Cut Elimination for a Logic with Generic Judgments and Induction

Cut Elimination for a Logic with Generic Judgmen ts and Induction Alw en Tiu Computer Sciences Lab oratory The Australian National Universit y Abstract. This paper presents a cu t -elimi nation p roof for the logic LG ω , whic h is an extension of a proof system for enco ding generic judgments, the logic F Oλ ∆ ∇ of Miller and Tiu, with an induction principle. The logic LG ω , just as F Oλ ∆ ∇ , features ex tensions of fi rst-order intuitionistic logic with fixed points and a “generic quantifier”, ∇ , which is used to reason ab out the dy namics of b indings in ob ject systems encod ed in the logic. A previous attemp t to extend F Oλ ∆ ∇ with an in d uction principle has b een un success ful in mod eling some b eha viours of bind ings in ind uctiv e specifications. It turns out that this p roblem can b e solved by relaxing some restrictions on ∇ , in particular by adding the axiom B ≡ ∇ x.B , where x is not free in B . W e show that by adopting the equiva riance principle, the presentatio n of the extended logic can b e muc h simplified. This p aper con tains the tec hnical proofs for the results stated in [14]; readers are encouraged t o consult [14] for motiv ations and examples for LG ω . 1 In tro duction This w ork aims at providing a framework for r easoning ab out sp ecifications o f deductive systems us ing higher- or der abstr act syntax [10]. Higher-orde r abstract syntax is a decla rativ e approach to enco ding syn tax with bindings using Ch urch’s simply typed λ -calculus. T he main idea is to suppor t the notions of α -equiv alence and substitutions in the ob ject syntax by op erations in λ -calculus, in particular α -con version and β - reduction. There ar e at lea st tw o approa c hes to higher -order abstract syntax. The funct io nal pr o gr amming appr oac h enco des the ob ject sy n tax as a data type, where the binding cons tr ucts in the ob ject langua ge are ma pped to functions in the functional la nguage. In this a pproach , terms in the ob ject language b ecome v alues of their cor responding types in the functional lang uage. The pr o o f se ar ch appro ac h enco des ob ject syntax as expressions in a log ic whose terms are s imply typed, and functions that act on the ob ject terms a re defined via relations , i.e., logic progra ms. There is a subtle difference b et ween this approach and the former; in the pro of s earc h approa ch, the s imple type s are inhabited by well-formed expressions , instead o f v alues a s in the functional approach (i.e., the abstra ction type is inhabited by functions). The pro of search a ppr oac h is often referred to as λ -tr e e syntax [7], to distinguish it fro m the functional a pproach . This paper concerns the λ -tree syn tax approa c h. Spec ific a tions which use λ -tr e e syntax are often for malized using hypothetical a nd generic judgments in int uitionistic lo gic. It is eno ugh to restrict to the fragment of first-orde r in tuitionistic log ic whose o nly formulas are those of hereditar y Harrop formulas, which w e will refer to as the H H logic. Consider for instanc e the problem of defining the data type for un typed λ -terms. One first in tro duces the fo llo wing constants: app : tm → tm → tm abs : ( tm → tm ) → tm where the t yp e tm denotes the syn tactic c ategory of λ -terms and ap p and a b s enco de application and abstraction, resp ectiv ely . The pr operty of b eing a λ -term is then defined via the following theory: ^ M ^ N ( lam M ∧ l am N ⇒ l am ( app M N )) & ^ M (( ^ x.lam x ⇒ l am ( M x )) ⇒ l am ( abs M )) where V is the universal quantifier and ⇒ is implication. Reasoning ab out ob ject sys tems e ncoded in H H is r educed to reasoning ab out the structure of pro ofs in H H . McDow ell and Miller formalize this kind of reasoning in the logic F Oλ ∆ I N [3], which is an extension of first-order intuitionistic logic with fixed p oin ts and natural num b ers induction. This is done b y enco ding the sequent calculus of H H inside F O λ ∆ I N and prov e prop erties abo ut it. W e r efer to H H as ob ject logic and F Oλ ∆ I N as meta logic. McDow ell and Miller c o nsidered differen t styles of enco dings and concluded that explicit representations of hypotheses and, more impo rtan tly , eig en v ar iables of the ob ject logic are required in order to capture some statemen ts about ob ject logic prov abilit y in the meta logic [4]. One t ypical example inv olves the use of hypo thetical a nd generic reasoning a s follows: Supp ose that the following for mula is prov able in H H . ^ x.p x s ⇒ ^ y .p y t ⇒ p x t. By ins pection on the inference rules of H H , o ne observes that this is only possible if s and t are syn tactically equal. This o bserv ation comes from the fact that the righ t introduction rule for universal quan tifier, re ading the rule bo ttom-up, introduce s new constants, o r eigenv ariables. The q ua n tified v ariables x and y will b e replaced by distinct e ig en v a r iables and henc e the o nly ma tching hypothesis for p x t would b e p x s , and therefore s a nd t has to be equal. L e t ⊢ H H F denote the prov a bilit y of the formula F in H H . Then in the meta logic, we would wan t to b e a ble to prove the sta temen t: ∀ s ∀ t. ( ⊢ H H ^ x.p x s ⇒ ^ y .p y t ⇒ p x t ) ⊃ s = t. The question is then how we would intrepret the ob ject logic eigenv ar iables in the meta logic. It is demon- strated in [4] that the existing quantifiers in F Oλ ∆ I N cannot b e used to ca pture the b eha viour s of ob ject logic eigenv ariable s directly . McDow ell and Miller then resort to a no n-logical enco ding technique (in the sense that no logica l connectiv es are used) which has so me similar flav or to the use of deBruijn indices. The use of this enco ding technique, howev er, has a consequence that substitutions in the o b ject lo gic ha s to b e formalized explicitly . Motiv a ted b y the above men tioned limitation of F O λ ∆ I N , Miller and Tiu later intro duce a new quan tifier ∇ to F O λ ∆ I N which allows one to move the binders from the ob ject lo gic to the meta logic. A gener ic judgment in the o b ject logic, for instance ⊢ H H V x.G x is reflected in the meta logic as ∇ x. ⊢ H H G x. This meta logic, called F Oλ ∆ ∇ [8], allows one to perfor m case analy ses o n the prov a bilit y of the ob ject logic . Tiu later extended F Oλ ∆ ∇ with induction and co-induction rules , resulting in the logic Linc [13]. How ever, some inductive prop erties about the ob ject logic a r e not pro v able in Linc. F or exa mple, the fact that ⊢ H H V x.G x implies ∀ t. ⊢ H H G t (that is , the extensional prop ert y of universal quan tification) is not prov able in Linc. As it is shown in [13], this is partly caused b y the fact that B ≡ ∇ x.B , where x is not fre e in B , is not prov a ble in Linc or F Oλ ∆ ∇ . In this pap er we present the logic LG ω , which is a n extension of F O λ ∆ ∇ with natur al nu mber induction and with the axiom sc hemes: ∇ x ∇ y .B x y ⊃ ∇ y ∇ x.B x y and B ≡ ∇ x.B (1) where x is not free in B in the second scheme. W e s ho w tha t inductiv e properties of λ -tree syntax s pecifications can be stated direc tly and in a purely logical fa s hion, and proved in L G ω . R elation t o nominal lo gic In formulating the pro of system for L G ω , it turns out that we can simplify the presentation a lo t if we a dopt the idea of e quivariant pr e dic ates fro m nominal lo gic [11]. That is, prov ability of a predicate is inv ariant under p erm utations of names . This is technically done b y introducing a countably infinite set of name co ns tan ts into the logic, and change the iden tity r ule of the logic to allow e q uiv alence under permutations of name cons tan ts: π .B = π ′ .B ′ Γ, B − B ′ id where π and π ′ are p erm utations on names. LG ω is in fact very close to no minal logic, when we consider only the b eha viour s of lo gical connectives. In particular, the quantifier ∇ in LG ω shares the same prop erties, in relatio n to other connec tiv es of the log ic, with the N quantifier in nominal logic . Howev er, there are tw o 2 impo rtan t differences in our approach. First, we do not a ttempt to redefine α -conv ersion and substitutions in LG ω in terms of pe r m utations (or swapping ) and the notion of fr eshness as in nominal logic. Name swapping and freshness c o nstrain ts a re no t part o f the s yn tax of LG ω . These notions a re prese n t only in the meta theory o f the logic. In L G ω , for exa mple, v a riables are alwa ys consider e d to hav e empt y supp ort, that is, π .x = x for every p ermutation π . This is beca use we restrict substitutions to the “closed” o nes, in the sense that no name co nstan ts can app ear in the substitutions. A r estricted for m of op en substitutions can b e recov ered indirec tly at the meta theory of LG ω . The fact that v ariables ha ve empty supp ort a llo ws one to work with p erm utation free formulas and terms. So in LG ω , we can prove that p x a ⊃ p x b , where a a nd b are names, without using explicit ax ioms o f permutations and freshness . In nominal logic, one would prove this by using the swapping axiom p x a ⊃ p (( a b ) .x ) (( a b ) .b ), wher e ( a b ) de no tes a swapping of a and b , and then show that ( a b ) .x = x . The latter mig h t not b e v a lid if x is substituted by a , for example. The v alidity o f this formula in nominal logic w ould therefor e depend on the as sumption on the suppor t of x . The s e cond difference between LG ω and nominal logic is that LG ω allows close d terms (a gain, in the sense that no name constants a ppear in them) o f t yp e name, while in nominal logic, allowing such terms would lead to an inconsistent theory in nominal logic [1 1]. As an example, the t yp e tm in the enco ding of λ -terms mentioned previously can be tre a ted as a nominal t yp e in LG ω . This has an imp ortant co ns equence that we do not need to redefine the no tio n of substitutions for the enco ded λ -terms. F or example, we ca n define the (lazy) ev aluation relation on un typed λ -terms as the theory: ev al ( abs M ) ( abs M ) ≡ ⊤ ev al ( app M N ) V ≡ ev al M ( abs P ) ∧ e val ( P N ) V without ha ving to e xplicitly define substitutions on terms o f t yp e tm inside LG ω . Substitutions in the ob ject language in this case is mo delled b y β -reduction in the meta-language of L G ω . Outline of t he p ap er Section 2 introduces the lo g ic LG , whic h is an extens ion of fir st order in tuitionistic logic with a no tion of name p erm utation and the ∇ -quantifier. LG serves as the core logic for a more express iv e logic, LG ω , which is obtained b y a dding rules for fixed points, equality a nd induction to LG. Sec tion 3 examines sev era l proper ties of deriv ations, in particula r, those that concern preserv ation of prov ability under several o perations on sequents, e.g ., substitutions. Section 4 defines the cut r eduction, used in the cut- elimination pro of. The cut elimination pro of itself is an adaptation of the cut-elimination pro of of F O λ ∆ I N by McDow ell and Miller [3], which makes use o f the reducibility technique. Section 5 defines the nor malizabilit y and the reducibilit y rela tions whic h ar e crucial to the cut elimination proo f in Section 6. Finally , in Section 7, we show that the pro of system LG is actually equiv alent to F O λ ∆ ∇ (without fixed p oin ts a nd equality) with non-logica l rules corresp onding to the axioms g iv en in (1) a b ov e. This paper co n tains the technical pro ofs for the r esults stated in [1 4]; rea der s are encourage d to consult [14] for motiv ations and examples for LG ω . 2 A logic for generic judgmen t s W e first define the co re fr agmen t of the logic LG ω which do es no t hav e fixed p oint r ules or induction. The starting p oin t is the logic F O λ ∇ int ro duced in [8]. F Oλ ∇ is an extension of a subset of Ch urch’s Simple Theory of Types in which for mulas are given the t yp e o . The co re fr a gmen t of LG ω , which w e refer to as LG , shares the same set o f connectives as F O λ ∇ , na mely , ⊥ , ⊤ , ∧ , ∨ , ⊃ , ∀ τ , ∃ τ and ∇ τ . The type τ in the quantifiers is restricted to that which do es no t cont ain the t yp e o. Hence the logic is essentially fir s t-order. W e a bbreviate ( B ⊃ C ) ∧ ( C ⊃ B ) as B ≡ C. The s equen ts of F O λ ∆ ∇ are expressions of the for m Σ ; σ 1 ⊲ B 1 , . . . , σ n ⊲ B n − σ 0 ⊲ B 0 where Σ is a signature, i.e., a set o f eigen v ariables scop ed ov er the s equen t and σ i is a loca l signature, i.e., lis t of v ariables loca lly scop ed ov er B i . The introduction rules for ∇ , r eading the rules b ottom-up, intro duce new 3 lo cal v ar iables to the lo cal signatures, just as the r igh t in tro duction rule o f ∀ int ro duces new e ig en v a r iables to the signatur e. The expres sion σ i ⊲ B i is called a lo cal judgment, a nd is identified up to r e naming of v ariables in σ i . This enforces a limited notion o f equiv ar iance: for example a ⊲ pa − b ⊲ pb is prov able, since b oth loca l judgment s are equiv alent up to renaming o f lo cal signatures. How ever, the judgments ( a, c ) ⊲ p a and b ⊲ p b are considered distinct judgments, a nd so are ( a, b ) ⊲ q a b and ( b, a ) ⊲ q a b . These restrictions are relaxed in LG. The sequent presen tation of LG ca n b e simplified, that is, without using the lo cal signatures, if we emplo y the equiv arianc e pr inciple. F or this purpos e, w e intro duce a distinguished set of ba se t yp es, called n omi nal typ es , which is denoted with N . Nominal types are ranged o ver by ι . W e r estrict the ∇ quantifier to nominal t yp es. F or eac h nominal t yp e ι ∈ N , we a s sume an infinite n um b er of constan ts o f that type. These consta n ts are ca lled nominal c onstants . W e denote the family of nominal constants b y C N . The r ole of the nominal constants is to enforce the notion o f equiv aria nce: prov ability of formulas is in v ariant under p ermutations of nominal constans. Dep e nding on the applica tio n, we might a lso ass ume a set of non-nominal constants, which is denoted by K . W e assume the usual notion o f c a pture-a voiding s ubstitut ions. Substitutions are ranged ov er by θ a nd ρ . Applicatio n of substitutions is written in a postfix notation, e.g., tθ is an applicatio n of θ to the term t . Given t wo substitutions θ and θ ′ , we denote their co mposition by θ ◦ θ ′ which is defined as t ( θ ◦ θ ′ ) = ( tθ ) θ ′ . A signatur e is a set of v a riables. A substitution θ r espects a given signa ture Σ if there exists a set o f typed v ar iables Σ ′ such that fo r ev ery x : τ in the domain of θ , it holds that K ∪ Σ ′ ⊢ θ ( x ) : τ . W e denote by Σ θ the minimal set of v a riables satisfying the ab ov e condition. W e as sume that v ariables, free or b ound, a re of a differen t syntactic categ ory from constants. Definition 1. A p ermutation on C N is a bije ction fr om C N to C N . The p ermutations on C N ar e r ange d over by π . Applic ation of a p ermu tation π to a nominal c onst ant a is denote d with π ( a ) . We s hal l b e c onc erne d only with p ermutations which r esp e ct typ es, i.e., for every a : ι , π ( a ) : ι. F urther, we shal l also r estrict to p ermu tations which ar e finite, that is, the set { a | π ( a ) 6 = a } is fi nite. A pp lic ation of a p ermutation to an arbitr ary term (or formula), written π .t , is define d as fol lows: π .a = π ( a ) , if a ∈ C N . π .c = c, if c 6∈ C N . π .x = x π . ( M N ) = ( π .M ) ( π .N ) π . ( λx .M ) = λx. ( π.M ) A p ermutation involving only two nominal c onst ants is c al le d swapping . W e use ( a b ) , wher e a and b ar e c ons t ants of t he same typ e, to denote the swapping { a 7→ b, b 7→ a } . The su pp ort of a term (or for m ula) t , written supp ( t ), is the set of nominal constants app e aring in it. It is clear from the ab o ve definition that if su pp ( t ) is e mpty , then π.t = t for all π . The definition of Σ -substitution implies tha t for every θ and for every x ∈ dom ( θ ), θ ( x ) has empt y suppo rt. Therefor e Σ -substitutions a nd per m utations comm ute, that is, ( π.t ) θ = π . ( tθ ) . A sequent in L G ω is an expressio n of the form Σ ; Γ − C where Σ is a signa ture. The free v aria bles of Γ and C are among the v a riables in Σ . The inference rules for the core fra gmen t of L G ω , i.e., the logic LG , is given in Figure 1. In the rules, the typing judgmen t Σ , K , C N ⊢ t : τ denotes the typabilit y of t : τ , given the t yping context Σ ∪ K ∪ C N in Church’s simple type system. In the ∇L and ∇R rules, a denotes a nominal co nstan t. In the ∃L and ∀R rules, we use r aising [6] to enco de the dep endency of the quant ified v a riable o n the supp ort of B , since we do not allo w Σ -subs titu tions to mention any nominal co nstan ts. In the rules, the v aria ble h has its type ra ised in the following wa y: suppo se ~ c is the list c 1 : ι 1 , . . . , c n : ι n and the qua n tified v aria ble x is of type τ . Then the v ariable h is of t yp e: ι 1 → ι 2 → . . . → ι n → τ . This raising tech nique is similar to that of F O λ ∆ ∇ , and is us e d to enco de explicitly the minimal supp ort of the quantified v ariable. Its us e prevents one fro m mixing the scop es of ∀ (dually , ∃ ) a nd ∇ . That is, it pr ev ents the formula ∀ x ∇ y .p x y ≡ ∇ y ∀ x.p x y , and its dual, to b e prov ed. Lo oking at the introduction rules for ∀ and ∃ , one might notice the asymmetry b et w een the left and the right intro duction r ules. The left r ule fo r ∀ a llo ws ins tan tiations with terms containing any nominal constants while the rais e d v ariable in the right introductio n rule of ∀ ta k es in to ac coun t only those which are in the supp ort o f the quan tified for m ula. Ho wev er, we will see that we can extend the dep endency of the 4 raised v aria ble to a n arbitrar y num b er of fresh nominal consta n ts not in the supp ort without affecting the prov ability of the sequent (see Lemma 17 a nd Lemma 18). π .B = π ′ .B ′ Σ ; Γ , B − B ′ id π Σ ; ∆ 1 − B 1 · · · Σ ; ∆ n − B n Σ ; B 1 , . . . , B n , Γ − C Σ ; ∆ 1 , . . . , ∆ n , Γ − C mc Σ ; Γ , B , B − C Σ ; Γ , B − C c L Σ ; Γ , ⊥ − C ⊥L Σ ; Γ − ⊤ ⊤R Σ ; Γ , B i − C Σ ; Γ , B 1 ∧ B 2 − C ∧L , i ∈ { 1 , 2 } Σ ; Γ − B Σ ; Γ − C Σ ; Γ − B ∧ C ∧R Σ ; Γ , B − C Σ ; Γ, D − C Σ ; Γ , B ∨ D − C ∨L Σ ; Γ − B i Σ ; Γ − B 1 ∨ B 2 ∨R , i ∈ { 1 , 2 } Σ ; Γ − B Σ ; Γ , D − C Σ ; Γ , B ⊃ D − C ⊃ L Σ ; Γ , B − C Σ ; Γ − B ⊃ C ⊃ R Σ , K , C N ⊢ t : τ Σ ; Γ , B [ t/x ] − C Σ ; Γ , ∀ τ x.B − C ∀L Σ , h ; Γ − B [ h ~ c/x ] Σ ; Γ − ∀ x.B ∀R , h 6∈ Σ , supp ( B ) = { ~ c } Σ ; Γ , B [ a/x ] − C Σ ; Γ , ∇ x.B − C ∇L , a 6∈ supp ( B ) Σ ; Γ − B [ a/x ] Σ ; Γ − ∇ x .B ∇R , a 6∈ supp ( B ) Σ , h ; Γ , B [ h ~ c/x ] − C Σ ; Γ , ∃ x.B − C ∃L , h 6∈ Σ , sup p ( B ) = { ~ c } Σ , K , C N ⊢ t : τ Σ ; Γ − B [ t/x ] Σ ; Γ − ∃ τ x.B ∃R Fig. 1. The inference rules of LG W e now extend the logic LG with a pro of theoretic notion of equality and fixed p oints, following on w ork s by Hallnas and Schro eder-Heister [2,12], Girard [1] a nd McDow ell and Miller [3]. The equa lit y r ules are as follows: { Σ θ ; Γ θ − C θ | ( λ ~ c.t ) θ = β η ( λ ~ c.s ) θ } Σ ; Γ , s = t − C eq L Σ ; Γ − t = t eq R where supp ( s = t ) = { ~ c } in the eq L rule. In the eq L rule, the substitution θ is a u nifier of λ ~ c.s and λ ~ c .t . W e sp ecify the premise of the r ule as a set to mean tha t ev ery element of the set is a premise. Since the terms s and t ca n b e ar bitrary higher -order terms, in genera l the set of their unifiers can b e infinite. How ever, in some restricted ca s es, e.g., when λ ~ c.s and λ ~ c.t are higher-or der p attern terms [5,9], if b oth terms are unifiable, then ther e exists a most gener al unifier. The applications we are considering ar e those which sa tisfy the higher-or der pattern restrictions. Definition 2. T o e ach atomic formula, we asso ciate a fi x e d p oint e quation, or a de finitio n clause , fol lowing the terminolo gy of F O λ ∆ ∇ . A definition clause is written ∀ ~ x.p ~ x △ = B wher e the fr e e variabl es of B ar e among ~ x. The pr e dic ate p ~ x is c al le d the head of the definition clause, and B is c al le d the b o dy . A definition is a set of definition clauses. We often omit the outer quantifiers when r eferring to a definition clause. The intro duction rules for defined atoms are as follo ws: Σ ; Γ , B [ ~ t/~ x ] − C Σ ; Γ , p ~ t − C def L , p ~ x △ = B Σ ; Γ − B [ ~ t/~ x ] Σ ; Γ − p ~ t def R , p ~ x △ = B In order to pro ve the cut-elimination theorem and the consistency of LG ω , w e allow only definition clauses which satisfy a n e quivarianc e pr eserving condition and a certain positivity co ndition, so as to guarantee the existence of fixed p oints. Definition 3. We asso ciate with e ach pr e dic ate s ymb ol p a natur al numb er, the level of p . Given a formula B , its level lv l ( B ) is define d as fol lows: 5 1. l v l ( p ¯ t ) = lv l ( p ) 2. l v l ( ⊥ ) = l vl ( ⊤ ) = 0 3. l v l ( B ∧ C ) = l v l ( B ∨ C ) = max( l v l ( B ) , lv l ( C )) 4. l v l ( B ⊃ C ) = max( l vl ( B ) + 1 , l vl ( C )) 5. l v l ( ∀ x.B ) = l v l ( ∇ x.B ) = lv l ( ∃ x.B ) = lv l ( B ) . A definition clause p ~ x △ = B is str atifie d if lv l ( B ) ≤ l v l ( p ) and B has no fr e e o c curr en c es of nominal c onstants. We c onsider only definition clauses which ar e str atifie d. An example that violates the first restr iction in Definition 3 is the definition p △ = p ⊃ ⊥ . In [12], Schroeder - Heister shows that admitting this definition in a logic with contraction leads to inco ns istency . T o see why we need the second restriction on name constants, consider the definition q x △ = ( x = a ) , wher e a is a nomina l constant. Let b be a nominal constant different fr om a . Using this definition, w e w ould b e able to derive ⊥ : − a = a eq R − q a def R q a − q b id π b = a − ⊥ eq L q b − ⊥ def L q a − ⊥ cut − ⊥ cut In ex amples and applications, w e often express definition clauses with patterns in the heads. Let us consider, for e x ample, a definition clause for lists. W e first introduce a type l st to denote lists of elements of t yp e α , and the consta nts nil : l st :: : α → l st → lst which deno te the empt y list and a constructor to build a list fro m an elemen t of t yp e α a nd another list. The latter will be written in the infix notation. The definitio n clause for lists is as follo ws. l ist L △ = L = ni l ∨ ∃ α A ∃ lst L ′ .L = ( A :: L ′ ) ∧ list L ′ . Using patterns, the above definition of lists ca n b e re written as l ist nil △ = ⊤ . l ist ( A :: L ) △ = l ist L . W e shall often work direc tly with this patter ne d no tation for de finitio n clauses. F or this purp ose, we int ro duce the no tion of p att erne d definitions . A p atterne d definition clause is written ∀ ~ x.H △ = B wher e the free v aria bles of H and B are among ~ x. The stratification of definitions in Definition 3 applies to patterned definitions as well. Since the patter ned definition cla uses a re not allowed to hav e free o ccurrence s of nominal constants, in matching the hea ds of the cla us es with an atomic formula in a sequent, we need to r a ise the v ar iables of the clauses to account for nominal c o nstan ts that are in the supp ort of the intro duced formula. Given a patterned definition cla use ∀ x 1 . . . ∀ x n .H △ = B its raised clause with resp ect to the list of constant s c 1 : ι 1 . . . c n : ι n is ∀ h 1 . . . ∀ h n .H [ h 1 ~ c/x 1 , . . . , h n ~ c/x n ] △ = B [ h 1 ~ c/x 1 , . . . , h n ~ c/x n ] . The in tro duction rules fo r patterned definitions are { Σ θ ; B θ, Γ θ − C θ } θ Σ ; A, Γ − C def L Σ ; Γ − B θ Σ ; Γ − A def R In the def L rule, B is the b ody of the rais e d patterned clause ∀ x 1 . . . ∀ x n .H △ = B and ( λ ~ c.H ) θ = ( λ ~ c .A ) θ where { ~ c } is the support of A. In the def R r ule, we matc h A with the head of the clause, i.e., λ ~ c.A = ( λ ~ c.H ) θ . These patterned rules can b e derived using the non-patterned definition rules and the equalit y rules, as shown in [13], 6 Natur al n umb er induction. W e intro duce a type nt to denote natural num b ers, with the us ual constants z : nt (zero ) and s : nt → nt (the successor function), and a sp ecial predicate nat : nt → nt → o. The rules for natural n umber induction are the same as those in F O λ ∆ I N [3], whic h are the int ro duction rules fo r the predicate nat . − D z j ; D j − D ( s j ) Σ ; Γ, D I − C Σ ; Γ , nat I − C nat L Σ ; Γ − nat z nat R Σ ; Γ − nat I Σ ; Γ − nat ( s I ) nat R The lo gic LG extended with the equality , definitio ns a nd induction rules is referred to as LG ω . 3 Prop erties of deriv at ion s In this section we examine several pro perties of the ∇ -quantifier and deriv a tions in L G ω that are useful in the cut elimination pr oof. The s e prop erties concer n the tr ansformation of deriv ations , in pa rticular, they state that prov a bilit y is pre serv ed under Σ -substitutions, p ermutations and a restricted for m of name substitutions. W e firs t lo ok a t the pro perties of the ∇ quantifier in relatio n to o ther connectives. The pr o of of the following prop osition is s traigh tforward b y insp ection on the rules of LG. Prop osition 4. The fol lowing formulas ar e pr ovable in LG : 1. ∇ x. ( B x ∧ C x ) ≡ ∇ x.B x ∧ ∇ x.C x. 2. ∇ x. ( B x ⊃ C x ) ≡ ∇ x.B x ⊃ ∇ x.C x. 3. ∇ x. ( B x ∨ C x ) ≡ ∇ x.B x ∨ ∇ x.C x. 4. ∇ x.B ≡ B , pr ovide d that x is not fr e e in B . 5. ∇ x ∇ y .B xy ≡ ∇ y ∇ x.B xy . 6. ∀ x.B x ⊃ ∇ x.B x. 7. ∇ x.B x ⊃ ∃ x.B x. The formulas (1) – (3) are prov a ble in F O λ ∇ . The pr oposition is true also in nominal lo gic with ∇ repla ced by N . Definition 5. Given a derivation Π with pr emise derivations { Π i } i ∈I wher e I is some index set , the me a- sur e ht ( Π ) , the height of Π , is define d as t he le ast upp er b ound of { ht ( Π i ) + 1 } i ∈I . W e now define some transfor mations of deriv a tions: weak ening of h yp otheses, substitutions on deriv ations , per m utations and re stricted na me subs titu tions. In the following definitions we omit the signatur e s in the sequents if it is clear from context which sig natures we refer to. W e deno te with id the identit y function on C N . Definition 6. W ea k ening of hypotheses . L et Π b e a derivatio n of Σ ; Γ − C . L et ∆ b e a multiset of formulas whose fr e e variables ar e among Σ . We define the derivation w ( ∆, Π ) of Σ ; Γ , ∆ − C as fol lows: 1. I f Π ends with eq L  Π θ Σ θ ; Γ ′ θ − C θ  θ Σ ; s = t, Γ ′ − C eq L then w ( ∆, Π ) is  w ( ∆θ , Π θ ) Σ θ ; Γ ′ θ, ∆θ − C θ  θ Σ ; s = t, Γ ′ , ∆ − C eq L 7 2. I f Π ends with nat L Π 1 − D z Π 2 D i − D ( s i ) Π 3 D I , Γ ′ − C nat I , Γ ′ − C nat L then w ( ∆, Π ) is Π 1 − D z Π 2 D i − D ( s i ) w ( ∆, Π 3 ) D I , Γ ′ , ∆ − C nat I , Γ ′ , ∆ − C nat L 3. I f Π ends with the mc rule Π 1 ∆ 1 − B 1 . . . Π n ∆ n − B n Π ′ B 1 , . . . , B n , Γ ′ − C ∆ 1 , . . . , ∆ n , Γ ′ − C mc then w ( ∆, Π ) is Π 1 ∆ 1 − B 1 . . . Π n ∆ n − B n w ( ∆, Π ′ ) B 1 , . . . , B n , Γ ′ , ∆ − C ∆ 1 , . . . , ∆ n , Γ ′ , ∆ − C mc 4. I f Π ends with any other rule and has pr emise derivations Π 1 , . . . , Π n then w ( ∆, Π ) ends with the same rule with pr emise derivations w ( ∆, Π n ) , . . . , w ( ∆, Π n ) . Definition 7. Substitutions on deriv ations. If Π is a derivation of Σ ; Γ − C and θ is a Σ -substitution, then we define t he derivation Π θ of Σ θ ; Γ θ − C θ as fol lows: 1. S upp ose Π ends with eq L :  Π ρ Σ ρ ; Γ ′ ρ − C ρ  ρ Σ ; s = t, Γ ′ − C eq L wher e e ach ρ is a unifier of λ ~ c.s and λ ~ c.t . Observe that if ρ ′ is a unifier of ( λ ~ c .s ) θ and ( λ ~ c .t ) θ , t he n θ ◦ ρ ′ is a unifier of λ ~ c .s and λ ~ c.t . Thus Π θ is the derivation:  Π θ ◦ ρ ′ Σ θ ρ ′ ; ∆θ ρ − C θ ρ  ρ ′ Σ ; sθ = tθ , ∆θ − C θ eq L 2. S upp ose Π ends with ∀R : Π 1 Σ ; Γ − B [ h ~ c/ x ] Σ ; Γ − ∀ x.B ∀R , wher e { ~ c } = supp ( ∀ x.B ) . L et { ~ d } b e the supp ort of ( ∀ x.B ) θ , which might b e smal ler than { ~ c } . L et ρ b e the substitution [ λ ~ c.h ′ ~ d/h ] wher e h ′ is a new variable n ot alr e ady in Σ and not among the fr e e variables in θ . We c an assume without loss of gener ality that x is not fr e e in θ , henc e (( B [ h ~ c/x ]) ρ ) θ = ( B [ h ′ ~ d/x ]) θ = ( B θ )[ h ′ ~ d/x ] . Then Π θ is Π 1 ( ρ ◦ θ ) Σ θ , h ′ ; Γ θ − ( B θ )[ h ′ ~ d/x ] Σ θ ; Γ θ − ( ∀ x.B ) θ ∀R 3. S upp ose Π ends with ∃L : this c ase is du al to t he pr evious one. 4. I f Π ends with any other rule and has pr emise derivations Π 1 , . . . , Π n , then Π θ ends with t he same rule and has pr emise derivations Π 1 θ , . . . , Π n θ. 8 Definition 8. L et Π b e a pr o of of Σ ; B 1 , . . . , B n − B 0 and let ~ π = π 0 , . . . , π n b e a list of p ermutations. We define a derivation h ~ π i .Π of Σ ; π 1 .B 1 , . . . , π n .B n − π 0 .B 0 as fol lows: 1. S upp ose t ha t Π ends with id π π .B j = π ′ .B 0 Σ ; B 1 , . . . , B n − B 0 id π . Obverse that π .π − 1 j .π j .B = π ′ .π − 1 0 .π 0 .B ′ . H en c e h ~ π i .Π ends with the same rule. 2. S upp ose Π ends with mc : Π 1 ∆ 1 − D 1 . . . Π m ∆ m − D m Π ′ D 1 , . . . , D m , ∆ m +1 − B 0 B 1 , . . . , B n − B 0 mc wher e ∆ 1 , . . . , ∆ m +1 ar e p artitions of B 1 , . . . , B n . Supp ose t hat for e ach i ∈ { 1 , . . . , m + 1 } , ∆ i = B i 1 , . . . , B ik i for some index k i . L et ~ π ( i ) , for i ∈ { 1 , . . . , m } , b e the p ermutations id, π i 1 , . . . , π ik i . L et ~ π ( m + 1) b e the p ermutations π 0 , i d, . . . , id | {z } m , π ( m +1)1 , . . . π ( m +1) k m +1 We denote with ∆ ′ i the list π i 1 .B ij , . . . , π ik i .B ik i . Then h ~ π i .Π is the derivation h ~ π (1) i .Π 1 ∆ ′ 1 − D 1 . . . h ~ π ( m ) i .Π m ∆ ′ m − D m h ~ π ( m + 1 ) i .Π ′ D 1 , . . . , D m , ∆ ′ m +1 − π 0 .B 0 π 1 .B 1 , . . . , π n .B n − π 0 .B 0 mc 3. S upp ose Π ends with ∇R : Π 1 Σ ; B 1 , . . . , B n − B [ a/x ] Σ ; B 1 , . . . , B n − ∇ ι x.B ∇R wher e a : ι 6∈ supp ( B ) . L et d : ι b e a nominal c onstant such that d 6∈ sup p ( B ) and π 0 ( d ) = d . Such a c ons t ant exists sinc e supp ( B ) is finite and π 0 is a finite p ermutation. Th us π 0 . ( a d ) .B 0 [ a/x ] = π 0 .B 0 [ d/x ] . Then h ~ π i .Π is the derivation: h π 0 . ( a d ) , . . . , π n i .Π 1 Σ ; π 1 .B 1 , . . . , π n .B n − π 0 .B [ d/x ] Σ ; π 1 .B 1 , . . . , π n .B n − π 0 . ( ∇ x.B ) ∇R 4. S upp ose Π ends with ∇L : this c ase is analo gous to pr evious one. 5. S upp ose Π ends with c L : Π ′ B 1 , . . . , B j , B j . . . , B n − B 0 B 1 , . . . , B j , . . . , B n − B 0 c L then h ~ π i .Π is h π 1 , . . . , π j , π j , . . . , π n i .Π ′ π 1 .B 1 , . . . , π j .B j , π j .B j . . . , π n .B n − π 0 .B 0 π 1 .B 1 , . . . , π j .B j , . . . , π n .B n − π 0 .B 0 c L 6. I f Π ends with any other ru le and has pr emise derivations Π 1 , . . . , Π m , then h ~ π i .Π ends with the same rule and has pr emise deriva tions h ~ π i .Π 1 , . . . , h ~ π i .Π m . 9 Definition 9. L et Π b e a pr o of of Σ , x : ι ; B 1 , . . . , B n − B 0 and let ~ a = a 0 , . . . , a n b e a list of nominal c on- stants such that a i 6∈ supp ( B i ) . We define a derivation r ( x, h ~ a i , Π ) of Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] , as fol lows: 1. S upp ose Π is π .B j = π ′ .B 0 Σ , x ; B 1 , . . . , B n − B 0 id π . L et d : ι b e a nominal c onstant which is not in the supp ort of B j and B 0 , and π ( d ) = d and π ′ ( d ) = d . Then r ( x, ~ a, Π ) is π . ( a j d ) .B 1 [ a 1 /x ] = π ′ . ( a 0 d ) .B 0 [ a 0 /x ] Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] id π 2. S upp ose Π ends with mc : Π 1 Σ , x ; ∆ 1 − D 1 . . . Π m Σ , x ; ∆ m − D m Π ′ Σ , x ; D 1 , . . . , D m , ∆ m +1 − B 0 Σ , x ; B 1 , . . . , B n − B 0 mc wher e ∆ 1 , . . . , ∆ m +1 is a p art ition of B 1 , . . . , B n . Supp ose t hat for e ach i ∈ { 1 , . . . , m + 1 } , ∆ i = B i 1 , . . . , B ik i for some index k i . L et ~ d = d 1 , . . . , d m b e a list of nominal c onstants such that d i 6∈ sup p ( D i ) . L et f ( i ) , for i ∈ { 1 , . . . , m } b e the list d i , a i 1 , . . . , a ik i and let f ( m + 1) b e the list a 0 , ~ d, a ( m +1)1 , . . . , a ( m +1) k ( m +1) . L et ∆ ′ i b e the list B i 1 [ a i 1 /x ] , . . . , B ik i [ a ik i /x ] and let Γ b e the list D 1 [ d 1 /x ] , . . . , D m [ d m /x ] , ∆ ′ m +1 . Then r ( x, ~ a, Π ) is the derivation r ( x, f (1) , Π 1 ) Σ ; ∆ ′ 1 − D 1 [ d 1 /x ] . . . r ( x, f ( m ) , Π m ) Σ ; ∆ ′ m − D m [ a m /x ] r ( x, f ( m + 1) , Π ′ ) Σ ; Γ − B 0 [ a 0 /x ] Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] mc 3. S upp ose Π is Π 1 Σ , x ; B 1 , . . . , B n − B [ c/y ] Σ , x ; B 1 , . . . , B n − ∇ y .B ∇R . If a 0 6 = c then r ( x, ~ a , Π ) is r ( x, ~ a , Π 1 ) Σ , x ; B 1 , . . . , B n − B [ c/y ] Σ , x ; B 1 , . . . , B n − ∇ y .B ∇R . If a 0 = c , then we swap c with a fr esh c onstant. L et d : ι b e a nominal c onstant not in t he supp ort of B [ c/y ] . We apply the swapping ( c d ) to the c onclusion of the end s e quent of Π 1 ac c or ding to the c ons t ruction in Definition 8 to get a pr o of Π 2 of Σ , x ; B 1 , . . . , B n − B 0 [ d/y ] . The derivation r ( x, ~ a , Π ) is c onstructe d as fol lows: r ( x, ~ a , Π 2 ) Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B [ a 0 /x, d/y ] Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − ∇ y .B [ a 0 /x ] ∇R 4. I f Π ends with ∇L apply the same c onstruction as in the pr evious c ase. 10 5. S upp ose Π ends with ∀R Π 1 Σ , x, h ; B 1 , . . . , B n − B [ h ~ c/y ] Σ , x ; B 1 , . . . , B n − ∀ y .B ∀R . L et θ = [ λ ~ c .h ′ ~ cx/h ] wher e h ′ is a variable not in Σ . Apply the c onstruction in Definition 7 to get the pr o of Π θ of Σ , x, h ′ ; B 1 , . . . , B n − B [ h ′ ~ ax/y ] Then r ( x, ~ a, Π ) is r ( x, ~ a , Π θ ) Σ , h ′ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B [ a 0 /x, ( h ′ ~ ca 0 ) /y ] Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − ∀ y .B [ a 0 /x ] ∀R . 6. I f Π ends with ∃L , apply the same c onstruction as in the pr evious c ase. 7. S upp ose Π ends with ∃R : Π 1 Σ , x ; B 1 , . . . , B n − B [ t/y ] Σ , x ; B 1 , . . . , B n − ∃ y .B ∃R . If a 0 6∈ sup p ( B [ t/y ]) then r ( x, ~ a , Π ) is r ( x, ~ a , Π 1 ) Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B [ a 0 /x, t/ y ] Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − ∃ y .B [ a 0 /x ] ∃R . If a 0 ∈ supp ( B [ t/y ] , we exchange it with a fr esh c onstant. L et d b e a nominal c onst ant distinct fr om a 0 and not in the supp ort of B [ t/y ] . Then (( a 0 d ) .B [ t/y ])[ a 0 /x ] = B [( a 0 d ) .t/y , a 0 /x ] . We first apply the c ons t ruction in Definition 8 to Π 1 to get a derivation Π 2 of Σ , x ; B 1 , . . . , B n − B [( a 0 d ) .t/y , a 0 /x ] . The derivation r ( x, ~ a, Π ) is thus r ( x, ~ a , Π 2 ) Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B [( a 0 d ) .t/y , a 0 /x ] Σ ; B 1 [ a 1 /x ] , . . . , B [ a n /x ] − ∃ y .B [ a 0 /x ] ∃R . 8. S upp ose Π ends with eq L :  Π θ ( Σ , x ) θ ; B 2 θ, . . . , B n θ − B 0 θ  θ Σ , x ; s = t, B 2 , . . . , B n − B 0 eq L wher e e ach θ is a unifier of ( λ ~ c.s, λ ~ c .t ) and { ~ c } = su pp ( s = t ) . We n e e d to show t hat for e ach unifier of ( λa 1 λ ~ c.s [ a 1 /x ] , λa 1 λ ~ c.t [ a 1 /x ]) ther e is a c orr esp onding unifier for λ ~ c.s and λ ~ c.t. We c an assume without loss of gener ality that x is not in the domain of ρ . We first show the c ase wher e x is not fr e e in ρ . It is cle ar that in t his c ase ρ is a unifier of λ ~ c.s and λ ~ c.t . Ther efor e we apply t he pr o c e dur e r e cursively to t he pr emise derivation Π ρ , to get the derivation r ( x, ~ a , Π ρ ) of Σ ρ ; ( B 2 [ a 2 /x ]) ρ, . . . , ( B n [ a n /x ]) ρ − ( B 0 [ a 0 /x ]) ρ. In the other c ase, wher e x is fr e e in the ra nge of ρ , we show that it c an b e r e duc e d to the pr evious c ase. First we define a substitut ion ρ ′ to b e the substitut io n ρ wher e x is r eplac e d by a n ew variable u which is not fr e e in ρ . Cle arly ρ ′ is also a u nifier of λa 1 λ ~ c .s [ a 1 /x ] and λa 1 λ ~ c.t [ a 1 /x ] . Mor e over, it is mor e gener al than ρ , sinc e ρ = [ x/u ] ◦ ρ ′ . Ther efor e we c an apply the c onstruction in the pr evious c ase to get a derivation r ( x, ~ a , Π ρ ′ ) and apply t he substitu tion [ x/u ] t o to this derivatio n, using the pr o c e dur e in Definition 7, to get a derivation of Σ ρ ; ( B 2 [ a 2 /x ]) ρ, . . . , ( B n [ a n /x ]) ρ − ( B 0 [ a 0 /x ]) ρ. 11 The derivation r ( x, ~ a , Π ) is then c onstru cte d as fol lows  Π ′ ρ Σ ρ ; ( B 2 [ a 2 /x ]) ρ, . . . , ( B n [ a n /x ]) ρ − ( B 0 [ a 0 /x ]) ρ  ρ Σ ; s [ a 1 /x ] = t [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] eq L wher e e ach Π ′ ρ is c onstructe d as explaine d ab ove. 9. I f Π ends with c L : Π ′ B 1 , . . . , B j , B j , . . . , B n − B 0 B 1 , . . . , B j , . . . , B n − B 0 c L then r ( x, ~ a, Π ) is r ( x, ( a 0 , . . . , a j , a j , . . . , a n ) , Π ′ ) B 1 [ a 1 /x ] , . . . , B j [ a j /x ] , B j [ a j /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] B 1 [ a 1 /x ] , . . . , B j [ a j /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] c L 10. If Π ends with any other r u le and has pr emise deriva tions Π 1 , . . . , Π n , then r ( x, ~ a , Π ) ends with t he same rule and has pr emise derivations r ( x, ~ a , Π 1 ) , . . . , r ( x, ~ a , Π n ) . Lemma 10. F or any derivation Π of Σ ; Γ − C and any multiset of Σ -formulas ∆ , w ( ∆, Π ) is a derivation of Σ ; Γ, ∆ − C and ht ( w ( ∆, Π )) ≤ ht ( Π ) . Lemma 11. F or any derivatio n Π of Σ ; Γ − C and any Σ -substitut io n θ , Π θ is a derivation of Σ θ ; Γ θ − C θ and ht ( Π θ ) ≤ ht ( Π ) . Lemma 12. F or any derivation Π of B 1 , . . . , B n − B 0 and p ermutations ~ π = π 0 , . . . , π n , h ~ π i .Π is a deriva- tion of π 1 .B 1 , . . . , π n .B n − π 0 .B 0 and ht ( h ~ π i .Π ) ≤ ht ( Π ) . Lemma 13. F or any derivation Π of Σ , x ; B 1 , . . . , B n − B 0 and any list of nominal c onstants ~ a = a 0 , . . . , a n such that a i 6∈ supp ( B i ) , r ( x, ~ a , Π ) is a derivation of Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] and ht ( r ( x, ~ a, Π )) ≤ ht ( Π ) . Lemma 14. Substitutions. L et Π b e a pr o of of Σ ; Γ − C and let θ b e a Σ -substitut io n. Then ther e exists a pr o of Π ′ of Σ θ ; Γ θ − C θ su ch that ht ( Π ′ ) ≤ ht ( Π ) . Pr o of. F o llo ws immediately fro m Lemma 11. ⊓ ⊔ Lemma 15. Perm utations. L et Π b e a pr o of of Σ ; B 1 , . . . , B n − B 0 . Then ther e exists a pr o of Π ′ of Σ ; π 1 .B 1 , . . . , π n .B n − π 0 .B 0 such t ha t ht ( Π ′ ) ≤ ht ( Π ) . Pr o of. F o llo ws immediately fro m Lemma 12. ⊓ ⊔ Lemma 16. Restricted name substitutions. L et Π b e a pr o of of Σ , x : ι ; B 1 , . . . , B n − B 0 . Then ther e exists a pr o of of Π ′ of Σ ; B 1 [ a 1 /x ] , . . . , B n [ a n /x ] − B 0 [ a 0 /x ] , wher e a i 6∈ supp ( B i ) for e ach i ∈ { 0 , . . . , n } , such that ht ( Π ′ ) ≤ ht ( Π ) . Pr o of. F o llo ws immediately fro m Lemma 13. ⊓ ⊔ The next t wo lemmas ar e crucia l to the cut-elimination pro of: they allow one to reintro duce the symmetr y betw een ∀L and ∀R , and dually , b et w een ∃L and ∃R rules. 12 Lemma 17. Suppor t extension. L et Π b e a pr o of of Σ , h ; Γ − B [ h ~ a/x ] wher e { ~ a } = supp ( B ) , h 6∈ Σ and h is not fr e e in Γ and B . L et ~ c b e a list of nominal c onstants not in t he su pp ort of B . Then ther e exists a pr o of Π ′ of Σ , h ′ ; Γ − B [ h ′ ~ a ~ c/x ] wher e h ′ 6∈ Σ . Pr o of. Suppo se ~ c is the list of constant s c 1 : ι 1 , . . . , c n : ι n . Le t ~ y = y 1 : ι 1 , . . . , y n : ι n be a list of distinct v ariables not app earing in Σ ∪ { h, h ′ } . W e first apply the substitution [ λ ~ a.h ′ ~ c~ x/h ] to the sequent Σ , h ; Γ − B [ h ~ a/ x ] . By Lemma 14, there is a pro of Π 1 of Σ , h ′ , ~ y ; Γ − B [ h ′ ~ a~ y /x ] The deriv a tion Π ′ is then obtained b y rep eatedly a pplying Lemma 16 to Π 1 to change ~ y into ~ c . ⊓ ⊔ Lemma 18. Suppor t extension. L et Π b e a pr o of of Σ , h ; B [ h ~ a/x ] , Γ − C wher e { ~ a } = supp ( B ) , h 6∈ Σ and h is not fr e e in Γ , B and C . L et ~ c b e a list of nominal c onstants not in the supp ort of B . Then ther e exists a pr o of Π ′ of Σ , h ′ ; B [ h ′ ~ a ~ c/x ] , Γ − C wher e h ′ 6∈ Σ . Pr o of. Use the same construction a s in the pro of o f Lemma 17. ⊓ ⊔ 4 Cut reduction W e define a r e duction rela tion betw een deriv a tions, following closely the reduction relation in [3]. F or simplicity of presentation, we shall omit the s ignatures in the seq uen ts in the following reduction o f cuts when the s ignatures are not changed by the reduction or whe n it is clear from context which s ig natures should be ass igned to the sequent s. The redex is alw ays a deriv ation Ξ ending with the multicut rule Π 1 Σ ; ∆ 1 − B 1 · · · Π n Σ ; ∆ n − B n Π Σ ; B 1 , . . . , B n , Γ − C Σ ; ∆ 1 , . . . , ∆ n , Γ − C mc . W e refer to the formulas B 1 , . . . , B n pro duced by the mc as cut formulas . If n = 0, Ξ r educes to the premise deriv ation Π . F or n > 0 we sp ecify the reduction relation based on the last rule of the premise deriv ations. If the rightmost pr emise deriv a tion Π ends with a left rule a cting on a cut formula B i , then the las t rule of Π i and the last r ule of Π tog ether determine the reduction rules that apply . W e classify these r ules according to the following criteria : we call the rule a n essential case when Π i ends with a r igh t rule; if it ends with a left rule, it is a left-c ommutative case; if Π i ends with the id rule, then we have an axiom case; a multicut case arises when it ends with the mc rule. When Π do es not end with a left rule ac ting on a cut formula, then its last rule is alone sufficient to determine the re ductio n rules that apply . If Π ends in a rule a cting on a formula other than a cut for m ula, then we call this a right-c ommu tative case. A st ru ctur al case results when Π ends with a contraction or weakening on a cut formula. If Π ends with the id rule, this is also a n axiom case; similarly a multicut case arises if Π ends in the mc rule. F or simplicity of pres en tation, we always show i = 1. Essential c ases: ∧R / ∧ L : If Π 1 and Π are Π ′ 1 ∆ 1 − B ′ 1 Π ′′ 1 ∆ 1 − B ′′ 1 ∆ 1 − B ′ 1 ∧ B ′′ 1 ∧R Π ′ B ′ 1 , B 2 , . . . , B n , Γ − C B ′ 1 ∧ B ′′ 1 , B 2 , . . . , B n , Γ − C ∧L , then Ξ reduces to Π ′ 1 ∆ 1 − B ′ 1 Π 2 ∆ 2 − B 2 · · · Π n ∆ n − B n Π ′ B ′ 1 , B 2 , . . . , B n , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc . 13 The case for the other ∧L rule is symmetric. ∨R / ∨ L : If Π 1 and Π are Π ′ 1 ∆ 1 − B ′ 1 ∆ 1 − B ′ 1 ∨ B ′′ 1 ∨R Π ′ B ′ 1 , B 2 , . . . , B n , Γ − C Π ′′ B ′′ 1 , B 2 , . . . , B n , Γ − C B ′ 1 ∨ B ′′ 1 , B 2 , . . . , B n , Γ − C ∨L , then Ξ reduces to Π ′ 1 ∆ 1 − B ′ 1 Π 2 ∆ 2 − B 2 · · · Π n ∆ n − B n Π ′ B ′ 1 , B 2 , . . . , B n , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc . The case for the other ∨R rule is symmetric. ⊃ R / ⊃ L : Supp ose Π 1 and Π are Π ′ 1 B ′ 1 , ∆ 1 − B ′′ 1 ∆ 1 − B ′ 1 ⊃ B ′′ 1 ⊃ R Π ′ B 2 , . . . , B n , Γ − B ′ 1 Π ′′ B ′′ 1 , B 2 , . . . , B n , Γ − C B ′ 1 ⊃ B ′′ 1 , B 2 , . . . , B n , Γ − C ⊃ L . Let Ξ 1 be  Π i ∆ i − B i  i ∈{ 2 ..n } Π ′ B 2 , . . . , B n , Γ − B ′ 1 ∆ 2 , . . . , ∆ n , Γ − B ′ 1 mc Π ′ 1 B ′ 1 , ∆ 1 − B ′′ 1 ∆ 1 , . . . , ∆ n , Γ − B ′′ 1 mc . Then Ξ r educes to Ξ 1 . . . − B ′′ 1  Π i ∆ i − B i  i ∈{ 2 ..n } Π ′′ B ′′ 1 , { B i } i ∈{ 2 ..n } , Γ − C ∆ 1 , . . . , ∆ n , Γ , ∆ 2 , . . . , ∆ n , Γ − C mc c L ∆ 1 , . . . , ∆ n , Γ − C . W e use the double horizontal lines to indicate that the relev an t inference rule (in this case, c L ) may need to be a pplied zer o or more times. ∀R / ∀L : Suppose Π 1 and Π are Π ′ 1 Σ , h ; ∆ 1 − B ′ 1 [( h ~ c ) /x ] Σ ; ∆ 1 − ∀ x.B ′ 1 ∀R Π ′ Σ ; B ′ 1 [ t/x ] , B 2 , . . . , B n , Γ − C Σ ; ∀ x.B ′ 1 , B 2 , . . . , B n , Γ − C ∀L , where { ~ c } = supp ( B ′ 1 ) . Let { ~ d } = supp ( B ′ 1 [ t/x ]) \ supp ( B ′ 1 ) . Apply Lemma 17 to get a deriv ation Π ′′ 1 of Σ , h ′ ; ∆ 1 − B ′ 1 [( h ~ c ~ d ) /x ] . The deriv atio n Ξ reduces to Π ′′ 1 [ λ ~ c ~ d .t/h ′ ] Σ ; ∆ 1 − B ′ 1 [ t/x ]  Π i Σ ; ∆ i − B i  i ∈{ 2 ..n } Π ′ . . . − C Σ ; ∆ 1 , . . . , ∆ n , Γ − C mc . 14 ∃R / ∃L : Suppose Π 1 and Π are Π ′ 1 Σ ; ∆ 1 − B ′ 1 [ t/x ] Σ ; ∆ 1 − ∃ x.B ′ 1 ∃R Π ′ Σ , h ; B ′ 1 [( h ~ c ) /x ] , B 2 , . . . , B n , Γ − C Σ ; ∃ x.B ′ 1 , B 2 , . . . , B n , Γ − C ∃L , where { ~ c } = su pp ( B ′ 1 ) . Let { ~ d } = supp ( B ′ 1 [ t/x ]) \ supp ( B ′ 1 ) . Apply Lemma 18 to Π ′ to get a deriv ation Π ′′ of Σ , h ′ ; ∆ 1 − B ′ 1 [( h ′ ~ c ~ d ) /x ] . Then Ξ re duce s to Π ′ 1 Σ ; ∆ 1 − B ′ 1 [ t/x ] . . . Π ′′ [ λ ~ c ~ d.t/h ′ ] Σ ; B ′ 1 [ t/x ] , B 2 , . . . , Γ − C Σ ; ∆ 1 , . . . , ∆ n , Γ − C mc . ∇R / ∇L : Suppose Π 1 and Π are Π ′ 1 ∆ 1 − B ′ 1 [ a/x ] ∆ 1 − ∇ x.B ′ 1 ∇R Π ′ B ′ 1 [ b/x ] , . . . , B n , Γ − C ∇ x.B ′ 1 , . . . , B n , Γ − C ∇L . Apply the co ns truction in Definition 8 to to Π ′ 1 to swap a with b to get a deriv ation Π ′′ 1 of ∆ 1 − B ′ 1 [ b/x ] . Ξ reduces to Π ′′ 1 ∆ 1 − B ′ 1 [ b/x ] . . . Π ′ B ′ 1 [ b/x ] , . . . , B n , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc . nat R /nat L : Supp ose Π 1 is ∆ 1 − n at z nat R and Π is Π ′ − D z Π ′′ D j − D ( s j ) Π ′′′ D z , B 2 , . . . , B n , Γ − C nat z , B 2 , . . . , B n , Γ − C nat L . Then Ξ r educes to w ( ∆ 1 , Π ′ ) ∆ 1 − D z  Π i ∆ i − B i  i ∈{ 2 ...n } Π ′′′ D z , B 2 , . . . , B n , Γ − C ∆ 1 , ∆ 2 , . . . , ∆ n , Γ − C mc nat R /nat L : Supp ose Π 1 is Π ′ 1 ∆ − nat I ∆ 1 − n at ( s I ) nat R and Π is Π ′ − D z Π ′′ D j − D ( s j ) Π ′′′ D ( s I ) , B 2 , . . . , B n , Γ − C nat ( s I ) , B 2 , . . . , B n , Γ − C nat L Let Ξ 1 be Π ′ 1 ∆ 1 − n at I Π ′ − D z Π ′′ D j − D ( s j ) D I − D I id π nat I − D I nat L ∆ 1 − D I mc. 15 Suppo se { ~ c } = supp ( I ) . W e apply the pro cedures in Definition 7 and Definition 9 to Π ′′ to obtain the deriv ation Π • of h ; D ( h ~ c ) − D ( s ( h ~ c )) . Let Ξ 2 be Ξ 1 ∆ 1 − D I Π • [ λ ~ c.I / h ] D I − D ( s I ) ∆ 1 − D ( s I ) mc. Then Ξ r educes to Ξ 2 ∆ 1 − D ( s I ) Π 2 ∆ 2 − B 2 . . . Π n ∆ n − B n Π ′′′ D ( s I ) , B 2 , . . . , B n , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc. eq L / eq R : If Π 1 and Π are Σ ; ∆ 1 − t = t eq R  Π θ Σ θ ; Γ θ − C θ  θ Σ ; t = t, Γ − C eq L then Ξ reduces to Π 2 Σ ; ∆ 2 − B 2 . . . Π n Σ ; ∆ n − B n w ( ∆ 1 , Π ǫ ) Σ ; ∆ 1 , B 2 , . . . , B n , Γ − C Σ ; ∆ 1 , . . . , ∆ n , Γ − C mc where ǫ is the empty substitution. def R / def L : Supp ose Π 1 and Π are Π ′ 1 ∆ 1 − B [ ~ t/~ x ] ∆ 1 − p ¯ t def R Π ′ B [ ~ t/~ x ] , B 2 , . . . , Γ − C p ~ t, B 2 , . . . , Γ − C def L . Then Ξ r educes to Π ′ 1 ∆ 1 − B [ ~ t/~ x ] Π 2 ∆ 2 − B 2 . . . Π n ∆ n − B n Π ′ B [ ~ t/~ x ] , . . . , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc . L eft-c ommut ative c ases: •L / ◦ L : Supp ose Π ends with a left r ule other than c L acting on B 1 and Π 1 is  Π i 1 ∆ i 1 − B 1  ∆ 1 − B 1 •L , where •L is a n y left rule except ⊃ L , e q L , or nat L . Then Ξ reduces to      Π i 1 ∆ i 1 − B 1  Π j ∆ j − B j  j ∈{ 2 ..n } Π B 1 , . . . , B n , Γ − C ∆ i 1 , ∆ 2 , . . . , ∆ n , Γ − C mc      ∆ 1 , ∆ 2 , . . . , ∆ n , Γ − C •L . 16 ⊃ L / ◦ L : Suppose Π ends with a left rule other than c L a cting on B 1 and Π 1 is Π ′ 1 ∆ ′ 1 − D ′ 1 Π ′′ 1 D ′′ 1 , ∆ ′ 1 − B 1 D ′ 1 ⊃ D ′′ 1 , ∆ ′ 1 − B 1 ⊃ L . Let Ξ 1 be Π ′′ 1 D ′′ 1 , ∆ ′ 1 − B 1 Π 2 ∆ 2 − B 2 · · · Π n ∆ n − B n Π B 1 , . . . , B n , Γ − C D ′′ 1 , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C mc . Then Ξ r educes to w ( ∆ 2 ∪ . . . ∪ ∆ n ∪ Γ , Π ′ 1 ) ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − D ′ 1 Ξ 1 D ′′ 1 , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C D ′ 1 ⊃ D ′′ 1 , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C ⊃ L . nat L / ◦ L : Suppose Π ends with a left rule other than c L a cting on B 1 and Π 1 is Π 1 1 − D 1 z Π 2 1 D 1 j − D 1 ( s j ) Π 3 1 D 1 I , ∆ ′ 1 − B 1 nat I , ∆ ′ 1 − B 1 nat L Let Ξ 1 be Π 3 1 D 1 I , ∆ ′ 1 − B 1 Π 2 ∆ 2 − B 2 . . . Π n ∆ n − B n Π B 1 , . . . , B n − C D 1 I , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C mc Then Ξ r educes to Π 1 1 − D 1 z Π 2 1 D 1 j − D 1 ( s j ) Ξ 1 D 1 I , ∆ ′ 1 , ∆ 2 , . . . , ∆ 2 , Γ − C nat I , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C nat L eq L / ◦ L : If Π ends with a left rule o ther than c L acting on B 1 and Π 1 is  Π θ ∆ ′ 1 θ − B 1 θ  θ s = t, ∆ ′ 1 − B 1 eq L then Ξ reduces to        Π θ ∆ ′ 1 θ − B 1 θ Π 2 θ ∆ 2 θ − B 2 θ . . . Π n θ ∆ n θ − B n θ Π θ B 1 θ, . . . , B n θ, Γ θ − C θ ∆ ′ 1 θ, ∆ 2 θ, . . . , ∆ n θ, Γ θ − C θ mc        θ s = t, ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C eq L Rig ht-c ommutative c ases: − / ◦ L : Supp ose Π is  Π i B 1 , . . . , B n , Γ i − C  B 1 , . . . , B n , Γ − C ◦L , 17 where ◦L is any left rule o ther than ⊃ L , eq L , or nat L (but including c L ) acting on a formula other than B 1 , . . . , B n . The der iv ation Ξ reduces to      Π 1 ∆ 1 − B 1 · · · Π n ∆ n − B n Π i B 1 , . . . , B n , Γ i − C ∆ 1 , . . . , ∆ n , Γ i − C mc      ∆ 1 , . . . , ∆ n , Γ − C ◦L , − / ⊃ L : Suppose Π is Π ′ B 1 , . . . , B n , Γ ′ − D ′ Π ′′ B 1 , . . . , B n , D ′′ , Γ ′ − C B 1 , . . . , B n , D ′ ⊃ D ′′ , Γ ′ − C ⊃ L . Let Ξ 1 be Π 1 ∆ 1 − B 1 · · · Π n ∆ n − B n Π ′ B 1 , . . . , B n , Γ ′ − D ′ ∆ 1 , . . . , ∆ n , Γ ′ − D ′ mc and Ξ 2 be Π 1 ∆ 1 − B 1 · · · Π n ∆ n − B n Π ′′ B 1 , . . . , B n , D ′′ , Γ ′ − C ∆ 1 , . . . , ∆ n , D ′′ , Γ ′ − C mc . Then Ξ r educes to Ξ 1 ∆ 1 , . . . , ∆ n , Γ ′ − D ′ Ξ 2 ∆ 1 , . . . , ∆ n , D ′′ , Γ ′ − C ∆ 1 , . . . , ∆ n , D ′ ⊃ D ′′ , Γ ′ − C ⊃ L . − /nat L : Supp ose Π is Π ′ − D z Π ′′ D j − D ( s j ) Π ′′′ B 1 , . . . , B n , D I , Γ ′ − C B 1 , . . . , B n , nat I , Γ ′ − C nat L Let Ξ 1 be Π 1 ∆ 1 − B 1 . . . Π n ∆ n − B n Π ′′′ B 1 , . . . , B n , D I , Γ ′ − C ∆ 1 , . . . , ∆ n , D I , Γ ′ − C mc, then Ξ reduces to Π ′ − D z Π ′′ D j − D ( s j ) Ξ 1 ∆ 1 , . . . , ∆ n , D I , Γ ′ − C ∆ 1 , . . . , ∆ n , nat I , Γ ′ − C nat L − / eq L : If Π is  Π ρ B 1 ρ, . . . , B n ρ, Γ ′ ρ − C ρ  B 1 , . . . , B n , s = t, Γ ′ − C eq L , then Ξ reduces to       Π i ρ ∆ i ρ − B i ρ  i ∈{ 1 ..n } Π ρ B i ρ, . . . , Γ ′ ρ − C ρ ∆ 1 ρ, . . . , ∆ n ρ, Γ ′ ρ − C ρ mc      ∆ 1 , . . . , ∆ n , s = t, Γ ′ − C eq L . 18 − / ◦ R : If Π is  Π i B 1 , . . . , B n , Γ i − C i  B 1 , . . . , B n , Γ − C ◦R , where ◦R is a n y r igh t r ule, then Ξ reduces to      Π 1 ∆ 1 − B 1 · · · Π ′ n ∆ n − B n Π i B 1 , . . . , B n , Γ i − C i ∆ 1 , . . . , ∆ n , Γ i − C i mc      ∆ 1 , . . . , ∆ n , Γ − C ◦R . Multicut c ases: mc/ ◦ L : If Π ends with a left rule other than c L acting on B 1 and Π 1 ends with a m ulticut and reduces to Π ′ 1 , then Ξ reduces to Π ′ 1 ∆ 1 − B 1 Π 2 ∆ 2 − B 2 · · · Π n ∆ n − B n Π B 1 , . . . , B n , Γ − C ∆ 1 , . . . , ∆ n , Γ − C mc . − /mc : Suppo se Π is  Π j { B i } i ∈ I j , Γ j − D j  j ∈{ 1 ..m } Π ′ { D j } j ∈{ 1 ..m } , { B i } i ∈ I ′ , Γ ′ − C B 1 , . . . , B n , Γ 1 , . . . , Γ m , Γ ′ − C mc , where I 1 , . . . , I m , I ′ partition the for m ulas { B i } i ∈{ 1 ..n } among the premise deriv ations Π 1 , . . . , Π m , Π ′ . F or 1 ≤ j ≤ m let Ξ j be  Π i ∆ i − B i  i ∈ I j Π j { B i } i ∈ I j , Γ j − D j { ∆ i } i ∈ I j , Γ j − D j mc . Then Ξ r educes to n Ξ j . . . − D j o j ∈{ 1 ..m }  Π i ∆ i − B i  i ∈ I ′ Π ′ . . . − C ∆ 1 , . . . , ∆ n , Γ 1 , . . . Γ m , Γ ′ − C mc . Structura l c ase: − / c L : If Π is Π ′ B 1 , B 1 , B 2 , . . . , B n , Γ − C B 1 , B 2 , . . . , B n , Γ − C c L , then Ξ reduces to Π 1 ∆ 1 − B 1  Π i ∆ i − B i  i ∈{ 1 ..n } Π ′ B 1 , B 1 , B 2 , . . . , B n , Γ − C ∆ 1 , ∆ 1 , ∆ 2 , . . . , ∆ n , ∆ n , Γ − C mc c L ∆ 1 , ∆ 2 , . . . , ∆ n , Γ − C . 19 Axiom c ases: id π / ◦ L : Supp ose Π ends with either nat L or eq L on B 1 and Π 1 ends with the id π rule: π 1 .B = π 2 .B 1 ∆ ′ 1 , B − B 1 id π Then it is the case that B = π − 1 1 .π 2 .B 1 . Apply the constr uction in Definition 8 to Π to get a deriv ation Π ′ of B , B 2 , . . . , B n , Γ − C . The der iv ation Ξ r e duces to Π 2 ∆ 2 − B 2 · · · Π n ∆ n − B n w ( ∆ ′ 1 , Π ′ ) B , ∆ ′ 1 , B 2 , . . . , B n , Γ − C B , ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C mc . − /id π : If Π ends with the id π rule with a matching formula in Γ , i.e., there exis ts C ′ ∈ Γ such that π .C ′ = π ′ .C for some permutations π and π ′ , then then Ξ reduces to ∆ 1 , . . . , ∆ n , Γ − C id π If Π ends with the id π rule but C does not ma tc h any for m ula in Γ , then C must match one of the cut formulas, s ay B 1 , i.e., there exists pe r m utations π 1 and π 2 such that π 1 .B 1 = π 2 .C . That is, C = π − 1 2 .π 1 .B 1 . In this case, we first apply the per m utation π − 1 2 .π 1 to Π 1 according to the construction in Definition 8 to get a deriv atio n Π ′ 1 of ∆ 1 − π − 1 2 .π 1 .B 1 . Ξ then reduces to w ( ∆ 2 ∪ . . . ∪ ∆ n ∪ Γ , Π ′ 1 ) . ⊓ ⊔ An inspectio n of the rules of the logic and this definition will reveal that every der iv ation ending with a m ulticut has a reduct. Beca use we use a multiset as the left side o f the sequent, there may b e ambiguit y as to w he ther a formu la oc c ur ring on the left side o f the rightmost premise to a m ulticut r ule is in fact a cut formula, and if so, which of the left premise s corresp onds to it. As a result, several of the reduction rules may apply , and so a deriv ation may hav e multiple r educts. 5 Normalizabilit y and reducibilit y W e now define tw o prop erties of deriv ations: normaliza bilit y and reducibility . Each of these prop erties im- plies that the deriv ation can b e reduced to a cut-free deriv a tion of the same end-sequent. In the following, substitutions mean Σ - substitutions for some signature Σ . The definitions are similar to those by McDow ell and Miller [3]. How ever, since the cut reduction in o ur case inv olves several transfo rmations of deriv ations, other than substitutions and weakening, we need to build this transfo rmations into the definitions of nor- malizability and reducibility . Definition 19. A height -pre s erving (HP) tr ansformation T is a finite se quenc e of tr ansformations F 1 , . . . , F n wher e e ach F i is one of the tr ansformations describ e d in Definition 6, Definition 7, Definition 8 and Defini- tion 9. The nu mb er n is t he order of T . The appli c ation of T to Π is define d as fol lows: T 0 ( Π ) = Π T i +1 ( Π ) = F i +1 ( T i ( Π )) T ( Π ) = T n ( Π ) Note tha t a height-preserving transformation may not be defined for all deriv ations, and that it may b e the ident ity transforma tio n (i.e., it does nothing). Height-preserving transformations are ranged over by T , F , G and H . Lemma 20. L et T b e a height-pr eserving tr ansformation. F or any derivation Π , if T ( Π ) is define d, then ht ( T ( Π )) ≤ ht ( Π ) . 20 Definition 21. We define the s et of normalizable deriva tions to b e the smal lest set that satisfies the fol lowing c onditions: 1. I f a derivation Π ends with a multicut, then it is normalizable if for every height-pr eserving tr ansforma- tion T such that T ( Π ) is define d, ther e is a normalizable r e duct of T ( Π ) . 2. I f a deri vation ends with any rule other than a multicut, then it is normaliza ble if the pr emise derivations ar e normalizable. These clauses assert that a given derivation is normalizable pr ovide d c ertain ( p erhaps infinitely many) other derivations ar e normalizable. If we c al l these other derivatio ns t he pr e de c essors of the given derivation, then a derivation is normalizable if and only if t he tr e e of the derivation and its su c c essive pr e de c essors is wel l-founde d. In this c ase, the wel l-founde d t re e is c al le d the no rmalization of the deriva tion. The set o f nor malizable deriv ations is not empty; the cut-free pro ofs, for instance, a re norma lizable. Since a normaliz a tion is well-founded, it has an asso ciated induction principle: for any prop erty P of deriv ations, if for every deriv ation Π in the nor ma lization, P holds for every predecessor of Π implies that P holds for Π , then P ho lds for every deriv ation in the nor malization. Lemma 22. If ther e is a normalizab le derivation of a se quent, then ther e is a cut-fr e e derivation of the se quent. Pr o of. Let Π b e a normalizable deriv a tion of the sequen t Γ − B . W e show b y induction on the no rmalization of Π that there is a cut-fr e e deriv ation of Γ − B . 1. If Π ends with a m ulticut, then any of its reducts is one of its predeces sors and so is norma lizable. One of its r educt, via the empty transformation, is also a deriv ation of Γ − B , s o by the induction h yp othesis this sequent has a cut-free deriv atio n. 2. Suppose Π ends with a rule other than multicut. Since w e a re given that Π is nor malizable, b y definition the premise deriv ations are normalizable. These premise deriv ations are the predecess ors of Π , so by the induction hypothesis ther e are cut-free deriv ations of the premises. Thus there is a cut-free deriv a tion of Γ − B . ⊓ ⊔ The ne x t four lemmas are also prov ed by induction on the normaliza tio n of deriv atio ns . Lemma 23. If Π is a normalizable derivation, t hen for any substitu tion θ su ch that Π θ is define d, Π θ is normalizable. Lemma 24. If Π is normalizable, then for any multiset of formulas ∆ , if w ( ∆, Π ) is define d, then w ( ∆, Π ) is normalizable. Lemma 25. If Π is normalizable, then for any p ermu tations ~ π such that h ~ π i .Π is define d, h ~ π i .Π is nor- malizable. Lemma 26. If Π is normalizable, then for any nominal c onstants ~ a such that r ( x, ~ a , Π ) is define d, r ( x, ~ a , Π ) is normalizable. Lemma 27. If Π is normalizable, t hen for any height-pr eserving tr ansformation T such t ha t T ( Π ) is de- fine d, T ( Π ) is normalizable. Definition 28. The level of a s e quent Γ − C is the level of C . The level of a derivation Π is the level of its ro ot se quent. The definition of r educibilit y for deriv ations is done by induction on the lev el of deriv ations: in defining the reducibility of level- i deriv ations, we assume that the reducibility of deriv atio ns of level j , for all j < i is already defined. In the follo wing definition, when we apply a transformation T to a deriv ation Π o f B 1 , . . . , B n − B 0 , we use the notation T ( B i ) to deno te the formula in the ro ot sequent of T ( Π ) that results from applying the tr ansformation to B i . 21 Definition 29. Reducibilit y . F or any i , we define t he set of reducible i -level derivations to b e the smal lest set of i -level derivations that satisfies the fol lo wing c onditions: 1. I f a deriva tion Π en ds with a multicut then it is r e ducible if for every height-pr eserving tr ansformation T such that T ( Π ) is define d, t her e is a r e ducible r e duct of T ( Π ) . 2. S upp ose t he derivatio n en ds with t he implic ation right ru le Π B , Γ − C Γ − B ⊃ C ⊃ R Then the derivation is r e ducible if Π is r e ducible and for every height-pr eserving t r ansformation T such that T ( Π ) is define d, multiset of formulas ∆ and r e ducible derivation Π ′ of ∆ − B ′ , wher e B ′ = T ( B ) , the derivation Π ′ ∆ − B ′ T ( Π ) B ′ , Γ ′ − C ′ ∆, Γ ′ − C ′ mc is r e ducible. 3. I f the deriva tion ends with the imp lic ation left rule or the nat rule, then it is r e ducible if t he right pr emise derivation is r e ducible and the other pr emise derivations ar e normaliza ble. 4. I f the derivation ends with any other ru le, then it is r e ducible if the pr emise deriva tions ar e r e ducible. These clauses assert that a given derivation is r e ducible pr ovide d c ertain other derivations ar e r e ducible. If we c al l these other derivatio ns the pr e de c essors of the given derivation, then a derivation is r e ducible only if the tr e e of the derivation and its suc c essive pr e de c essors is wel l founde d. In t hi s c ase, t he wel l founde d tr e e is c al le d the reduction of the deriva tion. Lemma 30. If a deriva tion is r e ducible, then it is normalizabl e. Pr o of. By induction on the reduction o f the deriv a tion. ⊓ ⊔ Lemma 31. If a derivation Π is re ducible, then for any height-pr eserving T such that T ( Π ) is define d, T ( Π ) is r e ducible. Pr o of. By induction on the reduction o f Π and Lemma 27. 6 Cut elimination In the following, when we mention T ( Π ) we assume implicitly tha t it is defined. W e s ha ll also use the notation B T to denote T ( B ), that is the application of the trans formation to the formula B . Similarly , the m ultiset T ( ∆ ) will b e wr itten ∆ T . W e dr op the subscript T if it is clear from c o n text which transformatio n we refer to. Lemma 32. F or any deriva tion Π of Σ ; B 1 , . . . , B n , Γ − C and r e du cible deri vations Π 1 , . . . , Π n of Σ ; ∆ 1 − C 1 , . . . , Σ ; ∆ n − C n , wher e n ≥ 0 , and for any tr ansformations T 1 , . . . , T n , T such that T i ( Π i ) is define d and T i ( C i ) = T ( B i ) , the derivation Ξ T 1 ( Π 1 ) Σ ′ ; ∆ 1 T 1 − B 1 T . . . T n ( Π n ) Σ ′ ; ∆ n T n − B n T T ( Π ) Σ ′ ; B 1 T , . . . , B n T , Γ T − C T Σ ′ ; ∆ 1 T 1 , . . . , ∆ n T n , Γ T − C T mc is r e ducible. 22 Pr o of. The pro of is by induction on ht ( Π ) with sub ordinate induction on n and on the reductions of Π 1 , . . . , Π n . Since the pro of doe s not dep end o n the or der of the inductions on reductions, when we need to distinguish of one the Π i ’s we shall refer to it as Π 1 without loss of g e neralit y . W e need to show that for every T ′ , the der iv ation every reduct of T ′ ( Ξ ) is r educible. If n = 0 then T ′ ( Ξ ) reduces to T ′ ( T ( Π )) . Since reducibilit y is preserved by heig h t-preserving transforma tio n, it suffices to consider the case wher e T and T ′ are the iden tit y tra ns formation, that is, we need only to show that Π is reducible. This is prov ed b y case analys is on the last rule of Π . F or each case, the results follo w from the outer induction h yp othesis and Definition 29. The case with ⊃ R requires that heig h t-preserving transforma tio ns do not increa se the height of the deriv a tions (see Le mma 20). In the cases for ⊃ L a nd nat L w e need the additional information that reducibilit y implies normaliza bility (see Lemma 30). F or n > 0, w e ana lyze all poss ible reductio ns that apply to T ′ ( Ξ ) and show that every reduct of T ′ ( Ξ ) is reducible. W e supp ose tha t T ′ ( Ξ ) is of the following form: F 1 ( Π 1 ) ∆ 1 F 1 − C 1 F 1 . . . F n ( Π n ) ∆ n F n − C n F n F ( Π ) B 1 F , B n F , Γ F − C F ∆ 1 F 1 , . . . , ∆ n F n , Γ F − C F mc where B i F = C i F i . In several ca ses below, w e often omit the subscripts F or F i when it is clear from context which tra nsformations we refer to. W e also often switch betw een B i F and C i F i to make the inference fig ur es more readable. Most cases follow immedia tely fro m the inductiv e h yp othesis and Definition 29 and Lemma 30, Lemma 31 and Lemma 20. W e show here the in teresting cases. ⊃ R / ⊃ L : Supp ose Π 1 and Π are Π ′ 1 ∆ 1 , B ′ 1 − B ′′ 1 ∆ 1 − B ′ 1 ⊃ B ′′ 1 ⊃ R Π ′ B 2 , . . . , Γ − B ′ 1 Π ′′ B ′′ 1 , B 2 , . . . , Γ − C B ′ 1 ⊃ B ′′ 1 , B 2 , . . . , B n , Γ − C ⊃ L . Let Ξ 1 be the deriv ation F 2 ( Π 2 ) ∆ 2 − B 2 . . . F n ( Π n ) ∆ n − B n F n ( Π ′ ) B 2 , . . . , B n , Γ − B ′ 1 ∆ 2 , . . . , ∆ n , Γ − B ′ 1 mc Then Ξ 1 is reducible by induction hypothesis since F a nd F i preserve r educibilit y (Lemma 31) and do not increase the height of deriv a tions (Lemma 20). Since we a re g iv en that Π 1 is reducible, by Definition 29, the deriv ation Ξ 2 Ξ 1 ∆ 2 , . . . , ∆ n , Γ − B ′ 1 F 1 ( Π ′ 1 ) B ′ 1 , ∆ 1 − B ′′ 1 ∆ 1 , . . . , ∆ n , Γ − B ′′ 1 mc is reducible as w ell. Therefore, the reduct o f T ′ ( Ξ ) Ξ 2 . . . − B ′′ 1  F i ( Π i ) ∆ i − B i  i ∈{ 2 ..n } F ( Π ′′ ) B ′′ 1 , { B i } i ∈{ 2 ..n } , Γ − C ∆ 1 , . . . , ∆ n , Γ , ∆ 2 , . . . , ∆ n , Γ − C mc c L ∆ 1 , . . . , ∆ n , Γ − C . is reducible b y the outer induction h yp othesis and Definition 29. 23 ∀R / ∀L : Supp ose Π 1 and Π are Π ′ 1 Σ , h ; ∆ 1 − B [ h ~ c/x ] Σ ; ∆ 1 − ∀ x.B ∀R Π ′ Σ ; B [ t/x ] , B 2 , . . . , B n , Γ − C Σ ; ∀ x.B , B 2 , . . . , B n , Γ − C ∀L Applying the transformation F 1 to Π 1 (and similarly , F to Π ) migh t require several trans fo rmation be do ne on the premise of the deriv ation, e.g., to av oid clashes of nominal constants, etc., so let us suppo se that F 1 ( Π 1 ) and F ( Π ) are of the following shap es: G 1 ( Π ′ 1 ) Σ ′ , h ; ∆ 1 − D [ h ′ ~ d/x ] Σ ′ ; ∆ 1 − ∀ x.D ∀R G ( Π ′ ) Σ ′ ; D [ s/x ] , B 2 , . . . , B n , Γ − C Σ ′ ; ∀ x.D , B 2 , . . . , B n , Γ − C ∀L where ∀ x.D = ∀ x.B and D [ s/x ] = B [ t/x ] . If the supp ort of D [ s/x ] is large r than { ~ d } , then the reduction rule for ∀R / ∀L require s further tra nsformations be a pplied to G 1 ( Π ′ 1 ), i.e., as is descr ibed in Lemma 17. So let us s upp ose that this transformation is applied, resulting in a deriv ation G ′ 1 ( Π ′ 1 ) Σ ′ , f ; ∆ 1 − D [ f ~ e/x ] . Then T ′ ( Ξ ) reduces to G ′ 1 ( Π ′ 1 )[ λ ~ e.s/f ] Σ ′ ; ∆ 1 − D [ s/x ] F 2 ( Π 2 ) ∆ 2 − B 2 . . . F n ( Π n ) ∆ 2 − B 2 G ( Π ′ ) Σ ′ ; D [ s/x ] , . . . , Γ − C Σ ′ ; ∆ 1 , . . . , ∆ n , Γ − C mc which is reducible b y the outer induction hypothesis. nat R /nat L : Supp ose Π 1 and Π are Π ′ 1 ∆ 1 − n at M ∆ 1 − n at M nat R Π ′ − D z Π ′′ D j − D ( s j ) Π ′′′ D ( s M ) , B 2 , . . . , B n , Γ − C nat ( s I ) , B 2 , . . . , B n , Γ − C nat L then F 1 ( Π 1 ) and F ( Π ) are F 1 ( Π ′ 1 ) ∆ 1 − n at I ∆ 1 − n at I nat R Π ′ − D z Π ′′ D j − D ( s j ) F ( Π ′′′ ) D ( s I ) , B 2 , . . . , B n , Γ − C nat ( s I ) , B 2 , . . . , B n , Γ − C nat L Note that the deriv ations Π ′ and Π ′′ are not affected by the tr a nsformation F since D is a closed term with no occur rences of nominal constants and j in Π ′′ is a new eige nv ar iable. Let Ξ 1 be the deriv ation F 1 ( Π ′ 1 ) ∆ 1 − n at I Π ′ − D z Π ′′ D j − D ( s j ) D I − D I id π nat I − D I nat L ∆ 1 − D I mc . Since the height of the r igh t premise is no larg er than ht ( Π ), and Π ′ 1 is a predecessor of Π 1 , Ξ 1 is reducible by inductio n on the reduction of Π 1 . Le t { ~ c } be the supp ort of I . W e construct the der iv ation Π • of h ; D ( h ~ c ) − D ( s ( h ~ c )) from Π ′′ using the proc e dures describ ed in Definition 7 and Definition 9. Let Ξ 2 be Ξ 1 ∆ 1 − D I Π • [ λ ~ c.I / h ] D I − D ( s I ) ∆ 1 − D ( s I ) mc. 24 Since ht ( Π • [ λ ~ c.I / h ]) ≤ h t ( Π ′′ ), by the outer induction hypo thesis, Ξ 2 is a lso reducible. Therefor e the reduct of T ′ ( Ξ ) Ξ 2 ∆ 1 − D ( s I ) F 2 ( Π 2 ) ∆ 2 − B 2 . . . F n ( Π n ) ∆ n − B n F ( Π ′′′ ) D ( s I ) , B 2 , . . . , Γ − C ∆ 1 , . . . , ∆ 2 , Γ − C mc is reducible b y the outer induction h yp othesis. eq L / ◦ L : Suppose Π 1 is  Π θ ∆ 1 θ − B 1 θ  θ s = t, ∆ 1 − B 1 eq L then F 1 ( Π 1 ) is  Π • ρ ∆ 1 θ − B 1 θ  ρ s = t, ∆ 1 − B 1 eq L where ea c h Π • ρ is o btained from some Π θ by the transfo rmations descr ibed in Definition 6, Definition 7, Definition 8 and Definition 9. W e denote with f ( ρ ) the s ubstitut ion θ such that Π • ρ is constructed out of Π θ . Thus we can write each Π • ρ as the deriv ation F ρ ( Π f ( ρ ) ) for some transformatio n F ρ . The r educt of T ′ ( Ξ )          F ρ ( Π f ( ρ ) ) ∆ ′ 1 ρ − B 1 ρ F 2 ( Π 2 ) ρ ∆ 2 ρ − B 2 ρ . . . F n ( Π n ) ρ ∆ n ρ − B n ρ F ( Π ) ρ B 1 ρ, . . . , B n ρ, Γ ρ − C ρ ∆ ′ 1 ρ, ∆ 2 ρ, . . . , ∆ n ρ, Γ ρ − C ρ mc          ρ s = t, ∆ ′ 1 , ∆ 2 , . . . , ∆ n , Γ − C eq L Each pr e mis e de r iv ation of the a bov e deriv ation is reducible by the induction hypothesis o n the reduction of Π 1 , since eac h Π f ( ρ ) is a predecessor o f Π 1 . The r educt of T ′ ( Ξ ) is therefore re ducible by Definition 29. − / ⊃ R : Supp ose Π is F ( Π ′ ) B 1 , . . . , B n , Γ , C 1 − C 2 B 1 , . . . , B n , Γ − C 1 ⊃ C 2 ⊃ R then F 1 ( Π ) F ( Π ′ ) B 1 , . . . , B n , Γ , C 1 − C 2 B 1 , . . . , B n , Γ − C 1 ⊃ C 2 ⊃ R Let Ξ 1 be F 1 ( Π 1 ) ∆ 1 − B 1 . . . F n ( Π n ) ∆ n − B n F ( Π ′ ) B 1 , . . . , B 1 , Γ , C 1 − C 2 ∆ 1 , . . . , ∆ n , C 1 − C 2 which is reducible b y the outer induction hypothesis. Let Ξ 2 be the deriv ation Ξ 1 ∆ 1 , . . . , ∆ n , Γ , C 1 − C 2 ∆ 1 , . . . , ∆ n , Γ − C 1 ⊃ C 2 ⊃ R , 25 which is the reduct of T ′ ( Ξ ) . T o show that Ξ 2 is r educible, w e need to show that for any T ′′ , a nd for any deriv ation Π ′′ of ∆ − D , where D = T ′′ ( C 1 ), the deriv ation Ξ 3 Π ′′ ∆ − D T ′′ ( Ξ 2 ) D , ∆ 1 G 1 , . . . , ∆ n G n , Γ G − C 2 G ∆, ∆ 1 G 1 , . . . , ∆ n G n , Γ G − C 2 G mc is reducible. Here the tr a nsformations G i and G are transfor mations asso ciated with the premise deriv atio ns in T ′′ ( Ξ 2 ) . Ξ 3 is reducible if for any transformatio n H , every reduct of the deriv ation H ( Ξ 3 ) is reducible. The reduct of H ( Ξ 3 ) in this case is: H ′ ( Π ′′ ) ∆ − D H 1 ( Π 1 ) ∆ 1 − B 1 . . . H n ( Π n ) ∆ n − B n H ′′ ( Π ′ ) D , B 1 , . . . , B n , Γ − C 2 ∆, ∆ 1 , . . . , ∆ n , Γ − C 2 mc where H 1 , . . . , H n and H ′′ are trans fo rmations applied to the pre mis e s of H ( T ′′ ( Ξ 2 )) and H ′ is the transfo r - mation applied to the left premise of H ( Ξ 3 ) . This deriv ation is reducible by the outer induction hypo thesis. ⊓ ⊔ Corollary 33. Every derivation is r e ducible. Pr o of. This result follows immediately from Lemma 32 with n = 0 . ⊓ ⊔ Theorem 34. The cut rule is admissible in LG ω . Pr o of. F o llo ws immediately fro m Cor ollary 33, Lemma 30 and Lemma 22. ⊓ ⊔ Corollary 35. The lo gic LG ω is c onsistent , i.e., it is not the c ase that b oth A and A ⊃ ⊥ ar e pr ovable. 7 Corresp ondence b et wee n LG and F O λ ∇ W e now show that the fo rm ulation o f LG is equiv a le n t to F O λ ∇ extended with the axiom schemes of na me per m utations and weakening: ∇ x ∇ y .B x y ⊃ ∇ y ∇ x.B x y and B ≡ ∇ x.B (2) where x is not free in B in the second scheme. Sequents in F O λ ∇ are expressions of the for m Σ ; σ 1 ⊲ B 1 , . . . , σ n ⊲ B n − σ 0 ⊲ B 0 . Σ is the signatur e of the sequent, σ i is a list of v ariables loca lly scop ed ov er B i , a nd is referred to as lo c al signatur e . The expression σ i ⊲ B i is c alled a lo c al judgment , o r judgment fo r short. In [8], lo cal judgments are consider ed equa l mo dulo renaming of their lo cal signatures , e.g., ( a, b ) ⊲ P a b is equal to ( c, d ) ⊲ P c d. Lo cal judgment s are ra nged over b y scr ipted capital letters, e.g., B , D , etc. F or the purpos e of proving the corres p ondence with LG , how ever, we will make this renaming step explicit, b y including the rules: ~ y ⊲ B ′ , Γ − C ~ x ⊲ B , Γ − C α R , λ~ x.B ≡ α λ~ y .B ′ Γ − ~ y ⊲ B ′ Γ − ~ x ⊲ B α L , λ~ x .B ≡ α λ~ y .B ′ The inference rules of F O λ ∇ are given in Figure 2. W e now consider the c orresp ondence b etw een L G with F O λ ∇ extended with the following axiom sc hemes: ∇ x ∇ y .B x y ≡ ∇ y ∇ x.B x y . (3) 26 Σ ; σ ⊲ B , Γ − σ ⊲ B id Σ ; ∆ − B Σ ; B , Γ − C Σ ; ∆, Γ − C cut Σ ; σ ⊲ B , σ ⊲ C , Γ − D Σ ; σ ⊲ B ∧ C, Γ − D ∧L Σ ; Γ − σ ⊲ B Σ ; Γ − σ ⊲ C Σ ; Γ − σ ⊲ B ∧ C ∧R Σ ; σ ⊲ B , Γ − D Σ ; σ ⊲ C, Γ − D Σ ; σ ⊲ B ∨ C, Γ − D ∨L Σ ; Γ − σ ⊲ B Σ ; Γ − σ ⊲ B ∨ C ∨R Σ ; σ ⊲ ⊥ , Γ − B ⊥L Σ ; Γ − σ ⊲ C Σ ; Γ − σ ⊲ B ∨ C ∨R Σ ; Γ − σ ⊲ B Σ ; σ ⊲ C , Γ − D Σ ; σ ⊲ B ⊃ C, Γ − D ⊃ L Σ ; σ ⊲ B , Γ − σ ⊲ C Σ ; Γ − σ ⊲ B ⊃ C ⊃ R Σ , σ ⊢ t : γ Σ ; σ ⊲ B [ t/x ] , Γ − C Σ ; σ ⊲ ∀ γ x.B , Γ − C ∀L Σ , h ; Γ − σ ⊲ B [( h σ ) /x ] Σ ; Γ − σ ⊲ ∀ x.B ∀R Σ , h ; σ ⊲ B [( h σ ) /x ] , Γ − C Σ ; σ ⊲ ∃ x.B , Γ − C ∃L Σ , σ ⊢ t : γ Σ ; Γ − σ ⊲ B [ t/x ] Σ ; Γ − σ ⊲ ∃ γ x.B ∃R Σ ; ( σ, y ) ⊲ B [ y /x ] , Γ − C Σ ; σ ⊲ ∇ x B , Γ − C ∇L Σ ; Γ − ( σ, y ) ⊲ B [ y /x ] Σ ; Γ − σ ⊲ ∇ x B ∇R Σ ; B , B , Γ − C Σ ; B , Γ − C c L Σ ; Γ − C Σ ; B , Γ − C w L Σ ; Γ − σ ⊲ ⊤ ⊤R Fig. 2. The core inference rules o f F O λ ∇ . B ≡ ∇ x.B , pr o vided that x is not free in B . (4) W e can equiv alently state these t wo axio ms as the following inference rules: ( ~ x, b, a, ~ y ) ⊲ B , Γ − C ( ~ x, a, b, ~ y ) ⊲ B , Γ − C p L Γ − ( ~ x , b, a, ~ y ) ⊲ B Γ − ( ~ x , a, b, ~ y ) ⊲ B p L ( ~ x, a, ~ y ) ⊲ B , Γ − C ( ~ x~ y ) ⊲ B , Γ − C ss L , a 6∈ { ~ x, ~ y } Γ − ( ~ x, a, ~ y ) ⊲ B Γ − ( ~ x~ y ) ⊲ B ss R , a 6∈ { ~ x, ~ y } ( ~ x~ y ) ⊲ B , Γ − C ( ~ x, a, ~ y ) ⊲ B , Γ − C ws L , a 6∈ supp ( B ) Γ − ( ~ x~ y ) ⊲ B Γ − ( ~ x , a, ~ y ) ⊲ B ss R , a 6∈ supp ( B ) Implicit in the ab o ve rules is the assumption that v ariables in lo cal signa tur es are consider ed as sp ecial constants, much like the nominal consta nts in LG . The supp ort of B , within a lo cal signature σ , is defined similarly as it is in LG : it is the set { a ∈ σ | a o ccurs in B . } . The logical system with the inference rule s in Fig ur e 2 together with α R , α L , p L , p R , ss L , s s R , w s L and w s R is referred to as F O λ ∇ + . In relating LG and F O λ ∇ + , we map the lo cal signatures to nominal constants, a nd vice versa. In the following, giv en a formula B , we ass ume a par ticular enumeration o f the nominal constants app earing in B ba sed the left-to-right o rder of their appea rance in B . Lemma 36. If the se qu en t Σ ; B 1 , . . . , B n − B 0 is pr ovable in LG then the se quent Σ ; ~ c 1 ⊲ B 1 , ~ c n ⊲ B n − ~ c 0 ⊲ B 0 wher e ~ c i is an enumer ation of supp ( B i ) , is pr ovable in F O λ ∇ + . Pr o of. Suppo se that Π is a pro of of Σ ; B 1 , . . . , B n − B 0 . W e construct a pro of Π ′ of Σ ; ~ c 1 ⊲ B 1 , ~ c n ⊲ B n − ~ c 0 ⊲ B 0 by induction on ht ( Π ) . W e consider some int ere s ting cases here: 27 – Suppose Π ends with id π : π .B i = π ′ .B 0 Γ ′ , B i − B 0 id π The p erm utations π a nd π ′ can b e imitated by a series of rena ming ( α R and α L rules). The deriv atio n Π ′ is therefore constructed b y applying a series o f α R , α L , follow ed by the id rule. – Suppose Π ends with ⊃ R : in this c a se we supp ose that B 0 = C ⊃ D . Π 1 B 1 , . . . , B n , C − D B 1 , . . . , B n − C ⊃ D ⊃ R By induction h yp othesis w e hav e a deriv ation Π 2 of ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n , ~ a ⊲ C − ~ b ⊲ D W e first hav e to weaken the signatures ~ a and ~ d to ~ c 0 befo re applying the in tro duction rule for ⊃ . That is, Π ′ is the deriv a tion Π 2 ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n , ~ a ⊲ C − ~ b ⊲ D ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n , ~ c 0 ⊲ C − ~ c 0 ⊲ D ∗ ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ C ⊃ D ⊃ R Here the star ‘* ’ denotes a series of applications of w s L , ws R , p L and p R . – Suppose Π is Π 1 B 1 , . . . , B n − C [ t/x ] B 1 , . . . , B n − ∃ x.C ∃R It is pos sible that t contains new consta n ts that a re not in the supp ort of C. Suppose ~ d is an enumeration of the suppo r t of C [ t/x ]. The deriv ation Π ′ is constructed as follows Π 2 ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ d ⊲ C [ t/x ] ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ d ⊲ ∃ x.C ∃R ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ ∃ x.C ∗ where Π 2 is obtained from induction hypothesis applied to Π 1 , and the rule ‘*’ denotes a series of applications of ss R (for introducing new constants) a nd p R (for rear ranging the o rder of the lo cal signature). – F or other cases, the constructio n of Π ′ follows the sa me pattern as in the prev io us cas e s, i.e., by induction hypothesis, follow ed by s ome rear ranging, extension, or weak ening o f lo cal sig natures. ⊓ ⊔ Lemma 37. If the se qu en t Σ ; ~ c 1 ⊲ B 1 , ~ c n ⊲ B n − ~ c 0 ⊲ B 0 is pr ovable in F O λ ∇ + then the se quen t Σ ; B 1 , . . . , B n − B 0 is pr ovable in LG Pr o of. Suppo se Π is a deriv ation of ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ B 0 W e construct a deriv atio n Π ′ of B 1 , . . . , B n − B 0 by induction on ht ( Π ). W e sho w her e the interesting cases; the other cases follow immediately from induction hypo thes is: 28 – If Π ends with i d , ⊤R , or ⊥L then Π ′ ends with the sa me rule. – Suppose Π is Π 1 ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ d ⊲ B ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ B 0 α L By induction hypothesis, there is a deriv a tio n Π 2 of B 1 , . . . , B n − B . T o get Π ′ apply the pro cedure in Definition 8 to Π 2 to rename B to B 0 . – Suppose Π is Π 1 ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ C [( h ~ c 0 ) /x ] ~ c 1 ⊲ B 1 , . . . , ~ c n ⊲ B n − ~ c 0 ⊲ ∀ x.C ∀R By induction h yp othesis, there is a deriv ation Π 2 of B 1 , . . . , B n − C [( h ~ c 0 ) /x ] . Suppos e { ~ d } = supp ( C ) . Then Π ′ is Π 2 [ λ ~ c 0 .h ′ ~ d/h ] B 1 , . . . , B n − C [ h ′ ~ d/x ] B 1 , . . . , B n − ∀ x.C ∀R – If Π ends with ∃L , apply the sa me construction as in the pre v ious case . ⊓ ⊔ Theorem 38. L et F b e a formula which c ontains no o c curr enc es of n omina l c onstants. 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