INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Universal proof theory: semi-analytic rules and uniform interpolation

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1 INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Unvesal poof theoy: sem-analytc ules and unfom ntepolaton Amhossen Akba Tabataba Raheleh Jalal Pepnt No PRAHA 2018

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3 Unvesal Poof Theoy: Sem-analytc Rules and Unfom Intepolaton Amhossen Akba Tabataba, Raheleh Jalal Insttute of Mathematcs Academy of Scences of the Czech Republc Septembe 26, 2018 Abstact In [7] and [8], Iemhoff ntoduced a connecton between the exstence of a temnatng sequent calcul of a cetan knd and the unfom ntepolaton popety of the supe-ntutonstc logc that the calculus captues. In ths pape, we wll genealze ths elatonshp to also cove the substuctual settng on the one hand and a much moe poweful class of ules on the othe. The esulted elatonshp then povdes a unfom method to establsh unfom ntepolaton popety fo the logcs FL e, FL ew, CFL e, CFL ew, IPC, CPC and the K and KD-type modal extensons. Moe nteestngly though, on the negatve sde, we wll show that no extenson of FL e can enjoy a cetan natual type of temnatng sequent calculus unless t has the unfom ntepolaton popety. It excludes almost all supe-ntutonstc logcs and the logcs K4 and S4 fom havng such a easonable calculus. 1 Intoducton Poof systems ae and always have been the man tool n any nvestgaton of the behavo of the mathematcal theoes fom seachng fo the consstency poofs and fndng the possble decson pocedues to captung the admssble ules and extactng the actual pogams fom gven poofs. Followng ths huge effectveness, a techncal appoach has emeged to fst desgn and The authos ae suppoted by the ERC Advanced Gant (FEALORA). 1

4 then study the appopate poof systems taloed fo the sole use n povng the popetes of a gven nteestng theoy. In ths espect, poof systems have been teated as the second ank ctzens contay to the ndependent nteestng mathematcal objects that they could have been. Fotunately, n the ecent yeas, alongsde ths nstumentalst appoach, anothe appoach has been also emeged; an appoach that s moe nteested n the geneal behavo of the poof systems than the possble techncal use n poof theoy, although t happens to bng ts own futs n the latte aspect, as well (see [7], [8], [3]). Ths geneal appoach wdens the poof theoetc hozon wth ts own stuctual poblems ncludng the exstence poblem (when does a theoy have a cetan type of poof system?), the equvalence poblem (when ae two poof systems equvalent?) and the chaactezaton poblem (s thee any chaactezaton of the poof systems elatve to a natual equvalence elaton?). Imtatng the tem unvesal algeba fo the genec study of the algebac stuctues, we wll call ths appoach the unvesal poof theoy 1 whch focuses on the model theoetc style nvestgaton of the dffeent possble poof systems n the most geneal fom. As the fst step n ths so-called unvesal poof theoy and followng the spt of [7] and [8], we begn wth the most basc poblem of the knd, the exstence poblem, addessng the exstence of the natual sequent style poof systems fo a gven popostonal o modal logc. Fo ths pupose, we have to develop some stong elatonshps between the exstence of some sot of poof systems and some egulaty condtons of the logc. One loose example of such a elatonshp s the elatonshp between the exstence of a temnatng calculus fo a logc and ts decdablty. Why these elatonshps ae mpotant? Because they educe the exstence poblem patally o completely to the egulaty condtons of the logc that ae calculus-ndependent and pobably moe amenable to ou tools. Agan usng ou loose example, we know that f a logc s not decdable, t can not have a temnatng calculus; a fact whch solves the exstence poblem negatvely. Ths pape s devoted to one of ths knd of elatonshps and to explan how, we have to bowse the hstoy a lttle bt, fst. The stoy begns wth Ptts semnal wok, [9], n whch he ntoduced a poof theoetc method to pove the unfom ntepolaton popety fo the popostonal ntutonstc logc. Hs technque s bult on the followng two man deas: Fst he extended the noton of unfom ntepolaton fom a logc to ts sequent calculus n a way that the unfom p-ntepolants fo a sequent ae oughly the best 1 We ae gateful to Masoud Memazadeh fo ths elegant temnologcal suggeston. 2

5 left and ght p-fee fomulas that f we add them to the left o ght sde of the sequent, they make the sequent povable. Ths educes the task of povng unfom ntepolaton fo the logc, to the task of fndng these new unfom ntepolants fo all sequents. Fo the latte, he assgned two sets of p-fee fomulas to any sequent usng the stuctue of the fomulas occued n the sequent tself. To defne these sets, though, he needed the second cucal tool of the game namely the temnatng calculus fo IPC, ntoduced n [4] by Dyckhoff. The temnatng calculus povdes a well-founded ode on sequents on whch we can defne the sets that we have mentoned befoe, ecusvely. Late, as wtnessed n [8], Iemhoff ecognzed that the man pont n the fst pat of Ptts agument s flexble enough to apply on any ule wth a cetan geneal fom. Ths obsevaton then lets he to lft the technque fom the ntutonstc logc to any extenson of the ntutonstc logc pesented wth a genec temnatng calculus consstng of that cetan sot of ules that she calls centeed ules. These ules ae vey natual ules to consde and they ae oughly the ules wth one man fomula n the consequence such that the ule espects both the sde of ths man fomula and the occuence of atoms n t,.e. f the man fomula occued n the left-sde (ght-sde) of the consequence, all non-contextual fomulas n the pemses should also occu n the left-sde (ght-sde) and any occuence of any atom n these fomulas should also occu n the man fomula. The usual conjuncton and dsjuncton ules ae the pototype examples of these ules whle the mplcaton ules ae the non-examples snce they clealy do not espect the sde of the man fomula. As we explaned, the nvestgatons n [8] lead to an exctng elatonshp between the exstence of a temnatng calculus consstng only of the centeed ules fo a logc and the unfom ntepolaton popety of the logc. Iemhoff used ths elatonshp fst n a postve manne to pove the unfom ntepolaton fo some well-known supe-ntutonstc and supe-ntutonstc modal logcs ncludng IPC, CPC, K and KD and the ntutonstc vesons. And then she swtched to the negatve pat to show that no centeed extenson of the ntutonstc logc can have a temnatng centeed calculus unless t has the unfom ntepolaton popety. Snce unfom ntepolaton s a ae popety fo a logc, t excludes almost all logcal systems, ncludng all supe-ntutonstc logcs except the seven logcs wth the unfom ntepolaton popety fom havng a temnatng centeed calculus. Now we ae eady to explan what we wll pusue n ths pape. Ou ap- 3

6 poach s a genealzaton of the mentoned elatonshp between the exstence of a temnatng calculus consstng of cetan sot of ules and the unfom ntepolaton popety. Ou esults ae the genealzaton of the esults n [7] and [8], n the followng two aspects. Fst we use a much moe geneal class of ules that we wll call sem-analytc and context-shang sem-analytc ules. These ules can be defed oughly as the centeed ules elaxng the sde pesevng condton. Theefoe, they cove a vast vaety of ules ncludng centeed ules, mplcaton ules, non-context shang ules n substuctual logcs and so many othes. Second, we lowe the base logc fom the ntutonstc logc to the basc substuctual logc FL e. It helps to povde a unfom method to establsh the unfom ntepolaton popety whch s applcable smultaneously fo FL e, FL ew, CFL e, CFL ew and the K and KD modal extensons on the one hand and the ntutonstc and classcal logcs and the modal extensons on the othe. (Fo the classcal modal case see [2], fo the sub-stuctual logcs see [1] and fo ntutonstc and ntutonstc modal logcs see [9] and [8].) It also sets the scene to povde the same chaactezaton fo any sem-analytc extensons of FL e f we fst povde a temnatng calculus fo them. Whle t s vey appealng to develop a geneal method to pove unfom ntepolaton, the man applcaton of ou nvestgaton belongs to the negatve sde of the elatonshp. Applyng ou esult negatvely, we can also push the esult n [8] futhe to show that the logcs wthout unfom ntepolaton popety can not even have ou moe geneal type of temnatng calcul. Fo nstance, usng the well-known esult on the chaactezaton of all supe-ntutonstc logcs, [6], we know that except IPC, LC, KC, Bd 2, Sm, GSc and CPC, non of the supe-ntutonstc logcs have a temnatng calculus consstng of sem-analytc and context-shang sem-analytc ules togethe wth the centeed axoms. The same also goes fo the modal logcs K4 and S4. 2 Pelmnaes In ths secton we wll cove some of the pelmnaes needed fo the followng sectons. The defntons ae smla to the same concepts n [8], but they have been changed wheneve t s needed. Fst, note that all of the fnte objects that we wll use hee can be epesented by a fxed easonable bnay stng code. Theefoe, by the length of any object O ncludng fomulas, poofs, etc. we mean the length of ths 4

7 stng code and we wll denote t by O. Defnton 2.1. Let L and L 1 be two languages. By a tanslaton t : L Ñ L 1, we mean an assgnment whch assgns a fomula ϕ C p pq P L 1 to any logcal connectve Cp pq P L such that any p has at most one occuence n ϕ C p pq. It s possble to extend a tanslaton fom the basc connectves of the language to all of ts fomulas n an obvous compostonal way. We wll denote the tanslaton of a fomula ϕ by ϕ t and the tanslaton of a multset Γ, by Γ t tϕ t ϕ P Γu. Note that fo any tanslaton t we have ψ t ď Op1q ψ whch shows that all tanslatons ae polynomally bounded. In ths pape, we wll wok wth a fxed but abtay language L that s augmented by a tanslaton t : t^, _, Ñ,, 0, 1u Y L Ñ L that fxes all logcal connectves n L. Fo ths eason and w.l.o.g, we wll assume that the language aleady ncludes the connectves t^, _, Ñ,, 0, 1u. In addton, wheneve we nvestgate the mult-concluson systems we always assume that the tanslaton expands to nclude `. Example 2.2. The usual language of classcal popostonal logc s a vald language n ou settng. In ths case, thee s a canoncal tanslaton that sends fuson, addton, 1 and 0 to conjuncton, dsjuncton, J and K, espectvely. In ths pape, wheneve we pck ths language, we assume that we ae wokng wth ths canoncal tanslaton. By a sequent, we mean an expesson of the fom Γ ñ, whee Γ and ae multsets of fomulas n the language, and t s ntepeted as Γ Ñ Ř. By a sngle-concluson sequent Γ ñ we mean a sequent that ď 1, and we call t mult-concluson othewse. We denote multsets by captal Geek lettes such as Σ, Γ, Π, and Λ. Howeve, sometmes we use the ba notaton fo multsets to make eveythng smple. Fo nstance, by ϕ, we mean a multset consstng of fomulas ϕ. Meta-language s the language wth whch we defne the sequent calcul. It extends ou gven language wth the fomula symbols (vaables) such as ϕ and ψ. A meta-fomula s defned as the followng: Atomc fomulas and fomula symbols ae meta-fomulas and f ϕ s a set of meta-fomulas, then Cp ϕq s also a meta-fomula, whee C P L s a logcal connectve of the language. Moeove, we have nfntely many vaables fo meta-multsets and we use captal Geek lettes agan fo them, wheneve t s clea fom the context whethe t s a multset o a meta-multset vaable. A meta-multset s a 5

8 multset of meta-fomulas and meta-multset vaables. By a meta-sequent we mean a sequent whee the antecedent and the succedent ae both metamultsets. We use meta-multset vaable and context, ntechangeably. Fo a meta-fomula ϕ, by V pϕq we mean the meta-fomula vaables and atomc constants n ϕ. A meta-fomula ϕ s called p-fee, fo an atomc fomula o meta-fomula vaable p, when p R V pϕq. Let us ecall some of the notons elated to sequent calcul and some of the mpotant systems that we wll use thoughout the pape. Fo a sequent S pγ ñ q, by S a we mean the antecedent of the sequent, whch s Γ, and by S s we mean the succedent of the sequent, whch s. And, the multplcaton of two sequents S and T s defned as S T ps a Y S a ñ T s Y T s q. By a ule we mean an expesson of the fom S 1,, S n S 0 whee S s ae meta-sequents. By an nstance of a ule, we mean substtutng multsets of fomulas fo ts contexts and substtutng fomulas fo ts meta-fomula vaables. A ule s backwad applcable to a sequent S, when the concluson of the ule s S. By a sequent calculus G, we mean a set of ules. A sequent S s devable n G, denoted by G $ S, f thee exsts a tee wth sequents as labels of the nodes such that the label of the oot s S and n each node the set of the labels of the chlden of the node togethe wth the label of the node tself, consttute an nstance of a ule n the system. Ths tee s called the poof of S n G whch s sometmes called a tee-lke poof to emphasze ts tee-lke fom. Now consde the followng set of ules: Identty: Contextual Axoms: ϕ ñ ϕ 6

9 Γ ñ J, Γ, K ñ Context-fee Axoms: ñ 1 0 ñ Rules fo 0 and 1: Γ ñ Γ, 1 ñ L1 Γ ñ Γ ñ 0, R0 Conjuncton Rules: Γ, ϕ ñ L^ Γ, ϕ ^ ψ ñ Γ ñ ϕ, Γ ñ ψ, R^ Γ ñ ϕ ^ ψ, Dsjuncton Rules: Γ, ϕ ñ Γ, ψ ñ L_ Γ, ϕ _ ψ ñ Γ ñ ϕ, Γ ñ ϕ _ ψ, R_ Γ ñ ψ, Γ ñ ϕ _ ψ, R_ Fuson Rules: Γ, ϕ, ψ ñ L Γ, ϕ ψ ñ Γ ñ ϕ, Σ ñ ψ, Λ R Γ, Σ ñ ϕ ψ,, Λ Implcaton Rules: Γ ñ ϕ, Σ, ψ ñ Λ L Ñ Γ, Σ, ϕ Ñ ψ ñ, Λ Γ, ϕ ñ ψ, R Ñ Γ ñ ϕ Ñ ψ, The system FL e conssts of the sngle-concluson veson of all of these ules. In the mult-concluson case defne CFL e wth the same ules as FL e, ths tme n the full mult-concluson veson and add ` to the language and the followng ules to the systems: Rules fo `: Γ, ϕ ñ Σ, ψ ñ Λ L` Γ, Σ, ϕ ` ψ ñ, Λ Γ ñ ϕ, ψ, Γ ñ ϕ ` ψ, R` Moeove, we have the followng addtonal ules that we wll use late: 7

10 Weakenng ules: Γ ñ Γ, ϕ ñ Lw Γ ñ Γ ñ ϕ, Rw Note that n the sngle-concluson cases, n the ule prwq, s empty. Contacton ules: Γ, ϕ, ϕ ñ Lc Γ, ϕ ñ Γ ñ, ϕ, ϕ Rc Γ ñ ϕ, The ule prcq s only allowed n mult-concluson systems. If we consde the logc FL e and add the weakenng ules (contacton ules), the esulted system s called FL ew (FL ec ). The same also goes fo CFL ew and CFL ec. We also have the followng ule: Context-shang left mplcaton: Γ ñ ϕ Γ, ψ ñ Γ, ϕ Ñ ψ ñ Fnally, note that Γ and ae multsets eveywhee, theefoe the exchange ule s bult n and hence admssble n ou system. Moeove, note that the calcul defned n ths secton ae wtten n the gven language whch can be any extenson of the language of the system tself. Fo nstance, FL e s the calculus wth the mentoned ules on ou fxed language that can have moe connectves than t^, _,, Ñ, J, K, 1, 0u. Defnton 2.3. We wll defne the sequent calculus fo ntutonstc logc, whch was fst ntoduced by Dyckhoff n [4]. Γ, p ñ p At, Γ, K ñ LK Γ, ϕ, ψ ñ L^ Γ, ϕ ^ ψ ñ Γ ñ ϕ, Γ ñ ψ, R^ Γ ñ ϕ ^ ψ, Γ, ϕ ñ Γ, ψ ñ L_ Γ, ϕ _ ψ ñ Γ ñ ϕ, ψ Γ ñ ϕ _ ψ R_ 8

11 Γ, ϕ ñ ψ R Ñ Γ ñ ϕ Ñ ψ Γ, p, ψ ñ L1 Ñ Γ, p, p Ñ ψ ñ Γ, ϕ Ñ pψ Ñ γq ñ L2 Ñ Γ, ϕ ^ ψ Ñ γ ñ Γ, ϕ Ñ γ, ψ Ñ γ ñ L3 Ñ Γ, ϕ _ ψ Ñ γ ñ Γ, ψ Ñ γ ñ ϕ Ñ ψ Γ, γ ñ L4 Ñ Γ, pϕ Ñ ψq Ñ γ ñ whee p s an atom. Stuctual ules and the cut ule ae admssble n the system and we have ď 1. Defnton 2.4. A calculus s temnatng f fo any sequent S, the numbe of ules whch ae backwad applcable to S ae fnte. Moeove, thee s a well-founded ode on the sequents such that the ode of the followng ae less than the ode of S: the pemses of a ule whose concluson s S; subsequents of S, and any sequent S 1 of the fom pγ, Π ñ, Λq, whee S s of the fom pγ, lπ ñ, lλq. Note that Π Y Λ must be non-empty. Defnton 2.5. Let L and L 1 be two logcs such that L L Ď L L 1. We say L 1 s an extenson of L f L $ A mples L 1 $ A. Defnton 2.6. Let G and H be two sequent calcul such that L G Ď L H. We say H s an extenson of G f G $ Γ ñ mples H $ Γ ñ. It s called an axomatc extenson, f the povable sequents n G ae consdeed as axoms of H, to whch H adds some ules. Defnton 2.7. Let G be a sequent calculus and L be a logc wth the same language as G s. We say G s a sequent calculus fo the logc L when: G $ Γ ñ f and only f L $ p Γ Ñ Ř q. Note that f the calculus s sngle-concluson, by Ř, we mean f s a sngleton, and 0 f s empty. Theefoe, n ths case we do not need the ` opeato. Theoem 2.8. Let L be a logc and G a sngle-concluson (mult-concluson) sequent calculus fo L. If L extends FL e (CFL e ), then cut s admssble n G. 9

12 Poof. Assume that G $ Γ ñ A, and G $ Γ 1, A ñ 1. Hence L $ Γ Ñ A ` p Ř q and L $ p Γ 1 q A Ñ p Ř 1 q. Snce L extends FL e (CFL e ) and n ths theoy the fomula mples the fomula Γ Ñ A ` p ă qs p Γ 1 q A Ñ p ă 1 qs p Γq p Γ 1 q Ñ p ă q ` p ă 1 qs the last fomula s povable n L whch mples G $ Γ, Γ 1 ñ, 1. Defnton 2.9. We say a logc L has Cag ntepolaton popety f fo any fomulas ϕ and ψ f L $ ϕ Ñ ψ, then thee exsts fomula θ such that L $ ϕ Ñ θ and L $ θ Ñ ψ and V pθq Ď V pϕq X V pψq. Defnton We say a logc L has the unfom ntepolaton popety f fo any fomulas ϕ and any atomc fomula p, thee ae two p-fee fomulas, the and the p-post-ntepolant Dpϕ, such that pq L Ñ ϕ, pq Fo any p-fee fomula ψ f L $ ψ Ñ ϕ then L $ ψ pq L $ ϕ Ñ Dpϕ, and pvq Fo any p-fee fomula ψ f L $ ϕ Ñ ψ then L $ Dpϕ Ñ ψ. 3 Sem-analytc Rules In ths secton we wll ntoduce a class of ules whch we wll nvestgate n the est of ths pape. Fst let us begn wth the sngle-concluson case n whch all sequents have at most one succedent. Defnton 3.1. A ule s called a left sem-analytc ule f t s of the fom xxπ j, ψ js ñ θ js y s y j xxγ, ϕ ñ y y Π 1,, Π m, Γ 1,, Γ n, ϕ ñ 1,, n whee Π j, Γ and s ae meta-multset vaables and ď V p ϕ q Y ď V p ψ js q Y ď V p θ js q Ď V pϕq j,s j,s, and t s called a ght sem-analytc ule f t s of the fom 10

13 xxγ, ϕ ñ ψ y y Γ 1,, Γ n ñ ϕ whee Γ s ae meta-multset vaables and ď V p ϕ q Y ď V p ψ q Ď V pϕq,, Moeove, a ule s called a context-shang sem-analytc ule f t s of the fom xxγ, ψ s ñ θ s y s y xxγ, ϕ ñ y y Γ 1,, Γ n, ϕ ñ 1,, n whee Γ and s ae meta-multset vaables and ď V p ϕ q Y ď V p ψ s q Y ď V p θ s q Ď V pϕq,,s,s We wll call the condtons fo the vaables n all the sem-analytc ules, the occuence pesevng condtons. Note that n the left ule, fo each we have ď 1, and snce the sze of the succedent of the concluson of the ule must be at most 1, t means that at most one of s can be non-empty. Fo the mult-concluson case, we defne a ule to be left mult-concluson sem-analytc f t s of the fom xxγ, ϕ ñ ψ, y y Γ 1,, Γ n, ϕ ñ 1,, n wth the same occuence pesevng condton as above and the same condton that Γ s and s ae meta-multset vaables. A ule s defned to be a ght mult-concluson sem-analytc ule f t s of the fom xxγ, ϕ ñ ψ, y y Γ 1,, Γ n ñ ϕ, 1,, n agan wth the smla occuence pesevng condton and the same condton that Γ s and s ae meta-multset vaables. Wheneve t s clea fom the context, we wll omt the phase mult-concluson. A ule s called modal sem-analytc f t has one of the followng foms: Γ ñ ϕ lγ ñ lϕ K Γ ñ lγ ñ D 11

14 whee Γ s a meta-multset vaable. Note that we always have the condton that wheneve the ule pdq s pesent, the ule pkq must be pesent, as well. In the case of the modal ules, we use the conventon that lh H. By the notaton xx y y we mean fst consdeng the sequents angng ove and then angng ove. Fo nstance, xxγ, ϕ ñ ψ y y s shot fo the followng set of sequents whee 1 ď ď m and 1 ď ď n: Γ 1, ϕ 11 ñ ψ 11,, Γ 1, ϕ 1m1 ñ ψ 1m1, Γ 2, ϕ 21 ñ ψ 21,, Γ 2, ϕ 2m2 ñ ψ 2m2,. Γ n, ϕ n1 ñ ψ n1,, Γ n, ϕ nmn ñ ψ nmn. xxγ, ϕ ñ y y and xxπ j, ψ js ñ θ js y s y j ae defned smlaly. Both n the sngle-concluson and mult-concluson case, a ule s called sem-analytc, f t s ethe a left sem-analytc ule, a ght sem-analytc ule o t s of the fom of a sem-analytc modal ule. In all the sem-analytc ules, the meta-vaables and atomc constants occung n the meta-fomulas of the pemses of the ule, should also occu n the meta-fomulas n the consequence. Because of ths condton, we call these ules sem-analytc. Ths occuence pesevng condton s a weake veson of the analycty popety n the analytc ules, whch demands the fomulas n the pemses to be sub-fomulas of the fomulas n the consequence. Example 3.2. A genec example of a left sem-analytc ule s the followng: whee Γ, ϕ 1, ϕ 2 ñ ψ Γ, θ ñ η Π, µ 1, µ 2, µ 3 ñ Γ, Π, α ñ V pϕ 1, ϕ 2, ψ, θ, η, µ 1, µ 2, µ 3 q Ď V pαq and a genec example of a context-shang left sem-analytc ule s: whee Γ, θ ñ η Γ, µ 1, µ 2, µ 3 ñ Γ, α ñ V pθ, η, µ 1, µ 2, µ 3 q Ď V pαq Moeove, fo a genec example of a ght sem-analytc ule we can have 12

15 whee Γ, ϕ ñ ψ Γ, θ 1, θ 2 ñ η Π, µ 1, µ 2, ñ ν Γ, Π ñ α V pϕ, ψ, θ 1, θ 2, η, µ 1, µ 2, νq Ď V pαq Hee ae some emaks. Fst note that n any left sem-analytc ule thee ae two types of pemses; the type whose ght hand-sde ncludes meta-mult vaables and the type whose ght hand-sde ncludes meta-fomulas. Ths s a cucal pont to consde. Any left sem-analytc ule allows any knd of combnaton of shang/combnng contexts n any type. Howeve, between two types, we can only combne the contexts. The case n whch we can shae the contexts of the two types s called context-shang sem-analytc ule. Ths should explan why ou second example s called context-shang left sem-analytc whle the fst s not. The eason s the fact that the two types shae the same context n the second ule whle n the fst one ths stuaton happens n just one type. The second pont s the pesence of contexts. Ths s vey cucal fo almost all the aguments n ths pape, that any sequent pesent n a sem-analytc ule should have meta-multset vaables as left contexts and n the case of left ules, at least one meta-multset vaable fo the ght hand-sde must be pesent. Example 3.3. Now fo moe concete examples, note that all the usual conjuncton, dsjuncton and mplcaton ules fo IPC ae sem-analytc. The same also goes fo all the ules n sub-stuctual logc FL e, the weakenng and the contacton ules and some of the well known estcted vesons of them ncludng the followng ules fo exponentals n lnea logc: Γ,!ϕ,!ϕ ñ Γ,!ϕ ñ Γ ñ Γ,!ϕ ñ Fo a context-shang sem-analytc ule, consde the followng ule n the Dyckhoff calculus fo IPC (see [4]): Γ, ψ Ñ γ ñ ϕ Ñ ψ Γ, γ ñ Γ, pϕ Ñ ψq Ñ γ ñ Example 3.4. Fo a concete non-example consde the cut ule; t s not sem-analytc because t does not peseve the vaable occuence condton. Moeove, the followng ule n the calculus of KC: Γ, ϕ ñ ψ, Γ ñ ϕ Ñ ψ, 13

16 n whch should consst of negaton fomulas s not a mult-concluson sem-analytc ule, smply because the context s not fee fo all possble substtutons. The ule of thumb s that any ule n whch we have sde condtons on the contexts s not sem-analytc. Defnton 3.5. A sequent s called a centeed axom f t has the followng fom: p1q Identty axom: (ϕ ñ ϕ) p2q Context-fee ght axom: (ñ ᾱ) p3q Context-fee left axom: ( β ñ) p4q Contextual left axom: (Γ, ϕ ñ ) p5q Contextual ght axom: (Γ ñ ϕ, ) whee n 2-5, the vaables n any pa of elements n ᾱ, β, ϕ ae equal and Γ and ae meta-multset vaables. A sequent s called context-fee centeed axom f t has the fom p1q, p2q o p3q. Example 3.6. It s easy to see that the axoms gven n the pelmnaes ae examples of centeed axoms. Hee ae some moe examples: 1 ñ, ñ 0 ϕ, ϕ ñ, ñ ϕ, ϕ Γ, J ñ, Γ ñ, K whee the fst fou ae context-fee whle the last two ae contextual. 4 Unfom Intepolaton In ths secton we wll genealze the nvestgatons of [8] to also cove the substuctual settng and sem-analytc ules. We wll show that any extenson of a sequent calculus by sem-analytc ules peseves unfom ntepolaton f the esulted system tuns out to be temnatng. Ou method hee s smla to the method used n [8]. As a fst step, let us genealze the noton of unfom ntepolaton fom logcs to sequent calcul. The followng defnton offes thee vesons of such a genealzaton, each of whch sutable fo dffeent foms of ules. 14

17 Defnton 4.1. Let G and H be two sequent calcul. G has H-unfom ntepolaton f fo any sequent S and T whee T s H and any atom p, thee exst p-fee fomulas IpSq and JpT q such that pq S pipsq ñq s devable n H. pq Fo any p-fee multset Γ, f S pγ ñq s devable n G then Γ ñ IpSq s devable n H. pq T pñ JpT qq s devable n H. pvq Fo any p-fee multsets Γ and, f T pγ ñ q s devable n G then JpT q, Γ ñ s devable n H. Smlaly, we say G has weak H-unfom ntepolaton f nstead of pq we have p 1 q Fo any p-fee multset Γ, f S pγ ñq s devable n G then Jp Sq, Γ ñ IpSq s devable n H whee S ps a ñq. We say G has stong H-unfom ntepolaton f nstead of pq we have p 2 q Fo any p-fee multsets Γ and, f S pγ ñ q s devable n G then Γ ñ IpSq, s devable n H. Note that n the case of the stong unfom ntepolaton, T s can be nonempty, and we have mult-concluson ules. We call IpSq a left p-ntepolant (weak p-ntepolant, stong p-ntepolant) of S and JpT q a ght p-ntepolant (weak ght p-ntepolant, stong ght p- ntepolant) of T n G elatve to H. The system H has unfom ntepolaton popety (weak unfom ntepolaton popety, stong unfom ntepolaton popety) f t has H-unfom ntepolaton (weak H-unfom ntepolaton, stong H-unfom ntepolaton). Theoem 4.2. If G s a sequent calculus wth (weak/stong) unfom ntepolaton and complete fo a logc L extendng (FL e /CFL e ) FL e, L has the unfom ntepolaton popety. Poof. Fst note that snce G s complete fo L, L $ ϕ Ñ ψ ff G $ ϕ ñ ψ. Hence we can ewte the defnton of the unfom ntepolaton usng the sequent system G. Now pck S pñ Aq. By unfom ntepolaton popety of G, thee s a p-fee fomula IpSq such that S pipsq ñq and fo any p-fee Σ f S pσ ñq, then Σ ñ IpSq. It s clea that IpSq woks as the p-pe-ntepolant of A, because fstly IpSq ñ A and secondly f B ñ A 15

18 then B ñ IpSq fo any p-fee B. The same agument also woks fo the p-post-ntepolant. In the case of weak unfom ntepolaton, fst note that by defnton f T pñq then pñ JpT qq. Secondly, note that snce G s complete fo L, the calculus should admt the cut ule by Theoem 2.8. Now we clam that IpSq woks agan. The eason now s that f B ñ A fo a p-fee B, then Jp Sq, B ñ IpSq. Snce S T and we have the cut ule, B ñ A. The case fo stong unfom ntepolaton s smla to the ntepolaton case. In the followng theoem, we wll check the unfom ntepolaton popety fo a set of centeed axoms. It can also be consdeed as an example to show how ths noton woks n pactce. Theoem 4.3. Let G and H be two sequent calcul such that evey povable sequent n G s also povable n H and G conssts only of fnte centeed axoms. Then: pq If H extends FL e, then G has H-unfom ntepolaton. pq If H extends FL e and has the left weakenng ule, then G has weak H-unfom ntepolaton. pq If both G and H ae mult-concluson and H extends CFL e, then G has stong H-unfom ntepolaton. Poof. To pove pat pq of the theoem, we have to fnd IpSq and JpT q fo gven sequents S pσ ñ Λq and T pπ ñq such that the fou condtons n the Defnton 4.1 hold. We wll denote ou IpSq and JpT q and DpT, espectvely. Fst, we wll pove pq and we wll nvestgate the case DpT, fst. Fo that pupose, defne DpT as the followng p Π p q Js ^ 0 ^ K whee Π p s the subset of Π consstng of all p-fee fomulas and by Π p we mean ϕ 1 ϕ k, whee tϕ 1,, ϕ k u Π p. Note that J appeas n the fst conjunct only when Π Π p s non-empty. Moeove, 0 only appeas as a conjunct when T s of the fom axom 3 (whch s β ñ) and β Π, and K only appeas as a conjuncton when T s of the fom of axom 4 (whch s Σ, ϕ ñ Λ) and we have ϕ Ď Π. Fst, we have to show that Π ñ DpT holds n H. Note that Π s of the fom Π p Y pπ Π p q. By defnton, fo evey ψ P Π p we have ψ ñ ψ 16

19 and hence usng the ule pr q we have Π p ñ Π p holds n H (note that snce H extends FL e, t has the ule pr q). On the othe hand, usng the axom fo J we have Π Π p ñ J and then usng the ule pr q we have Π p, Π Π p ñ p Π p q J, whch s Π ñ p Π p q J. The fomula 0 appeas as a conjunct when T s of the fom axom 3 and β Π, whch means that n ths case Π ñ s an nstance of axom 3 and t holds n H. Hence, usng the ule pr0q we have Π ñ 0. The fomula K appeas as a conjunct when T s of the fom axom 4 and ϕ Ď Π. Hence, Π ñ K s an nstance of axom 4 when we let to be K. Now, we have to show that f fo p-fee sequents C and D f Π, C ñ D s povable n G, then DpT, C ñ D s povable n H. Theefoe, Π, C ñ D s of the fom of one of the centeed axoms and we have fve cases to consde: p1q If Π, C ñ D s of the fom of the axom ϕ ñ ϕ. Then, snce D ϕ, t means that ϕ s p-fee. Thee ae two cases; fst, f Π ϕ and C H, then Π p ϕ and snce Π Π p H, we do not have J n the conjunct. Hence, Π ñ ϕ and usng the ule pl^q we have DpT ñ D. Second, f Π H and C ϕ, then Π p 1 and snce Π Π p H, then J does not appea n the fst conjunct n the defnton of DpT. Hence, snce C ñ D s equal to ϕ ñ ϕ and ths s of the fom of the axom 1, usng the ule pl1q we have 1, ϕ ñ ϕ and usng pl^q we have DpT, C ñ D. p2q If Π, C ñ D s of the fom of the axom ñ ᾱ. Then, snce D ᾱ, t means that ᾱ s p-fee and Π C H. Hence, lke the above case Π p 1 and we do not have J n the defnton, ethe. Agan, usng the ule pl1q we have 1 ñ ᾱ and by pl^q we have DpT ñ ᾱ. p3q If Π, C ñ D s of the fom of the axom p β ñq. Then thee ae two cases; fst f β Π, then we must have 0 as one of the conjuncts n the defnton of DpT. We have C D H and 0 ñ s an axom n H and usng the ule pl^q we have DpT ñ. Second, f Π Ĺ β, snce we have β Π, C and C s p-fee, and we have ths condton that fo any two fomulas n β they have the same vaables, we have Π s p-fee, as well, whch means evey fomula n Π s p-fee and Π Π p and J does not appea n the defnton of DpT. Hence, usng the ule pl q on Π, C ñ, we have Π p, C ñ and by the ule pl^q we have DpT, C ñ. p4q If Π, C ñ D s of the fom of the axom Γ, ϕ ñ, then thee ae two cases; fst f ϕ Ď Π, then by defnton of DpT, K s one of the 17

20 conjuncts. Theefoe, snce K, C ñ D s an nstance of an axom n H, usng the ule pl^q we have DpT, C ñ D s devable n H. Second, f ϕ Ę Π, then at least one of the elements n ϕ s n C and hence t s p-fee. Theefoe, by the condton that fo any two fomulas n ϕ they have the same vaables, ϕ s p-fee. Hence, thee can not be any element of ϕ pesent n Π Π p and hence ϕ Ď Π p, C. Theefoe, we have Π p, C ñ D because t s of the fom of the axom Γ, ϕ ñ of G and hence t s povable n H. Theefoe, usng the axom pl q we have p Π p q J, C ñ D and by pl^q, DpT, C ñ D. (Note that t s possble that Π Π p s empty. It s easy to show that n ths case the clam also holds. It s enough to dop J n the last pat of the poof.) p5q Consde the case whee Π, C ñ D s of the fom of the axom Γ ñ ϕ,. Then, snce ϕ Ď D, we have DpT, C ñ D s an nstance of the same axom Γ ñ ϕ, when we substtute Γ by DpT, C. Now, we wll nvestgate the fo S of the fom Σ ñ Λ. as the followng p Σ p Ñ Kqs _ p β Σqs _ ϕ _ 1 _ J whee n the fst dsjunct, Σ p means the p-fee pat of Σ, the second dsjunct appeas wheneve thee exsts an nstance of an axom of the fom p3q n G whee Σ Ď β, Λ H and β s p-fee. The thd dsjunct appeas f Σ H and Λ ϕ whee ϕ s p-fee. The fouth dsjunct appeas f Σ ñ Λ equals to one of the nstances of the axom p1q, p2q, o p3q n G. And fnally, the ffth dsjunct appeas when ϕ Ď Σ fo an nstance of ϕ n axom p4q n G o ϕ Ď Λ fo an nstance of ϕ n axom p5q n G. Fst we have to show that ñ Λ. Fo ths pupose, we have to pove that fo any possble dsjunct X, we have Σ, X ñ Λ. Fo the fst dsjunct note that Σ p ñ Σ p and Σ Σ p, K ñ Λ. Hence, Σ, p Σ p Ñ Kq ñ Λ. Fo the second dsjucnt, we have Σ Ď β and Λ H. Theefoe Σ, p β Σq ñ Λ by the axom p3q tself. Fo the thd dsjunct, note that Σ H and Λ ϕ whee ϕ s p-fee. Hence Σ, ϕ ñ Λ by axom p1q. Fo the fouth dsjunct, note that Σ ñ Λ s an axom tself and hence Σ, 1 ñ Λ. Fnally, fo the ffth dsjunct, note that Σ ñ Λ s an nstance of the axoms p4q o p5q whch means f we also add J to the left hand-sde of the sequent, t emans povable. 18

21 Now we have to pove that f Σ, C ñ Λ then C Fo ths pupose, we wll check all possble axomatc foms fo Σ, C ñ Λ. p1q If Σ, C ñ Λ s an nstance of the axom p1q, thee ae two possble cases. Fst f Σ H and C ϕ and Λ ϕ. Then ϕ wll be p-fee and hence ϕ appeas as a dsjunct. Snce C ñ ϕ, we have C Fo the second case, f Σ ϕ and C H then Σ ñ Λ s an nstance of the axom p1q whch means that 1 s a dsjunct Snce pñ 1q and C H we have C p2q If Σ, C ñ Λ s an nstance of the axom p2q. Then Σ C H and Λ ᾱ. Theefoe, 1 s a dsjunct and snce C H we have C p3q If Σ, C ñ Λ s an nstance of the axom p3q. Then thee ae two cases to consde. Fst f Σ β. Then C H and Λ H. By defnton, 1 s a dsjunct and agan lke the pevous cases C Second f Σ Ĺ β. Then β X C s non-empty. Pck ψ P β X C. ψ s p-fee, snce any pa of the elements n β have the same vaables, β s p-fee. Now by defnton, p β Σq s a dsjunct Snce C β Σ, we have C p4q If Σ, C ñ Λ s an nstance of the axom p4q. Smla to the pevous case, thee ae two cases. If ϕ Ď Σ, then by defnton J s a dsjunct and thee s nothng to pove. In the second case, at least one the elements of ϕ s n C and hence p-fee. Snce any pa of the elements n ϕ have the same vaables, ϕ s p-fee. We can patton Σ, C to Σ p, C, pσ Σ p q. Snce evey element of pσ Σ p q has p, and ϕ s p-fee, the whole ϕ should belong to Σ p, C. Theefoe, by the axom p4q tself, Σ p, C ñ K whch mples C ñ p Σ p Ñ Kq. By defnton p Σ p q Ñ K s a dsjunct and hence C p5q If Σ, C ñ Λ s an nstance of the axom p5q. Then ϕ Ď Λ. By defnton J s a dsjunct and theefoe, thee s nothng to pove. Fo pq, note that usng the pat pq we have fomulas DpT fo any sequents S and T (T s H) wth the condtons of H-unfom ntepolaton. The condtons fo the weak H-unfom ntepolaton s the same except fo the second pat of the left weak p-ntepolant whch demands that f Σ, C ñ Λ, then Dp S, C If we use the same unfom ntepolants, we satsfy all the condtons of weak H-unfom ntepolaton. The eason s that except the mentoned condton, all of the othes ae the same as the 19

22 condtons fo H-ntepolaton and fo the othe condton, we can ague as follows: By Σ, C ñ Λ, we have C and by the left weakenng ule we wll have Dp S, C Fo pq, fst note that povng the exstence of the ght ntepolants s enough. It s suffcent to DpS and usng the assumpton that CFL e s admssble n H to educe the condtons to DpS. Now defne DpS fo any S pσ ñ Λq as: p Σ p q Js ^ pk ` p ă Λ p qqs ^ 0 ^ K whee by Σ p we mean ψ 1 ψ, whee tψ 1,, ψ u Π p and Ř Λ p s defned smlaly. Note that n p Σ p q Js the fomula J appeas ff Σ Σ p, and K appeas n the second conjunct ff Λ Λ p. The thd conjunct appeas f Σ ñ Λ s an nstance of an axom of the foms p1q, p2q and p3q n G and the fouth conjunct appeas f Σ ñ Λ s an nstance of an axom of the foms p4q, p5q n G. Fst, we have to show that Σ ñ DpS, Λ. Fo that pupose, we have to check that fo any conjunct X we have Σ ñ X, Λ. Fo the fst conjunct, f Σ Σ p then note that Σ p ñ Σ p and Σ Σ p ñ J, Λ theefoe Σ ñ p Σ p q Js, Λ If Σ Σ p, then thee s no need fo J and the clam s clea by Σ ñ Σ p. Fo the second conjunct, f Λ Λ p note that Ř Λ p ñ Λ p and Σ, K ñ Λ Λ p, hence Σ, K ` p ă Λ p qs ñ Λ hence Σ ñ pk ` p ă Λ p qqs, Λ If Λ Λ p, smla to the case befoe, thee s no need fo K. The cases fo the thd and the fouth conjuncts ae smla to the smla cases n the poof of pq. Now we want to pove that f Σ, C ñ Λ, D, then DpS, C ñ D. Fo ths pupose, we wll check all the cases one by one: p1q If Σ, C ñ Λ, D s an nstance of the axom p1q, we have fou cases to check. 20

23 If ϕ P C and ϕ P D, then Σ Λ H and C D ϕ. Hence Σ p 1. Theefoe, snce 1, C ñ D we have DpS, C ñ D. If ϕ P C and ϕ R D then Σ H and Λ ϕ. Theefoe, ϕ s p-fee and hence Λ p ϕ. Snce D H and Λ ϕ, we have, ϕ, C ñ D. Theefoe, p Ř Λ p q, C ñ D. If ϕ R C and ϕ P D. Ths case s smla to the pevous case. If ϕ R C and ϕ R D then Σ Λ ϕ and C D H. Hence, by defnton, we have 0 as a conjunct n DpS. Snce 0 ñ, we wll have DpS, C ñ D. p2q If Σ, C ñ Λ, D s an nstance of the axom p2q. Then Σ C H. Thee ae two cases to consde. If Λ ᾱ. Then by defnton 0 appeas n DpS. Snce D H and p0 ñq we have C, DpS ñ D. If Λ Ĺ ᾱ, then D Xᾱ s non empty. Theefoe, thee exsts a p-fee fomula n ᾱ. Snce the vaables of any pa n ᾱ ae equal, ᾱ s p-fee. Theefoe, Λ Ď ᾱ s p-fee, hence Λ Λ p (and K does not appea n the second conjunct). Snce pñ Λ, Dq, we have pñ Ř Λ, Dq theefoe p p Ř Λ p q ñ Dq whch mples pdps ñ Dq. p3q If Σ, C ñ Λ, D s an nstance of the axom p3q. Ths case s smla to the pevous case p2q. p4q If Σ, C ñ Λ, D s an nstance of the axom p4q. Thee ae two cases to consde. If ϕ Ď Σ. Then by defnton K s a conjunct n DpS and theefoe thee s nothng to pove. Fo the second case, f ϕ Ę Σ, then ϕ X C s non-empty. Hence, ϕ has a p-fee element. Snce the vaables of any pa n ϕ ae equal, ϕ s p-fee. Snce ϕ Ď Σ p, C, Σ Σ p and ϕ s p-fee, we should have ϕ Ď Σ p, C. Theefoe, f Σ Σ p, by the axom p4q tself, J, Σ p, C ñ D. Snce p Σ p q J s a conjunct n DpS, we wll have DpS, C ñ D. Note that f Σ Σ p, then we wll use Σ p, C ñ D nstead of J, Σ p, C ñ D. p5q If Σ, C ñ Λ, D s an nstance of the axom p5q. Ths case s smla to the pevous case The Sngle-concluson Case In ths secton, we assume that fo any sequent S Γ ñ we have ď 1. We wll show how the sngle-concluson sem-analytc and context-shang 21

24 sem-analytc ules peseve the unfom ntepolaton popety. Fo ths pupose, we wll nvestgate these two knds of ules sepaately. Fst we wll study the sem-analytc ules and then we wll show n the pesence of weakenng and context-shang mplcaton ules, we can also handle the context-shang sem-analytc ules Sem-analytc Case Let us begn ght away wth the followng theoem whch s one of the man theoems of ths pape. Theoem 4.4. Let G and H be two sequent calcul and H extends FL e. If H s a temnatng sequent calculus axomatcally extendng G wth only sem-analytc ules, then f G has H-unfom ntepolaton popety, then so does H. Poof. Fo any sequent U and V whee V s H and any atom p, we defne two p-fee fomulas, denoted and DpV and we wll pove that they meet the condtons fo the left and the ght p-ntepolants of U and V, espectvely. We defne them smultaneously and the defnton uses ecuson on the ank of sequents whch s specfed by the temnatng condton of the sequent calculus H. If V s the empty sequent we defne DpV as 1 and othewse, we defne DpV as the followng p ľ DpS q^p ľ ľ ľ js q q Ñ ł DpS 1 sq^pldpv 1 q^pd G pv q. j 1 pa LR s In the fst conjunct, the conjuncton s ove all non-tval pattons of V S 1 S n and anges ove the numbe of S s, n ths case 1 ď ď n. In the second conjunct, the fst bg conjuncton s ove all left sem-analytc ules that ae backwad applcable to V n H. Snce H s temnatng, thee ae fntely many of such ules. The pemses of the ule ae xxt js y s y j, xxs y y 1 and xs 1 y and the concluson s V, whee T js pπ j, ψ js ñ θ js q and S pγ, ϕ ñ q whch means that S s ae those who have context n the ght sde of the sequents ( ) n the pemses of the left sem-analytc ule. (Note that pckng the block xs 1 y s abtay and we nclude all conjuncts elated to any choce of xs 1 y.) The conjunct ldpv 1 appeas n the defnton wheneve V s of the fom plγ ñq and we consde V 1 to be pγ ñq. And fnally, snce G has the H-unfom ntepolaton popety, by defnton thee exsts JpV q as ght p-ntepolant of V. We choose one such JpV q and 22

25 denote t as D G pv and nclude t n the defnton. If U s the empty sequent as 0. Othewse, as the followng p ł p DpS 1 qq _ p ł ľ ľ js q qsq 1 j pa LR _p ł RRp qq _ pl@pu 1 q _ p@ G puq. In the fst dsjunct, the bg dsjuncton s ove all pattons of U S 1 S n such that fo each 1 we have S s H and S 1 U. (Note that n ths case, f S s H t may be possble that fo one 1 we have S U. Then the fst dsjunct of the defnton must be DpU But ths does not make any poblem, snce the defnton of DpU s po to the defnton In the second dsjunct, the bg dsjuncton s ove all left sem-analytc ules that ae backwad applcable to U n H. Snce H s temnatng, thee ae fntely many of such ules. The pemses of the ule ae xxt js y s y j and xxs y y and the concluson s U. In the thd dsjunct, the bg dsjuncton s ove all ght sem-analytc ules backwad applcable to U n H. The pemse of the ule s xxs y y and the concluson s U. The fouth dsjunct s on all sem-analytc modal ules wth the esult U and the pemse U 1. And fnally, snce G has the H-unfom ntepolaton popety, by defnton thee exsts IpUq as left p-ntepolant of U. We choose one such IpUq and denote t G pu and nclude t n the defnton. To pove the theoem we use nducton on the ode of the sequents and we pove both and DpV smultaneously. Fst we have to show that pq V pñ DpV q s devable n H. pq U p@pu ñq s devable n H. We show them usng nducton on the ode of the sequents U and V. When povng pq, we assume that pq holds fo sequents whose succedents ae empty and wth ode less than the ode of V and pq holds fo any sequent wth ode less than the ode of V. We have the same condton fo U when povng pq. To pove pq, note that f V s the empty sequent, then by defnton DpV 1 and hence pq holds. Fo the est, we have to show that V pñ Xq 23 s

26 s devable n H fo any X that s one of the conjuncts n the defnton of DpV. Then, usng the ule pr^q t follows that V pñ DpV q. Snce V s of the fom Γ ñ, we have to show Γ ñ X s devable n H. In the case that the conjunct s p Ź DpS q, we have to show that fo pa any non-tval patton S 1 S n of V we have Γ ñ DpS s devable n H. Snce the ode of each S s less than the ode of Ť V and S s pγ ñq fo 1 ď ď n whee n Γ Γ, we can use the nducton hypothess and we have Γ ñ DpS. Usng the ght ule fo p q we have Γ 1,, Γ n ñ DpS whch s Γ ñ DpS. Fo the second conjunct n the defnton of DpV, we have to check that fo evey left sem-analytc ule we have ľ ľ V pñ js q q Ñ ł DpS 1 sq. j 1 s s devable n H. Theefoe, V s the concluson of a left sem-analytc ule such that the pemses ae xxt js y s y j, xxs y y and xs 1 y and hence the ode of all of them ae less than the ode of V. We can easly see that the clam holds snce by nducton hypothess we can js to the left sde of the sequents T js and S fo 1. And agan by nducton hypothess we can add DpS 1 to the ght sde of the sequents S 1. Then usng the ules L^, L and R_ the clam follows. What we have sad so fa can be seen pecsely n the followng: 1 Note that xxt js y s y j s of the fom xxπ j, ψ js ñ θ js y s y j and xxs y y s of the fom xxγ, ϕ ñy y and V s of the fom Π 1,, Π m, Γ 1,, Γ n, ϕ ñ Usng nducton hypothess we have fo evey 1 ď j ď m pπ j1, ψ j1 ñ θ j1 q,, pπ js, ψ js ñ θ js q, fo evey 1 ă ď n we have and fo 1 we have 1, ϕ 1 ñq,, ϕ ñq, pγ 1, ϕ 11 ñ DpS 11 q,, pγ 1, ϕ 1 ñ DpS 1 q, 24

27 Hence, usng the ule pl^q, fo evey 1 ď j ď m we have pπ j, ľ js, ψ j1 ñ θ j1 q,, pπ j, ľ js, ψ js ñ θ js q, and fo evey 1 ă ď n we have pγ, ϕ 1 ñq, pγ, ϕ ñq, and usng the ule pr_q, fo 1 we have pγ 1, ϕ 11 ñ ł DpS 1 q,, pγ 1, ϕ 1 ñ ł DpS 1 q Substtutng all these thee n the ognal left sem-analytc ule (we can do ths, snce n the ognal ule, thee ae contexts, s n the ght hand sde of the sequents S 1 s), we conclude Π, Γ, ϕ, x ľ js y j, x y 1 ñ ł DpS 1. whee Π Π 1,, Π m, Γ Γ 1,, Γ n, x js y j 1s,, Ź s s s and x y 1 2,, n. Now, usng the ule pl q we have ľ ľ Π, Γ, ϕ, js q q ñ ł DpS 1. j 1 s And fnally, usng the ule R Ñ we conclude ľ ľ Π, Γ, ϕ ñ js q q Ñ ł j 1 s DpS 1 ms Consde the conjunct ldpt 1. In ths case, T must have been of the fom plγ ñq and T 1 of the fom pγ ñq. By defnton, the ode of T 1 s less than the ode of T. Hence, by nducton hypothess we have T 1 pñ DpT 1 q o n othe wods Γ ñ DpT 1. Now, we use the ule K and we have lγ ñ ldpt 1 whch means T pñ ldpt 1 q. The last case s D G pv. We have to show V pñ D G pv q s povable n H whch s the case snce G has H-unfom ntepolaton popety and by Defnton 4.1 pat pq thee exsts p-fee fomula J such that V pñ Jq s devable n H. We chose one such J and call t D G pv, hence V pñ D G pv q n H by defnton. 25

28 To pove pq, note that f U s the empty sequent, then by 0 and hence pq holds. Fo the est, we have to show that U px ñq s devable n H fo any X that s one of the dsjuncts n the defnton Then, usng the ule pl_q t follows that U p@pu ñq. Snce U s of the fom Γ ñ, we have to show Γ, X ñ s devable n H. In the case that the dsjunct s p pap Ž DpS 1 qq we have to pove 1 that fo any pattons of U S 1 S n such that S s H fo each 1 and S 1 U, we have U pp 1 DpS 1 q ñq. Fst, consde the case that non of S s ae equal to U (o n othe wods, S s H); then the ode of each S s less than the ode of S and we can use the nducton hypothess. Snce fo 1 the succedent of each S s empty, we have S pγ ñq and pγ ñ DpS q and usng the ule R we have pγ 2,, Γ n ñ 1 Hence usng the ule L Ñ we conclude DpS q. And fo S 1 Γ 1 ñ we have Γ 1 ñ. Γ 1,, Γ n, DpS 1 ñ 1 and the clam follows. In the case that U s H, t s possble that fo 1, one of S s s equal to U. In ths case what appeas n the defnton s DpU 1 whch s equvalent to DpU Ñ 0. But, we can do ths, snce we defned DpU po to the defnton and we have poved U pñ DpUq po to the case that we ae checkng now. In the case that the dsjunct s p LRp js q qsq, we have j s to pove that fo any left sem-analytc ule that s backwad applcable to U n H we have U pp j Ź s js q p Ź q ñq. The pemses of the ule ae xxt js y s y j and xxs y y and the concluson s U. Snce the odes of all T js s and S s ae less than the ode of U we can use the nducton hypothess and have T js p@pt js ñq and S p@ps ñq. Usng the ule pl^q fo context shang sequents (when j s fxed and s fxed we have context shang sequents) and then usng the ule pl q fo non context shang sequents (when s and ae fxed and we ae angng ove j and ) and then applyng the same left ule we can pove the clam. The poof s smla to the second case of pq and pecsely t goes as the followng: Usng nducton hypothess we have fo evey 1 ď j ď m pπ j1, ψ j1 ñ θ j1 q,, pπ js, ψ js ñ θ js q, 26

29 and fo evey 1 ď ď n we have 1, ϕ 1 ñ q,, ϕ ñ q, Hence, usng the ule pl^q, fo evey 1 ď j ď m we have pπ j, ľ js, ψ j1 ñ θ j1 q,, pπ j, ľ js, ψ js ñ θ js q, and fo evey 1 ď ď n we have pγ, ϕ 1 ñ q,, pγ, ϕ ñ q, Substtutng these two n the ognal left sem-analytc ule, we conclude Π, Γ, ϕ, x js y j, x y ñ, s and usng the ule pl q we have ľ ľ Π, Γ, ϕ, js q q ñ. j s In the case that the dsjunt s p RRp qq, we have to pove that fo any ght sem-analytc ule backwad applcable to U n H, we have U p Ź ñq. In ths case the pemses of the ule ae xxs y y, whee S pγ, ϕ ñ ψ q and the concluson s U pγ 1,, Γ n ñ ϕq. Snce the ode of each S s less than the ode of S, we can use the nducton hypothess and fo evey 1 ď ď n we have 1, ϕ 1 ñ ψ 1 q,, ϕ ñ ψ q, Usng the ule L^ we have pγ, ϕ 1 ñ ψ 1 q,, pγ, ϕ ñ ψ q, and substtutng t n the ognal ght ule, we conclude Γ, x y ñ ϕ, and usng the ule pl q we have ľ ñ ϕ. 27

30 Fo the case that the dsjunct s l@pu 1 we have that U s the concluson of a sem-analytc modal ule and the pemse s U 1. Hence, U s of the fom plγ ñ l q and U 1 s of the fom pγ ñ q. Snce the ode of U 1 s less than the ode of U, we can use the nducton hypothess and we have 1 ñ q. Now, usng the ule K we can conclude plγ, l@pu 1 ñ l q whch s equvalent to U pl@pu 1 ñq. And fnally, fo the case that the dsjunct G pu we have to show that U p@ G pu ñq holds n H, whch does snce G has H-unfom ntepolaton popety and by Defnton 4.1 pat pq thee exsts p-fee fomula I such that U pi ñq s devable n H. We choose one such I and call G pu and hence we have U p@ G pu ñq n H by defnton. So fa we have poved pq and pq. We want to show that H has H- unfom ntepolaton. Theefoe, based on the Defnton 4.1, we have to pove the followng, as well: pq Fo any p-fee multsets C and D, f V p C ñ Dq s devable G then DpV, C ñ D s devable n H, whee C C 1,, C k and D ď 1. pvq Fo any p-fee multset C, f U p C ñq s devable n G then C s devable n H, whee C C 1,, C k. Recall that V s of the fom pγ ñq and U s of the fom pγ ñ q. We wll pove pq and pvq smultaneously usng nducton on the length of the poof and nducton on the ode of U and V. Moe pecsely, fst by nducton on the ode of U and V and then nsde t, by nducton on n, we wll show: Fo any p-fee multsets C and D, f V p C ñ Dq has a poof n G wth length less than o equal to n, then DpV, C ñ D s devable n H. Fo any p-fee multset C, f U p C ñq has a poof n G wth length less than o equal to n, then C s devable n H. Whee by the length we mean countng just the new ules that H adds to G. Fst note that fo the empty sequent and fo pq, we have to show that f C ñ D s vald n G, then C, 1 ñ D s vald n H, whch s tval by the ule pl1q. Smlaly, fo pvq, f C ñ s vald n G, then C ñ 0 s vald n H, whch s tval by the ule pr0q. Fo the base of the othe nducton, note that f n 0, fo pq t means that Γ, C ñ D s vald n G. By Defnton 4.1 pat pvq, D G pv, C ñ D and 28

31 hence DpV, C ñ D s povable n H. Fo pvq, t means that Γ, C ñ s vald n G. Theefoe, agan by Defnton 4.1, C G pu and hence C s povable n H. Fo n 0, to pove pq, we have to consde the followng cases: The case that the last ule used n the poof of V p C ñ Dq s a left sem-analytc ule and ϕ P C (whch means that the man fomula of the ule, ϕ, s one of C s). Theefoe, V p C ñ Dq pπ, Γ, X, Ȳ, ϕ ñ q s the concluson of a left sem-analytc ule and V s of the fom pπ, Γ ñq and C p X, Ȳ, ϕq and we want to pove pdpv, X, Ȳ, ϕ ñ q. Hence, we must have had the followng nstance of the ule xxπ j, X j, ψ js ñ θ js y s y j xxγ, Ȳ, ϕ ñ y y Π, Γ, X, Ȳ, ϕ ñ whee Ť j Π j Π, Ť Γ Γ, Ť j X j X, Ť Ȳ Ȳ and Ť. Consde T js pπ j ñq and S pγ ñq. Snce T js s do not depend on the suffx s, we have T j1 T js and we denote t by T j. And, snce S s do not depend on, we have S 1 S and we denote t by S. Theefoe, T 1,, T m, S 1,, S n s a patton of V. Fst, consde the case that t s a non-tval patton. Then the ode of all of them ae less than the ode of V and snce the ule s sem-analytc and ϕ s p-fee then ψ js, θ js and ϕ ae also p-fee. Hence, we can use the nducton hypothess to get: DpT j, ψ js, X j ñ θ js, DpS, ϕ, Ȳ ñ If we let tdpt j, X j u and tdps, Ȳu be the contexts n the ognal left sem-analytc ule, we have the followng xxdpt j, ψ js, X j ñ θ js y s y j xxdps, ϕ, Ȳ ñ y y DpT 1,, DpT m, DpS 1,, DpS n, X, Ȳ, ϕ ñ Usng the ule pl q we have p j DpT j q p DpS q, X, Ȳ, ϕ ñ. Theefoe usng the ule pl^q, we have pdpv, C ñ Dq. 29

32 If T 1,, T m, S 1,, S n s a tval patton of V, t means that one of them equals V and all the othes ae empty sequents. W.l.o.g. suppose T 1 V pσ ñq and the othes ae empty. Then we must have had the followng nstance of the ule: xxσ, ψ js, X j ñ θ js y s y j xx ϕ, Ȳ ñ y y Σ, X, Ȳ, ϕ ñ Theefoe, V p ψ js, X j ñ θ js q fo evey j and s ae pemses of V p C ñ Dq, and hence the length of the tees ae smalle than the length of the poof tee of V p C ñ Dq, and snce the ule s sem-analytc and ϕ s p-fee then ψ js and θ js ae also p-fee. Hence, fo all of them we can use the nducton hypothess (nducton on the length of the poof), and we have DpV, ψ js, X j ñ θ js. Substtutng tdpv, X j u, t X j u, tȳu and t u as the contexts of the pemses n the ognal left ule we have whch s pdpv, C ñ Dq. xxdpv, ψ js, X j ñ θ js y s y j xx ϕ, Ȳ ñ y y DpV, X, Ȳ, ϕ ñ Consde the case whee the last ule used n the poof of V p C ñ Dq s a left sem-analytc ule and ϕ R C. Theefoe, V p C ñ Dq pπ, Γ, X, Ȳ, ϕ ñ q s the concluson of a left sem-analytc ule and V s of the fom pπ, Γ, ϕ ñq and C p X, Ȳ q and we want to pove pdpv, X, Ȳ ñ q. Hence, we must have had the followng nstance of the ule xxπ j, X j, ψ js ñ θ js y s y j xxγ, Ȳ, ϕ ñy y 1 xγ 1, Ȳ1, ϕ 1 ñ y p:q Π, Γ, X, Ȳ, ϕ ñ whee Ť j Π j Π, Ť Γ Γ, Ť j X j X and Ť Ȳ Ȳ. Snce, Xj s and Ȳ s ae n the context postons n the ognal ule, we can consde the same substton of meta-sequents and meta-fomulas as above n the ognal ule, except that we do not take X j s and Ȳ s as contexts. Moe pecsely, we each the followng nstance of the ognal ule: 30