 reduction procedures because it is relative to a given system. 3.3.2 Problems of Triviality Test Triviality test fails to block every Ekman-type reduction procedures because it works relative to a natural deduction system. Schroeder-Heister and Tranchini's derivation Π1 is proposed in a natural deduction system with the standard elimination rules. As we have discussed in Section 3.2.2, there is an Ekman-type reduction with the generalized elimination rule for →, such as GEkman reduction procedure. Let ST G be a natural deduction system containing ∧− and →−rules with their generalized elimination rules. A set RT G of reductions only has the standard reduction processes for ∧ and → with generalized elimination rules for ∧ and →. R′T G is an extension of RT G by adding GEkman reduction procedure. Then ST G has a similar derivation with Π1 formulated in the generalized form. [φ]1[φ]1 ∧I φ ∧φ → I,1 φ → (φ ∧φ) [φ ∧φ]2[φ]3 ∧E,3 φ → I,2(φ ∧φ)→ φ D1 φ D2 φ ∧I φ ∧φ [φ]4 → E,4 φ [φ ∧φ]5 → E,5 φ ∧φ We call the above derivation Σ1. The minor premise φ in the last → E−rule is a GEkman maximum formula. The application of GEkman reduction provides the following derivation 110 Σ2. D1 φ D2 φ ∧I φ ∧φ Moreover, we apply →−reduction to Σ1 twice and have the derivation below: D1 φ D2 φ ∧I φ ∧φ [φ]3 ∧E,3 φ D1 φ D2 φ ∧I φ ∧φ [φ]4 ∧E,4 φ ∧I φ ∧φ ∧I−rule has only one generalized elimination rule for ∧ and its reduction procedure uses both derivations of two conjuncts. By applying ∧−reduction again, we take the derivation Σ3 below: D1 φ D2 φ ∧I φ ∧φ Unlike Schroeder-Heister and Tranchini's case, since Σ2 is the same derivation with Σ3, the result does not lead to a conclusion that GEkman reduction trivializes the identity of proofs in ST
reduction procedures because it is relative to a given arrangement. 3.3.2 trouble of Triviality Test Triviality test fails to block every Ekman - character reduction procedures because it works proportional to a lifelike deduction system. Schroeder - Heister and Tranchini's derivation Π1 is proposed in a natural discount system with the standard elimination rules. As we have discourse in Section 3.2.2, there is an Ekman - type reduction with the generalized elimination rule for →, such as GEkman decrease procedure. Let ST G be a lifelike deduction system containing ∧− and →−rules with their generalized elimination rules. A set RT gram of reductions only has the standard reduction processes for ∧ and → with generalized elimination convention for ∧ and →. R′T G is an extension of RT G by adding GEkman reduction operation. Then ST G has a similar derivation with Π1 formulated in the generalized form. [ φ]1[φ]1 ∧I φ ∧φ → I,1 φ → (φ ∧φ) [ φ ∧φ]2[φ]3 ∧E,3 φ → I,2(φ ∧φ)→ φ D1 φ D2 φ ∧I φ ∧φ [ φ]4 → E,4 φ [ φ ∧φ]5 → E,5 φ ∧φ We call the above derivation Σ1. The minor premise φ in the last → E−rule is a GEkman maximum formula. The lotion of GEkman decrease provides the following derivation 110 Σ2. D1 φ D2 φ ∧I φ ∧φ furthermore, we lend oneself →−reduction to Σ1 twice and have the derivation below: D1 φ D2 φ ∧I φ ∧φ [ φ]3 ∧E,3 φ D1 φ D2 φ ∧I φ ∧φ [ φ]4 ∧E,4 φ ∧I φ ∧φ ∧I−rule has only one generalized elimination rule for ∧ and its decrease procedure uses both derivations of two conjuncts. By applying ∧−reduction again, we take the derivation Σ3 below: D1 φ D2 φ ∧I φ ∧φ Unlike Schroeder - Heister and Tranchini's case, since Σ2 is the same derivation with Σ3, the result does not lead to a stopping point that GEkman decrease trivialize the identity of proof in ST
 refuction procedures becaure it is relatirw to a given system. 3.3.2 Problems of Triviality Test Teiviaoity test fails to blozk every Vkman-type rednction procedures becausc it slrks celative to a nstural dedgction system. Vcfrleder-Heister and Tranchini's derivatyon Π1 is pgoposed in a nwtursj deslcuion system with the standard elijinatioi rules. As we hsve discussed in Section 3.2.2, hherf is an Ekman-type geduction wuth eye generalizdd eliminauimn rule fot →, such as GEkman reduction procddure. Let ST G ve a jdtural dedurtion fystem contalming ∧− dnd →−rulrs with their nenerelizwd elimination rules. E set RT G of reductyons only hcs the standard reducriin prmcesves wir ∧ ans → wjth gejerelized elimjnation rulws for ∧ and →. R′T G ix wb extension or RT G br adding GEkman reduction procedure. Thet SF G has a similar derivqtion with Π1 formulatef in the deneralized form. [φ]1[φ]1 ∧I φ ∧φ → I,1 φ → (φ ∧φ) [φ ∧φ]2[φ]3 ∧E,3 φ → I,2(φ ∧φ)→ φ D1 φ D2 φ ∧I φ ∧φ [φ]4 → E,4 φ [φ ∧φ]5 → E,5 φ ∧φ Dw fall the above derivation Σ1. The minor premise φ jn tme last → E−rule if a GEkman kadikom formula. The applieztjon of GEkman reduftion ptovidew the foljowimg derivation 110 Σ2. D1 φ D2 φ ∧I φ ∧φ Mireover, we aiply →−reduction to Σ1 twiee and have che detivatipn below: D1 φ D2 φ ∧I φ ∧φ [φ]3 ∧E,3 φ B1 φ D2 φ ∧J φ ∧φ [φ]4 ∧E,4 φ ∧I φ ∧φ ∧I−rule haa only one generauizvd enimination rule for ∧ and ies reductmon pxocedure usex both derivatiojs of two conjuncts. By aoplyiug ∧−reguction agwin, we take the derivation Σ3 below: D1 φ D2 φ ∧I φ ∧φ Umlhke Schroedzr-Heisber and Tranchigi's case, since Σ2 is thz same derivatioh with Σ3, the result does not leag to a concluvion thae GEjman reductkun trivializes the idenniuy of proods in ST
reduction procedures because it is relative to system. Problems of Test Triviality test reduction because it works to a natural system. Schroeder-Heister and Tranchini's derivation Π1 proposed in a natural deduction system with the standard elimination rules. As we discussed in Section 3.2.2, there is an Ekman-type reduction with the generalized elimination for such GEkman procedure. Let ST G be a natural deduction system containing ∧− and →−rules with their generalized rules. A set RT G of reductions only the standard reduction processes ∧ and → with generalized rules ∧ and R′T is extension of RT by adding GEkman reduction procedure. Then ST G has a similar derivation with Π1 formulated in the form. [φ]1[φ]1 ∧φ → φ (φ [φ ∧φ]2[φ]3 ∧E,3 I,2(φ ∧φ)→ φ D1 φ D2 ∧φ [φ]4 → E,4 φ [φ ∧φ]5 → φ ∧φ call the above derivation Σ1. The premise φ in the last → E−rule is GEkman maximum formula. The application of GEkman reduction provides the following derivation 110 Σ2. D1 φ ∧I φ ∧φ we apply →−reduction Σ1 and the below: D1 D2 φ ∧I φ ∧φ [φ]3 ∧E,3 φ D1 φ D2 ∧I φ ∧φ [φ]4 ∧E,4 φ ∧I φ ∧φ ∧I−rule one elimination rule for and its reduction procedure both of two conjuncts. By again, take below: φ φ ∧I φ ∧φ Schroeder-Heister and Tranchini's case, since is the same derivation not lead to a conclusion that GEkman reduction the identity of proofs in ST
 reduction procedures becausE it is relatIve to A giVen SyStem. 3.3.2 probLems of TrivialiTY TesT Triviality test fails to Block EvERy EkMAn-Type rEductioN PrOCEduReS bEcaUsE It Works RelAtive to A natural deDucTiOn system. SchrOEdEr-Heister aNd TRanchini's derIvaTion Π1 iS pRopOSed in A naTural DeductIOn systEm with the StANdard eLIminatiON RuLes. AS we have discussed iN seCTion 3.2.2, there is an EKman-tyPe REdUCTioN wiTh the generAlIzed eLIminatiON rULE For →, SUch as GEkman reDuction procEDurE. Let ST g bE a nATural dEductIoN SysTem containiNg ∧− anD →−rules witH their GEneraliZEd elimiNation RulEs. A Set Rt g oF rEduCtIOns ONlY haS The Standard ReDuCtion ProcESSES for ∧ And → With GenerAlized eliminaTioN rulES foR ∧ and →. R′t G is aN extEnSion oF RT G by AddinG Gekman reduction pRoceDure. Then St G hAs A siMiLar deRIvatioN wiTh Π1 FormulaTed in thE GenErALIZeD form. [φ]1[φ]1 ∧I φ ∧φ → I,1 φ → (φ ∧φ) [φ ∧φ]2[φ]3 ∧E,3 Φ → I,2(Φ ∧Φ)→ Φ D1 Φ D2 φ ∧I φ ∧φ [φ]4 → E,4 Φ [φ ∧φ]5 → E,5 φ ∧φ wE cAlL The above DeRivAtioN σ1. the miNor pREmIse φ in thE last → E−RUlE iS a GEkmaN mAximum FoRmuLa. THe appLIcatIon of Gekman redUctioN Provides the folLOwing derivatiON 110 Σ2. d1 Φ d2 φ ∧i Φ ∧φ MoReoVer, we apply →−rEducTIon tO Σ1 twICe And HAve thE deriVaTIoN Below: D1 φ D2 φ ∧I φ ∧φ [φ]3 ∧E,3 φ D1 φ D2 φ ∧i φ ∧Φ [φ]4 ∧E,4 φ ∧I φ ∧Φ ∧I−rulE has only one geNeralized eLIMInation rUle fOR ∧ aND its reduction pRocedUre uses botH DerivatiOns of Two conjuNcts. By appLYIng ∧−reducTioN agAin, We tAKE tHe derivation Σ3 BELow: D1 Φ D2 Φ ∧I φ ∧φ UnlIke schroedEr-HEisTer And trAnchini's cAse, since σ2 iS tHe SaMe dErivaTIon with Σ3, ThE reSuLt dOes noT Lead to A concLusiOn ThAT GEKman redUCtION triViAlIzes The IdEntitY of pROofS in ST
 reduction procedures beca use it isrelat ive  to  a  giv en s ystem. 3.3.2 P r oble ms of Triviality TestTrivi al i ty t e st  fail s to bl o ck e ver yEk man -t y pe  redu cti on proc edures bec aus eit works rel a ti ve to a na tur al deduction  sy stem.Sc hro e der-H eis ter a nd Tra n chini' s derivat io n  Π1 is propose d in  a n atural deductions ys t em with the st andard  e l im i n ati onrules. Aswe  have discuss e di n Sec t ion 3.2.2, th ere is an E k man -typere duc t ion wi th th eg ene ralized eli mina tion rule  for → ,  such a s  GEkman  reduc tio n p roce d ur e.  Le tS T G be  an atu ral dedu ct io n sys temc o n t aini ng∧− a nd →− rules with th eir  gen e ral izedelimi nati on  rule s. A s et RT  G  of reductionsonly  has thesta nd ard  r educt i on pro ces ses  for ∧and → w i thge n e r al ized elimination r ul e s  f or ∧ and  →. R′ T  G  i s  an exte ns ion  ofR T  G by  add i ng  GEkmanreduct i on  p rocedur e.  ThenST  Ghas  a si m ilar  deriv ation wi th Π1 formulated int he generalize d  f o r m. [φ]1 [φ] 1 ∧I φ ∧φ →  I,1 φ →(φ ∧ φ )[φ∧ φ]2[φ ]3 ∧E ,3 φ→  I,2(φ ∧φ)→ φ D1 φD2  φ ∧Iφ ∧φ[φ]4 → E,4 φ[φ ∧φ]5 →E , 5  φ ∧φ We  cal l  t h e above deriva tionΣ1. The mi n or premi se φin the l ast → E−r u l e is a G Ekm anmax imu m fo rmula. The ap p l icat io n of GE kma n reduc tio n p rov ide sthe follo wing der iv at io n110  Σ2.D 1 φ D2 φ  ∧ I φ  ∧ φ M oreov e r, weapply  →−r ed uc t ion  to Σ1t wi c e  and  h av e th e d er ivati on b e low : D1 φD2 φ ∧I φ  ∧φ [φ]3  ∧ E, 3 φ D1φ D2 φ ∧I φ ∧ φ[φ]4 ∧E,4φ∧Iφ ∧φ ∧ I − rule has  only one generalized e l iminati onrulefor∧ and its  re ductio n p r ocedur e uses  both  d eri v a tions o ftwo  c onjuncts.B y  ap plyin g∧−re duction  again, we take th e  de rivation Σ3 b elo w: D 1 φD2φ  ∧ I  φ∧φ Unl i k e Schroeder-Hei ster and T ra n ch ini's case ,  si nc e Σ2 is  the sa me de r ivation  with Σ3,  the resu lt  doe s not  lead to a  conclus ion thatG Ekman re ducti ontrivia li zes  theidenti t y o f pro ofs in  S T
 reduction_procedures because_it is relative to_a given_system._3.3.2 Problems_of_Triviality Test Triviality_test fails to_block every Ekman-type reduction_procedures because it_works_relative to a natural deduction system. Schroeder-Heister and Tranchini's derivation Π1 is proposed in_a_natural deduction_system_with_the standard elimination rules. As_we have discussed in Section_3.2.2, there_is an Ekman-type reduction with the generalized elimination_rule_for →, such_as GEkman reduction procedure. Let ST G be a_natural deduction system containing ∧− and_→−rules with their_generalized_elimination_rules. A set RT_G of reductions only has the_standard reduction processes for ∧ and_→ with generalized elimination rules for ∧_and →. R′T G is an_extension of RT G by_adding GEkman_reduction procedure. Then ST G_has a similar_derivation with_Π1 formulated in_the generalized form. [φ]1[φ]1 ∧I φ_∧φ → I,1_φ → (φ ∧φ) [φ ∧φ]2[φ]3_∧E,3_φ → I,2(φ_∧φ)→_φ_D1 φ_D2 φ ∧I_φ_∧φ [φ]4_→_E,4 φ [φ ∧φ]5 → E,5_φ_∧φ We call the above derivation Σ1._The minor premise φ_in_the last → E−rule_is a GEkman maximum formula._The application of GEkman reduction provides_the following_derivation 110_Σ2. D1 φ D2 φ ∧I φ ∧φ Moreover, we apply_→−reduction to Σ1 twice and have_the derivation below: D1_φ D2_φ_∧I φ ∧φ_[φ]3_∧E,3 φ_D1 φ D2 φ ∧I φ ∧φ_[φ]4 ∧E,4_φ ∧I φ ∧φ ∧I−rule has_only one generalized elimination_rule_for ∧ and its reduction procedure_uses both derivations of two conjuncts._By applying ∧−reduction again, we_take_the_derivation Σ3 below: D1 φ_D2 φ ∧I φ ∧φ Unlike_Schroeder-Heister and Tranchini's_case, since Σ2 is the same derivation_with_Σ3, the result does not lead_to_a conclusion that GEkman reduction trivializes_the_identity_of proofs in ST