 Since, by Prawitz's thesis, a proper reduction does not affect the identity of proofs and, by strong normalization theorem, a derivation representing the same proof has a unique normal derivation, we may conclude that ⊵ is not a proper reduction procedure. From this perspective, Schroeder-Heister and Tranchini (2017, p. 575) propose Triviality test for a proper reduction process that a newly added reduction procedure should not trivialize the identity of proofs. A natural requirement of the addition of a new reduction could be that of not trivializing identity of proof, in the sense that it should always be possible to exhibit two derivations of the same conclusion belonging to two distinct equivalence classes. (Schroeder-Heister and Tranchini, 2017, p. 575) We say that, for any derivations D1 and D2 which represent different proofs, i.e. D1 ≁D2, in S(R), a reduction process ⊵ trivializes the identity of proofs in S iff it is not possible to show in S(R′) that D1 ≁ D2; otherwise, ⊵ does not trivialize the identity of proofs in S. Then, their Triviality test can be summarized as below: Triviality Test: Let S be any natural deduction system. Let ⊵ be a newly added reduction procedure not in R and R′ be an extension of R by adding ⊵. ⊵ is a proper reduction procedure for S iff ⊵ does not trivialize the identity of proofs in S. In order to show that Ekman reduction fails to pass Triviality test, Schroeder-Heister and Tranchini (2017, pp. 575-577) provide an example that Ekman reduction trivializes the identity of proofs. Let ST be a natural deduction system containing ∧− and →−rules with their standard elimination rules. ST has a set RT of standard reductions for ∧ and →. We say that R′T is an extension of RT by adding Ekman reduction ⊵E. If Ekman reduction ⊵E is a proper reduction, it does not trivialize the identity of proofs. Schroeder-Heister and Tranchini's example shows that it is not possible to show that two reduced derivations from Π1 in ST (R′T ) which represent different proofs are not equivalent. 108 Suppose that, for any two derivations D1 and D2 of φ
Since, by Prawitz's thesis, a proper reduction does not affect the identity of proofs and, by impregnable standardization theorem, a derivation representing the like proof have a unique normal ancestry, we may reason that ⊵ is not a proper reduction procedure. From this perspective, Schroeder - Heister and Tranchini (2017, p. 575) propose Triviality examination for a proper reduction process that a newly add decrease procedure should not trivialize the identity of proofs. A lifelike requirement of the accession of a fresh reduction could be that of not trivializing identity of proof, in the common sense that it should always be possible to expose two derivations of the like conclusion belong to two distinct equivalence classes. (Schroeder - Heister and Tranchini, 2017, p. 575) We allege that, for any derivations D1 and D2 which represent different proofs, i.e. D1 ≁D2, in S(R), a reduction procedure ⊵ trivializes the identity of proofs in S iff it is not possible to show in S(R′) that D1 ≁ D2; otherwise, ⊵ does not trivialize the identity of proofs in S. Then, their Triviality test can be summarized as below: pettiness Test: Let south be any lifelike deduction system. Let ⊵ be a newly added reduction procedure not in R and R′ be an extension of radius by adding ⊵. ⊵ is a proper reduction procedure for S iff ⊵ does not trivialize the identity of proofs in S. In order to show that Ekman decrease fails to pass Triviality test, Schroeder - Heister and Tranchini (2017, pp. 575 - 577) provide an example that Ekman reduction trivialize the identity of proof. Let ST be a natural deduction system containing ∧− and →−rules with their standard elimination rules. ST has a located RT of standard reductions for ∧ and →. We say that R′T is an extension of RT by adding Ekman decrease ⊵E. If Ekman reduction ⊵E is a proper reduction, it does not trivialize the identity of proofs. Schroeder - Heister and Tranchini's example shows that it is not potential to show that two reduced derivations from Π1 in ST (R′T) which represent unlike proofs are not equivalent. 108 Suppose that, for any two derivation D1 and D2 of φ
 Sijce, by Prawitz's thesis, a proper reduction does not arfect thd identity of proofs and, by dteong bormalization theorem, x derivatpon reprewentmng the same proof has a uniqus norkel derivation, wg may concluge that ⊵ is nod x 'roper reduction procedure. From this perspevtlve, Schroeder-Hgisteg wnd Fgakchini (2017, p. 575) propose Triviality teat for e proper reductoon process that a newly afded reduction proceduge should nit twuvialize the identity of proofs. A hatural requirement of the addigion pf a new rgbyctlmn could be that of not trivlslizinc identoty of proof, ik the sebse that it should alxays be possible to gxhibit twm berivations of the sane concnusimn bdooneinf vo fwo didtiict equivalsnce classew. (Schroeder-Heister snq Tranchini, 2017, p. 575) We swy that, for any derivations D1 and D2 which relresent different proofw, i.e. D1 ≁D2, in S(R), a reduftion prosess ⊵ trivializes the identity of proofs in S iff it ia nou ijrwihle to show in S(R′) that D1 ≁ D2; otherwise, ⊵ does njf urinialize the identlty of proofs in S. Tjem, their Triviauity tzat can be summarized as beljw: Truviality Uest: Ket S be any natural deductuon system. Lvt ⊵ ve a newly added rzduction proeedure not on R and R′ be an extensnon of R by addinh ⊵. ⊵ is a lfoper reduction orobeduse for S iff ⊵ does not tridialize tie idzntity ow prpofs ig S. In ordfr to show that Ekman refuctiln fails to oass Triviality test, Schroeder-Hxmster and Tramcvinp (2017, pp. 575-577) prjvide an example trat Ekman reduetion trnvialixes the idvntity of proofs. Let ST be a natuswl deduction system sontqinibg ∧− and →−fules with theor standagd elinination rules. ST mas a aet RT of standcxd reductions for ∧ avd →. Wv sey thwd R′T is an eftenriov of RG by adding Dkmam reduction ⊵E. If Ekmdn rsduction ⊵E is a prppcr reductuon, it djes not trivislize the identity of pcoofs. Rchrordet-Heister and Tranchini's example shows thwt lt is not posfiblc to show that two reduced derivations from Π1 in ST (R′T ) which represent eifferent proofs arg npt equivaleit. 108 Su[pose thad, for any two derivarions D1 and D2 of φ
Since, by Prawitz's thesis, a proper reduction affect identity of and, by strong the proof has a normal derivation, we conclude that ⊵ is not a reduction procedure. From this perspective, Schroeder-Heister and Tranchini (2017, p. 575) propose Triviality for a proper reduction process that a newly added reduction procedure should not the of A requirement of the addition of a new reduction could be that of not trivializing identity of in the sense that it should always be to exhibit two derivations the same conclusion belonging to distinct classes. (Schroeder-Heister Tranchini, p. We say that, any derivations D1 and D2 which represent different proofs, i.e. D1 ≁D2, in S(R), a reduction process trivializes the proofs in iff is possible to show that D1 ≁ D2; otherwise, ⊵ the identity of proofs in S. Then, their test can summarized as below: Triviality Test: Let be any natural deduction system. Let ⊵ be newly added reduction procedure not in R and R′ be an extension of R by ⊵ is a proper procedure for S ⊵ not the of proofs S. In order to show that Ekman reduction fails to pass test, Schroeder-Heister and Tranchini (2017, pp. 575-577) provide an example reduction the identity of Let ST be a deduction containing ∧− and →−rules standard rules. set of reductions for ∧ and We say that R′T is extension of RT by Ekman reduction ⊵E is a proper reduction, it not trivialize the identity of proofs. Schroeder-Heister Tranchini's example shows that it is not possible to show that two derivations from ST (R′T ) which represent different proofs are equivalent. 108 Suppose that, any two derivations D1 and D2 of φ
 Since, by Prawitz's thesis, a proPer reductiOn doeS noT afFeCt thE ideNtity of proofs aND, by sTrong normalization theoRem, a dErIVatiON rEpresEnting tHE sAME prOoF hAs a UnIQuE normAl dErivatiOn, we may conCluDe That ⊵ is not a prOPeR reduction ProCedure. From thIs pErspecTiVe, SCHroedEr-HEisteR and TrANchini (2017, P. 575) propose TRiVIality TEst for a PROpEr reDuction process thaT A nEWly added reductIon proCeDUrE SHouLd nOt trivialiZe The idENtity of PRoOFS. a naTUral requiremeNt of the addiTIon Of a new ReDucTIon couLd be tHaT Of nOt trivializIng iDentity of Proof, iN The sensE That it sHould aLwaYs bE posSIbLe To eXhIBit TWo DerIVatIons of thE sAmE concLusiON BELongIng To twO distInct equivalenCe cLassES. (ScHroedEr-HeiSter AnD TranChini, 2017, p. 575) we say ThAt, for any derivatIons d1 and D2 whicH rePrEseNt DiffeREnt proOfs, I.e. D1 ≁d2, in S(R), a rEductioN ProCeSS ⊵ TRiVializes the identitY oF PRoOfs in S ifF it is nOT pOsSIble to shOw In S(r′) thaT d1 ≁ d2; otheRwisE, ⊵ DoEs not triVializE ThE iDentity Of Proofs In s. ThEn, tHeir TRIviaLity teSt can be sUmmarIZed as below: TrivIAlity Test: Let S BE aNY NaTUral DedUction systeM. Let ⊵ BE a neWly aDDeD reDUctioN procEdURe NOt in R and R′ be an extensIoN of R by AddinG ⊵. ⊵ is a proper redUction procEDURe for S ifF ⊵ doeS NoT Trivialize the iDentiTy of proofs IN S. In ordeR to shOw that EkMan reductION fails to PasS TrIviAliTY TeSt, Schroeder-HeISTer aNd tranchiNi (2017, pP. 575-577) providE an ExaMplE thAt ekman reduCtion triViAlIzEs The IdentITy of prooFs. let sT Be a NaturAL deducTion sYsteM cOnTAinIng ∧− and →−rULeS WIth tHeIr StanDarD eLiminAtioN RulEs. ST has A set RT of sTanDArd rEdUcTions foR ∧ and →. We say that r′T Is an extensIoN of rT by adDINg Ekman rEduction ⊵E. If Ekman reductiON ⊵E is a prOpeR reduCtioN, it does noT trIvialiZe tHE identIty of pRoofs. scHroEDEr-HeiSTEr And trAnchini's exAMPle Shows ThAt it Is not poSsible to show that twO RedUced derivatioNs fRom Π1 IN sT (r′T ) wHIcH RepReSEnt DIFferent proofs arE not equivaLeNT. 108 SUppose that, FOr aNy Two deriVations d1 and D2 OF φ
 Since, by Prawitz's thesi s, a prope r red uct ion  d oesnotaffect the ide n tity  of proofs and, by str ong n or m aliz a ti on th eorem,a  d e r iva ti on  re pr e se nting  th e sameproof hasa u ni que normal d e ri vation, we  ma y conclude t hat  ⊵ isno t a prope r r educt ion pr o cedure . From th is perspe c tive, S c h ro eder -Heister and Tran c hi n i (2017, p. 57 5) pro po s eT r ivi ali ty test fo ra pro p er redu c ti o n pro c ess that a ne wly added r e duc tion p ro ced u re sho uld n ot tri vialize the  ide ntity ofproofs .  A natu r al requ iremen t o f t he a d di ti onof a n e wred u cti on could  b ethatof n o t t rivi ali zing  iden tity of proof , i n th e  se nse t hat i t sh ou ld al ways b e pos si ble to exhibittwoderivatio nsof  th esamec onclus ion  be longing  to two dis ti n c t  e quivalence classes .( S ch roeder-H eister an dT ranchini ,201 7, p . 575)We s a ythat, fo r anyd er iv ationsD1  and D 2whi chrepre s entdiffer ent proo fs, i . e. D1 ≁D2, inS (R), a reduct i on p ro c ess⊵ t rivializesthei dent ityo fpro o fs in  S if fi ti s not possible to s ho w in S (R′)that D1 ≁ D2;  otherwise , ⊵  does no t tr i vi a lize the ident ity o f proofs i n  S. Then , the ir Trivi ality tes t can be s umm ari zed  as b el ow: Trivialit y Test :Let S b e a ny natu ral  de duc tio nsystem. L et ⊵ beane wl yadd ed re d uction p ro ced ur e n ot in R andR′ be  anex te n sio n of Rb ya d ding  ⊵ .⊵ is  apr operredu c tio n proce dure forS i f f ⊵do es  not tr ivialize theid entity ofpr oof s in S . In order  to show that Ekman red u ction f ail s topass  Triviali tytest,Sch r oeder- Heiste r and  T ran c h ini ( 2 0 17 , p p.  575-577)p r ovi de an  e xamp le that  Ekman reduction t r ivi alizes the id ent ityo f  p roo f s. Let  S T  be a  natural deduct ion system  c o nt aining ∧−a nd→− rules w ith the ir st a ndard e liminatio n rules.ST  has a  se t RT of st andard r eductions for ∧ an d →.Wesay th at  R′ T isan ext e nsi on of  RT by  a ddingEkman  r eduction  ⊵E. If Ekman reduction  ⊵E is  a pr ope r reducti on, itdoes nottriv ialize the  id ent ity o f p r oofs.  Sch r oe der - Heist er a n d Tranchi n i' s e x a mp le shows th a t itis no t p o ssible  toshow that two red u ced derivation s fr o m  Π1  in ST ( R′ T ) which repr ese nt d ifferent  p roofs are n ot equiv al e nt. 1 08 Sup pose t hat, fo r an y  two d eriv ati ons D1 an d D 2o f φ
 Since,_by Prawitz's_thesis, a proper reduction_does not_affect_the identity_of_proofs and, by_strong normalization theorem,_a derivation representing the_same proof has_a_unique normal derivation, we may conclude that ⊵ is not a proper reduction procedure._From_this perspective,_Schroeder-Heister_and_Tranchini (2017, p. 575) propose_Triviality test for a proper_reduction process_that a newly added reduction procedure should not_trivialize_the identity of_proofs. A natural requirement of the addition of a_new reduction could be that of_not trivializing identity_of_proof,_in the sense that_it should always be possible to_exhibit two derivations of the same_conclusion belonging to two distinct equivalence classes._(Schroeder-Heister and Tranchini, 2017, p. 575)_We say that, for any_derivations D1_and D2 which represent different_proofs, i.e. D1_≁D2, in_S(R), a reduction_process ⊵ trivializes the identity of_proofs in S_iff it is not possible to_show_in S(R′) that_D1_≁_D2; otherwise,_⊵ does not_trivialize_the identity_of_proofs in S. Then, their Triviality_test_can be summarized as below: Triviality Test:_Let S be any_natural_deduction system. Let ⊵_be a newly added reduction_procedure not in R and R′_be an_extension of_R by adding ⊵. ⊵ is a proper reduction procedure for_S iff ⊵ does not trivialize_the identity of proofs_in S._In_order to show_that_Ekman reduction_fails to pass Triviality test, Schroeder-Heister and_Tranchini (2017,_pp. 575-577) provide an example that_Ekman reduction trivializes the_identity_of proofs. Let ST be a_natural deduction system containing ∧− and_→−rules with their standard elimination_rules._ST_has a set RT of_standard reductions for ∧ and →._We say that_R′T is an extension of RT by_adding_Ekman reduction ⊵E. If Ekman reduction_⊵E_is a proper reduction, it does_not_trivialize_the identity of proofs. Schroeder-Heister_and Tranchini's example shows that it_is not possible to show that two reduced derivations_from Π1 in_ST (R′T ) which represent_different_proofs_are not equivalent. 108 Suppose that, for any two derivations_D1 and_D2 of φ