rawitz (1971, p. 256) says, 'two different normal derivations are never equivalent.' Therefore, if a reduction process is proper, it does not affect the identity of proofs. We summarize his thesis in the following way. Prawitz's thesis A proper reduction does not affect the identity of proofs represented by 106 derivations in the same equivalence class. (i.e. derivations representing different proofs are not equivalent.) Schroeder-Heister and Tranchini (2017, pp. 574-575) take the conjecture for the identity of proofs and Prawitz's thesis. Especially, they suggest Triviality test to evaluate whether a newly added reduction makes derivations representing different proofs belonging an equivalent class. For instance, let us consider the following two derivations with the same assumptions and the same conclusion but the assumptions are discharged at different places. [φ]1 →I,1 φ → φ → I, /0 φ → (φ → φ) D φ → E φ → φ [φ]1 → I, /0 φ → φ → I,1 φ → (φ → φ) D φ → E φ → φ For their view, the two derivations above belong to two different equivalence classes in the sense of Prawitz' equivalent relation. In the case of the empty discharge, the reduction process for → has the following form. D1 ψ → I, /0 φ → ψ D2 φ → E ψ ▷→( /0) D1 ψ ▷→( /0)−reduction is an instance of ▷→. Then, the reduced derivations below are obtained by ▷→ respectively. [φ]1 → I,1 φ → φ D φ → I, /0 φ → φ The two normal derivations are different. Provided that the equivalent derivations representing the same proof must have the same normal derivation, if two derivations are equivalent, they should have the same normal derivation, but they do not. Let ⊵ be a reduction procedure not in a set R of reductions and R′ be an extension of R by adding ⊵. We use an abbreviation 'S(R)' for a natural deduction system S relative to 107 R. Let us assume that the strong normalization theorem is provable for S(R) and consider the case that there is a derivation Π in S such that Π has a unique normal derivation in S(R) but has two normal derivations in S(R′).
rawitz (1971, p. 256) says,' two different normal derivations are never equivalent.' Therefore, if a decrease procedure is proper, it does not affect the identity of proofs. We sum up his thesis in the following room. Prawitz's thesis A proper reduction does not involve the identity of proofs represented by 106 derivation in the same equivalence course. (i.e. derivations representing different proofs are not equivalent .) Schroeder - Heister and Tranchini (2017, pp. 574 - 575) drive the conjecture for the identity of proofs and Prawitz's thesis. Especially, they suggest pettiness test to evaluate whether a newly added decrease makes derivations represent different proof belonging an equivalent class. For instance, let us consider the follow two derivations with the same assumptions and the same conclusion but the assumption are discharged at different places. [ φ]1 →I,1 φ → φ → I, /0 φ → (φ → φ) D φ → E φ → φ [ φ]1 → I, /0 φ → φ → I,1 φ → (φ → φ) D φ → E φ → φ For their view, the two derivations above belong to two unlike equality classes in the common sense of Prawitz' equivalent relation back. In the casing of the empty discharge, the reduction procedure for → has the following form. D1 ψ → I, /0 φ → ψ D2 φ → E ψ ▷→ (/0) D1 ψ ▷→ (/0)−reduction is an instance of ▷→. Then, the reduced derivation below are obtained by ▷→ respectively. [ φ]1 → I,1 φ → φ D φ → I, /0 φ → φ The two normal derivations are different. Provided that the equivalent derivation representing the same proof must have the same normal deriving, if two derivations are equivalent, they should get the like normal derivation, but they do not. Let ⊵ be a reduction procedure not in a set R of reductions and R′ be an elongation of R by adding ⊵. We use an abbreviation' S(R)' for a natural subtraction system S relative to 107 R. permit us assume that the potent normalization theorem is demonstrable for S(R) and consider the case that there be a derivation Π in S such that Π has a unique normal ancestry in S(R) but has two normal derivation in S(R′).
rawltz (1971, p. 256) says, 'two differekt normal derivajiins arx never equivaldnt.' Therefore, if a reduction peocesw is proper, it does nog affect nhe identuty id proofs. Wx summarlve hia thevms in the folloeing way. Psawitz's thesis A pxoper reduction does not affect the ydentitu lf proofs reprgsenttd fy 106 svrlvations in the same equivalence class. (p.e. derivations relresenting different proofd arf not equivalent.) Sfhroeder-Heiwter qnd Tranchink (2017, pp. 574-575) takt che conjectore for the identity of proofs avd Prcwitz's thesus. Esogcially, they suggvst Trivialitn test do evaliate whether a nexly qdded reduction makes derivations reprefenting dhfyerent proofs belongibg an exuivdleng clxss. Fpr instajce, let us cohsider the dollowing two derivstyins with the aame afstmptions and the same conclusion but tht asshmptions are discharged at different places. [φ]1 →I,1 φ → φ → I, /0 φ → (φ → φ) D φ → E φ → φ [φ]1 → I, /0 φ → φ → I,1 φ → (φ → φ) D φ → E φ → φ For their view, ghe tru dfrivations above belong to two different equidzltncv classes in the fense of Prswlta' equivalent rglation. In the case of the elpty difcharte, the requctoon process for → has the foolowing form. E1 ψ → I, /0 φ → ψ D2 φ → E ψ ▷→( /0) D1 ψ ▷→( /0)−reductnon is an imstance of ▷→. Then, the rebuced serivations below ars obtained by ▷→ rerpebtivaly. [φ]1 → I,1 φ → φ D φ → I, /0 φ → φ The tro normal dernvations are diffewent. Provifed tmdt the equivalent ferivctionv represenhing the same proof must have tix same normal darination, if two cerivations awe equivalent, jhey shoujd haxe the samv normal verivation, btt they do nod. Let ⊵ be a rxduction [rocwdurw not iv a set R of recuctions cud R′ be ab extension of R bn addkhg ⊵. We use an ayyrtvuation 'S(R)' for s nxtuwap veducehon system S relxtixr to 107 R. Let us awrume that the strong norkaliaation theorem is lrjvable fir S(R) anq consider thr case that there ls a verivavion Π on F such that Π has a unique normzl derivahiok in S(R) but hws tqo normal dexivations in S(R′).
rawitz (1971, p. 256) says, 'two different are equivalent.' Therefore, a reduction process affect identity of proofs. summarize his thesis the following way. Prawitz's thesis A reduction does not affect the identity of proofs represented by 106 derivations in same equivalence class. (i.e. derivations representing different proofs are not equivalent.) Schroeder-Heister and (2017, 574-575) the for the identity of proofs and Prawitz's thesis. Especially, they suggest Triviality test to evaluate whether newly added reduction makes derivations representing different proofs an equivalent class. For let us consider the following derivations the same and same but the assumptions discharged at different places. [φ]1 →I,1 φ → φ → I, /0 φ → (φ → φ) φ → → φ → /0 → φ → → (φ → φ) D φ → φ For their view, the two derivations belong to different equivalence classes in the sense Prawitz' equivalent relation. In the case of the discharge, the reduction process for → has the following form. D1 ψ → I, /0 ψ D2 φ → ψ ▷→( /0) ψ /0)−reduction an of ▷→. the reduced derivations below are obtained by ▷→ respectively. [φ]1 → φ → φ D φ → I, /0 φ → two derivations are different. that the equivalent derivations the proof must have the derivation, two they have same normal derivation, but do not. Let ⊵ be reduction procedure not in and R′ be an extension of R by ⊵. We use an abbreviation 'S(R)' for natural deduction system S relative to 107 R. Let us assume that strong normalization provable for S(R) and consider the case that is a derivation Π S such that Π has a unique normal derivation S(R) has two derivations in S(R′).
rawitz (1971, p. 256) says, 'two different noRmal derivaTions Are NevEr EquiValeNt.' Therefore, if a REducTion process is proper, it dOes noT aFFect THe IdentIty of prOOfS. wE suMmArIze HiS ThEsis iN thE followIng way. PrawItz'S tHesis A proper REdUction does Not Affect the ideNtiTy of prOoFs rEPreseNteD by 106 deRivatiONs in thE same equiVaLEnce clASs. (i.e. derIVAtIons Representing diffeREnT Proofs are not eqUivaleNt.) sChROEdeR-HeIster and TrAnChini (2017, PP. 574-575) take thE CoNJECtuRE for the identiTy of proofs aND PrAwitz's ThEsiS. especiAlly, tHeY SugGest TrivialIty tEst to evalUate whETher a neWLy added ReductIon MakEs deRIvAtIonS rEPreSEnTinG DifFerent prOoFs BelonGing AN EQUivaLenT claSs. For Instance, let us ConSideR The FolloWing tWo deRiVatioNs with The saMe Assumptions and tHe saMe conclusIon BuT thE aSsumpTIons arE diSchArged at DiffereNT plAcES. [Φ]1 →i,1 φ → Φ → I, /0 φ → (φ → φ) D φ → E φ → φ [φ]1 → I, /0 φ → φ → I,1 φ → (φ → φ) d φ → e Φ → Φ FOr their vIew, the TWo DeRIvations AbOve BeloNG To two DiffEReNt equivaLence cLAsSeS in the sEnSe of PrAwItz' EquIvaleNT relAtion. IN the case Of the EMpty discharge, tHE reduction proCEsS FOr → HAs thE foLlowing form. d1 ψ → I, /0 φ → Ψ d2 φ → E ψ ▷→( /0) d1 ψ ▷→( /0)−reDUcTioN Is an iNstanCe OF ▷→. THEn, the reduced derivatIoNs beloW are oBtained by ▷→ respEctively. [φ]1 → I,1 Φ → Φ d Φ → I, /0 φ → φ The tWo noRMaL Derivations are DiffeRent. ProvidED that the EquivAlent derIvations rEPResentinG thE saMe pRooF MUsT have the same nORMal dErIvation, If tWo derivAtiOns Are EquIvAlent, they Should haVe ThE sAmE noRmal dERivation, BuT thEy Do nOt. Let ⊵ BE a reduCtion ProcEdUrE Not In a set R OF rEDUctiOnS aNd R′ bE an ExTensiOn of r By aDding ⊵. We Use an abbrEviATion 's(R)' FoR a naturAl deduction syStEm S relativE tO 107 R. LEt us asSUMe that thE strong normalization theORem is prOvaBle foR S(R) aNd consideR thE case tHat THere is A derivAtion π iN S sUCH that π HAs A unIqUe normal deRIVatIon in s(R) But hAs two noRmal derivations in S(r′).
rawitz (1971, p. 256) says , 'two dif feren t n orm al  der ivat ions are never equi valent.' Therefore, if  a re du c tion pr ocess  is pro p er , itdo es  no ta ff ect t heidentit y of proof s.We  summarize h i sthesis inthe  following w ay.  Prawi tz 'st hesis  Aprope r redu c tion d oes not a ff e ct the identit y of  pro ofs represented b y  1 0 6 derivationsin the  s a me e qui val ence class .(i.e. derivat i on s r epr e senting diffe rent proofs are  not e qu iva l ent.)Schro ed e r-H eister andTran chini (20 17, pp .  574-57 5 ) takethe co nje ctu re f o rth e i de n tit y  o f p r oof s and Pr aw it z's t hesi s . E spec ial ly,theysuggest Trivi ali ty t e stto ev aluat e wh et her a  newly  adde dreduction makes  der ivationsrep re sen ti ng di f ferent  pr oof s belon ging an equ iv a l e nt  class. For instan ce , le t us con sidert he  f o llowingtw o d eriv a t ionswith th e same a ssumpt i on sand the  s ame co nc lus ion  butt he a ssumpt ions are  disc h arged at diffe r ent places. [ φ ]1 → I, 1  φ →  φ→ I, /0 φ →  (φ→  φ)D φ→  E  φ→  φ [φ ]1 →I, /0 φ → φ → I,1 φ → (φ→φ) D φ  → Eφ → φ For the ir view, t h e two deri vati o ns above belong t o two  different equivale nce c lasses i n the sen s e  of Praw itz ' e qui val e n trelation. Int h e ca se  of the  em pty dis cha rge , t here duction p rocess f or  →  h as  th e fol l owing fo rm . D 1ψ →  I, / 0  φ → ψ  D2 φ  → E  ψ  ▷ → ( / 0) D1 ψ ▷→ ( /0)− re du ctio n i san in stan c e o f ▷→. T hen, thered u cedde ri vations  below are ob ta ined by ▷→  r esp ective l y . [φ]1 →  I,1 φ → φ D φ → I, /0φ  → φ Th e t wo no rmal  derivati ons  are d iff e rent.Provid ed th at  th e equiv a l en t d er ivations r e p res entin gthesame pr oof must have thes ame  normal deriv ati on,i f  t wod er i vat io n s a r e  equivalent, th ey shouldha v ethe same n o rma lderivat ion, bu t the y  do not . Let ⊵ b e a reduc ti on p r o ced ure not in  a set R  of reduc t ionsa nd  R′ b e a n exte ns ion  of R  by ad d ing  ⊵. W e usean  abbre viati on  'S(R)'for a natural deduction  syste m S r ela tive to 1 07R . L et us ass umethat the s tro ngnorma liz a tiontheo r em  is prova blef or S(R) a n dcon s i de r the caset h a t t hereisa  deriv atio n Π in S such tha t  Π has a uniqu e no r m alder i vati on  in S(R) but h astw o normal d er ivations in  S(R′).
rawitz (1971,_p. 256)_says, 'two different normal_derivations are_never_equivalent.' Therefore,_if_a reduction process_is proper, it_does not affect the_identity of proofs._We_summarize his thesis in the following way. Prawitz's thesis A proper reduction does not_affect_the identity_of_proofs_represented by 106 derivations in_the same equivalence class. (i.e._derivations representing_different proofs are not equivalent.) Schroeder-Heister and Tranchini_(2017,_pp. 574-575) take_the conjecture for the identity of proofs and Prawitz's_thesis. Especially, they suggest Triviality test_to evaluate whether_a_newly_added reduction makes derivations_representing different proofs belonging an equivalent_class. For instance, let us consider_the following two derivations with the same_assumptions and the same conclusion but_the assumptions are discharged at_different places._[φ]1 →I,1 φ → φ_→ I, /0_φ →_(φ → φ)_D φ → E φ →_φ [φ]1 →_I, /0 φ → φ →_I,1_φ → (φ_→_φ)_D φ_→ E φ_→_φ For_their_view, the two derivations above belong_to_two different equivalence classes in the sense_of Prawitz' equivalent relation._In_the case of the_empty discharge, the reduction process_for → has the following form._D1 ψ_→ I,_/0 φ → ψ D2 φ → E ψ ▷→( /0)_D1 ψ ▷→( /0)−reduction is an_instance of ▷→. Then,_the reduced_derivations_below are obtained_by_▷→ respectively._[φ]1 → I,1 φ → φ D_φ →_I, /0 φ → φ The_two normal derivations are_different._Provided that the equivalent derivations representing_the same proof must have the_same normal derivation, if two_derivations_are_equivalent, they should have the_same normal derivation, but they do_not. Let ⊵_be a reduction procedure not in a_set_R of reductions and R′ be_an_extension of R by adding ⊵._We_use_an abbreviation 'S(R)' for a_natural deduction system S relative to_107 R. Let us assume that the strong normalization_theorem is provable_for S(R) and consider the_case_that_there is a derivation Π in S such that Π_has a_unique normal derivation_in S(R) but has two normal derivations in S(R′).