, which is ana logous to our co unterexample, would not be a counterexample in the presence of r enaming. 1.9 Cut elimination Our aim in this section is to establish the following theo rem for GQDS . Cut-Elimina tio n Theorem. F or every variable-pur e Gentzen term t t her e is a variable-pur e and cut-fr e e Gentz en term t ′ such that t = t ′ . The pro of of this theorem is obtained by mo difying and expanding the pro of of the Cut-Elimination Theorem fo r GDS in [7] (Section 7.7). W e pre s uppose below the termino logy introduced in this previous pro of. The Q - r ank of cut Q x X ( f , g ) is n 1 + n 2 when f has a subterm Q R x,X f ′ of depth n 1 and g has a subterm Q L x,X g ′ of depth n 2 . The r ank of a topmos t cut cu t X ( f , g ) is either its ∧ -r ank, or ∨ -rank, or p -rank, o r Q -rank depending o n X . The c omplexity of a topmost cut cut X ( f , g ) is ( m, n ) where m ≥ 1 is the sum of the num ber o f predic a te letters and o ccurrences of qua n tifier prefixes in X and n ≥ 0 is the rank of this cut. Every form set o f the for m Q x X is co ns idered to be b oth of colour ∧ and colo ur ∨ . In the pr o of w e have the following additiona l ca ses. W e consider only the most c o mplicated case s , and leav e out the remaining simpler cases, whic h ar e dealt with analogous ly . ( ∀ 1) If our topmost cut is cut ∀ x X ( ∀ R x,X f , ∀ L x,X g ) : U ∧ Y ⊢ Z ∨ W 21 for f : U ⊢ X x u ∨ Z and g : X x v ∧ Y ⊢ W , with co mplexit y ( m, 0 ) where m > 1, then we use the equa tio n cut ∀ x X ( ∀ R x,X f , ∀ L x,X g ) = cu t X x v ([ f ] u v , g ) , in which the c ut on the right-hand side is of low er co mplexit y than the to pmost cut on the le ft-hand side. T o deriv e this equation w e us e essentially the eq uations ( ∀ β r e d ) (see Section 1.5 ) together with natur a lit y and functorial equa tions. W e pro ceed ana logously when the topmost cut we start from is cut ∃ x X ( ∃ R x,X f , ∃ L x,X g ) . Suppo se for the cases below tha t X is of colour ∧ . ( ∀ 2) If our topmost cut is cut X ( ∀ R x,V f , g ) : U ∧ Y ⊢ ∀ x V ∨ Z ∨ W for f : U ⊢ X ∨ V x u ∨ Z and g : X ∧ Y ⊢ W , with complexity ( m, n ) where m, n ≥ 1, then we use the equa tio n cut X ( ∀ R x,V f , g ) = ∀ R x,V cut X ([ f ] u v , g ) with v b eing a v ariable new for the left-hand side. By the streng thened v ersion of Renaming Eliminatio n from the preceding s ection, there is a v a riable-pure, cut- free a nd r enaming-free Gent zen term f ′ such that [ f ] u v = f ′ , and the complex it y of cut X ( f ′ , g ) is ( m, n − 1). This equatio n is derived as follows: cut X ( ∀ R x,V f , g ) = cut X ( ∀ R x,V [ f ] u v , g ), by ( ∀ R r en ) for v new for the left-hand side (see Sectio n 1.7 ), = ∀ R x,V (([ ι ∀ y V ] y v ∨ 1 Z ∨ W ) ◦ cut X ( ∀ R x,V [ f ] u v , g )), by ( ∀ η r e d ) (see Sectio n 1 .5), = ∀ R x,V ( cut X (([ ι ∀ x V ] x v ∨ 1 Z ) ◦ ∀ R x,V [ f ] u v , g ), b y functoria l and naturality equations, = ∀ R x,V ( cut X ([ f ] u v , g )), by ( ∀ β r e d ). W e needed to rename the eigenv a riable u by a new v in order to ensure that the proviso for the e ig en v ariable is satisfied in the second line for the ∀ R x,V op eration newly int ro duced. ( ∀ 3) If our topmost cut is cut X ( ∀ L x,V f , g ) : ∀ x V ∧ U ∧ Y ⊢ Z ∨ W for f : V x y ∧ U ⊢ X ∨ Z and g : X ∧ Y ⊢ W , then we use the straightforw ard equation 22 cut X ( ∀ L x,V f , g ) = ∀ L x,V cut X ( f , g ). W e hav e a lso the straig h tforward equation cut X ( ∃ R x,V f , g ) = ∃ R x,V cut X ( f , g ), and the equatio n cut X ( ∃ L x,V f , g ) = ∃ L x,V cut X ([ f ] u v , g ), prov ed like the analo gous equation in cas e ( ∀ 2). These equa tions enable us to settle the remaining cases when X is of colour ∧ . When X is of co lour ∨ , we pro ceed in a dual manner. By Renaming E limination, we need no t cons ider ca ses when in o ur to pmost cut cut X ( f , g ) either f or g is o f the form [ h ] y z . 1.10 In v ert ibilit y in GQDS The results we are go ing to prove in this section corre s pond to inv e r ting rules in deriv ations , i.e. passing fro m conclus ions to premises. This inv ertibility is g ua r- anteed b y the p ossibility to p ermute rules, i.e . change their o rder in deriv ations, and we show for that p ermuting that it is covered by the eq uations of GQDS . (Perm utation of rules is a theme trea ted in [16], but without considering equa- tions b etw een der iv a tions.) Besides the equations men tioned in [7] (beginning of Section 7.8) we will need the e quations of GQDS of the following fo r m: ( ξ Q S ) ξ X 1 ,X 2 ( Q S x,X f 1 , f 2 ) = Q S x,X ξ X 1 ,X 2 ( f ′ 1 , f 2 ), for X x y being a s ubfor m set of the source o r targ et of f 1 , and f ′ 1 being f 1 when Q S ∈ {∀ L , ∃ R } , a nd [ f 1 ] y v with v new for the left-hand side when Q S ∈ {∀ R , ∃ L } . These eq uations ar e either straightforw ard to derive, or when Q S ∈ {∀ R , ∃ L } we derive them by imitating the deriv ation o f the eq uation of case ( ∀ 2) of the preceding se c tion, with the help of the equa tions ( Qβ re d ) and ( Qη r e d ) (see the end of Section 1.5 ). W e will a lso need for the end of the s ection the following equations, whos e deriv ations are not difficult to find: for Q S ∈ {∀ L , ∃ R } , ( Q S Q S ) Q S y ,Y Q S x,X f = Q S x,X Q S y ,Y f , ( ∃ R ∀ L ) ∃ R y ,Y ∀ L x,X f = ∀ L x,X ∃ R y ,Y f , for Q ∈ {∀ , ∃} , S ∈ { L, R } , and the proviso for the e igen v ariable being satisfied, ( Q R Q L ) Q R y ,Y Q L x,X f = Q L x,X Q R y ,Y f , ( ∃ S ∀ S ) ∃ S y ,Y ∀ S x,X f = ∀ S x,X ∃ S y ,Y f . 23 The Inv er tibilit y L emmata for ∧ and ∨ are formulated as in [7] (Section 7.8). Only l et ( X ) is the set of pr e dic ate letters occur ring in the for m set X . These lemma ta hold also when we replace thro ughout “cut-free Gentzen term” by “v ar iable-pure, cut-free and renaming-free Gentzen term”. They are proved as in [7], with additional cases cov ered by the equations ( ξ Q S ). The fo llowing invertibilit y lemmata are easy consequenc e s of the equa tions ( ∀ η r e d ) and ( ∃ η r e d ) (see the end of Section 1.5 ). Inver tibility Lemma f or ∀ R . If f is a varia ble-pur e Gentzen term of the typ e U ⊢ ∀ x X ∨ Z or U ⊢ ∀ x X , then ther e is a variable-pur e Gentzen term f ′ of t he typ e U ⊢ X x u ∨ Z or U ⊢ X x u r esp e ctively such t hat ∀ R x,X f ′ = f . Inver tibility Lemma for ∃ L . If f is a varia ble-pur e Gent zen term of the typ e ∃ x X ∧ Z ⊢ U or ∃ x X ⊢ U , then ther e is a variable-pur e Gentzen term f ′ of t he typ e X x u ∧ Z ⊢ U or X x u ⊢ U r esp e ctively such that ∃ L x,X f ′ = f . Before for m ulating the remaining inv ertibilit y lemmata for ∀ L and ∃ R we m ust int ro duce a num ber of notions concer ning o ccurr ences of v aria bles within the types of v ar iable-pure, cut- fr ee and renaming- free Gent zen terms. Although many o f the notions in tro duced make sense also for other Gentzen terms, w e need these notions only in the c on text of v ar iable-pure, cut-free a nd r enaming- free Gentzen terms. The ess en tial ass ertions using these notions, which we need for o ur results, need not hold for a ll Gentzen terms. Let α and β , s ometimes with indices , stand for o ccurrence s of individual v ariable s in a form set, and let γ , sometimes with indices, stand for an o ccurre nc e of a qua n tifier prefix in a form set. Let α and β b e different o ccurrences of the v ariable x in the for m set X , a nd let γ b e an o ccurrence of Q x in X . Then we say that α and β are simult ane ously b ound by γ when X has a subform set γ Y such that α and β are free in Y . W e say that a predicate letter P o c curs in the typ e of the Gentzen term f : X 1 ⊢ X 2 when it o ccurs in X 1 or X 2 . Because of diversification (see the beg inning of Section 1.5), every n -ary predica te letter P that o ccurs in the type of f o ccurs exa ctly o nc e in X i in a subform set P α i 1 . . . α i n of X i , for i ∈ { 1 , 2 } . Here α 1 j and α 2 j , for 1 ≤ j ≤ n , ar e not necessa rily o ccurr ences of the same v ariable . W e say that the pair ( P α 1 1 . . . α 1 n , P α 2 1 . . . α 2 n ) is a formula c ouple of f , and we say that ( α 1 j , α 2 j ) is a c ouple of f . W e say that it is the P j - c ouple o f f when we wan t to stress from whic h formula couple and from whic h place j in it originates. If ( α 1 , α 2 ) is a co uple of f , then α i , for i ∈ { 1 , 2 } , is fr e e in ( α 1 , α 2 ) when it is free in X i , and analog o usly with “free” r eplaced by “b ound”, “universally bo und” and “ex isten tially b ound”. F or example, if f is a Gentzen term of type ∀ x Rux ∧ P y ⊢ ∃ z ( Ruz ∧ P z ) , then ( R ux, Ruz ) and ( P y , P z ) are the formula couples of f , and the R 1 -couple 24 of f consists of the o ccurrenc e s of u in R ux and Ruz , the R 2 -couple of f consis ts of the o ccurrence of x in Rux a nd the o ccurr ence of z in Ru z , a nd the P 1 -couple of f consists of the o ccurrence of y in P y and the o ccurrence of z in P z in the t yp e o f f . A left bridge betw een the differen t co uples ( α 1 , α 2 ) and ( β 1 , β 2 ) of f such that α 1 and β 1 are o ccurrences of the same v ar iable x is an occ ur rence γ of a quantifier prefix Q x in X 1 such that α 1 and β 1 are simultaneously bo und by γ . W e define analogously a right bridge b y r eplacing α 1 , β 1 and X 1 with α 2 , β 2 and X 2 resp ectively . A bridge is a left bridge or a r igh t bridge. F or f as ab ov e, we have that the o ccurrence of ∃ z in the type of f is the rig ht bridge be tween the R 2 -couple and the P 1 -couple of f , while the o ccurrence of ∀ x is not a bridge at all. Two different couples of f ar e bridge able when there is a bridge betw een them (there might be b oth br idges, as , for exa mple, b et w een the R 1 -couple and R 2 -couple of a Gentzen term o f type ∀ x Rxx ⊢ ∃ y Ry y ). F or n ≥ 2, a s equence of couples ( α 1 1 , α 2 1 ) , ( α 1 2 , α 2 2 ) , . . . , ( α 1 n , α 2 n ) of f suc h that ( α 1 j , α 2 j ) a nd ( α 1 j +1 , α 2 j +1 ), w he r e 1 ≤ j ≤ n − 1, are bridgea ble is called a bridge able chain of c ouples . F or every bridg eable chain o f couples w e can find a t leas t one s equence of bridges γ 1 , . . . , γ n − 1 that ensure its bridgea bilit y . W e say that ( α 1 1 , α 2 1 ) and ( α 1 n , α 2 n ) are cluster e d when there is a bridg e able chain o f couples in which ( α 1 1 , α 2 1 ) and ( α 1 n , α 2 n ) are resp e ctiv ely the first and last member . A s e quence of br idg es ensuring the bridge abilit y of this bridgeable chain o f couples is said to en s ur e the clustering of ( α 1 1 , α 2 1 ) a nd ( α 1 n , α 2 n ). A set C of co uples of f is a cluster of f when there is a couple ( α 1 , α 2 ) in C such that for every co uple ( β 1 , β 2 ) of f different from ( α 1 , α 2 ) we have that ( β 1 , β 2 ) ∈ C iff ( β 1 , β 2 ) is clustered with ( α 1 , α 2 ). If f is o f type ∀ x Rux ∧ P y ⊢ ∃ z ( Ruz ∧ P z ), then w e hav e tw o clusters of f : one singleton c onsisting o f the R 1 -couple, and ano ther whose elements a re the R 2 -couple and P 1 -couple o f f . If f is of type ∀ x Rxx ∧ P y ⊢ ∃ z ( Ry z ∧ P z ), then we hav e just one clus ter o f f , since the R 1 -couple, R 2 -couple and P 1 -couple of f make a bridg eable chain o f couples. Let P b e a predicate letter o ccurr ing in the t yp e o f a cut-free Gentzen term f , a nd let S u b P ( f ) b e the s et of s ubterms of f in whose type P o ccurs. F or every member f ′ of S ub P ( f ) there is a for m ula couple ( P α 1 1 . . . α 1 n , P α 2 1 . . . α 2 n ) of f ′ . The set of all the couples ( α 1 , α 2 ) such that there is a member f ′ of S u b P ( f ) with ( α 1 , α 2 ) the P j -couple of f ′ is ca lled an ar c of f . F or exa mple, in the following picture: 25 1 Ruy : R u ✐ y ⊢ R u ✐ y ∀ L x,Rux 1 Ruy : ∀ x Ru ✐ x ⊢ Ru ✐ y 1 P y : P ✐ y ⊢ P ✐ y ∧ Ruy ,P y ( ∀ L x,Rux 1 Ruy , 1 P y ) : ∀ x Ru ✐ x ∧ P ✐ y ⊢ R u ✐ y ∧ P ✐ y ∃ R z ,Ruz ∧ P z ∧ Ruy ,P y ( ∀ L x,Rux 1 Ruy , 1 P y ) : ∀ x Ru ✐ x ∧ P ✐ y ⊢ ∃ z ( Ru ✐ z ∧ P ✐ z ) ❅ ❅ ❅ ❅ P P P P P ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❏ ❏ ❍ ❍ ❍ ✓ ✏ ✑ ✑ ✑ ✑ ✑ ✑ ▲ ▲ ▲ ▲ ▲ ▲ ❏ ❏ ❏ ❏ ❏ ❏ ❍ ❍ ❍ ❍ ❍ ❍ ✓ ✏ ✎ ☞ ✓ ✏ the encir c led o ccurrences of v a riables connected by lines make tw o arcs o f the v ariable - pure, cut-free a nd re na ming-free Gentzen ter m in the last line. W e say that an ar c is the P j - ar c when we want to stre s s from which formula couples and from whic h place j in them it originates. In o ur ex ample a bov e, we hav e drawn the R 2 -arc and the P 1 -arc. The b ott om of the P j -arc of f is the j -th co ordina te ( α 1 j , α 2 j ) o f the fo r m ula couple ( P α 1 1 . . . α 1 n , P α 2 1 . . . α 2 n ) o f f . In the exa mple ab ov e, the b ottom of the R 2 -arc is ( x, z ) in the last line and the bo ttom of the P 1 -arc is ( y , z ) in the la st line. Two arcs of f a re cluster e d w he n their b ottoms are c lustered. In the example ab o ve, the tw o ar c s are clustered. If we delete the last line, then we obtain t wo arcs o f ∧ Ruy ,P y ( ∀ L x,Rux 1 Ruy , 1 P y ) that are not clustere d. A s et A o f ar c s of f is an ar c-cluster of f when there is an a rc a in A such that for every a r c b o f f differen t from a we have that b ∈ A iff b is cluster ed with a . The b ottoms of the a rcs in an arc-cluster of f make a cluster o f f . W e call this cluster the b ottom clu s ter of the arc- c luster. In our e xample, we hav e an arc-cluster , whose b ottom cluster is made of the bottom of the R 2 -arc and the botto m o f the P 1 -arc. All o ccurrences of v ariables that ar e free in the couples of a n a rc, or of a n arc-cluster , of a cut- fr ee and rena ming -free Gentzen term f are o ccurrences o f the same v ariable. (Here cut-fre e do m and r enaming-freedom is essential.) This v ariable is called the fr e e variable of the arc, or of the ar c-cluster. The free v ariable of the a rc-cluster in our example is y . F or S ∈ { L, R } , supp ose we have a cut-free a nd rena ming-free Gentzen term f that has a s ubterm Q S x,X g for x free in X and X x y o ccurring in the t yp e of g . W e say that Q S x,X g b elongs to an ar c-cluster of f when the o ccurrences of y in X x y in the t ype of g that hav e replaced x in X b elong to couples in this arc-cluster . W e say that Q S x,X g b elongs to a cluster o f f whe n it belo ngs to an arc-cluster whose b ottom cluster is this cluster. In our example ab ov e, the Gent zen term in the last line: ∃ R z ,Ruz ∧ P z ∧ Ruy ,P y ( ∀ L x,Rux 1 Ruy , 1 P y ) belo ngs to the arc -cluster co nsisting of the R 2 -arc and P 1 -arc, a nd the Gentzen 26 term ∀ L x,Rux 1 Ruy in the second line b elongs to the arc- cluster consisting of the R 1 -arc, which is no t drawn in the picture. The subterms of f that belong to a n arc-cluster of f are ca lled the gates of that arc-cluster, and analogous ly with “arc-cluster ” r eplaced b y “ cluster”. A gate belong ing to a cluster may corr espond to bridges in br idgeable chains of couples in this cluster, but it need not c orresp ond to such a bridg e. In our example ab ov e, the Gentzen ter m in the las t line corres p onds to a bridge, but ∀ L x,Rux 1 Ruy do es not. A gate is called an eigengate when it is either of the ∀ R or of the ∃ L t yp e. As a conseq uence of the proviso for the eigenv ariable, we obtain that if a clus ter has an eigeng a te Q S x,X g , then e v ery other ga te o f that cluster is a subterm o f g . This implies that ev ery cluster has at mo s t one eig engate. As ano ther co nsequence of the proviso for the eigenv a r iable, we have the following rema r k. Eigenga te Remark. If we have an eigengate in a clu ster, then for every c ouple ( α 1 , α 2 ) in this cluster b oth α 1 and α 2 ar e b ound. A couple ( α 1 , α 2 ) can b e of the fo llo wing six kinds, dep ending on whether the o ccurrences of v ar iables in it ar e universally b ound, free or ex isten tially b ound: α 1 α 2 name o f kind universally bo und u niversally b ound ( ∀ , ∀ ) universally bo und free ( ∀ , ∅ ) universally bo und existentially b ound ( ∀ , ∃ ) free free ( ∅ , ∅ ) free e x isten tially b ound ( ∅ , ∃ ) existentially b ound existen tially bound ( ∃ , ∃ ) The kinds not mentioned—namely , ( ∅ , ∀ ), ( ∃ , ∀ ) and ( ∃ , ∅ )—ar e not po ssible. If a ( ∅ , ∅ ) couple occur s in a cluster , then this cluster is a singleton. If a ( ∀ , ∀ ) couple o r a ( ∃ , ∃ ) couple o ccurs in a cluster, then this clus ter has an eigeng ate. T ogether with ( ∀ , ∀ ) couples in a clus ter we can find only ( ∀ , ∀ ) couples a nd ( ∀ , ∃ ) couples, and analogo usly toge ther with ( ∃ , ∃ ) couples in a cluster we can find only ( ∃ , ∃ ) couples and ( ∀ , ∃ ) couples. This is a consequence of the Eig engate Remark and of the fact that a cluster can hav e only one eig engate. Couples of the ( ∀ , ∃ ), ( ∀ , ∅ ) a nd ( ∅ , ∃ ) kind, ca n be joined together in a cluster without eigengate. In the b ottom clus ter in our example ab ov e, we have a ( ∀ , ∃ ) couple ( x, z ) and a ( ∅ , ∃ ) couple ( y , z ). This cluster has no eigengate. A Gentzen term f is eigendi versifie d when it is v ariable-pure , cut-fre e a nd renaming-free , a nd, mor eo ver, for ev ery arc-clus ter o f f that has an eigengate the free v aria ble o f this arc- cluster is different from the free v aria ble o f an y other arc-cluster of f . (Eige ndiversification is inspired by [9 ], Section II I.3.10.) W e hav e the fo llo wing for GQDS . 27 Eigendiversifica tion Lemma. F or every variable-pur e, cut-fr e e and r enaming- fr e e Gentzen term f t her e is an eigendiversifie d Gentzen term f ′ such that f = f ′ . Proof. This lemma is proved by repla cing the fr e e v ar iable of an arc-cluster of f that has an eigengate by a v a riable new for f . B y doing that for every a rc- cluster of f that has an eig engate we obtain f ′ , which differs from f just in the indices of identit y arrows a nd of Gent zen op erations. The equations ( Q S r en ) of Section 1 .7 gua rant ee that f ′ = f . T ake for example a subterm of f of the form ∀ R x,X g : Y ⊢ ∀ x X ∨ Z for g : Y ⊢ X x u ∨ Z , and supp ose that in f ′ we hav e instead a t the same place a subterm ∀ R x,X g ′ : Y ⊢ ∀ x X ∨ Z for g ′ : Y ⊢ X x u ′ ∨ Z where u ′ is new for f . Since u is not free in the type of ∀ R x,X g , so u is not fr e e in the type of g ′ , and by ( ∀ R r en ) we ha ve that ∀ R x,X g ′ = ∀ R x,X [ g ′ ] u ′ u . By the s trengthened version of Renaming Elimination (see Section 1 .8 ), we obtain that [ g ′ ] u ′ u = g . ⊣ Next we g iv e the following inductive definition of the notio n of subform of a form set, which ex tends the notion of subfor m set (see the be ginning of Section 1.5): X is a subfor m of X ; if Y is a subfor m of X x y , then Y is a subform of Q x X ; if X a nd Y ar e subfor ms of X ′ and Y ′ resp ectively , then X ξ Y is a subform of X ′ ξ Y ′ ; if X is a subform of Y , then X is a subform of Y ξ Z . (Note that Y ξ Z is the same fo rm set as Z ξ Y .) W e s ay that a Gentzen term is ∀ x X - r e gular when it do es not hav e subterms of one o f the following tw o forms: ( a ) ∀ R z ,Z ∀ L x,X g for g of the type X x y ∧ U ⊢ Z z y ∨ V , or o f one of the thre e types obtained by omitting ∧ U or ∨ V , ( b ) ∃ L z ,Z ∀ L x,X g for g o f the type X x y ∧ Z z y ∧ Y ⊢ V or X x y ∧ Z z y ⊢ V . W e can then prove the following lemma. Lemma ∀ L . If f 1 : ∀ x X ∧ U 1 ⊢ Z 1 is an eigendiversifie d Gentzen term and f 2 : X x u ∧ U 2 ⊢ Z 2 is a variable-pur e Gentzen term such that U 1 and Z 1 ar e sub- forms of U 2 and Z 2 r esp e ctively, then f 1 is ∀ x X - r e gular. The same holds if in al l the typ es ab ove we omit ∧ U 1 and ∧ U 2 , or just ∧ U 1 . Proof. Suppo s e f 1 is not ∀ x X - regular. W e will consider only the ca se when f 1 has a subterm of the form ∀ R z ,Z ∀ L x,X g for g : X x y ∧ U ⊢ Z z y ∨ V . When f 1 has a subterm o f the form mentioned in the remaining cases of ( a ) or in cas e 28 ( b ), we pro ceed analogo usly . Let Z ′ be the subform set o f the ta rget Z 1 of f 1 containing exa ctly the same predica te letters as Z , a nd let γ be the o ccurr e nce of ∀ x at the b eginning of ∀ x X in the sour ce o f f 1 . By the assumption that f 1 is eigendiversified, for α an o ccurrence o f x in X and β an o ccurrence of z in Z ′ , either (1) w e hav e a couple ( α, β ) of f 1 , or (2) w e hav e tw o clustered couples ( α, α ′ ) a nd ( β ′ , β ) of f 1 with a seque nce of bridges γ 1 , . . . , γ n − 1 different from γ that ensure their clustering. If we ha ve clustered co uples a s in (2), but γ o ccurs in γ 1 , . . . , γ n − 1 , then let γ j , for 1 ≤ j ≤ n − 1, b e the rightmost o ccurre nce of γ in γ 1 , . . . , γ n − 1 . The bridge γ j is betw een ( α j , β j ) and ( α j +1 , β j +1 ), and α j +1 is an o ccurr ence of x in X in the source of f 1 . If j = n − 1, then we hav e (1), and if j < n − 1, then we have (2 ). By Renaming Eliminatio n and the Cut-Elimination Theorem (see Sections 1.8-9), we may assume that f 2 is cut-free and r enaming-free. If w e hav e (1), then we should hav e a ( ∅ , ∀ ) couple of f 2 , which is impo ssible. If we have (2), then in ( α, α ′ ) we have that α is univ ersally b ound. Since U 1 and Z 1 are subforms of U 2 and Z 2 resp ectively , there should b e a bridgeable chain of couples of f 2 whose fir s t member is of the k ind ( ∅ , ∅ ) or ( ∅ , ∃ ), and whose last member is o f the kind ( ∀ , ∀ ). The br idg es ensuring the br idgeabilit y of this chain of couples of length n cor respo nd to γ 1 , . . . , γ n − 1 . How ever, a bridg eable chain of couples of the kind ab ov e canno t exist, as we said after the Eigenga te Remar k. (All couples in a bridge able chain of c ouples b elong to the s a me cluster.) ⊣ There is an a na logous lemma that should b e called L emma ∃ R . It inv olves ∃ R instead of ∀ L (whic h engenders the notion of ∃ x X - regularity). W e ca n now finally s ta te the following lemmata. Inver tibility Lemma f o r ∀ L . If f 1 : ∀ x X ∧ U 1 ⊢ Z 1 is an eigendiversifie d Gentzen term, and ther e is a variable-pur e Gentzen term f 2 : X x y ∧ U 2 ⊢ Z 2 wher e U 1 and Z 1 ar e subforms of U 2 and Z 2 r esp e ctively, then ther e is an eigen- diversifie d Gentzen term f ′ 1 : X x y ∧ U 1 ⊢ Z 1 such that ∀ L x,X f ′ 1 = f 1 . The s ame holds if in al l the t yp es ab ove we omit ∧ U 1 and ∧ U 2 , or just ∧ U 1 . Inver tibility Lemma for ∃ R . If f 1 : Z 1 ⊢ ∃ x X ∨ U 1 is an eigendiversifie d Gentzen term, and ther e is a variable-pur e Gentzen term f 2 : Z 2 ⊢ X x y ∨ U 2 wher e U 1 and Z 1 ar e subforms of U 2 and Z 2 r esp e ctively, then ther e is an eigen- diversifie d Gentzen term f ′ 1 : Z 1 ⊢ X x y ∨ U 1 such that ∃ R x,X f ′ 1 = f 1 . The s ame holds if in al l the t yp es ab ove we omit ∨ U 1 and ∨ U 2 , or just ∨ U 1 . These t wo lemmata are prov ed b y induction on the co mplexit y of f 1 with the help o f the equations ( ξ Q S ), ( Q S Q S ), ( ∃ R ∀ L ), ( Q R Q L ) and ( ∃ S ∀ S ), from the beg inning of the section, and Lemmata ∀ L and ∃ R . Without the L emmata 29 ∀ L and ∃ R we would not b e able to apply the eq uations ( Q R Q L ) and ( ∃ S ∀ S ). If we end up with f ′ 1 of the type X x y ′ ∧ U 1 ⊢ Z 1 , o r Z 1 ⊢ X x y ′ ∨ U 1 resp ectively , where y ′ is differe n t from y , then we pr oceed in the spirit of the pro o f of the Eigendiversification Lemma by us ing the equations ( Q T r en ) of Section 1 .7. 1.11 Pro of of QDS Coherence W e a re now rea dy to prov e that GQ DS is a preor der (see the b eginning of Sec- tion 1 .5 ). Supp o se we have tw o arrow terms of GQDS of the sa me type. These arrow terms are equal to tw o Gen tzen terms f 1 and f 2 by the new Gent zeniza- tion Lemma of Section 1.5. Let x 1 , . . . , x k be the v a riables o ccurring b ound in the types o f f 1 and f 2 . By the V ar iable-Purification Lemma of Section 1 .6, w e hav e that f i = h i 2 ◦ f ′ i ◦ h i 1 , for i ∈ { 1 , 2 } , where f ′ i is v aria ble-pure, while h i 1 and h i 2 are isomorphisms of GQDS . By choosing the same new v a riables x ′ 1 , . . . , x ′ k bo th for f 1 and f 2 , we obtain that h 1 j = h 2 j for j ∈ { 1 , 2 } . So if f ′ 1 = f ′ 2 , w e will b e able to derive f 1 = f 2 . The Gen tzen terms f ′ 1 and f ′ 2 are v aria ble-pure, and hence by Rena ming Elimination and the Cut-E limination Theorem (see Sec tio ns 1.8-9) we c a n as- sume that they are cut-free and renaming-fre e. By the Eigendiversification Lemma of the pre ceding s ection, we ca n assume that they ar e moreover eig en- diversified. Let the quant ity of a Gentzen term of GQDS be the sum of the num be r of predicate letters in its source (which is equal to the num b er o f predicate letters in its ta rget) with the n umber o f o ccurrence s o f quantifier pr efixes in its source and tar get. Then we pro ceed by induction on the qua n tit y of f ′ 2 , which is equal to the quan tit y of f ′ 1 , in order to show tha t f ′ 1 = f ′ 2 . In the basis of this induction, if n = 1, then f ′ 1 = f ′ 2 = 1 A , where A is ato mic. In the induction step we apply the inv ertibility lemma ta of the preceding section (cf. [7], end of Section 7.9). So GQDS is a preo rder. And, a s we explained at the b eginning of Section 1.5, w e have proved there by QDS Coherence. 2 Coherence of QPN ¬ 2.1 The categories QP N ¬ and QPN In this sec tio n we intro duce the categor y QPN ¬ (here PN comes fro m “pr oo f net”), which cor resp o nds to the m ultiplicativ e frag ment without prop ositional constants o f c lassical linear first-or der predica te lo gic without mix. This ca t- egory extends with qua n tifiers the pr opo s itional ca tegory PN ¬ of [8] (Section 2.2). 30 The category QPN ¬ is defined as the categor y QDS in Section 1.2 sa ve that we make the following a dditions and changes. Instead of the lang uage L of Section 1 .1 we have the language L ¬ , which differs fro m L by having the additional unary connective ¬ . So in the definition of formula we hav e the additional clause if A is a formula, then ¬ A is a formula. The ob jects of the category QPN ¬ are the formulae of L ¬ . Let x n stand for the sequence x 1 , . . . , x n when n ≥ 1, and for the empt y sequence when n = 0. F or A a formula, let A x n y n stand for A x 1 ... x n y 1 ... y n when n ≥ 1, and for A when n = 0 ; so A x n y n is the r esult o f a ser ies of n substitutions. W e use Q x n as an a bbreviation for Q x n . . . Q x 1 when n ≥ 1 , a nd for the empty sequence when n = 0. When A is a formula containing free exa ctly the mut ually different v ar iables x n in or der of fir st o ccurr ence counting from the left, we say that x n is the fr e e- variable se quenc e of A . F or ex a mple, the fre e -v a riable s equence of ∀ y ( P y x ∧ ∃ x Rz xz ) is x, z (pro vided x , y and z ar e all m utually different). T o define the ar row ter ms o f QPN ¬ , in the inductiv e definition we had for the arrow terms of QDS we r eplace L b y L ¬ and assume in addition tha t for all formulae A and B of L ¬ , and for x n being the free-v a riable sequence o f B , the following primitive arr ow terms : ∆ ∀ B ,A : A ⊢ A ∧ ∀ x n ( ¬ B ∨ B ), Σ ∃ B ,A : ∃ x n ( B ∧ ¬ B ) ∨ A ⊢ A are a r row terms of QPN ¬ . In other words, ∀ x n ( ¬ B ∨ B ) is the universal closure of ¬ B ∨ B , a nd ∃ x n ( B ∧ ¬ B ) is the existential closure of B ∧ ¬ B . W e assume throughout the remaining text that Ξ ∈ { ∆ , Σ } . W e call the first index B of ∆ ∀ B ,A and Σ ∃ B ,A the cr own index, and the seco nd index A the s t em index. (W e need the stem A b ecause we la c k pro positiona l constants.) The r igh t c o njunct ∀ x n ( ¬ B ∨ B ) in the targ et of ∆ ∀ B ,A is the cr own of ∆ ∀ B ,A , and the left disjunct ∃ x n ( B ∧ ¬ B ) in the sour ce o f Σ ∃ B ,A is the cr own of Σ ∃ B ,A . W e hav e analo gous definitions o f crown a nd stem indices, a nd crowns, for Σ ∀ , ∆ ′∀ , Σ ′∀ , ∆ ∃ , Σ ′∃ and ∆ ′∃ , which will be intro duced later. (The symbo l ∆ should b e asso ciated with the La tin dexter , be c ause in ∆ ∀ B ,A , ∆ ′∀ B ,A , ∆ ∃ B ,A and ∆ ′∃ B ,A the crown is on the rig ht-hand side of the stem; a nalogously , Σ sho uld b e asso ciated with sinister .) Before we define the arrows of QPN ¬ , w e introduce a num b er of abbrevia - tions: for n = 0 and α ∈ { ι, γ } , α Q x n B = d f 1 B , 31 for n > 0, for ( α, Q ) ∈ { ( ι, ∀ ) , ( γ , ∃ ) } , α Q x n B = d f α Q x n − 1 B ◦ α Q x n Q x n − 1 B : Q x n B ⊢ B , for ( α, Q ) ∈ { ( ι, ∃ ) , ( γ , ∀ ) } , α Q x n B = d f α Q x n Q x n − 1 B ◦ α Q x n − 1 B : B ⊢ Q x n B , for n = 0, [ f ] x n y n = d f f , for n > 0, [ f ] x n y n = d f [[ f ] x n − 1 y n − 1 ] x n y n , for u n and v n not free in A , τ ∀ x n A, u n , v n = d f ∀ v n [ ι ∀ u n A x n u n ] u n v n ◦ γ ∀ v n ∀ u n A x n u n : ∀ u n A x n u n ⊢ ∀ v n A x n v n , τ ∃ x n A, v n , u n = d f γ ∃ v n ∃ u n A x n u n ◦ ∃ v n [ ι ∃ u n A x n u n ] u n v n : ∃ v n A x n v n ⊢ ∃ u n A x n u n . In QDS , for n = 0 we hav e τ Q x n A, u n , v n = 1 A , and for n > 0 w e hav e τ ∀ x n A, u n , v n = τ ∀ x n ∀ v n − 1 A,u n ,v n ◦ ∀ u n τ ∀ x n − 1 A, u n − 1 , v n − 1 , τ ∃ x n A, v n , u n = ∃ u n τ ∃ x n − 1 A, v n − 1 , u n − 1 ◦ τ ∃ x n ∃ v n − 1 A,v n ,u n . F or x n being the fr ee-v ariable sequence of B , we hav e als o the abbreviatio ns Σ ∀ B ,A = d f ˆ c A, ∀ x n ( ¬ B ∨ B ) ◦ ∆ ∀ B ,A : A ⊢ ∀ x n ( ¬ B ∨ B ) ∧ A , ∆ ∃ B ,A = d f Σ ∃ B ,A ◦ ˇ c ∃ x n ( B ∧¬ B ) ,A : A ∨ ∃ x n ( B ∧ ¬ B ) ⊢ A , ˆ ∆ B ,A = d f ( 1 A ∧ ι ∀ x n ¬ B ∨ B ) ◦ ∆ ∀ B ,A : A ⊢ A ∧ ( ¬ B ∨ B ), ˇ Σ B ,A = d f Σ ∃ B ,A ◦ ( ι ∃ x n B ∧¬ B ∨ 1 A ) : ( B ∧ ¬ B ) ∨ A ⊢ A , ˆ ∆ ′ B ,A = d f ( 1 A ∧ ˇ c B , ¬ B ) ◦ ˆ ∆ B ,A : A ⊢ A ∧ ( B ∨ ¬ B ), ˇ Σ ′ B ,A = d f ˇ Σ B ,A ◦ (ˆ c ¬ B ,B ∨ 1 A ) : ( ¬ B ∧ B ) ∨ A ⊢ A . T o define the arrows of QPN ¬ we assume in the inductive definition we had for the equa tions of QDS the following additional axiomatic eq ua tions: (∆ ∀ nat ) ( f ∧ 1 ∀ x n ( ¬ B ∨ B ) ) ◦ ∆ ∀ B ,A = ∆ ∀ B ,D ◦ f , (Σ ∃ nat ) f ◦ Σ ∃ B ,A = Σ ∃ B ,D ◦ ( 1 ∃ x n ( B ∧¬ B ) ∨ f ), ( ˆ b ∆ ∀ ) ˆ b ← A,B , ∀ x n ( ¬ C ∨ C ) ◦ ∆ ∀ C,A ∧ B = 1 A ∧ ∆ ∀ C,B , ( ˇ b Σ ∃ ) Σ ∃ C,B ∨ A ◦ ˇ b ← ∃ x n ( C ∧¬ C ) ,B ,A = Σ ∃ C,B ∨ 1 A , ( d Σ ∀ ) d ∀ x n ( ¬ A ∨ A ) ,B ,C ◦ Σ ∀ A,B ∨ C = Σ ∀ A,B ∨ 1 C , ( d ∆ ∃ ) ∆ ∃ A,C ∧ B ◦ d C,B , ∃ x n ( A ∧¬ A ) = 1 C ∧ ∆ ∃ A,B , ( ˇ Σ ˆ ∆) ˇ Σ A,A ◦ d A, ¬ A,A ◦ ˆ ∆ A,A = 1 A , ( ˇ Σ ′ ˆ ∆ ′ ) ˇ Σ ′ A, ¬ A ◦ d ¬ A,A, ¬ A ◦ ˆ ∆ ′ A, ¬ A = 1 ¬ A , 32 ( r en Ξ Q ) [Ξ Q B ,A ] x y = Ξ Q B ,A x y , for Ξ Q ∈ { ∆ ∀ , Σ ∃ } , (∆ τ ) ∆ ∀ B x n v n ,A = ( 1 A ∧ τ ∀ x n ¬ B ∨ B , u n , v n ) ◦ ∆ ∀ B x n u n ,A , (Σ τ ) Σ ∃ B x n v n ,A = Σ ∃ B x n u n ,A ◦ ( τ ∃ x n B ∧¬ B , v n , u n ∨ 1 A ). The equation ( r en α ) of Section 1.2 does not hold when α is ∆ ∀ or Σ ∃ , but instead we hav e the equations ( r en Ξ Q ) ab ov e. This defines the ca tegory QPN ¬ . In this list of axio ma tic equations the equations ( ˇ Σ ˆ ∆) a nd ( ˇ Σ ′ ˆ ∆ ′ ) ar e taken as they stand from [8] (Section 2.2), where they w ere used to axiomatize the category PN ¬ . The preceding fir st six a xiomatic equations of PN ¬ are o btained from the fir st six axiomatic equations of QPN ¬ ab o ve by replacing ∆ ∀ and Σ ∃ with ˆ ∆ a nd ˇ Σ r espectively , and by deleting quantifier prefixes. It is c le ar that we can derive these axiomatic equa tions of PN ¬ in QPN ¬ , and hence we hav e in QPN ¬ all the e q uations of PN ¬ , with A , B , C, . . . b eing formulae of the language L ¬ . The r eally new axiomatic equatio ns o f Q PN ¬ are o nly the last display ed ( r en Ξ Q ) a nd (Ξ τ ). W e hav e in QPN ¬ the additional a bbreviations ∆ ′∀ B ,A = d f ( 1 A ∧ ∀ x n ˇ c B , ¬ B ) ◦ ∆ ∀ B ,A : A ⊢ A ∧ ∀ x n ( B ∨ ¬ B ), Σ ′∃ B ,A = d f Σ ∃ B ,A ◦ ( ∃ x n ˆ c ¬ B ,B ∨ 1 A ) : ∃ x n ( ¬ B ∧ B ) ∨ A ⊢ A , Σ ′∀ B ,A = d f ˆ c A, ∀ x n ( B ∨¬ B ) ◦ ∆ ′∀ B ,A : A ⊢ ∀ x n ( B ∨ ¬ B ) ∧ A , ∆ ′∃ B ,A = d f Σ ′∃ B ,A ◦ ˇ c ∃ x n ( ¬ B ∧ B ) ,A : A ∨ ∃ x n ( ¬ B ∧ B ) ⊢ A , and a s in [8] (see Section 2.2) we have also the abbrevia tions ˆ Σ B ,A = d f ˆ c A, ¬ B ∨ B ◦ ˆ ∆ B ,A : A ⊢ ( ¬ B ∨ B ) ∧ A , ˇ ∆ B ,A = d f ˇ Σ B ,A ◦ ˇ c B ∧¬ B ,A : A ∨ ( B ∧ ¬ B ) ⊢ A , ˆ Σ ′ B ,A = d f ˆ c A,B ∨¬ B ◦ ˆ ∆ ′ B ,A : A ⊢ ( B ∨ ¬ B ) ∧ A , ˇ ∆ ′ B ,A = d f ˇ Σ ′ B ,A ◦ ˇ c ¬ B ∧ B ,A : A ∨ ( ¬ B ∧ B ) ⊢ A . Note that the e q uations (Ξ τ ) say that we could define our ar rows ∆ ∀ B ,A and Σ ∃ B ,A in ter ms of such arrows with the proviso that the set of v aria ble s free in the stem index A and the set v ariables free in the crown index B are disjoint. In Q DS and QPN ¬ for f : B ⊢ ∀ x n A and g : ∃ x n A ⊢ B such that the v ari- ables x n are not free in B we have the equations ∀ x n ( ι ∀ x n A ◦ f ) ◦ γ ∀ x n B = f , γ ∃ x n B ◦ ∃ x n ( g ◦ ι ∃ x n A ) = g , which genera lize the equations ( ∀ ext ) and ( ∃ ext ) o f Section 1.2 . These equa- tions, together with equatio ns of QDS a nalogous to the equations ( Qτ ι ) of Section 1.3 and the iso morphism o f τ Q u n , entail the following ca ncellation im- plications: 33 ( ∀ ι c anc ) if [ ι ∀ x n A ] x n y n ◦ f 1 = [ ι ∀ x n A ] x n y n ◦ f 2 , then f 1 = f 2 , ( ∃ ι c anc ) if g 1 ◦ [ ι ∃ x n A ] x n y n = g 2 ◦ [ ι ∃ x n A ] x n y n , then g 1 = g 2 , provided the v a riables y n are not free in the source of f 1 and f 2 and in the target of g 1 and g 2 . In QPN ¬ we hav e stem-incr e asing equations a nalogous to the s tem-increasing equations of [8] (Section 2.5; ˆ ∆ and ˇ Σ are replaced b y ∆ ∀ and Σ ∃ resp ectively , which ent ails fur ther adjustments). The equations ( ˆ b ∆ ∀ ), ( ˇ b Σ ∃ ), ( d Σ ∀ ) a nd ( d ∆ ∃ ) are such stem-incr easing equations (when read from righ t to left), a nd there a re further such equations for all the a rrows Ξ Q B ,A and Ξ ′ Q B ,A . W e hav e in QPN ¬ the following additional stem-increasing equations: ( ∀ ∆ ∀ ) ∀ x ∆ ∀ B ,A = ∀ x ( ι ∀ x A ∧ 1 ∀ x n ( ¬ B ∨ B ) ) ◦ γ ∀ x ∀ x A ∧∀ x n ( ¬ B ∨ B ) ◦ ∆ ∀ B , ∀ x A , ( ∃ ∆ ∀ ) ∃ x ∆ ∀ B ,A = ˆ θ ∃ x ← A, ∀ x n ( ¬ B ∨ B ) ◦ ∆ ∀ B , ∃ x A , ( ∀ Σ ∃ ) ∀ x Σ ∃ B ,A = Σ ∃ B , ∀ x A ◦ ˇ c ∃ x n ( B ∧¬ B ) , ∀ x A ◦ ˇ θ ∀ x → A, ∃ x n ( B ∧¬ B ) ◦ ∀ x ˇ c A, ∃ x n ( B ∧¬ B ) , ( ∃ Σ ∃ ) ∃ x Σ ∃ B ,A = Σ ∃ B , ∃ x A ◦ γ ∃ x ∃ x n ( B ∧¬ B ) ∨∃ x A ◦ ∃ x ( 1 ∃ x n ( B ∧¬ B ) ∨ ι ∃ x A ), which are derived with the help of the implications ( Qι c anc ), QDS equations, QDS Coherence a nd the naturality of ∆ ∀ and Σ ∃ in their stem index. W e introduce next a catego ry called QPN , for which w e will establis h in Section 2.6 that it is equiv alent to the category QPN ¬ . The category QPN is for us a n a uxiliary categ ory (though it is closer to the form ulation of linear logic in [10]). W e prov e coher ence for this category in Sectio n 2.5, and from that and the equiv a lence of QPN ¬ and QPN we infer coherence for QPN ¬ in Section 2.7. The categor y QPN is very much like QPN ¬ sav e that in its ob jects the negation connective ¬ is prefix e d o nly to a to mic formulae. The arrow terms ∆ ∀ B ,A and Σ ∃ B ,A are primitive only for the crown index B b e ing an atomic fo rm ula. Here is a more formal definition of QPN . F or P being the set o f letters that we used to genera te L and L ¬ in Sections 1.1 and 2.1 , let P ¬ be the set of predicate letter s {¬ P | P ∈ P } . The a rit y o f the new pr e dicate le tter ¬ P is the same as the a r it y of P . The o b jects of Q PN are the formulae of the first-orde r languag e L ¬ P generated fro m P ∪ P ¬ in the same way as L was genera ted from P in Section 1.1. T o define the arrow ter ms of QPN , in the inductiv e definitio n we had for the arrow terms of Q DS we r eplace L by L ¬ P , and we assume in addition that for every for m ula A of L ¬ P , for every pr edicate letter P ∈ P of ar ity n , and for x ′ n ′ being the free-v ariable sequenc e of P x n , ∆ ∀ P x n ,A : A ⊢ A ∧ ∀ x ′ n ′ ( ¬ P x n ∨ P x n ), Σ ∃ P x n ,A : ∃ x ′ n ′ ( P x n ∧ ¬ P x n ) ∨ A ⊢ A are primitive a rrow terms of Q PN . 34 T o define the a rrows of Q PN , we assume as additiona l axioma tic equations in the inductive definition we had for the equations of QDS all the additional axiomatic e q uations assumed ab ov e for Q PN , but restricted to the arrow terms ∆ ∀ P x n ,A and Σ ∃ P x n ,A whose crown index is atomic. This defines the catego ry QPN . 2.2 Dev elopmen t for QDS, QPN ¬ and QPN If β is a primitive arrow term of QPN ¬ except 1 B , then w e call β - terms of QPN ¬ the set of arr o w terms defined inductively as follows: β is a β -term; if f is a β -term, then for every A in L ¬ and a ll v ar iables x and y we have that 1 A ξ f , f ξ 1 A , Q x f and [ f ] x y are β -terms, provided [ f ] x y is defined. In a β -term the s ubterm β is ca lled the he ad of this β -term. F or example, the head of the ∆ ∀ B ,C -term 1 A ∧ ∀ x ([∆ ∀ B ,C ] y z ∨ 1 E ) is ∆ ∀ B ,C . W e define 1 - term s like β -terms; we just r eplace β in the definition ab ov e by 1 B . So 1 - terms are headless . An arrow term o f the for m f n ◦ . . . ◦ f 1 , where n ≥ 1 , with parentheses tied to ◦ asso ciated arbitrar ily , such that for every i ∈ { 1 , . . . , n } we hav e tha t f i is comp osition-free is c a lled factorize d . In a facto rized arrow term f n ◦ . . . ◦ f 1 the arrow ter ms f i are called factors . A factor that is a β -term for some β is called a he ade d factor. A factor ized arr o w term is called he ade d when each of its fac to rs is either hea ded or a 1 -term. A factorized arrow term f n ◦ . . . ◦ f 1 is called develop e d when f 1 is a 1 -term and if n > 1 , then ev ery factor o f f n ◦ . . . ◦ f 2 is headed. Analog ous definitions of β -ter m a nd develop e d ar row term can b e given for QDS . W e hav e the following lemma for QDS . Development Lemma. F or every arr ow term f ther e is a develo p e d arr ow term f ′ such t hat f = f ′ . Proof. This lemma would be ea sy to pr o ve by using the catego rial and func- torial equations together with the equation ( r en ◦ ) of Section 1.2 if for ( r en ◦ ), as for the other of these eq ua tions, we had that the right-hand s ide is defined whenever the left-hand side is defined. Since this need not b e the case, w e m ust firs t eliminate r enaming. This ca n b e a chieved by relying on the categor y GQDS , cut elimination and the res ult of Section 1 .8. W e a dapt the definitions of β -term and de velop ed ar row term to the ca tegory GQDS o f Se c tio n 1.5. F or every ar row term g of GQDS there is a Gen tzen term g ′ denoting the arrow g . By the V ar iable-Purification Lemma of Sectio n 1.6, we hav e that g ′ is eq ua l to h 2 ◦ g ′′ ◦ h 2 where h 1 and h 2 are comp ositions of τ -terms and g ′′ is a v ariable-pure Gen tzen term, which b y Renaming Elimination and the Cut-Elimina tion Theorem (see Sections 1.8 -9) we may ass ume to b e cut-free and renaming-free. Then by using the categoria l a nd functorial equations it is easy to o btain from h 2 ◦ g ′′ ◦ h 2 a developed a rrow term g ′′′ of GQDS e q ual to 35 the initial arrow term g . F or a n arbitra ry a rrow term f : A ⊢ B of Q DS we find a diversified arr o w term f ′ : A ′ ⊢ B ′ of QDS s uc h that f is a letter-fo r-letter subs titution instance of f ′ (see the b eginning of Sectio n 1.5). As in [7 ] (Sections 3.2-3) we pass by a functor H G from f to the arrow ter m H G f o f GQDS , which, as we ha ve shown ab ov e, is eq ua l to a developed arrow ter m ( H G f ) ′′′ of GQDS . By ap- plying a functor H in the opp osite directio n we o btain a develop ed arrow ter m H (( H G f ) ′′′ ) of QDS , which w e ca ll h . The type o f h is A ′′ ⊢ B ′′ , where A ′′ and B ′′ belo ng to the same form sets as A ′ and B ′ resp ectively . So b y Q DS Coherence we hav e that f ′ = j 2 ◦ h ◦ j 1 , where j 1 and j 2 are headed facto r ized arrow ter ms of QDS whose heads a re of the ξ b and ξ c kind. W e obtain the arrow term o f QDS equal to f as a letter- fo r-letter substitution instance of j 2 ◦ h ◦ j 1 . ⊣ By rely ing o n v ar ious r enaming equations of Q DS , we can prove a Refine d Development L emma for QDS , which differs from the Developmen t Lemma by requiring that in the develop e d arrow term f ′ renaming o ccurs only in s ubterms of the form [ ι Q x A ] x y for x different from y and free in A . With the help of the Refined Dev elopment Lemma for QDS , the stem-increas ing equations o f QPN ¬ (see the preceding section), together with the natura lit y of ∆ ∀ and Σ ∃ in their stem index and the equa tions ( r en Ξ Q ), we ca n prov e the Refined Development Lemma, and hence als o the Developmen t Lemma, for the ca teg ories QPN ¬ and QPN to o. The Refined Dev elopment Lemma is not o nly imp ortant b ecause of the ap- plications it will find la tter in this paper. It is also imp ortant b ecause we can conclude fr om it that renaming , except in [ ι Q x A ] x y for x different fro m y and free in A , is eliminable in QDS , QPN ¬ and QPN . This e limination of renaming is not straightforward, but is achieved in a roundab out wa y , involving cut elim- ination. The elimina bilit y of rena ming may p erhaps serve to explain why it is neglected a s a primitive rule of inference in lo gic. 2.3 Some prop erties of QDS In this section w e es ta blish some results concer ning the categ ory QDS o f Section 1.2, which w e will use to prov e cohere nc e for QPN and QPN ¬ . First we int ro duce a definition. Suppo se X is the n - th o ccurr ence of a pr edicate letter (co un ting fr o m the left) in a for m ula A of L , and Y is the m -th o ccurrence o f the same pr edicate letter in a formula B of L . Then we say that X and Y are tie d in an a rrow f : A ⊢ B of QDS w he n ( n − 1 , m − 1 ) ∈ Gf (see Section 1.5 ; note that to find the n -th o ccurrence we count starting fr om 1, but the or dinal n > 0 is { 0 , . . . , n − 1 } ). It is easy to establish that every o ccurrence of a predicate letter in A is tied to exactly o ne o ccurrence o f the sa me letter in B , and vice versa. This is related to matters ab o ut diversification mentioned at the beg inning o f Section 1.5. 36 F or the lemma b elow, let X in A and Y in B b e o ccurrences of the sa me predicate le tter tied in an arrow f : A ⊢ B of Q DS , and let S A and S B be t wo finite (p ossibly empty) se quences of qua n tifier pre fix es. Then by an easy induction on the complexity o f f we can prov e the following, which gener a lizes Lemma 2 o f Section 2.4 o f [8]. ∧∨ Lemma. It is imp ossible that A has a s u bformula S A X x n ∧ A ′ or A ′ ∧ S A X x n while B has a subformula S B Y y n ∨ B ′ or B ′ ∨ S B Y y n . F or the next lemma, for i ∈ { 1 , 2 } let X i in A and Y i in B b e o ccurrences o f the predica te letter P i tied in an a rrow f : A ⊢ B of QDS (here P 1 and P 2 may also be the sa me predicate letter). ∨∧ Lemma. F or every i, j ∈ { 1 , 2 } , it is imp ossible that A has a subformula X i y n ∨ X 3 − i z m while B has a subformula Y j u k ∧ Y 3 − j v l . This lemma , exactly a nalogous to Lemma 3 of Section 2.4 of [8], is a cor ollary of lemmata ex actly analo gous to Lemmata 3D and 3 C of Se c tio n 2.4 of [8], which are easily proved by induction on the complexity of the arrow term f . As a matter of fac t, the ∧∨ and ∨∧ Lemmata ab ov e could b e prov ed by suppo sing the con trary and deleting quan tifiers and individual v aria bles to gether with ar row terms and op erations o n arr o w terms inv o lving them, which w ould yield a r row terms contradicting Lemma 2 and Lemma 3 resp ectively of Sectio n 2.4 of [8]. The ∧∨ Lemma is related to the acyclicity c ondition of pr oof nets, while the ∨∧ Lemma is r elated to the c onnectedness condition (see [8], Sections 2.4, 7.1, a nd references therein). Next w e ca n prov e the following lemma. P-Q-R Lemma. L et f : A ⊢ B b e an arr ow of QDS , let X i for i ∈ { 1 , 2 , 3 } b e o c cu rr enc es of the pr e dic ate letters P , Q and R , r esp e ctively, in A , and let Y i b e o c curr enc es of P , Q and R , r esp e ctively, in B , such t hat X i and Y i ar e tie d in f . L et, mor e over, X 2 x 2 q ∨ X 3 x 3 r b e a subformula of A and Y 1 y 1 p ∧ Y 2 y 2 q a subformula of B . Then ther e is a d P z 1 p ,Q z 2 q ,R z 3 r -term h : A ′ ⊢ B ′ such that X ′ i ar e o c curr enc es of P , Q and R , r esp e ctively, in t he sour c e P z 1 p ∧ ( Q z 2 q ∨ R z 3 r ) of the he ad of h and Y ′ i ar e o c curr enc es of P , Q and R , r esp e ctively, in the t ar get ( P z 1 p ∧ Q z 2 q ) ∨ R z 3 r of t he he ad of h , su ch that for some arr ows f x : A ⊢ A ′ and f y : B ′ ⊢ B of QDS we have f = f y ◦ h ◦ f x in QDS , and X i is tie d to X ′ i in f x , while Y ′ i is tie d to Y i in f y . This lemma , exactly analog ous to the p-q-r Le mma of [8] (Section 2.4), is prov ed like this pre vious lemma by r elying on the Gentzenization of GQDS . 37 2.4 Some prop erties of QP N In this section, by rely ing on the results of the prece ding s ection, we establish some r esults concer ning the category QPN in tr oduced at the end of Section 2.1, which we will find useful for calculations later o n. F or these r esults we need to int ro duce the following. Let Q DS ¬ P be the c ategory defined as Q DS save that it is generated not by P , but b y P ∪ P ¬ (see the e nd of Section 2 .1). So the ob jects of QDS ¬ P are the formulae of L ¬ P , i.e. the ob jects of QPN . F or A and B formulae of L ¬ P , we define when an o ccurrence of the predicate letter P in A is tied to an o ccurrence of P in B in an a rrow f : A ⊢ B of QDS ¬ P analogo usly to what we had a t the b eginning of the pr eceding section. W e say that a finite (po ssibly empt y ) sequence S of quantifier prefixes is for eign to a formula B when the set of v ariables o ccurring in S is disjoint fr o m the set of free v ar iables of B . If S is fo reign to B , then there is an is omorphism j → S,B : B ⊢ S B o f QDS ¬ P ; defined in ter ms o f γ ∀ and ι ∃ . The inv erse of j → S,B is the a rrow j ← S,B : S B ⊢ B of QDS ¬ P defined in terms o f ι ∀ and γ ∃ . By QDS Coherence, these is o morphisms are unique. W e introduce next a generalizatio n of the arrow terms Ξ Q B ,A obtained by letting the arrow terms Ξ Q B ,A “absor b” v arious Q DS arr o w terms. These gen- eralized terms have the r igh t form for the fir st tw o technical results b elow—the Ξ Q -Perm utation Lemmata. F or S and S ¬ t wo indep enden t finite s equences of quantifier pre fix es b oth foreign to P x n , and for I b eing the se quence of indices P x n , A, y m , S, S ¬ , we hav e ∆ ∀ I = d f ( 1 A ∧ ( ∀ y m (( j → S ¬ , ¬ P x n ∨ j → S,P x n ) ◦ ι ∀ x n ¬ P x n ∨ P x n ) ◦ γ ∀ y m ∀ x n ( ¬ P x n ∨ P x n ) )) ◦ ∆ ∀ P x n ,A : A ⊢ A ∧ ∀ y m ( S ¬ ¬ P x n ∨ S P x n ), Σ ∃ I = d f Σ ∃ P x n ,A ◦ (( γ ∃ y m ∃ x n ( P x n ∧¬ P x n ) ◦ ∃ y m ( ι ∃ x n P x n ∧¬ P x n ◦ ( j ← S,P x n ∧ j ← S ¬ , ¬ P x n ))) ∨ 1 A ) : ∃ y m ( S P x n ∧ S ¬ ¬ P x n ) ∨ A ⊢ A . The analogo us abbrevia tions Σ ∀ I : A ⊢ ∀ y m ( S ¬ ¬ P x n ∨ S P x n ) ∧ A , ∆ ∃ I : A ∨ ∃ y m ( S P x n ∧ S ¬ ¬ P x n ) ⊢ A , ∆ ′∀ I : A ⊢ A ∧ ∀ y m ( S P x n ∨ S ¬ ¬ P x n ), Σ ′∃ I : ∃ y m ( S ¬ ¬ P x n ∧ S P x n ) ∨ A ⊢ A , Σ ′∀ I : A ⊢ ∀ y m ( S P x n ∨ S ¬ ¬ P x n ) ∧ A , ∆ ′∃ I : A ∨ ∃ y m ( S ¬ ¬ P x n ∧ S P x n ) ⊢ A are defined in ter ms of ∆ ∀ and Σ ∃ like the analog ous abbreviations of Section 2.1. The right conjunct ∀ y m ( S ¬ ¬ P x n ∨ S P x n ) in the target of ∆ ∀ I is the cr own 38 of ∆ ∀ I , and analo gously for Σ ∃ I and the other abbreviations , replacing “right” by “left”, “co njunct” b y “disjunct”, and “tar get” b y “source” , as appr opriate. Note that here y m is a n arbitrar y sequence of v ariables, a nd not ne c essarily the free-v ar iable sequence of P x n as in Ξ Q P x n ,A (see Sectio n 2 .1). The definition o f ∆ ∀ I ab o ve is of the form f ◦ ∆ ∀ P x n ,A . By QDS Coherence, instead of the arr o w term f of Q DS ¬ P we c o uld have used for this definitio n any o ther arrow term g of QDS ¬ P of the sa me type as f such that Gf = Gg , since we hav e g = f , a nd analogo usly for Σ ∃ I , etc. Let Ξ , Θ ∈ { ∆ , ∆ ′ , Σ , Σ ′ } , a nd let a Ξ Q I -term b e defined as a β -term in Sec- tion 2.2 sav e that β is replaced b y Ξ Q I , and the cla use “ if f is a Ξ Q I -term, then [ f ] x y is a Ξ Q I -term” is omitted. Then we hav e the fo llowing analo gue of the ˆ Ξ-Perm utation Lemma o f [8] (Section 2.5). Ξ ∀ -Permut a tion Lemma. L et g : C ⊢ D b e a Ξ ∀ P x n ,A, y m ,S,S ¬ -term of QPN such that X 1 and ¬ X 2 ar e r esp e ctively the o c curr enc es within D of the pr e d- ic ate let t ers P and ¬ P in the cr own of the he ad Ξ ∀ P x n ,A, y m ,S,S ¬ of g , and let f : D ⊢ E b e an arr ow term of QDS ¬ P such that we have an o c curr enc e Y 1 of P and an o c curr enc e ¬ Y 2 of ¬ P within a subformula of E of the form ∀ y ′ m ′ ( S ′ Y 1 x ′ n ∨ S ¬ ′ ¬ Y 2 x ′ n ) or ∀ y ′ m ′ ( S ¬ ′ ¬ Y 2 x ′ n ∨ S ′ Y 1 x ′ n ) , for S ′ and S ¬ ′ finite se- quenc es of quantifier pr efixes, and X i is tie d to Y i for i ∈ { 1 , 2 } in f . Then ther e is a Θ ∀ P x ′ n ,A ′ , y ′ m ′ ,S ′ ,S ¬ ′ -term g ′ : D ′ ⊢ E of QPN t he cr own of whose he ad is ∀ y ′ m ′ ( S ′ Y 1 x ′ n ∨ S ¬ ′ ¬ Y 2 x ′ n ) or ∀ y ′ m ′ ( S ¬ ′ ¬ Y 2 x ′ n ∨ S ′ Y 1 x ′ n ) , and ther e is an arr ow term f ′ : C ⊢ D ′ of QDS ¬ P such t hat in QPN we have f ◦ g = g ′ ◦ f ′ . Proof. W e pro ceed in principle a s fo r the pro of o f the ˆ Ξ-Permut ation Lemma in [8], with so me adjustmen ts a nd additions. W e app eal to the Refined Develop- men t Lemma for QPN ¬ (see Section 2.2), a nd we use the ∧∨ Lemma of the pre - ceding section to ascertain that cases inv o lving “problema tic” d A,S P x n ,S ¬ ¬ P x n - terms o r d A,S ¬ ¬ P x n ,S P x n -terms in the develop ed arr o w term f are excluded. W e rely then on equations analog ous to the equations mentioned in the pr oo f of the ˆ Ξ-Perm utation Lemma, wher e ˆ Ξ p,A is replaced by Ξ ∀ I , which entails fur - ther adjustments. Suc h equa tions, which a re either s tem- inc r easing, or related to the stem-increasing equations, or ar e simply consequence s of definitions, a re established with the help of the implications ( Qι c anc ) to gether with the equa - tions (Ξ τ ) (se e Section 2.1) and QDS Coherence. W e r ely also on the r emark we made b efore the lemma co ncerning the alterna tiv e definitions of Ξ ∀ I . ⊣ W e hav e a dual lemma, called the Ξ ∃ - Permutation L emma , analog ous to the ˇ Ξ-Perm utation Lemma o f [8] (Sectio n 2 .5), which inv olves Ξ ∃ I -terms instead o f Ξ ∀ I -terms. Next we hav e a lemma analogo us to the p- ¬ p-p Lemma of [8] (Section 2 .5). 39 P- ¬ P-P Lemma. L et X 1 , ¬ X 2 and X 3 b e o c curr enc es of the pr e dic ate letters P , ¬ P and P , r esp e ctively, in a formula A of L ¬ p , and let Y 1 , ¬ Y 2 and Y 3 b e o c cur- r enc es of P , ¬ P and P , r esp e ctively, in a formula B of L ¬ p . L et g 1 : A ′ ⊢ A b e a Ξ ∀ P x n ,A, y m ,S,S ¬ -term of Q PN such that ∀ y m ( S ¬ ¬ X 2 x n ∨ S X 3 x n ) or ∀ y m ( S X 3 x n ∨ S ¬ ¬ X 2 x n ) is the cr own of the he ad of g 1 , let g 2 : B ⊢ B ′ b e a Θ ∃ P x ′ n ,A ′ , y ′ m ′ ,S ′ ,S ¬ ′ - term of Q PN such that ∃ y ′ m ′ ( S ′ Y 1 x ′ n ∧ S ¬ ′ ¬ Y 2 x ′ n ) or ∃ y ′ m ′ ( S ¬ ′ ¬ Y 2 x ′ n ∧ S ′ Y 1 x ′ n ) is the cr own of the he ad of g 2 , and let f : A ⊢ B b e an arr ow t erm of QDS ¬ P such that X i and Y i ar e t ie d in f for i ∈ { 1 , 2 , 3 } . Then g 2 ◦ f ◦ g 1 is e qual in QPN to an arr ow term of QDS ¬ P . The pro of of this lemma is analogo us to the pro of of the p- ¬ p-p Lemma in [8]. W e use the P-Q-R Lemma o f the prece ding section a nd the Ξ Q -Perm utation Lemmata instead o f the p-q-r Lemma and the ξ Ξ-Perm utation Lemmata , and we a pply the equa tion ( ˇ Σ ˆ ∆) o f Section 2.1. W e establish in the same manner the ¬ P-P- ¬ P L emma , ana logous to the ¬ p-p- ¬ p Lemma of [8] (Section 2.5). The fo rm ulation o f the ¬ P-P- ¬ P Le mma is o btained from that of the P- ¬ P-P Lemma by re pla cing the se q uence P , ¬ P, P by the se q uence ¬ P , P , ¬ P , which entails that S and S ¬ , as well a s S ′ and S ¬ ′ , are p erm uted. The ¬ P-P- ¬ P Lemma is prov ed by applying the P-Q-R Lemma, the Ξ Q -Perm utation Lemmata and the equa tio n ( ˇ Σ ′ ˆ ∆ ′ ) o f Section 2.1. 2.5 QPN Coherence In [8] (Section 2.3) one can find a detailed definition of a ca tegory called Br , whose ob jects a re finite o r dinals, and whose arrows are gra phs sometimes called Kel ly-Mac L ane gr aphs (b ecause of [14]). These graphs may also be found in [2] (from whos e autho r the name of Br is der iv ed). W e define a functor G from QPN ¬ or QPN in to Br as we defined in [8] (Section 2.3) an ide ntically na med functor from the categor ies PN ¬ and PN into Br , without paying a ttention to v ariable s and quantifier prefixes. This means that GQ x A = GA (so that GA is the num ber of o ccurrences of predica te letters in the formula A ), the a rrow Gα for α b eing ι Q x A , γ Q x A , ˇ θ ∀ x → A,D and ˆ θ ∃ x ← A,D is a n identit y ar r o w, GQ x f = G [ f ] x y = Gf , while G ∆ ∀ B ,A and G Σ ∃ B ,A are like G ˆ ∆ B ,A and G ˇ Σ B ,A resp ectively . The category R el mentioned in Section 1.5 is a subcatego ry of Br , a nd G restricted to the QDS part of QPN ¬ and QPN coincides with the functor G from QDS to Re l . The theor ems that the functors G from Q PN ¬ or QPN into Br are faithful functors ar e called QPN ¬ Coher enc e and QPN Coher enc e resp ectiv ely . W e establish fir s t QPN Co herence, and QPN ¬ Coherence will b e derived from it in Section 2.7 . W e prov e QPN Coherence b y pro ceeding as for the pro of of PN Co her - ence in [8] (Section 2.7), through lemmata analogo us to the Confr on tation a nd Purification L emmata. Roughly sp eaking, the a nalogue of the Confro n tation Lemma s a ys tha t a ∆ ∀ P x n ,A -term, called a ∆ ∀ - factor , a nd a Σ ∃ P y n ,B -term, called 40 a Σ ∃ - factor , mutually tied in a direct manner throug h the crowns, whic h are called c onfr onte d factors, can b e p ermuted with the help of stem-incr easing a nd naturality e q uations so that they are ready to get eliminated by a pplying the P- ¬ P-P and ¬ P-P- ¬ P Lemmata of Section 2 .4. The analog ue o f the Purifi- cation Lemma states that this elimination can be pur sued until w e obtain an arrow term without confr on ted factors , such an ar row ter m b eing called pur e . F or the pro of of these ana logues o f the Confr o n tation and Purification Lem- mata w e need the Refined Developmen t Lemma for QPN o f Section 2.2. W e also need the stem-increas ing equations for ∆ ∀ and Σ ∃ (see Sectio n 2.1 ) and the naturality o f ∆ ∀ and Σ ∃ in the stem index. Where in the proo f o f the Purifi- cation Lemma in [8] (Section 2.7) we appe a led to Lemma 3, we now app e al to the ∨∧ Le mma of Section 2.3. Instead o f the p- ¬ p-p a nd ¬ p-p- ¬ p Lemmata we now have the P- ¬ P-P and ¬ P-P- ¬ P Lemmata. The equation (∆ τ ) of Section 2.1 is essential, together with the s tem- increas- ing eq ua tions and the naturality of ∆ ∀ and Σ ∃ in the s tem index, to guarantee that if ther e is a ∆ ∀ -factor in a pur e arr o w ter m f , then f = f ′ ◦ ∆ ∀ P x n ,A for any sequence of v ariable s x n . The equation (Σ τ ) of Section 2.1 is needed to es tablish an analo gous equation for Σ ∃ -factors. By pushing in this manner to the extreme right the ∆ ∀ -factors remaining in a pure ar row term, and to the extreme left the r emaining Σ ∃ -factors, and b y relying on QDS Co herence, we establish QPN Coherence. 2.6 The equiv alence of QPN ¬ and QPN T o pr o ve that QPN ¬ and QPN ar e equiv alent categ ories we pro ceed as in [8] (Section 2.6), w ith the following a djustments and additions. When we define the functor F from QPN ¬ to QPN we hav e the following new clauses on o b jects: F A = A , for A of the form P x n or ¬ P x n , F Q x A = Q x F A , F ¬∀ x A = ∃ x F ¬ A , F ¬∃ x A = ∀ x F ¬ A . On arr o ws we hav e first new clauses analog o us to the old cla uses where α is ι Q x , γ Q x , ˇ θ ∀ x → and ˆ θ ∃ x ← , while ˆ ∆ and ˇ Σ ar e replaced b y ∆ ∀ and Σ ∃ , the letter p is replaced by P x n , and so me further adjustments are made. W e ha ve mor eo ver the following new cla us es: if x is free in B , F ∆ ∀ ∀ x B ,A = ( 1 A ∧ ( ∀ x n − 1 (ˇ c ∃ x F ¬ B , ∀ x F B ◦ ˇ θ ∀ x → F B , ∃ x F ¬ B ◦ ∀ x (ˇ c F B , ∃ x F ¬ B ◦ ( ι ∃ x F ¬ B ∨ 1 F B ))) ◦ h )) ◦ F ∆ ∀ B ,A , 41 where h : ∀ x n − 1 . . . ∀ x i +1 ∀ x ∀ x i . . . ∀ x 1 ( F ¬ B ∨ F B ) ⊢ ∀ x n − 1 ∀ x ( F ¬ B ∨ F B ) is a n is o morphism of QDS ¬ P (see the preceding section) generalizing iso mor- phisms of the type ∀ x ∀ y C ⊢ ∀ y ∀ x C , if x is not free in B , F ∆ ∀ ∀ x B ,A = ( 1 A ∧ ∀ x n ( ι ∃ x F ¬ B ∨ γ ∀ x F B )) ◦ F ∆ ∀ B ,A , if x is free in B , for h as a bov e, F ∆ ∀ ∃ x B ,A = ( 1 A ∧ ( ∀ x n − 1 ( ˇ θ ∀ x → F ¬ B , ∃ x F B ◦ ∀ x ( 1 F ¬ B ∨ ι ∃ x F B )) ◦ h )) ◦ F ∆ ∀ B ,A , if x is not free in B , F ∆ ∀ ∃ x B ,A = ( 1 A ∧ ∀ x n ( γ ∀ x F ¬ B ∨ ι ∃ x F B )) ◦ F ∆ ∀ B ,A , and dual cla uses for F Σ ∃ ∀ x B ,A and F Σ ∃ ∃ x B ,A , F Q x f = Q x F f , F [ f ] x y = [ F f ] x y . This defines the functor F . F or f an arrow ter m of QPN ¬ we hav e that GF f coincides with Gf , where G in GF f is the functor G from QPN to Br , and G in Gf is the functor G from Q PN ¬ to Br (see the b eginning of the preceding section). T o show that, it is essential to check that GF ∆ ∀ B ,A and GF Σ ∃ B ,A coincide with G ∆ ∀ B ,A and G Σ ∃ B ,A resp ectively , which is done by induction on the complex ity of the crown index B . Then we can ea sily verify that F , as defined ab ov e, is indeed a functor. If f = g in QPN ¬ , then Gf = Gg , and henc e , as we hav e just s een, GF f = GF g . By QPN Coherenc e of the pr eceding section, we co nc lude tha t F f = F g in QPN . (T o verify that the functor F from PN ¬ to PN in Section 2.6 of [8 ] is a functor we could have pro ceeded analog ously , by establis hing PN Coherence first, b efore intro ducing the functor F . W e did not need the functor F to prove PN Co herence. This would make the exp osition in [8] somewha t simpler, and better organized.) W e define a functor F ¬ from Q PN to QPN ¬ by stipula ting that F ¬ A = A and F ¬ f = f . T o sho w that QPN ¬ and QPN are equiv alent categories via the functors F and F ¬ we pro ceed a s in [8] (Section 2 .6 ) with the follo wing additions. W e ha ve the following auxiliary definitions in QPN ¬ , for x n being the free-v ar iable seq uence of A (see Section 2.1), and y m being this sequence with x omitted (if x is free in A , then m = n − 1; otherw is e, x n is y m and m = n ): 42 q ∀ x → A = d f Σ ′∃ ∀ x A, ∃ x ¬ A ◦ ( ι ∃ y m ¬∀ x A ∧∀ x A ∨ 1 ∃ x ¬ A ) ◦ d ¬∀ x A, ∀ x A, ∃ x ¬ A ◦ ( 1 ¬∀ x A ∧ ( ˇ θ ∀ x → A, ∃ x ¬ A ◦ ∀ x (( 1 A ∨ ι ∃ x ¬ A ) ◦ ι ∀ x n A ∨¬ A ) ◦ γ ∀ x ∀ x n ( A ∨¬ A ) )) ◦ ∆ ′∀ A, ¬∀ x A : ¬∀ x A ⊢ ∃ x ¬ A , q ∀ x ← A = d f Σ ′∃ A, ¬∀ x A ◦ (( γ ∃ x ∃ x n ( ¬ A ∧ A ) ◦ ∃ x ( ι ∃ x n ¬ A ∧ A ◦ ( 1 ¬ A ∧ ι ∀ x A )) ◦ ˆ θ ∃ x ← ¬ A, ∀ x A ) ∨ 1 ¬∀ x A ) ◦ d ∃ x ¬ A, ∀ x A, ¬∀ x A ◦ ( 1 ∃ x ¬ A ∧ ι ∀ y m ∀ x A ∨¬∀ x A ) ◦ ∆ ′∀ ∀ x A, ∃ x ¬ A : ∃ x ¬ A ⊢ ¬∀ x A , and we hav e a nalogous definitions of q ∃ x → A : ¬∃ x A ⊢ ∀ x ¬ A , q ∃ x ← A : ∀ x ¬ A ⊢ ¬∃ x A . It can b e shown that q Q x → A is an is o morphism, with in verse q Q x ← A . Next in the inductive definitions of the isomorphisms i A : A ⊢ F A and i − 1 A : F A ⊢ A we hav e the following claus e s in a ddition to clauses in [8] (Section 2.6): i A = i − 1 A = 1 A , if A is P x n or ¬ P x n , i Q x A = Q x i A , i − 1 Q x A = Q x i − 1 A , i ¬∀ x A = ∃ x i ¬ A ◦ q ∀ x → A , i − 1 ¬∀ x A = q ∀ x ← A ◦ ∃ x i − 1 ¬ A , i ¬∃ x A = ∀ x i ¬ A ◦ q ∃ x → A , i − 1 ¬∃ x A = q ∃ x ← A ◦ ∀ x i − 1 ¬ A . W e can then ex tend the pro of of the Auxiliary Lemma of Sectio n 2.6 of [8] in order to establis h tha t for f : A ⊢ B we ha ve in QPN ¬ the equatio n f = i − 1 B ◦ F f ◦ i A . In this e x tended pro of, for the iso morphism n ← B : B ⊢ ¬¬ B of QPN ¬ , we need the following equation of Q PN ¬ : (∆ ∀ n ) ∆ ∀ ¬ B ,A = ( 1 A ∧ ∀ x n ( n ← B ∨ 1 ¬ B )) ◦ ∆ ′∀ B ,A , analogo us to the equa tio n ( ˆ ∆ n ) of [8] (Section 2.6, Pro of of the Auxiliary Lemma). T o derive (∆ ∀ n ) we use, analogously to what we had befor e for the deriv ation of ( ˆ ∆ n ), the s tem-increasing equa tion ( ∀ ∆ ∀ ) of Section 2.1 , the nat- urality of ∆ ∀ in the s tem index, the ¬ P-P- ¬ P Lemma of the preceding section and Q DS Coher ence. (In the deriv ation of ( ˆ ∆ n ) in the printed text of [8], Sec- tion 2.6 , P roo f of the Auxiliar y Lemma, “(with p r eplaced by A )” is a misprint for “(with p replaced by B )”.) W e derive s imila rly an equation analo g ous to the equation ( ˆ ∆ r ) of [8] (Section 2.6, Pro of of the Auxiliary Lemma) inv olving ∆ ∀ . W e need also the following equations of QPN ¬ , analogo us to the clauses defining F ∆ ∀ ∀ x B ,A ab o ve: if x is free in B , ∆ ∀ ∀ x B ,A = ( 1 A ∧ ( ∀ x n − 1 (( q ∀ x ← B ∨ 1 ∀ x B ) ◦ ˇ c ∃ x ¬ B , ∀ x B ◦ ˇ θ ∀ x → B , ∃ x ¬ B ◦ ∀ x (ˇ c B , ∃ x ¬ B ◦ ( ι ∃ x ¬ B ∨ 1 B ))) ◦ h )) ◦ ∆ ∀ B ,A , 43 where h : ∀ x n − 1 . . . ∀ x i +1 ∀ x ∀ x i . . . ∀ x 1 ( ¬ B ∨ B ) ⊢ ∀ x n − 1 ∀ x ( ¬ B ∨ B ) is an is o morphism of QDS ¬ P , if x is not free in B , ∆ ∀ ∀ x B ,A = ( 1 A ∧ ∀ x n (( q ∀ x ← B ∨ 1 ∀ x B ) ◦ ( ι ∃ x ¬ B ∨ γ ∀ x B ))) ◦ ∆ ∀ B ,A . The idea for the der iv a tion of these equatio ns is the same as the idea for the deriv ation of (∆ ∀ n ) above. W e need also e q uations ana logous to the claus es defining F ∆ ∀ ∃ x B ,A ab o ve. T o show tha t in QPN ¬ [ f ] x y = i − 1 B x y ◦ F [ f ] x y ◦ i A x y we need the equation [ i A ] x y = i A x y of QPN ¬ , which is established by induction on the complex ity of A . F or this induction we use the equation [ ˆ ∆ B ,A ] x y = ˆ ∆ B x y ,A x y and analo gous eq uations of QPN ¬ . The last displayed e q uation is es tablished with the help of the equatio ns ( r en ∆ ∀ ) a nd (∆ τ ) of Section 2.1. W e need also the equation [ q ∀ z → A ] x y = q ∀ z → A x y of QPN ¬ , for which we use the equatio ns (Ξ τ ) (see Section 2.1 ) and QDS Coherence. This suffices to establish that the categ ories QPN ¬ and QPN are equiv alent. 2.7 QPN ¬ Coherence As w e sa id at the beginning of Section 2.5, QPN ¬ Coher enc e is the theorem that the functor G from QPN ¬ to the catego ry Br is faithful. W e can then prov e QPN ¬ Coherence as follows. Proof o f Q PN ¬ Coherence. Suppo s e that for f a nd g arrows o f QPN ¬ of the same type we hav e Gf = Gg . Then, as we noted after the definition of the functor F from QPN ¬ to QPN in the prece ding section, we hav e GF f = GF g , and hence F f = F g in Q PN b y QPN Coherence of Section 2.5. It follows that f = g in Q PN ¬ by the equiv alence of the categories QPN ¬ and QPN established in the pre c eding section. ⊣ With QPN ¬ Coherence we can esta blish easily e quations of Q PN ¬ whose deriv ation ma y otherwise b e q uite demanding. W e hav e, for example, the fol- lowing equations in QPN ¬ : 44 ˇ θ ∀ x → A,D = (( ∀ x ( ˇ ∆ D,A ◦ d R A,D , ¬ D ) ◦ ˆ θ ∀ x ← A ∨ D , ¬ D ) ∨ 1 D ) ◦ d ∀ x ( A ∨ D ) , ¬ D,D ◦ ˆ ∆ D, ∀ x ( A ∨ D ) , ˆ θ ∃ x ← A,D = ˇ ∆ ′ D, ∃ x ( A ∧ D ) ◦ d R ∃ x ( A ∧ D ) , ¬ D,D ◦ (( ˇ θ ∃ x → A ∧ D , ¬ D ◦ ∃ x ( d A,D , ¬ D ◦ ˆ ∆ ′ D,A )) ∧ 1 D ) (see the end Section 1.2 for the definitions of ˆ θ ∀ x ← A ∨ D , ¬ D and ˇ θ ∃ x → A ∧ D , ¬ D ). These equations say that the distributivity a rrow terms ˇ θ ∀ x → A,D and ˆ θ ∃ x ← A,D are definable in QPN ¬ in terms of the remaining primitiv e ar row terms and op erations on arrow terms. If these distributivit y ar row terms are taken as defined when we int ro duce QPN ¬ , then the equa tions ( Q ξ θ ξ θ ) o f Section 1.2 b ecome sup erfluous as a xioms—they ca n b e der iv ed fro m the remaining axiomatic equations. W e define a contrav ar ian t endofunctor of QPN ¬ , i.e. a functor fro m QPN ¬ to QPN ¬ op , in the following manner, for f : A ⊢ B : ¬ f = d f ˇ Σ ′ B , ¬ A ◦ d ¬ B ,B ,A ◦ ( 1 ¬ B ∧ ( f ∨ 1 ¬ A )) ◦ ˆ ∆ ′ A, ¬ B : ¬ B ⊢ ¬ A, and we verify that this is indeed a contrav ar ian t functor b y pr o ceeding as in [8] (Section 2.8, wher e there is a lso an alternative definition o f ¬ f ). In the course of this verification, we es ta blish ea sily with the help of QPN ¬ Coherence that Ξ Q is a dinatural transfor mation in the crown index (se e [2 1], Section IX.4, for the notion of dinatur a l transfor mation). 3 Coherence of QMDS and QMPN ¬ 3.1 QMDS Coherence The catego ry QM DS is defined as the catego ry QDS in Section 1 .2 s ave that we have the additiona l primitive arrow ter ms m A,B : A ∧ B ⊢ A ∨ B for all form ulae A a nd B of L , a nd we as s ume the following a dditional equatio ns: ( m nat ) ( f ∨ g ) ◦ m A,B = m D,E ◦ ( f ∧ g ), fo r f : A ⊢ D and g : B ⊢ E , ( ˆ b m ) m A ∧ B ,C ◦ ˆ b → A,B ,C = d A,B ,C ◦ ( 1 A ∧ m B ,C ), ( ˇ b m ) ˇ b → C,B ,A ◦ m C,B ∨ A = ( m C,B ∨ 1 A ) ◦ d C,B ,A , ( c m ) m B ,A ◦ ˆ c A,B = ˇ c B ,A ◦ m A,B . The pro of-theor etical principle underlying m A,B is ca lled mix (see the Gent zen op eration b elow, and [7], Section 8 .1 , where references are given). T o obtain the functor G from QMDS to the catego ry R el (see Sec tio n 1.5 ), or to the c ategory Br (see Section 2.5 ), we extend the definition o f the functor G from QDS to Rel by a dding the clause that says that Gm A,B is an identit y arrow. T o prov e that this functor G is faithful—this result is ca lle d QMDS 45 Coher enc e —we extend the pr oof of QDS Coher e nce of the first part of this pap er. The Ge ntzenization of QMDS is obtained with the catego ry GQMDS , which has an additio na l Gentzen op eration f : U ⊢ Z g : Y ⊢ W mix ( f , g ) = dn ( f ∨ g ) ◦ m U,Y : U ∧ Y ⊢ Z ∨ W The Cut-Elimination Theorem is proved for GQMDS b y enla rging the pro of w e had for GQDS in Section 1.9 with an additio na l ca se dealt with in [7 ] (Section 8.4). The prepara tion for this Cut-Elimination Theorem inv olving v aria ble- purity is no t impe de d by the pr esence of mix . T o prov e the inv ertibilit y lemmata w e need fo r GQM DS we rely on the following equations o f GQMDS : ( mix Q S ) mix ( Q S x,X f 1 , f 2 ) = Q S x,X mix ( f ′ 1 , f 2 ), for f ′ 1 being a s for ( ξ Q S ) in Sectio n 1 .10 and S ∈ { L, R } . These equations are either straig h tforward to derive, or when Q S ∈ {∀ R , ∃ L } we derive them by imitating the deriv a tio n of the e q uation of case ( ∀ 2) o f Section 1 .9, with the help of the equa tions ( Qβ r e d ) and ( Qη r e d ) (se e the end of Sec tio n 1.5). T o prov e the new Inv ertibility Lemma ta for ∧ and ∨ we enlar ge the pro ofs o f such inv er tibility lemmata we had for GQDS in Section 1.10 with cases in volving mix covered b y the rema rks pr e c eding the Inv er tibilit y Lemma for mix in [7] (Section 8 .4). The pr oo fs of the new Inv ertibilit y Lemmata for ∀ R and ∃ L are taken ov er unchanged. T o pr o ve the new Inv ertibility Lemmata for ∀ L and ∃ R we use in additio n the equations ( mix ∀ L ) a nd ( mix ∃ R ) r espectively . W e need mor eov er a new Inv er tibilit y Lemma for mix , analog ous to the lemma with the sa me name in [7] (Section 8.4). The pr oof of this new lemma is bas ed on the pr oo f in [7] and on the equations ( mix Q S ). This s uffices to establish QMDS Coherence. 3.2 QMPN ¬ Coherence W e in troduce now the categ ory QMPN ¬ , which corr esponds to the m ulti- plicative fragment without prop ositiona l cons tan ts of classical linear first-o rder predicate logic with mix. The categor y QMPN ¬ is defined as the category QPN ¬ in Section 2.1 sav e that we hav e the additional primitiv e ar row terms m A,B : A ∧ B ⊢ A ∨ B for all formulae A and B of L ¬ , and we assume as addi- tional equations ( m nat ), ( ˆ b m ), ( ˇ b m ) and ( c m ) of the preceding section. T o obtain the functor G from QMPN ¬ to the c ategory Br w e ex tend what we had for the functor G fr om QPN ¬ to Br (see Section 2.5) with the clause that says that Gm A,B is an identit y arr o w. The theorem asserting that this functor is faithful is ca lled QMPN ¬ Coherence. 46 The categor y QMPN is de fined as the catego r y QPN a t the end of Section 2.1 sav e that we hav e the a dditional primitive a rrow terms m A,B for all ob jects A and B of QPN , and we assume the a dditional equations ( m nat ), ( ˆ b m ), ( ˇ b m ) and ( c m ). W e can pr ove that QMPN ¬ and QM PN are e q uiv a len t catego r ies as in Sectio n 2.6, with trivial a dditions. The pr oo f of Q MPN ¬ Coherence is then reduced to the pr oof of Q MPN Coherence, and the latter pro of can b e obta ined quite a nalogously to what we hav e in [8] (Sections 6.1 -2). Her e are some remarks co ncerning a dditions a nd changes. The pro blem here is that the ∧∨ Lemma o f Section 2 .3, which was used for proving the Ξ ∀ -Perm utation Lemma for Q PN in Section 2 .4, do es not hold for QMDS (the ∨∧ Lemma of Section 2 .3 holds for QMDS ). W e can nevertheless prov e a mo dified version of the Ξ ∀ -Perm utation Lemma, where we as sume that Y 1 and Y 2 o ccur within a subformula o f E of the for m ¬ P x ′ n ∧ ( Y 1 x ′ n ∨ ¬ Y 2 x ′ n ) or P x ′ n ∧ ( ¬ Y 2 x ′ n ∨ Y 1 x ′ n ). F o r the pro of of this modified v ersion of the Ξ ∀ - Perm utation Lemma we r ely on some auxiliar y r esults, which w e will now con- sider. Let us call qu asi-atomic formulae of L all formulae of the form S P x n for S a finite sequence of quantifier prefixes, i.e. formulae in which ∧ and ∨ do no t o ccur. F or X a particular o ccurrence of a pr edicate letter in a for m ula A such that there is a s ubform ula o f the form B ξ C or C ξ B of A where C is a quas i- atomic form ula in which X o ccur s , let A − X be obtaine d from A by repla cing the par ticula r subform ula B ξ C or C ξ B by B . When X o ccurs in A as w e hav e just said we say that X is deletable from A . F or i ∈ { 1 , 2 } , let A i be a formula o f L , let X i be an o ccurrence of the predicate le tter P deletable in A i , and let X 1 and X 2 be tied in the ar row f : A 1 ⊢ A 2 of QMDS (see the b eginning o f Sectio n 2 .3 for the mea ning of “tied”). The new version of Lemma 1 o f Section 6.1 of [8] then says tha t there is an a rrow term f − P : A − X 1 1 ⊢ A − X 2 2 of QM DS such that Gf − P is o btained from Gf by de le ting the pair co r resp onding to ( X 1 , X 2 ). In the pro of of Lemma 1 of Section 6.1 in the printed version o f [8] there is a n omission. The last sentence of the first para graph should b e replace d by: “If x i is not a pro per subformula of the subform ula B j , then d − q B 1 ,q, B 3 is m B 1 ,B 3 or f − q is 1 A − x i i .” The pro of of the new version of Lemma 1 is then ana logous to the o ld pro of with the addition in the induction step that when f is g ξ h or h ξ g , then f − P is g not only when h is equal to 1 P x n , but also when it is of a type B 1 ⊢ B 2 such that B i is a qua si- atomic s ubform ula of A i in which X i o ccurs. No te that this deleting lemma do es not hold for Q DS , b ecause we cannot cov er d − P B 1 ,S P x n ,B 3 . A c ontext Z is obtained fr om a fo r m ula of L by repla cing a particular oc- currence of an atomic subformula with a place ho lder ✷ . W e write Z ( A ) for the formula o btained by putting the formula A at the pla ce o f ✷ in Z , and we write Z ( f ) for the arrow term obtained by putting the arrow term f at the place of ✷ in Z and 1 B at the pla ce o f every atomic formula B in Z . F or X and 47 Y contexts, let f : X ( P x n ) ∧ ∀ y m B ⊢ Y ( P x n ∧ B ) b e a n arrow ter m of QMDS such that y m are a ll the v ariables fr ee in B , the display ed o ccurrences o f P in the source and target are tied in f , and the same holds for the k -th o ccurrence of pre dica te letter (coun ting from the left) in the displa yed occur rences of B in the so urce and target, for every k . Then, by successive a pplications of the new version o f Lemma 1, we obtain the ar row term f − B : X ( P x n ) ⊢ Y ( P x n ) o f QMDS such that the display ed o ccurrences o f P in the sour ce and tar get are tied in f . Let f † : X ( P x n ∧ ∀ y m B ) ⊢ Y ( P x n ∧ ∀ y m B ) be the arrow term of Q MDS ob- tained fr o m f − B by re pla cing P x n by P x n ∧ ∀ y m B in the indices of the primitiv e arrow ter ms of f − B at pla c e s corr esponding to the o ccurre nc e s display ed in the source and target (see [8], Section 6.1, for an example). The new v ersion o f Lemma 2 ∧ of Section 6.1 of [8] sho uld state the following: L et f and f † b e as ab ove. Then ther e is an arr ow term h X : X ( P x n ) ∧ ∀ y m B ⊢ X ( P x n ∧ ∀ y m B ) of QDS such that f = Y ( 1 P x n ∧ ι ∀ y m B ) ◦ f † ◦ h X in Q MDS . In the pro of of this new lemma, when w e define inductively h X , b esides clauses analogo us to the old clauses, w e sho uld hav e the additional clauses h ∀ x Z = ∀ x h Z ◦ ˆ θ ∀ x ← Z ( P x n ) , ∀ y m B , h ∃ x Z = ∃ x h Z ◦ ˆ θ ∃ x ← Z ( P x n ) , ∀ y m B (see the end Section 1.2 for the definition of ˆ θ ∀ x ← Z ( P x n ) , ∀ y m B ). There is an ana logous new version of Lemma 2 ∨ of Sectio n 6.1 of [8]. The pro of of QMPN Coherenc e then pro ceeds as in Sec tio n 6.2 o f [8]. This suffices to establish QMPN ¬ Coherence. 4 Concluding remarks In this pa per we hav e no t dealt with the multip licative prop ositional co nstan ts bec ause, as we s a id in the Int ro duction, they ra ise pr o blems for co herence understo o d as the existence of a faithful functor into the category Br . Star - autonomous categorie s have howev er unit ob jects corre sponding to the multi- plicative pro p ositiona l cons tan ts, and it would b e interesting to define a notion of sta r-autonomous categor y with quantifiers, i.e. with functor s corres ponding to quantifi ers. It w ould be des irable that o ur category QPN ¬ be isomorphic to a full s ubcatego ry of a category equiv a len t with the freely generated star - autonomous category with quantifiers. (W e are not lo oking for a full sub cat- egory of the freely genera ted star -autonomous category with quan tifiers, but 48 for a full s ubcatego ry of a catego ry e quivalent with this categ ory , b ecause of a difference in lang uage; see [8], Chapter 3). W e hav e pr oved an analo gous result, whic h concerns prop ositional linear logic, in [8] (Cha pter 4); we hav e prov ed na mely that the category PN ¬ is isomorphic to a full sub category of a ca tegory equiv alen t with the freely gen- erated star-a utonomous catego ry . This sho ws that our notion of catego ry for which PN ¬ is the freely generated one is the right notion of s tar-autonomous category without units. Besides [8], a sys tematic work devoted to sta r-autonomous ca tegories with- out units is [11]. It introduces a differently defined notion, for whic h it is suppo sed that it is equiv alent to ours (see also [1 2]). The freely ge ne r ated star-autonomous category with quantifiers should b e obtained by extending the freely ge ne r ated star-a utonomous ca tegory with as- sumptions inv olving quantifiers like thos e intro duced in this pap er. W e supp ose that the appro ac h of [8] could b e extended to prove for this catego ry the result men tioned in the first par agraph of this se ction. A cknow le dgement . W e are grateful to Phil Scott and to an anonymous referee for helpful suggestions. W ork on this pap er w as sup ported by the Ministry of Science of Serbia (Grants 144013 and 144029). References [1] R. Blute and P. Scott , Cate gory the ory for line ar lo gicians , Line ar L o gic in Computer Scienc e (T. Ehrhard et al., editors), Londo n Ma the- matical So ciety Lecture Notes, vol. 316 , Cambridge Univ ersity Press, Cam- bridge, 2004, pp. 3-64 [2] R. Brauer , On algebr as which ar e c onne cte d with the semisimple c ontin- uous gr oups , Annals of Mathematics , vol. 38 (1937), pp. 857-8 72 [3] G. B u rde and H. Z ieschang , Knots , de Gruyter, Berlin, 1985 [4] J. R.B. Cockett and R.A .G. Seel y , We akly distributive c ate gories , Journal of Pur e and Applie d Algebr a , vol. 114 (1997), pp. 13 3-173 [5] Dj. ˇ Cubri ´ c , On the semant ics of the u niversal quantifier , Annals of Pur e and Appli e d L o gic , vol. 87 (1997), pp. 2 0 9-239 [6] K. Do ˇ sen , Cu t Elimination in Cate gories , Kluwer, Dordr ec h t, 19 9 9 49 [7] K. Do ˇ sen and Z. Petri ´ c , Pr o of-The or etic al Coher enc e , KCL P ubli- cations (College Publicatio ns ), London, 2 004 (r e vised version av ailable at: ht tp://www.mi.sanu.ac.yu/ ∼ kosta/coh.p df ) [8] ——– , Pr o of-Net Cate gories , Polimetrica, Monza, 20 07 (preprint av ail- able a t: h ttp://www.mi.sanu.ac.yu/ ∼ kosta/pn.pdf, 2 005) [9] G. Gentzen , Untersuchungen ¨ ub er das lo gische Schließen , Mathemati- sche Zei tschrift , vol. 39 (1935 ), pp. 176 -210, 40 5-431 (Englis h trans- lation: Investigations into lo gic al de duction , The Col l e cte d Pap ers of Gerhar d Gentzen , M.E. Szab o, editor , North-Ho lland, Amsterda m, 1969, pp. 68- 131) [10] J.-Y. Girard , Line ar lo gic , The or eti c al Computer Scienc e , v ol. 50 (1987), pp. 1- 101 [11] R. Housto n , Line ar L o gic withou t Units , do ctoral disser ta tion, Uni- versit y of Manchester, 2 007 [12] R. H ouston, D. H ughes and A . S chalk , Mo del ling line ar lo gic without units (Pr eliminary r esults) , preprint, 2005 (a v ailable at: http://arXiv.org/ - math.CT/050 4 037) [13] B. Ja cobs , Cate goric al L o gic and T yp e The ory , Els evier, Amsterda m, 1999 [14] G.M. Kell y and S. Mac Lane , Coher enc e in close d c ate gories , Journal of Pur e and Applie d Algebr a , vol. 1 (197 1), pp. 97 -140, 219 [15] S.C. Kleene , Intr o du ction to Metamathematics , North-Ho lla nd, Amsterdam, 1 952 [16] ——–, Permut ability of infer enc es in Gentzen ’s c alculi LK and LJ, Two Pap er s on the Pr e dic ate C alculus, Memoir s of the Americ an Mathematic al So ciety , vol. 10 (1952), pp. 1 -26 [17] J. Lambek , De ductive syst ems and c ate gories I I: Standar d c onstructions and close d c ate gories , Cate gory The or y, Homolo gy The ory and thei r Applic ations I , Lecture Notes in Mathematics, vol. 86, Spr inger, Ber lin, 1969, pp. 76- 122 [18] F.W. La wvere , A djointness in foundations , D iale ctic a , vol. 2 3 (1969), pp. 281-29 6 [19] ——–, Equal ity in hyp er do ctrines and c ompr ehension schema as an ad- joint functor , Applic ations of Cate gori c al Algebr a (A. Heller, editor ), American Mathematical So ciety , Pr o vidence, 197 0, pp. 1 - 14 50 [20] F.W. La wvere and R. Rosebr ugh , Sets for Mathematics , Cam- bridge Univ ersity P r ess, Cambridge, 200 3 [21] S. Mac Lane , Cate gori es for the Working Mathematician , Spring- er, B erlin, 197 1 (ex panded second edition, 1998 ) [22] D. P a vlovi ´ c , Maps II: Chasing diagr ams in c ate goric al pr o of the ory , Jour- nal of the IGPL , vol. 4 (19 9 6), pp. 1 59-194 [23] D. Pra witz , Natur al De ducti on: A Pr o of-The or etic al Study , Alm- qvist & Wiksell, Sto ckholm, 1965 [24] W.V. Quine , Pr e dic ate functors r evisite d , The Jour nal of Symb olic L o gic , vol. 46 (1981), pp. 649-6 52 [25] R.A.G. Seel y , H yp er do ctrines, natur al de duction and the Be ck c ondition , Zeitschrift f¨ ur mathematische L o gik und Grund lagen der Math - ematik , vol. 2 9 (198 3), pp. 50 5-542 [26] ——–, Line ar lo gic, ∗ -autonomous c ate gories and c ofr e e c o algebr as , C ate- gories in Computer Scienc e and L o gic (J.W. Gray a nd A. Scedrov, editors), Contemporar y Mathematics, vol. 92 , American Ma thema tical So- ciety , Providence, 198 9, pp. 371-38 2 51