 has an application of an elimination rule with its maximum formula. On the other hand, it is not easy to characterize a 'detour' in terms of the length of a derivation. There are cases that the length of a normal derivation is longer than the length of an original derivation. It cannot be said that a derivation of a longer length has a detour and a derivation with a shorter length has no detour. Thus, we only use a 'detour reasoning' as defined through a maximum formula. Then, (T2) means that a proper reduction should neither introduce an application of an elimination rule containing a maximum formula nor increase the degree of maximum formulas. To check (T2), it is useful to translate a reduction in natural deduction to a one in sequent calculus. It is often claimed that a natural deduction system is isomorphic to a sequent calculus system if the generalized eliminations are used. (Cf. Tennant (2002), Negri and Von Plato (2001) and Von Plato (2011)). Especially, Negri and Von Plato (2001, Ch.1 and Ch. 8) propose an inductive definition of the translation algorithm between an intuitionistic natural deduction system with generalized elimination rules and an intuitionistic sequent calculus system with independent contexts. Let Γ, ∆, Θ be a finite multiset, i.e. a list with multiplicity but no order, of assumptions in sequent calculus. We use a binary derivation symbol ⇒ and 'Γ ⇒ φ' means that the antecedent Γ derives the succedent φ. A sequent calculus has the following left and right rules for → and structural rules, such as cut-, weakening, and contraction rules. Γ ⇒ φ ψ,∆ ⇒ σ L → φ → ψ,Γ,∆ ⇒ σ φ,Γ ⇒ ψ R → Γ ⇒ φ → ψ 119 Γ ⇒ φ φ,∆ ⇒ Ψ Cut Γ,∆ ⇒ Ψ Γ ⇒ Ψ Wk φ,Γ ⇒ ψ φ,φ,Γ ⇒ Ψ Ctr φ,Γ ⇒ Ψ φ in Cut−rule is called a cut-formula. Let us remind the reduction process for → in the style of generalized elimination rule. [φ]1 D1 ψ → I,1 φ → ψ D2
has an application of an elimination rule with its maximum rule. On the early hand, it is not easy to qualify a' detour' in terms of the distance of a derivation. There are case that the length of a normal derivation is longer than the length of an original derivation. It cannot be suppose that a derivation of a longer length has a detour and a derivation with a shorter duration has no detour. Thus, we only practice a' detour reasoning' as defined through a maximum recipe. Then, (T2) means that a proper decrease should neither introduce an lotion of an elimination rule containing a maximum formula nor increase the academic degree of maximum formulas. To check (T2), it is useful to translate a reduction in natural deduction to a one in sequent calculus. It is often claimed that a natural deduction system is isomorphic to a sequent tartar arrangement if the generalized eliminations are use. (Cf. Tennant (2002), Negri and Von Plato (2001) and Von Plato (2011) ). specially, Negri and Von Plato (2001, Ch.1 and Ch. 8) propose an inductive definition of the translation algorithm between an intuitionistic lifelike deduction arrangement with generalized elimination rules and an intuitionistic sequent calculus arrangement with independent contexts. Let Γ, ∆, Θ be a finite multiset, i.e. a list with multiplicity but no decree, of assumptions in sequent calculus. We use a binary derivation symbol ⇒ and' Γ ⇒ φ' means that the ancestor Γ deduce the succedent φ. A sequent calculus has the following left and right rules for → and structural rules, such as cut-, weakening, and compression rules. Γ ⇒ φ ψ,∆ ⇒ σ fifty → φ → ψ, Γ,∆ ⇒ σ φ, Γ ⇒ ψ R → Γ ⇒ φ → ψ 119 Γ ⇒ φ φ,∆ ⇒ Ψ Cut Γ,∆ ⇒ Ψ Γ ⇒ Ψ Wk φ, Γ ⇒ ψ φ, φ, Γ ⇒ Ψ Ctr φ, Γ ⇒ Ψ φ in Cut−rule is called a baseball swing - formula. get us remind the reduction procedure for → in the style of generalized elimination convention. [ φ]1 D1 ψ → I,1 φ → ψ D2
 had an application of an euimination rule with ivs maxijum formjla. On the other hand, it is iot wasy uj characterize a 'ddtour' in nerms of rhe owngth of a derivatljn. Tgcre axe cases that thg length of d normal derivdtkou is longer than the length of an oryginal cegivation. It cagnot fe szpd that a derivation of a longed lengti has a detour snd a derivation with a shlrteg length has no dehour. Thus, wg onjt use a 'detojr reasoning' as defines through a maximum formula. Thev, (T2) mzans that a peopft reduction whoulq neither inbgoduce dn applocation of an climiiatiin rule containing a kaximum formula not increase tke degree of maximum doemulav. To chezj (T2), it ix hseful to translate a reductiob in natural deductoog to a one in aequene salculus. It is often claimed that a natlral deduction system is isimorphic to a sequent calculus system if the generalized eliminations are used. (Wf. Teinxnt (2002), Kegrk ajd Von Plato (2001) and Von Plato (2011)). Especially, Negri znc Non Plato (2001, Ch.1 and Ch. 8) proposr wn ynductive defknitiou or the translation wlgoritrm berween an yntuotionistic natural deductiob system witk gwneralized eliminacion rules aud an ontuiyionistic sequent calcuuus aystem with independsvt contexts. Let Γ, ∆, Θ ne d finite multiset, i.e. a lise with muotipkicity cut mo ordqr, of assulptiokv in sequent calcupus. Wg use d binary dfrivation symbol ⇒ and 'Γ ⇒ φ' means viat the antecgdett Γ derives the xuccedent φ. A fequent calculos has thz folluwing left and rijht rules fow → and structgtal rules, suci as cut-, reakwnint, and cuvtraction rulex. Γ ⇒ φ ψ,∆ ⇒ σ L → φ → ψ,Γ,∆ ⇒ σ φ,Γ ⇒ ψ E → Γ ⇒ φ → ψ 119 Γ ⇒ φ φ,∆ ⇒ Ψ Cut Γ,∆ ⇒ Ψ Γ ⇒ Ψ Wk φ,Γ ⇒ ψ φ,φ,Γ ⇒ Ψ Ctr φ,Γ ⇒ Ψ φ in Cut−rule is cauleq w rut-fowkula. Let us semivd gne reauction progesr fot → in the style of ganerzlized elimination rmle. [φ]1 D1 ψ → U,1 φ → ψ D2
has an application of an elimination rule maximum On the hand, it is 'detour' terms of the of a derivation. are cases that the length of normal derivation is longer than the length of an original derivation. It cannot said that a derivation of a longer length has a detour and a with shorter has detour. Thus, we only use a 'detour reasoning' as defined through a maximum formula. Then, (T2) that a proper reduction should neither introduce an of an elimination rule a maximum formula nor increase degree maximum formulas. check it useful to translate reduction in natural deduction to a one in sequent calculus. It is often claimed that a natural system is a sequent system the eliminations are used. (2002), Negri and Von Plato (2001) (2011)). Especially, Negri and Von Plato (2001, Ch.1 Ch. 8) an inductive definition of the translation between an intuitionistic natural deduction system with generalized rules and an intuitionistic sequent calculus system with independent contexts. Let Γ, ∆, Θ be multiset, i.e. a list multiplicity but no of in calculus. use a derivation symbol ⇒ and 'Γ ⇒ φ' means that the antecedent derives the succedent φ. A sequent calculus has the following right for → and rules, such as cut-, and rules. Γ ⇒ φ σ → ⇒ φ,Γ ψ R → Γ φ → ψ 119 Γ φ φ,∆ ⇒ Ψ ⇒ Ψ Wk φ,Γ ⇒ ψ φ,φ,Γ ⇒ Ctr φ,Γ ⇒ Ψ φ in Cut−rule called a cut-formula. Let us remind the reduction process for → in style of rule. [φ]1 D1 ψ → I,1 φ → D2
 has an application of an elimiNation rule With iTs mAxiMuM forMula. on the other hand, IT is nOt easy to characterize a 'dEtour' In TErms OF tHe lenGth of a dERiVATioN. THeRe aRe CAsEs thaT thE length Of a normal dEriVaTion is longer THaN the length Of aN original derIvaTion. It CaNnoT Be saiD thAt a deRivatiON of a loNger lengtH hAS a detoUR and a deRIVaTion With a shorter lengtH HaS No detour. Thus, we Only usE a 'DEtOUR reAsoNing' as defiNeD throUGh a maxiMUm FORMulA. then, (T2) means thaT a proper redUCtiOn shouLd NeiTHer intRoducE aN AppLication of aN eliMination rUle conTAining a MAximum fOrmula Nor IncReasE ThE dEgrEe OF maXImUm fORmuLas. To cheCk (t2), iT is usEful TO TRAnslAte A redUctioN in natural dedUctIon tO A onE in seQuent CalcUlUs. It iS often ClaimEd That a natural dedUctiOn system iS isOmOrpHiC to a sEQuent cAlcUluS system If the geNEraLiZED ElIminations are used. (CF. TENNaNt (2002), Negri aNd Von PLAtO (2001) aND Von PlatO (2011)). ESpeCialLY, negri And VON PLato (2001, Ch.1 anD Ch. 8) proPOsE aN inductIvE definItIon Of tHe traNSlatIon algOrithm beTween AN intuitionistiC Natural deductIOn SYStEM witH geNeralized elIminATion RuleS AnD an INtuitIonisTiC SeQUent calculus system wItH indepEndenT contexts. Let Γ, ∆, θ be a finite MULTiset, i.e. a List WItH Multiplicity buT no orDer, of assumPTions in sEquenT calculuS. We use a biNARy derivaTioN syMboL ⇒ anD 'γ ⇒ Φ' mEans that the anTECedeNt γ deriveS thE succedEnt Φ. A sEquEnt CaLculus has The folloWiNg LeFt And Right RUles for → aNd StrUcTurAl rulES, such aS cut-, wEakeNiNg, ANd cOntractIOn RULes. Γ ⇒ Φ ψ,∆ ⇒ Σ L → Φ → ψ,Γ,∆ ⇒ σ Φ,Γ ⇒ ψ r → Γ ⇒ Φ → ψ 119 Γ ⇒ φ φ,∆ ⇒ ψ Cut γ,∆ ⇒ ψ Γ ⇒ Ψ wk φ,Γ ⇒ ψ φ,φ,γ ⇒ Ψ Ctr φ,Γ ⇒ Ψ φ In CUT−rulE iS cAlled a cUt-formula. Let uS rEmind the reDuCtiOn procESS for → in thE style of generalized elimINation rUle. [Φ]1 D1 ψ → I,1 φ → Ψ D2
 has an application of aneliminatio n rul e w ith  i ts m axim um formula. On theother hand, it is noteasyto char a ct erize  a 'det o ur ' inte rm s o ft he  leng thof a de rivation.The re  are cases t h at  the lengt h o f a normal d eri vation  i s l o ngertha n the  lengt h  of an  original  d e rivati o n. It c a n no t be  said that a deri v at i on of a longer  lengt hh as a  de tou r and a de ri vatio n  with a sh o r t erl ength has nodetour. Thu s , w e only  u sea  'deto ur re as o nin g' as defin ed t hrough amaximu m  formul a . Then,  (T2)mea nsthat apr ope rr edu c ti ons hou ld neith er  i ntrod ucea n a ppli cat ionof an  eliminationrul e co n tai ninga max imum  f ormul a norincre as e the degree of  max imum form ula s.  To  c heck( T2), i t i s u seful t o trans l ate  a r e du ction in natural d ed u c ti on to aone in se qu e nt calcu lu s.It i s often  cla i me d that a  natur a lde duction  s ystemis  is omo rphic to a  seque nt calcu lus s y stem if the ge n eralized elim i na t i on s  are  us ed. (Cf. Te nnan t  (20 02), Ne gri and V on Pl at o  ( 2 001) and Von Plato(2 011)).  Espe cially, Negri  and Von P l a t o (2001,  Ch. 1  a n d Ch. 8) propo se an  inductive definiti on of  the tra nslationa l gorithmbet wee n a n i n t ui tionistic nat u r al d ed uctionsys tem wit h g ene ral ize deliminati on rules  a nd  a nint uitio n istic se qu ent  c alc uluss ystemwithinde pe nd e ntcontext s .L e t Γ,  ∆ ,Θ be  afi nitemult i set , i.e.a list wi thm ulti pl ic ity but  no order, of  a ssumptions  i n s equent c alculus.  We use a binary deriva t ion sym bol  ⇒ an d 'Γ  ⇒ φ' mea nsthat t hea nteced ent Γderiv es  th e succe d e nt  φ.  A  sequent c a l cul us ha sthefollowi ng left and rightr ule s for → and s tru ctur a l  r ule s ,s uch  a s  cu t - , weakening, an d contract io n  r ules. Γ ⇒φ  ψ, ∆⇒ σ L →  φ → ψ, Γ,∆ ⇒ σ φ,Γ ⇒  ψ R → Γ⇒ φ → ψ 1 19  Γ ⇒ φ  φ, ∆ ⇒ Ψ CutΓ,∆ ⇒ ΨΓ ⇒ Ψ Wkφ ,Γ ⇒ψ  φ ,φ,Γ⇒ Ψ  Ctr φ ,Γ  ⇒Ψ φ i n Cut− r ule  is c alledacut-fo rmula .Let us r emind the reduction pro cess f or →inthe style  of gen eralizedelim ination ru le.  [φ ]1 D1  ψ→  I,1φ →ψ  D 2
 has_an application_of an elimination rule_with its_maximum_formula. On_the_other hand, it_is not easy_to characterize a 'detour'_in terms of_the_length of a derivation. There are cases that the length of a normal derivation_is_longer than_the_length_of an original derivation. It_cannot be said that a_derivation of_a longer length has a detour and a_derivation_with a shorter_length has no detour. Thus, we only use a_'detour reasoning' as defined through a_maximum formula. Then,_(T2)_means_that a proper reduction_should neither introduce an application of_an elimination rule containing a maximum_formula nor increase the degree of maximum_formulas. To check (T2), it is_useful to translate a reduction_in natural_deduction to a one in_sequent calculus. It_is often_claimed that a_natural deduction system is isomorphic to_a sequent calculus_system if the generalized eliminations are_used._(Cf. Tennant (2002),_Negri_and_Von Plato_(2001) and Von_Plato_(2011)). Especially,_Negri_and Von Plato (2001, Ch.1 and_Ch._8) propose an inductive definition of the_translation algorithm between an_intuitionistic_natural deduction system with_generalized elimination rules and an_intuitionistic sequent calculus system with independent_contexts. Let_Γ, ∆,_Θ be a finite multiset, i.e. a list with multiplicity but_no order, of assumptions in sequent_calculus. We use a_binary derivation_symbol_⇒ and 'Γ_⇒_φ' means_that the antecedent Γ derives the succedent_φ. A_sequent calculus has the following left_and right rules for_→_and structural rules, such as cut-,_weakening, and contraction rules. Γ ⇒_φ ψ,∆ ⇒ σ L_→_φ_→ ψ,Γ,∆ ⇒ σ φ,Γ_⇒ ψ R → Γ ⇒_φ → ψ_119 Γ ⇒ φ φ,∆ ⇒ Ψ_Cut_Γ,∆ ⇒ Ψ Γ ⇒ Ψ_Wk_φ,Γ ⇒ ψ φ,φ,Γ ⇒ Ψ_Ctr_φ,Γ_⇒ Ψ φ in Cut−rule_is called a cut-formula. Let us_remind the reduction process for → in the style_of generalized elimination_rule. [φ]1 D1 ψ →_I,1_φ_→ ψ D2