 on the right side derivation. It means that GEkman reduction fails to satisfy the requirements of a proper reduction via Translation test, because of (T2).4 If any proper reduction must not generate new cut-(or maximum) formula, GEkman reduction is not a proper reduction procedure because it is a detour-making process. 3.4.2 Does Crabbé Reduction Overgenerate? Schroeder-Heister and Tranchini (2018) introduce an example first observed by Marcel Crabbé which arises the problem of overgeneration and says that their Triviality test cannot block the reduction used in Crabbé's case. The case has rules for Zermelo's separation axiom. For our discussions include an evaluation of Crabbé's case via Translation test, we propose the rules for Zermelo's separation axiom in generalized form. D1 t ∈ s D2 φ[t/x] ∈Z It ∈ {x ∈ s|φ(x)} t ∈ {x ∈ s|φ(x)} [t ∈ s]1, [φ[t/x]]1 D3 ψ ∈Z E,1 ψ 4It is readily seen that ▷GEg−reduction fails to pass Translation test because of (T2). 123 The reduction procedure ▷∈Z for ∈Z I− and ∈Z E−rules are as below: D1 t ∈ s D2 φ[t/x] ∈Z It ∈ {x ∈ s|φ(x)} [t ∈ s]1, [φ[t/x]]1 D3 ψ ∈Z E,1 ψ ▷∈Z D1 t ∈ s D2 φ[t/x] D3 ψ We all ▷∈Z−reduction, 'Crabbé reduction.' We shall claim in this subsection that since Triviality test is not suitable for evaluating standard reductions, it does not apply to Crabbé's case. Also, it will be discussed that the culprit of the overgeneration occurred by Crabbé's case may not be Crabbé reduction but be the form of ∈Z I−rule. If Crabbé reduction is the culprit, then the requirements of a proper reduction through Translation test can solve the problem of overgeneration. Let us investigate Crabbé's case. For any set b, we define Zb as a set {
on the right side derivation. It means that GEkman decrease fail to satisfy the requirements of a proper decrease via Translation examination, because of (T2).4 If any proper reduction must not generate fresh cut-(or maximum) formula, GEkman decrease is not a proper decrease procedure because it is a detour - making summons. 3.4.2 Does Crabbé Reduction Overgenerate? Schroeder - Heister and Tranchini (2018) introduce an example foremost observed by Marcel Crabbé which arises the trouble of overgeneration and says that their pettiness trial cannot block the reduction used in Crabbé's character. The case has rules for Zermelo's separation maxim. For our discussions admit an evaluation of Crabbé's case via Translation test, we propose the convention for Zermelo's separation axiom in generalized form. D1 t ∈ s D2 φ[t / x ] ∈Z It ∈ { x ∈ s|φ(x) } t ∈ { x ∈ s|φ(x) } [ t ∈ s]1, [ φ[t / x]]1 D3 ψ ∈Z E,1 ψ 4It is readily seen that ▷GEg−reduction fails to guide Translation test because of (T2). 123 The reduction procedure ▷∈Z for ∈Z I− and ∈Z E−rules are as below: D1 t ∈ s D2 φ[t / x ] ∈Z It ∈ { x ∈ s|φ(x) } [ t ∈ s]1, [ φ[t / x]]1 D3 ψ ∈Z E,1 ψ ▷∈Z D1 t ∈ s D2 φ[t / x ] D3 ψ We all ▷∈Z−reduction,' Crabbé reduction.' We shall claim in this subsection that since Triviality test is not suitable for measure standard reductions, it does not apply to Crabbé's casing. besides, it will be hash out that the culprit of the overgeneration occurred by Crabbé's shell may not be Crabbé reduction but be the form of ∈Z I−rule. If Crabbé reduction is the culprit, then the prerequisite of a proper reduction through Translation test can solve the problem of overgeneration. Let us investigate Crabbé's sheath. For any set b, we define Zb as a set {
 on the right side derivatiun. It means thaj TEkman reducfion faius to satisfy the requiremenvs od a peoper reduction via Trxnslation test, bexaust of (T2).4 If any pro'sr redugcion jmst nmv generate new gut-(or maximgm) formula, GEkkav xeduction is not a proper reduction [rocedute because it is a dttotr-maipnn process. 3.4.2 Does Crabbé Reduction Kvergentrate? Schroeder-Heixter and Tranchini (2018) introdkce wn example first ohserved by Narcqo Crabbé whicf arises the problem or overgeneration and says that gheir Trivialiti fedj cannot blork the reduction used in Csabbé's csse. The case hss culew for Zermelo's separavion axiom. For our dyscussionv nnclude an evaluation od Cratbé's wase cia Trznxlztion hesv, we proposs the rules for Zermelo's separstyin axiom in gsneralyzqd form. D1 t ∈ s D2 φ[t/x] ∈Z It ∈ {x ∈ s|φ(x)} t ∈ {x ∈ s|φ(q)} [t ∈ s]1, [φ[t/x]]1 D3 ψ ∈Z E,1 ψ 4It is reaeily seen that ▷GEg−redoction faijs to pass Translation test because of (T2). 123 The redgctioi oroecqjee ▷∈Z for ∈Z I− and ∈Z E−rules are as below: D1 t ∈ s D2 φ[f/x] ∈Z It ∈ {x ∈ s|φ(x)} [t ∈ f]1, [φ[t/x]]1 D3 ψ ∈Z E,1 ψ ▷∈X C1 t ∈ s D2 φ[t/x] D3 ψ We all ▷∈Z−deduction, 'Crabbé refuction.' We syall claii in this subsection that since Triviality nest is not suitable flr evaluatiug stamdard reductions, it does not appmy to Crabbé'd case. Alau, it will be diszusxeg that the culprit of the jvergeneretion occurrdd bi Crabbé'f case may not na Crabbé reduction hut bg the xorm of ∈Z L−rule. If Crabbé reduction is the culprit, then jhe rexuiremencs of s proper redustion through Jranslatijn tert can solne the prmblem of ovqrgeneration. Ngt us investijate Crabfé's cqse. Dor any ret b, we definr Zb as a set {
on the right side derivation. It means reduction to satisfy requirements of a because (T2).4 If any reduction must not new cut-(or maximum) formula, GEkman reduction not a proper reduction procedure because it is a detour-making process. 3.4.2 Does Reduction Overgenerate? Schroeder-Heister and Tranchini (2018) introduce an example first observed by Marcel which the of and says that their Triviality test cannot block the reduction used in Crabbé's case. The case rules for Zermelo's separation axiom. For our discussions an evaluation of Crabbé's via Translation test, we propose rules Zermelo's separation in form. t ∈ s φ[t/x] ∈Z It ∈ {x ∈ s|φ(x)} t ∈ {x ∈ s|φ(x)} [t ∈ s]1, [φ[t/x]]1 D3 ∈Z E,1 is readily that fails pass Translation test (T2). 123 The reduction procedure ▷∈Z and ∈Z E−rules are as below: D1 t s D2 ∈Z It ∈ {x ∈ s|φ(x)} ∈ s]1, [φ[t/x]]1 D3 ψ ∈Z E,1 ψ D1 t ∈ s D2 φ[t/x] D3 ψ We all ▷∈Z−reduction, 'Crabbé reduction.' We shall this subsection that since test is not for standard it not apply Crabbé's case. Also, it will be discussed that the culprit of overgeneration occurred by Crabbé's case may not be Crabbé reduction the of ∈Z I−rule. Crabbé reduction is the then requirements of a proper Translation can of Let investigate Crabbé's case. For set b, we define Zb a set {
 on the right side derivation. IT means that gEkmaN reDucTiOn faIls tO satisfy the reqUIremEnts of a proper reduction Via TrAnSLatiON tEst, beCause of (t2).4 if ANY prOpEr RedUcTIoN must Not GeneratE new cut-(or mAxiMuM) formula, GEkmAN rEduction is Not A proper reducTioN proceDuRe bECause It iS a detOur-makINg procEss. 3.4.2 Does CrAbBÉ ReducTIon OverGENeRate? schroeder-Heister aND TRAnchini (2018) introduCe an exAmPLe FIRst ObsErved by MarCeL CrabBÉ which aRIsES THe pROblem of overgeNeration and SAys That thEiR TrIVialitY test CaNNot Block the redUctiOn used in CRabbé's CAse. The cASe has ruLes for zerMelO's sePArAtIon AxIOm. FOR oUr dIScuSsions inClUdE an evAluaTION Of CrAbbÉ's caSe via translation teSt, wE proPOse The ruLes foR ZerMeLo's seParatiOn axiOm In generalized foRm. D1 t ∈ S D2 φ[t/x] ∈Z It ∈ {x ∈ S|φ(x)} T ∈ {x ∈ S|φ(x)} [T ∈ s]1, [Φ[t/x]]1 D3 ψ ∈z e,1 ψ 4It is ReaDilY seen thAt ▷GEg−reDUctIoN FAIlS to pass Translation TeST BeCause of (T2). 123 the redUCtIoN ProcedurE ▷∈Z For ∈z I− anD ∈z e−ruleS are AS bElow: D1 t ∈ s D2 Φ[t/x] ∈Z It ∈ {X ∈ S|φ(X)} [t ∈ S]1, [φ[t/x]]1 D3 ψ ∈Z e,1 ψ ▷∈z D1 t ∈ s D2 φ[T/x] d3 ψ WE alL ▷∈Z−redUCtioN, 'CrabbÉ reductiOn.' We sHAll claim in this SUbsection that SInCE trIVialIty Test is not suItabLE for EvalUAtIng STandaRd redUcTIoNS, it does not apply to CrAbBé's casE. Also, It will be discuSsed that thE CULprit of tHe ovERgENeration occurrEd by CRabbé's case MAy not be CRabbé ReductioN but be the FORm of ∈Z I−ruLe. IF CrAbbÉ reDUCtIon is the culprIT, Then ThE requirEmeNts of a pRopEr rEduCtiOn Through TrAnslatioN tEsT cAn SolVe the PRoblem of OvErgEnEraTion. LET us invEstigAte CRaBbÉ'S caSe. For anY SeT B, We deFiNe zb as A seT {
 on the right side derivat ion. It me ans t hat  GE km an r educ tion fails tos atis fy the requirements of  a pr op e r re d uc tionvia Tra n sl a t ion  t es t,be c au se of  (T 2).4 If  any prope r r ed uction mustn ot  generatenew  cut-(or max imu m) for mu la, GEkma n r educt ion is not aproper re du c tion p r ocedure b ec ause  it is a detour-m a ki n g process. 3.4 .2 Doe sC ra b b é R edu ction Over ge nerat e ? Schro e de r - H eis t er and Tranch ini (2018)i ntr oducean  ex a mple f irstob s erv ed by Marce l Cr abbé whic h aris e s the p r oblem o f over gen era tion an dsay st hat th eir Tri vialityte st  cann ot b l o c k  the  re duct ion u sed in Crabbé 'scase .  Th e cas e has  rul es  forZermel o's s ep aration axiom.Forour discu ssi on s i nc ludea n eval uat ion  of Cra bbé's c a sevi a T ra nslation test, wepr o p os e the ru les fo r  Z er m elo's se pa rat iona x iom i n ge n er alized f orm. D 1  t  ∈  s D2 φ [t /x] ∈Z  I t ∈  {x  ∈ s| φ (x)}  t ∈ { x ∈ s|φ( x)} [ t  ∈ s]1, [φ[t/x ] ]1 D3 ψ ∈Z E, 1  ψ 4 It is r ead ily seen th at ▷ G Eg−r educ t io n f a ils t o pas sT ra n slation test becaus eof (T2 ). 12 3 The reducti on procedu r e ▷∈Z for∈Z I −  a n d ∈Z E−rules a re as  below: D1 t ∈ s D2  φ[t/ x] ∈Z It  ∈ {x ∈ s | φ (x)} [t∈ s ]1,  [φ [t/ x ] ]1  D3 ψ ∈Z E,1ψ ▷∈ZD1  t ∈ sD2φ[t/x]D3ψ W e a ll▷∈ Z−reducti on, 'Cra bb ére du cti on.'W e shallcl aim  i n t his s u bsecti on th at s in ce Tri viality te s t  isno tsuit abl efor e valu a tin g stand ard reduc tio n s, i tdo es notapply to Crab bé 's case. A ls o,it wil l be discu ssed that the culprit o f  the ov erg enera tion  occurred  by  Crabb é's case m ay not  be C ra bbé r educt i o nbut  b e the form o f ∈ Z I−r ul e. I f Crabb é reduction is the cul prit, then th e r equi r e me nts of a p ro p err e duction through  Translati on te st can sol v e t he  proble m of ov ergen e ration.  Let us i nvestigat eCrab b é 'scase. Forany setb, we def i ne Zb as  a se t { 
 on_the right_side derivation. It means_that GEkman_reduction_fails to_satisfy_the requirements of_a proper reduction_via Translation test, because_of (T2).4 If_any_proper reduction must not generate new cut-(or maximum) formula, GEkman reduction is not a_proper_reduction procedure_because_it_is a detour-making process. 3.4.2_Does Crabbé Reduction Overgenerate? Schroeder-Heister_and Tranchini_(2018) introduce an example first observed by Marcel_Crabbé_which arises the_problem of overgeneration and says that their Triviality test_cannot block the reduction used in_Crabbé's case. The_case_has_rules for Zermelo's separation_axiom. For our discussions include an_evaluation of Crabbé's case via Translation_test, we propose the rules for Zermelo's_separation axiom in generalized form. D1_t ∈ s D2 φ[t/x]_∈Z It_∈ {x ∈ s|φ(x)} t_∈ {x ∈_s|φ(x)} [t_∈ s]1, [φ[t/x]]1_D3 ψ ∈Z E,1 ψ 4It_is readily seen_that ▷GEg−reduction fails to pass Translation_test_because of (T2)._123_The_reduction procedure_▷∈Z for ∈Z_I−_and ∈Z_E−rules_are as below: D1 t ∈_s_D2 φ[t/x] ∈Z It ∈ {x ∈_s|φ(x)} [t ∈ s]1,_[φ[t/x]]1_D3 ψ ∈Z E,1_ψ ▷∈Z D1 t ∈_s D2 φ[t/x] D3 ψ We_all ▷∈Z−reduction,_'Crabbé reduction.'_We shall claim in this subsection that since Triviality test is_not suitable for evaluating standard reductions,_it does not apply_to Crabbé's_case._Also, it will_be_discussed that_the culprit of the overgeneration occurred by_Crabbé's case_may not be Crabbé reduction but_be the form of_∈Z_I−rule. If Crabbé reduction is the_culprit, then the requirements of a_proper reduction through Translation test_can_solve_the problem of overgeneration. Let_us investigate Crabbé's case. For any_set b, we_define Zb as a set {