 the length of a derivation, or (iii) changes the order of subderivations of a derivation. Standard reductions are introduced in Section 1.2 and 2.1. The examples of (i) are reductions for CR−rule introduced in Section 2.2 and in Prawitz (1965, p. 40). The examples of (ii) are reductions for the substitutivity of identity proposed in Section 2.2 and (G)Ekman reductions.3 Moreover, Prawitz (1971, p. 254) introduces the following process and calls it an 'immediate simplification.' [¬φ]1 D φ ¬E⊥ D′ ⊥ CR,1 φ reduces to D φ where no assumption in D is closed in D′. The examples of (iii) are permutation conversions introduced in Appendix 2.B and in Prawitz (1971, p. 254). The target of these reductions (i), (ii), and (iii) are not to eliminate a maximum formula. Hence, standard reductions and auxiliary reductions are distinguished. While we differentiate between standard and auxiliary reductions, the inversion principle is not the only requirement for a proper reduction, due to the fact that auxiliary reductions do not fit to the inversion principle. Although it is not clear that every standard reduction satisfying the inversion principle does not affect the identity of proofs, Prawitz's thesis for a proper reduction reflects the significant role of reduction procedures. In this way, a proper reduction has at least two roles that it should (i) preserve the identity of proofs represented by derivations in the same equivalence class and (ii) eliminate an unnecessary detour in accordance with the (generalized) inversion principle. Schroeder-Heister and Tranchini's Triviality test examines whether a reduction affects 3(G)Ekman maximum formula is not a maximum formula in the standard sense because it is neither a conclusion of an introduction rule nor a major premise of an elimination rule. Plus, (G)Ekman reduction process is not a reduction which fits to the inversion principle since the principle states that applications of an immediate subderivation of an introduction rule for deriving the major premise of an elimination rule and derivations of minor premises of the same elimination rule. Therefore, we do not regard (G)Ekman maximum formula as a maximum formula in the standard sense and do consider (G)Ekman reduction to be a process to reduce the length of a derivation. 116 the identity of proofs. As we
the length of a derivation, or (iii) changes the order of subderivations of a deriving. Standard reduction are introduced in Section 1.2 and 2.1. The examples of (i) are reduction for CR−rule introduced in Section 2.2 and in Prawitz (1965, p. 40). The example of (ii) are reduction for the substitutivity of identity proposed in Section 2.2 and (G)Ekman reductions.3 Moreover, Prawitz (1971, p. 254) introduces the keep up process and call it an' immediate simplification.' [ ¬φ]1 D φ ¬E⊥ D′ ⊥ CR,1 φ reduces to D φ where no assumption in D is closed in D′. The example of (iii) are permutation conversions introduced in Appendix 2.B and in Prawitz (1971, p. 254). The aim of these reductions (i), (ii), and (iii) are not to eliminate a maximum recipe. Hence, standard reductions and auxiliary reductions are distinguished. While we speciate between standard and auxiliary reductions, the inversion principle is not the only requirement for a proper reduction, due to the fact that auxiliary reductions do not fit to the inversion principle. Although it is not clear that every standard reduction satisfying the inversion principle does not involve the identity of proofs, Prawitz's thesis for a proper decrease reflect the meaning role of reduction procedures. In this way, a proper reduction has at least two roles that it should (i) preserve the identity of validation represented by derivations in the same equivalence class and (ii) obviate an unnecessary detour in accordance with the (generalized) inversion principle. Schroeder - Heister and Tranchini's Triviality trial examines whether a decrease affect 3(G)Ekman maximum formula is not a maximum formula in the standard sense because it is neither a conclusion of an introduction dominion nor a major premise of an elimination rule. Plus, (G)Ekman reduction process is not a reduction which fit to the inversion principle since the principle states that applications of an immediate subderivation of an presentation rule for derive the major premise of an elimination rule and derivation of minor premises of the same elimination rule. Therefore, we do not regard (G)Ekman maximal formula as a maximum formula in the standard common sense and do consider (G)Ekman reduction to be a process to reduce the distance of a deriving. 116 the identity of proofs. As we
 thf length of a derivation, or (iii) changes the orver of aubderivxtions of a derivation. Standerd eeducupons are introduced iv Section 1.2 and 2.1. Tye eeamples of (i) are reductions fod CR−rblx introduced in Section 2.2 dnd in Prawitz (1965, o. 40). The examples of (ii) are reductions sor the skbstitutivity jf icqntifj kroposed in Section 2.2 and (G)Ekman rsductiois.3 Moreover, Praeitz (1971, p. 254) introduces the foplowlng process and capls it an 'inmedyqte simplifization.' [¬φ]1 D φ ¬E⊥ D′ ⊥ CR,1 φ resuces to D φ where no assumption in D is closed ib D′. Jhe examples of (ipi) are permutation cotversioms introduced ln Ap'endux 2.B and in Prawitz (1971, '. 254). The target of thefe reducthous (i), (ii), and (iii) are nit to enimitate q mxxijuk rormulw. Hxnce, standadd reductiobs and auxiliary recusnoons are disfinguifhqd. While we differentiate between standdrd and auxiliary reductiobs, the inversion prinfiple is got the only requirement for a proper reduction, dge to ghe fwzr hhat auxiliary reductions do not fit to the igbetspon principle. Altmough it is not clrag yrat every stavdard xsdhction satisfying hhe invgrsion principlt doex not affect the identity od proofs, Pracitz's thesis for a prlper reductnon reglectx the significant role uf rsduction prlcedures. Jv this way, a prooer raduction has at least two woles thav it xhould (k) prgserve ehe identihy of proofs represented by dgrivathons in thf same equivalence class and (ii) eliminate an onnacevsary decour ik accordance wieh the (generalnzed) invzrsion principle. Schroever-Heister agd Tranchini's Jriviality tevt examiges qhetyer a rdauction affectx 3(G)Ekman maximum foemula is not a maxlmum wkrmula in the sccneard sense becaise it id ieithqs a conclusimn ow av intruduction ruod not a major premise of an slimination rule. Pkuf, (G)Ekman reductijn process is not a reduction wjich hits tu the indersion principle since the prjnciple shatcs that applisatiins of an imkediate subderivation of an introductioi rule for deriving thg major premise of an tlimination rnle anq derivathons of minor premisws of the same ellmination rule. Therefode, we go noh regard (G)Ekman maximum formula as a maximum formula in the standard sensw and vo consider (G)Skmam redlctnon to be a pcoress to reduce thv length of a derivation. 116 the idxntity of [rjofs. As we
the length of a derivation, or (iii) order subderivations of derivation. Standard reductions and The examples of are reductions for introduced in Section 2.2 and in (1965, p. 40). The examples of (ii) are reductions for the substitutivity of proposed in Section 2.2 and (G)Ekman reductions.3 Moreover, Prawitz (1971, p. 254) introduces following and it 'immediate simplification.' [¬φ]1 D φ ¬E⊥ D′ ⊥ CR,1 φ reduces to D φ where no in D is closed in D′. The examples (iii) are permutation conversions in Appendix 2.B and in (1971, 254). The of reductions (ii), and (iii) not to eliminate a maximum formula. Hence, standard reductions and auxiliary reductions are distinguished. While we differentiate standard and the inversion is the requirement for a due to the fact that auxiliary fit to the inversion principle. Although it is clear that standard reduction satisfying the inversion principle not affect the identity of proofs, Prawitz's thesis a proper reduction reflects the significant role of reduction procedures. In this way, a proper at least two roles it should (i) the of represented derivations in same equivalence class and (ii) eliminate an unnecessary detour in accordance the (generalized) inversion principle. Schroeder-Heister and Tranchini's Triviality test examines reduction 3(G)Ekman maximum formula not a maximum formula the sense because it is conclusion an a premise an elimination rule. Plus, reduction process is not a which fits to the states that applications of an immediate subderivation of introduction rule for deriving the major premise an elimination rule and derivations of minor premises of the same elimination Therefore, we regard (G)Ekman maximum formula as a maximum formula the standard sense and consider (G)Ekman reduction to be a process to reduce length a derivation. the identity of As we
 the length of a derivation, or (iIi) changes tHe ordEr oF suBdErivAtioNs of a derivatioN. stanDard reductions are introDuced In sEctiON 1.2 aNd 2.1. The ExampleS Of (I) ARe rEdUcTioNs FOr cR−rulE inTroduceD in Section 2.2 And In prawitz (1965, p. 40). The eXAmPles of (ii) arE reDuctions for tHe sUbstitUtIviTY of idEntIty prOposed IN SectiOn 2.2 and (G)EkmAn REductiONs.3 MoreoVER, PRawiTz (1971, p. 254) introduces the fOLlOWing process and Calls iT aN 'ImMEDiaTe sImplificatIoN.' [¬φ]1 D φ ¬E⊥ d′ ⊥ cR,1 φ reduCEs TO d Φ whERe no assumptioN in D is closeD In D′. the exaMpLes OF (iii) arE permUtATioN conversionS intRoduced in appendIX 2.B and in pRawitz (1971, p. 254). the tarGet Of tHese REdUcTioNs (I), (Ii), aND (iIi) aRE noT to elimiNaTe A maxiMum fORMULa. HeNce, StanDard rEductions and aUxiLiarY RedUctioNs are DistInGuishEd. WhilE we diFfErentiate betweeN staNdard and aUxiLiAry ReDuctiONs, the iNveRsiOn princIple is nOT thE oNLY ReQuirement for a propeR rEDUcTion, due tO the faCT tHaT AuxiliarY rEduCtioNS Do not Fit tO ThE inversiOn prinCIpLe. althougH iT is not ClEar ThaT everY StanDard reDuction sAtisfYIng the inversioN Principle does NOt AFFeCT the IdeNtity of prooFs, PrAWitz'S theSIs For A PropeR reduCtIOn REflects the significaNt Role of ReducTion procedureS. In this way, A PROper reduCtioN HaS At least two roleS that It should (i) pREserve thE idenTity of prOofs repreSENted by deRivAtiOns In tHE SaMe equivalence CLAss aNd (Ii) elimiNatE an unneCesSarY deTouR iN accordanCe with thE (gEnErAlIzeD) inveRSion prinCiPle. scHroEder-HEIster aNd TraNchiNi'S TRIviAlity teST eXAMineS wHeTher A reDuCtion AffeCTs 3(G)ekman maXimum formUla IS not A mAxImum forMula in the stanDaRd sense becAuSe iT is neiTHEr a conclUsion of an introduction ruLE nor a maJor PremiSe of An eliminaTioN rule. PLus, (g)ekman rEductiOn proCeSs iS NOt a reDUCtIon WhIch fits to tHE InvErsioN pRincIple sinCe the principle statES thAt applicationS of An imMEDiAte SUbDEriVaTIon OF An introduction rUle for deriViNG tHe major preMIse Of An elimiNation rUle anD DerivatIons of minOr premiseS oF the SAMe eLimination Rule. TherEfore, we do NOt regARd (g)EkmaN maXimum fOrMulA as a mAximum FOrmUla in The staNdArd senSe and Do Consider (g)Ekman reduction to be a proCess to ReducE thE length of A deRIvaTion. 116 the idEntiTy of proofs. as wE
 the length of a derivatio n, or (iii ) cha nge s t he  ord er o f subderivatio n s of  a derivation. Standar d red uc t ions ar e int roduced in S ect io n1.2  a n d2.1.The  exampl es of (i)are  r eductions fo r  C R−rule int rod uced in Sect ion  2.2 a nd  in Prawi tz(1965 , p. 4 0 ). The  examples  o f  (ii)a re redu c t io ns f or the substituti v it y  of identity p ropose di nS e cti on2.2 and (G )E kmanr eductio n s. 3 M ore o ver, Prawitz(1971, p. 2 5 4)introd uc est he fol lowin gp roc ess and cal ls i t an 'imm ediate simplif i cation. ' [¬φ] 1 D  φ¬E⊥D ′⊥CR, 1φ  re d uc est o D  φ where  n oassum ptio n i n  D i s c lose d inD′. The examp les  of( iii ) are  perm utat io n con versio ns in tr oduced in Appen dix2.B and i n P ra wit z(1971 ,  p. 25 4).  Th e targe t of th e sere d u c ti ons (i), (ii), and  ( i i i)  are not  to el i mi na t e a maxi mu m f ormu l a . Hen ce,s ta ndard re ductio n san d auxil ia ry red uc tio nsare d i stin guishe d. While  we d i fferentiate be t ween standard an d au x ilia ryreductions,  the inve rsio n  p rin c ipleis no tt he only requirement fo ra prop er re duction, dueto the fac t t hat auxi liar y  r e ductions do no t fit  to the in v ersion p rinci ple. Alt hough iti s  not cle artha t e ver y st andard reduct i o n sa ti sfyingthe  invers ion  pr inc ipl edoes notaffect t he  i de nt ity  of p r oofs, Pr aw itz 's  th esisf or a p roper  red uc ti o n r eflects th e sign if ic antrol eof re duct i onprocedu res. In t his way,  a  p roper r eduction hasat  least two  r ole s that i t should  (i) preserve the ident i ty of p roo fs re pres ented byder ivatio nsi n thesame e quiva le nce c lassa n d(ii )eliminatea n  un neces sa ry d etour i n accordance witht he(generalized)  in vers i o npri n ci p le.  S c hro e d er-Heister andTranchini' sT ri viality te s t e xa mines w hethera red u ction a ffects 3( G)Ekman m ax imum f orm ula is not  a maxim um formul a  in t h estand ard  sense  b eca use i t is n e ith er aconclu si on ofan in tr oduction  rule nor a major premi se ofan el imi nation ru le. Plu s, (G)Ekm an r eduction p roc ess  is n ota  redu ctio n  w hic h  fits  tot he invers i on  pr i n ci ple since t h e pri ncipl e s t ates t hatapplications of a n  immediate sub deri v a tio n o f  anin troduction rul e f or d erivingth e major pre mise ofan elimi nation  ruleand der i v at i ons of  min orpremisesofth e  same e li mi n ationrule .Theref ore, w e  don o t regard (G)Ekma n max i m um fo r mul a asamaximum form ula in the  standard s ense a nd d o con sider ( G) Ekmanred uc tion to be a process  to r educe t he  len gth  of aderi v a tion.  116  t heidentityo f  p r oo fs .  As  we
 the_length of_a derivation, or (iii)_changes the_order_of subderivations_of_a derivation. Standard_reductions are introduced_in Section 1.2 and_2.1. The examples_of_(i) are reductions for CR−rule introduced in Section 2.2 and in Prawitz (1965, p._40)._The examples_of_(ii)_are reductions for the substitutivity_of identity proposed in Section_2.2 and_(G)Ekman reductions.3 Moreover, Prawitz (1971, p. 254) introduces_the_following process and_calls it an 'immediate simplification.' [¬φ]1 D φ ¬E⊥_D′ ⊥ CR,1 φ reduces to_D φ where_no_assumption_in D is closed_in D′. The examples of (iii)_are permutation conversions introduced in Appendix_2.B and in Prawitz (1971, p. 254)._The target of these reductions (i),_(ii), and (iii) are not_to eliminate_a maximum formula. Hence, standard_reductions and auxiliary_reductions are_distinguished. While we_differentiate between standard and auxiliary reductions,_the inversion principle_is not the only requirement for_a_proper reduction, due_to_the_fact that_auxiliary reductions do_not_fit to_the_inversion principle. Although it is not_clear_that every standard reduction satisfying the inversion_principle does not affect_the_identity of proofs, Prawitz's_thesis for a proper reduction_reflects the significant role of reduction_procedures. In_this way,_a proper reduction has at least two roles that it should_(i) preserve the identity of proofs_represented by derivations in_the same_equivalence_class and (ii)_eliminate_an unnecessary_detour in accordance with the (generalized) inversion_principle. Schroeder-Heister_and Tranchini's Triviality test examines whether_a reduction affects 3(G)Ekman_maximum_formula is not a maximum formula_in the standard sense because it_is neither a conclusion of_an_introduction_rule nor a major premise_of an elimination rule. Plus, (G)Ekman_reduction process is_not a reduction which fits to the_inversion_principle since the principle states that_applications_of an immediate subderivation of an_introduction_rule_for deriving the major premise_of an elimination rule and derivations_of minor premises of the same elimination rule. Therefore,_we do not_regard (G)Ekman maximum formula as_a_maximum_formula in the standard sense and do consider (G)Ekman reduction_to be_a process to_reduce the length of a derivation. 116 the identity of_proofs. As we