 desired result in the system. Theorem 4.2.2. Let L be a language which has constants ∧, ∨, →, ∀, and ∃, but does not have ¬ and ⊥. Let S be a natural deduction system in L which has rules for ∧, ∨, →, ∀, and ∃. Let P be a set of production rules for S which only consists of closed derivations, and R be a set of reduction procedures for ∧, ∨, →, ∀, and ∃. For every derivation D in S, if D is P−valid relative to R, D is normalizable relative to R. Proof. Let D be any derivation in S and φ be any end-formula of D. Suppose that D is P−valid relative to R. The proof is by induction over the degree of φ. Since Definition 4.2.1 has four conditions, there are four cases that we should consider: (1) D is a closed derivation in P, (2) D is a closed canonical derivation, (3) D is a closed non-canonical derivation, and (4) D is an open derivation. 4Although Prawitz (1973, p. 236; 1974, p. 73; 2006, p. 515) does not consider extensions of R, we follow Schroeder-Heister (2006, 2015) and consider both extensions of R and P. Extensions of P and R are required because when the open derivation contain any expressions and inference rules that are not already given by P and R, we need to add the expressions and to assign reductions to the inference rules substituted for the open assumption of the derivation. 148 Induction basis: if d(φ) = 0, then φ is ⊥ or φ is α for an atomic formula α. It means that D is a closed derivation in P. Case (1) if D is a closed derivation in P, then it is in normal form. Hence, by Definition 1.2.5, D is normalizable relative to R. Induction hypothesis: let D′ be any immediate subderivation of D and φ ′ is an endformula of D′. Suppose that D′ is normalizable relative to R with d(φ ′) ⩽ n. We have to show that D is normalizable relative to R with d(φ)⩽ n+1. Case (2) if D is a closed canonical
desired result in the system. Theorem 4.2.2. Let L be a linguistic process which have constants ∧, ∨, →, ∀, and ∃, but does not have ¬ and ⊥. Let S be a natural subtraction system in L which has rule for ∧, ∨, →, ∀, and ∃. get P be a set of production rules for S which entirely consists of closed derivation, and radius be a set of reduction procedures for ∧, ∨, →, ∀, and ∃. For every derivation D in S, if D is P−valid proportional to R, D is normalizable relative to R. Proof. Let D be any ancestry in S and φ be any end - formula of D. think that D is P−valid relative to R. The validation is by induction over the academic degree of φ. Since Definition 4.2.1 has four conditions, there are four cases that we should study: (1) D is a closed derivation in P, (2) D is a shut canonic derivation, (3) D is a closed non - canonical derivation, and (4) D is an open derivation. 4Although Prawitz (1973, p. 236; 1974, p. 73; 2006, p. 515) does not consider extension of R, we follow Schroeder - Heister (2006, 2015) and think both extension of R and P. Extensions of P and R are required because when the open deriving contain any expressions and inference rules that are not already given by phosphorus and R, we need to add the expressions and to assign reductions to the inference rules substituted for the open assumption of the derivation. 148 Induction basis: if d(φ) = 0, then φ is ⊥ or φ is α for an nuclear formula α. It intend that D is a shut deriving in P. Case (1) if D is a closed derivation in P, then it is in normal mannequin. Hence, by Definition 1.2.5, D is normalizable relative to R. Induction hypothesis: lease D′ be any immediate subderivation of D and φ ′ is an endformula of D′. Suppose that D′ is normalizable relative to R with d(φ ′) ⩽ n. We consume to show that D is normalizable relative to R with d(φ)⩽ n+1. Case (2) if D is a closed canonic
 dedired result in the systtm. Theorem 4.2.2. Let L be a lenguage which hxs constants ∧, ∨, →, ∀, and ∃, but doxs nit hace ¬ and ⊥. Let S be a nagural dedlction sywtem un L which has rulcf fod ∧, ∨, →, ∀, end ∃. Let P be a set of prmduction rules fur S which only consists of closed dewivatioms, and R be a sej of gequctjon procedures for ∧, ∨, →, ∀, and ∃. For evsry derpvation D in S, if D is P−valid relative to R, D id normalizable relwtive to R. Kdoos. Let D be anh derivation in S and φ be any end-formula of D. Suppose that D is P−valud repdtive to R. Vhe prjof is by induction mver thr degree of φ. Slnce Vefibition 4.2.1 has four condmtions, there are fout cases thdt we should considwr: (1) D iv a wlosde ddribavioh in P, (2) D is a clossd canonicao derivation, (3) D is s soosed non-canohical qewivation, and (4) D is an open derivation. 4Anthkugh Prawitz (1973, p. 236; 1974, p. 73; 2006, p. 515) does not consider edtensions of R, we follow Schroeder-Heister (2006, 2015) and consider bmth eetdnsnins ow R and P. Extensions of P and R are required becwhst wmen the open derlvation contain anu fxltessions and ivferenes dules that are not alreadi giveb by P anq R, ee need to add the expressiins and to afwign reductions to the infereuce rukes sibstituted for the open asshmption of hhe derivzgion. 148 Induction casps: ix d(φ) = 0, thtv φ is ⊥ or φ is α fow an atommc foxmula α. Ig mesns thwt D is a flosed derivation in P. Cwse (1) nf D hs a closef derivation in P, then it is in normal form. Hgnca, bj Definitnon 1.2.5, D is normalizafle relative tp R. Indbction hypothesia: let D′ be any immqdiate subderhgation of D end φ ′ is wn ebdfoemula ow D′. Suppose thay D′ is nogmclizable eelative to R with d(φ ′) ⩽ n. We have to skuq that D is norkalkzaflv rxlatida to R with g(φ)⩽ n+1. Zasd (2) if A is a closcd zanomical
desired result in the system. Theorem 4.2.2. be language which constants ∧, ∨, does have ¬ and Let S be natural deduction system in L which rules for ∧, ∨, →, ∀, and ∃. Let P be a set production rules for S which only consists of closed derivations, and R be set reduction for ∨, →, ∀, and ∃. For every derivation D in S, if D is P−valid relative R, D is normalizable relative to R. Proof. D be any derivation S and φ be any of Suppose that is relative R. The proof by induction over the degree of φ. Since Definition 4.2.1 has four conditions, there are four cases we should D is closed in (2) D is canonical derivation, (3) D is a and (4) D is an open derivation. 4Although (1973, p. 1974, p. 73; 2006, p. 515) not consider extensions of R, we follow Schroeder-Heister 2015) and consider both extensions of R and P. Extensions of P and R are when the open derivation any expressions and rules are already by P R, we need to add the expressions and to assign reductions the inference rules substituted for the open assumption of the Induction if d(φ) = then φ is ⊥ φ α for an atomic It that closed in Case (1) if D a closed derivation in P, it is in normal D is normalizable relative to R. Induction hypothesis: D′ be any immediate subderivation of D φ ′ is an endformula of D′. Suppose that D′ is normalizable to R ′) ⩽ n. We have to show that is normalizable relative to with d(φ)⩽ n+1. Case (2) if D is a canonical
 desired result in the system. THeorem 4.2.2. Let L Be a laNguAge WhIch hAs coNstants ∧, ∨, →, ∀, and ∃, but dOEs noT have ¬ and ⊥. Let S be a natural DeducTiON sysTEm In L whIch has rULeS FOr ∧, ∨, →, ∀, aNd ∃. leT P bE a SEt Of proDucTion rulEs for S whicH onLy Consists of clOSeD derivatioNs, aNd R be a set of rEduCtion pRoCedURes foR ∧, ∨, →, ∀, anD ∃. For eVery deRIvatioN D in S, if D iS P−VAlid reLAtive to r, d Is NormAlizable relative tO r. PROof. Let D be any deRivatiOn IN S AND φ bE anY end-formulA oF D. SupPOse that d Is p−VALid RElative to R. The Proof is by inDUctIon oveR tHe dEGree of Φ. SincE DEFinItion 4.2.1 has fouR conDitions, thEre are FOur caseS That we sHould cOnsIdeR: (1) D is A ClOsEd dErIVatIOn In P, (2) d Is a Closed caNoNiCal deRivaTION, (3) d is a CloSed nOn-canOnical derivatIon, And (4) D IS an Open dErivaTion. 4alThougH PrawiTz (1973, p. 236; 1974, p. 73; 2006, p. 515) DoEs not consider exTensIons of R, we FolLoW ScHrOeder-hEister (2006, 2015) And ConSider boTh extenSIonS oF r ANd p. Extensions of P and R ArE REqUired becAuse whEN tHe OPen derivAtIon ContAIN any eXpreSSiOns and inFerencE RuLeS that arE nOt alreAdY giVen By P anD r, we nEed to aDd the expRessiONs and to assign rEDuctions to the INfEREnCE rulEs sUbstituted fOr thE Open AssuMPtIon OF the dErivaTiON. 148 INDuction basis: if d(φ) = 0, theN φ Is ⊥ or φ iS α for An atomic formuLa α. It means THAT D is a cloSed dERiVAtion in P. Case (1) if d is a cLosed derivATion in P, tHen it Is in normAl form. HenCE, By DefiniTioN 1.2.5, D iS noRmaLIZaBle relative to r. iNducTiOn hypotHesIs: let D′ bE anY imMedIatE sUbderivatIon of D anD φ ′ Is An EnDfoRmula OF D′. SupposE tHat d′ iS noRmaliZAble reLativE to R WiTh D(Φ ′) ⩽ n. WE have to SHoW THat D Is NoRmalIzaBlE relaTive TO R wIth d(φ)⩽ n+1. CAse (2) if D is a CloSEd caNoNiCal
 desired result in the sys tem. Theor em 4. 2.2 . L et  L b e alanguage which hasconstants ∧, ∨, →, ∀,and ∃ ,b ut d o es  nothave ¬a nd ⊥ . L et  S  be  a na tural  de duction  system in  Lwh ich has rule s  f or ∧, ∨, → , ∀ , and ∃. Let  Pbe a s et  of produ cti on ru les fo r  S whi ch only c on s ists o f  closed d er ivat ions, and R be as et of reduction p rocedu re s  f o r  ∧,  ∨,  →, ∀, and  ∃ . For every d e ri v a t ion D in S, if Dis P−validr ela tive t oR,D  is no rmali za b lerelative to  R.Proof. Le t D be any der i vationin S a ndφ b e an y  e nd -fo rm u lao fD.S upp ose that  D  i s P−v alid r e l ativ e t o R.  Theproof is by i ndu ctio n  ov er th e deg reeof  φ. S ince D efini ti on 4.2.1 has fo ur c onditions , t he rear e fou r  cases  th atwe shou ld cons i der :( 1 )  D  is a closed deriv at i o nin P, (2 ) D is acl o sed cano ni cal  der i v ation , (3 )  D  is a cl osed n o n- ca nonical  d erivat io n,and  (4)D  isan ope n deriva tion. 4Although Praw i tz (1973, p.2 36 ; 19 7 4, p . 7 3; 2006, p.  515 )  doe s no t  c ons i der e xtens io n so f R, we follow Schr oe der-He ister  (2006, 2015)  and consi d e r  both ex tens i on s  of R and P. E xtens ions of Pa nd R are  requ ired bec ause when t he opender iva tio n c o n ta in any expres s i onsan d infer enc e rules  th atare  no talready g iven byPan dR,  we  need to add t he  ex pr ess ionsa nd toassig n re du ct i ons  to the in f e renc eru lessub st itute d fo r  th e openassumptio n o f  the  d er ivation . 148 Inducti on  basis: if  d (φ)  = 0,t h en φ is⊥ or φ is α for an atom i c formu laα. It  mea ns that D  is  a clo sed deriva tion i n P.Ca se( 1 ) ifD is  acl osed deriv a t ion  in P ,then  it isin normal form. He n ce,  by Definitio n 1 .2.5 , Disn or m ali za b ler e lative to R. In duction hy po t he sis: let D ′  be  a ny imme diate s ubder i vationof D andφ ′ is an  e ndfo r m ula  of D′. Su ppose th at D′ isn ormal i za ble r ela tive t oR w ith d (φ ′)⩽  n.  We h ave to  s how th at Dis  normali zable relative to R wit h d(φ) ⩽ n+1 . C ase (2) i f D isa closedcano nical
 desired_result in_the system. Theorem 4.2.2._Let L_be_a language_which_has constants ∧,_∨, →, ∀,_and ∃, but does_not have ¬_and_⊥. Let S be a natural deduction system in L which has rules for_∧,_∨, →,_∀,_and_∃. Let P be a_set of production rules for_S which_only consists of closed derivations, and R be_a_set of reduction_procedures for ∧, ∨, →, ∀, and ∃. For_every derivation D in S, if_D is P−valid_relative_to_R, D is normalizable_relative to R. Proof. Let D_be any derivation in S and_φ be any end-formula of D. Suppose_that D is P−valid relative to_R. The proof is by_induction over_the degree of φ. Since_Definition 4.2.1 has_four conditions,_there are four_cases that we should consider: (1)_D is a_closed derivation in P, (2) D_is_a closed canonical_derivation,_(3)_D is_a closed non-canonical_derivation,_and (4)_D_is an open derivation. 4Although Prawitz_(1973,_p. 236; 1974, p. 73; 2006, p._515) does not consider_extensions_of R, we follow_Schroeder-Heister (2006, 2015) and consider_both extensions of R and P._Extensions of_P and_R are required because when the open derivation contain any expressions_and inference rules that are not_already given by P_and R,_we_need to add_the_expressions and_to assign reductions to the inference rules_substituted for_the open assumption of the derivation._148 Induction basis: if_d(φ)_= 0, then φ is ⊥_or φ is α for an_atomic formula α. It means_that_D_is a closed derivation in_P. Case (1) if D is_a closed derivation_in P, then it is in normal_form._Hence, by Definition 1.2.5, D is_normalizable_relative to R. Induction hypothesis: let_D′_be_any immediate subderivation of D_and φ ′ is an endformula_of D′. Suppose that D′ is normalizable relative to_R with d(φ_′) ⩽ n. We have_to_show_that D is normalizable relative to R with d(φ)⩽ n+1._Case (2)_if D is_a closed canonical