 derivation, since D′ is normalizable, trivially D is normalizable relative to R. Case (3) if D is a closed non-canonical derivation, then, by Definition 4.2.1, D reduces to a P−valid canonical derivation relative to R. By the case (2), D is normalizable relative to R. Case (4) if D is an open derivation, then, by Definition 4.2.1, for any P′ ⊇ P and R′ ⊇R, a closed derivation D1 φ1,..., Dn φn D ψ is P′−valid relative to R′. By the cases (1), (2), and (3), since every list of closed derivation Di φi (1 ⩽ i ⩽ n) is normalizable relative to R′, D is normalizable relative to R. The result shows that at least in a particular system containing rules for ∧, ∨, →, ∀, and ∃, proof-theoretic validity implies normalizability. There is a further strengthened concept, strong-P−validity, which implies strong normalizability. Definition 4.2.3. (Inductive Definition of Stong P−Validity) Let P be a set of production rules and R be a set of reduction procedures. (1) Every closed derivation in P is strongly P−valid relative to R (for every R). 149 (2) A closed canonical derivation D is strongly P−valid relative to R iff all immediate subderivations of D are strongly P−valid relative to R. (3) A closed non-canonical derivation D is strongly P−valid relative to R iff every D′, such that D▷D′, is strongly P−valid relative to R. (4) An open derivation φ1,...,φn D ψ, where all open assumptions of D are among φ1,...,φn, is strongly P−valid relative to R iff for every P′ ⊇ P and R′ ⊇ R, and for every list of closed derivation Di φi (1 ⩽ i ⩽ n), which are strongly P′−valid relative to R′, D1 φ1,..., Dn φn D ψ is strongly P′−valid relative to R′. Theorem 4.2.4. Let S be a natural deduction system having rules for ∧, ∨, →, ∀, and ∃. Let
derivation, since D′ is normalizable, trivially D is normalizable relative to R. Case (3) if D is a closed non - basic deriving, then, by Definition 4.2.1, D reduces to a P−valid canonical derivation proportional to R. By the case (2), D is normalizable relative to R. Case (4) if D is an open deriving, then, by Definition 4.2.1, for any P′ ⊇ P and R′ ⊇R, a closed derivation D1 φ1, ..., Dn φn D ψ is P′−valid proportional to R′. By the case (1), (2), and (3), since every tilt of closed derivation Di φi (1 ⩽ i ⩽ n) is normalizable proportional to R′, D is normalizable relative to R. The result shows that at least in a particular system containing rules for ∧, ∨, →, ∀, and ∃, proof - theoretical validity implies normalizability. There is a far strengthened concept, strong - P−validity, which imply strong normalizability. Definition 4.2.3. (Inductive Definition of Stong P−Validity) permit P be a set of production rules and R be a hardening of reduction procedures. (1) Every closed derivation in P is strongly P−valid relative to R (for every R). 149 (2) A closed canonical derivation D is strongly P−valid relative to R iff all immediate subderivations of D are strongly P−valid proportional to R. (3) A shut non - canonical derivation D is strongly P−valid proportional to R iff every D′, such that D▷D′, is powerfully P−valid relative to R. (4) An open derivation φ1,... ,φn D ψ, where all open assumptions of D are among φ1,... ,φn, is strongly P−valid relative to R iff for every P′ ⊇ P and R′ ⊇ radius, and for every list of closed derivation Di φi (1 ⩽ i ⩽ n), which are strongly P′−valid relative to R′, D1 φ1, ..., Dn φn D ψ is strongly P′−valid proportional to R′. Theorem 4.2.4. Let S be a natural deduction system having rules for ∧, ∨, →, ∀, and ∃. Let
 degivation, since D′ is normxlizable, trivially D iv normzlizable relative to R. Case (3) if D is a clostb non-canonical derivxtion, thej, by Defunitmon 4.2.1, D reduces to a P−vallb cankkical verivation relajive to R. By the case (2), D iv voxmalizable relative to R. Case (4) if D ys an olej derivation, tren, nr Derpnltion 4.2.1, for any P′ ⊇ P and R′ ⊇R, a clksed degivation D1 φ1,..., Dn φn C ψ is P′−valid relative to R′. By hhe cases (1), (2), and (3), slnce every oist if closed defivation Dp φi (1 ⩽ i ⩽ n) js normalizable relative to R′, D is nprmalizablg depdtive to R. Vhe refult shows tmst at neast im a particular syvten containing rules foc ∧, ∨, →, ∀, and ∃, proof-theotetic valigicy implies normalizabuluty. Tvere is x fufthtr xtdengthfnev concept, sfrong-P−validuty, which implies surogt normalizabimity. Dqfynition 4.2.3. (Inductive Definition of Stong K−Valisity) Let P be a set of production rules and T be a set of reduction procedures. (1) Every closed derivation in P ks wtvongut O−valid relative to R (for every R). 149 (2) A closed cwhomibal derivation D ls strongly P−valid rflsjive to R iff xll immedjate subderivationd of D wre srrongly P−dalic relative to R. (3) A closed nin-canonical beruvation D is stronyly P−valid rzlativg to R iff every D′, such that B▷D′, is atrongly P−vwlid relafkve to R. (4) An opev dvrivdtion φ1,...,φn D ψ, where all open wssumptiois of D are xmonb φ1,...,φn, is strongly O−valid relative to R iff for gvery [′ ⊇ P and R′ ⊇ R, and for every list of closev derivation Ci φi (1 ⩽ i ⩽ n), wkich ave strongly P′−vajid relative tp R′, D1 φ1,..., Bn φn D ψ is stronfly P′−vanid relativq to R′. Theorek 4.2.4. Let S be a natural dedyctiin systdo having rules for ∧, ∨, →, ∀, cud ∃. Let
derivation, since D′ is normalizable, trivially D relative R. Case if D is by 4.2.1, D reduces a P−valid canonical relative to R. By the case D is normalizable relative to R. Case (4) if D is an open then, by Definition 4.2.1, for any P′ ⊇ P and R′ ⊇R, a derivation φ1,..., φn ψ is P′−valid relative to R′. By the cases (1), (2), and (3), since every list closed derivation Di φi (1 ⩽ i ⩽ is normalizable relative to D is normalizable relative to The shows that least a system containing rules ∧, ∨, →, ∀, and ∃, proof-theoretic validity implies normalizability. There is a further strengthened concept, strong-P−validity, implies strong 4.2.3. (Inductive of P−Validity) P be a production rules and R be a procedures. (1) Every closed derivation in P is P−valid relative R (for every R). 149 (2) closed canonical derivation D is strongly P−valid relative R iff all immediate subderivations of D are strongly P−valid relative to R. (3) A derivation D is strongly relative to R every such D▷D′, strongly P−valid to R. (4) An open derivation φ1,...,φn D ψ, where all assumptions of D are among φ1,...,φn, is strongly P−valid relative iff every P′ ⊇ and R′ ⊇ R, for list of closed derivation (1 i are P′−valid to R′, D1 φ1,..., φn D ψ is strongly relative to R′. Theorem natural deduction system having rules for ∧, ∨, ∀, and ∃. Let
 derivation, since D′ is normaliZable, triviAlly D Is nOrmAlIzabLe reLative to R. Case (3) iF d is a Closed non-canonical deriVatioN, tHEn, by dEfInitiOn 4.2.1, D reduCEs TO A P−vAlId CanOnICaL deriVatIon relaTive to R. By tHe cAsE (2), D is normalizABlE relative tO R. CAse (4) if D is an opEn dErivatIoN, thEN, by DeFinItion 4.2.1, For any p′ ⊇ p and R′ ⊇R, A closed deRiVAtion D1 Φ1,..., dn φn D ψ iS p′−VaLid rElative to R′. By the caSEs (1), (2), ANd (3), since every liSt of clOsED dERIvaTioN Di φi (1 ⩽ i ⩽ n) is nOrMalizABle relaTIvE TO r′, D iS Normalizable rElative to R. THE reSult shOwS thAT at leaSt in a PaRTicUlar system cOntaIning ruleS for ∧, ∨, →, ∀, anD ∃, Proof-thEOretic vAliditY imPliEs noRMaLiZabIlITy. THErE is A FurTher streNgThEned cOncePT, STRong-p−vaLidiTy, whiCh implies stroNg nOrmaLIzaBilitY. DefiNitiOn 4.2.3. (inducTive DeFinitIoN of Stong P−ValidiTy) LeT P be a set oF prOdUctIoN ruleS And R be A seT of ReductiOn proceDUreS. (1) EVERY cLosed derivation in P Is STRoNgly P−valId relaTIvE tO r (for everY R). 149 (2) a clOsed CANonicAl deRIvAtion D is StrongLY P−VaLid relaTiVe to R iFf All ImmEdiatE SubdErivatIons of D aRe strONgly P−valid relaTIve to R. (3) A closed NOn-CANoNIcal DerIvation D is sTronGLy P−vAlid RElAtiVE to R iFf eveRy d′, SuCH that D▷D′, is strongly P−vAlId relaTive tO R. (4) An open derivAtion φ1,...,φn D ψ, WHERe all opeN assUMpTIons of D are amonG φ1,...,φn, iS strongly P−VAlid relaTive tO R iff for Every P′ ⊇ P anD r′ ⊇ r, and for eVerY liSt oF clOSEd Derivation Di φI (1 ⩽ I ⩽ N), whiCh Are stroNglY P′−valid RelAtiVe tO R′, D1 Φ1,..., DN φn D ψ is stRongly P′−vAlId ReLaTivE to R′. THEorem 4.2.4. Let s bE a nAtUraL deduCTion syStem hAvinG rUlES foR ∧, ∨, →, ∀, and ∃. Let
 derivation, since D′ is n ormalizabl e, tr ivi all yD is  nor malizable rela t iveto R. Case (3) if D is  a cl os e d no n -c anoni cal der i va t i on,  t he n,by De finit ion  4.2.1,  D reduces  to  a  P−valid can o ni cal deriva tio n relative t o R . By t he  ca s e (2) , D  is n ormali z able r elative t oR . Case (4) ifD is  anopen derivation,t he n , by Definitio n 4.2. 1, fo r any  P′  ⊇ P and R ′⊇R, a closedd er i v a tio n  D1 φ1,..., D n φn D ψ is P′− validre lat i ve toR′. B yt hecases (1),(2),  and (3),  since every l i st of c losedder iva tion Di  φ i ( 1⩽  i⩽  n ) i s  no rmalizab le  r elati ve t o R ′ , Disnorm aliza ble relativetoR. T h e r esult  show s th at  at l east i n a p ar ticular systemcont aining ru les  f or∧,  ∨, → ,  ∀, an d ∃ , p roof-th eoretic val id i t y  i mplies normalizabi li t y .There is  a fur t he rs trengthe ne d c once p t , str ong- P −v alidity,  which im pl ies str on g norm al iza bil ity.D efin ition4.2.3. ( Induc t ive Definition of Stong P−Va l id i t y) LetP b e a set ofprod u ctio n ru l es  an d  R be  a se to fr eduction procedures .(1) Ev ery c losed derivat ion in P i s s tronglyP−va l id relative to R(forevery R).1 49 (2) A  clos ed canon ical deri v a tion D i s s tro ngl y P − v al id relative t o R if fall imm edi ate sub der iva tio nsof  D are st rongly P −v al id  r ela tivet o R. (3)  A  cl os ednon-c a nonica l der ivat io nD  is  strong l yP − vali dre lati veto  R if f ev e ryD′, suc h that D▷ D′, is s tr on gly P−v alid relative  t o R. (4) A nope n deri v a tion φ1, ...,φn D ψ, where all o p en assu mpt ionsof D  are amon g φ 1,..., φn, is str onglyP−val id  re l a tivet o  R  if ffor everyP ′  ⊇P and  R ′ ⊇R, andfor every list ofc los ed derivation  Di  φi( 1  ⩽  i⩽  n ) , w hi c h a r e  strongly P′−va lid relati ve to  R′, D1 φ1 , ... ,Dn φn D  ψ is s trong l y P′−va lid relat ive to R′ .Theo r e m 4 .2.4. LetS be a n atural de d uctio n  s ystem  ha ving r ul esfor ∧ , ∨, → ,  ∀,  and∃. Let 
 derivation,_since D′_is normalizable, trivially D_is normalizable_relative_to R._Case_(3) if D_is a closed_non-canonical derivation, then, by_Definition 4.2.1, D_reduces_to a P−valid canonical derivation relative to R. By the case (2), D is_normalizable_relative to_R._Case_(4) if D is an_open derivation, then, by Definition_4.2.1, for_any P′ ⊇ P and R′ ⊇R, a_closed_derivation D1 φ1,...,_Dn φn D ψ is P′−valid relative to R′._By the cases (1), (2), and_(3), since every_list_of_closed derivation Di φi_(1 ⩽ i ⩽ n) is_normalizable relative to R′, D is_normalizable relative to R. The result shows_that at least in a particular_system containing rules for ∧,_∨, →,_∀, and ∃, proof-theoretic validity_implies normalizability. There_is a_further strengthened concept,_strong-P−validity, which implies strong normalizability. Definition_4.2.3. (Inductive Definition_of Stong P−Validity) Let P be_a_set of production_rules_and_R be_a set of_reduction_procedures. (1)_Every_closed derivation in P is strongly_P−valid_relative to R (for every R). 149_(2) A closed canonical_derivation_D is strongly P−valid_relative to R iff all_immediate subderivations of D are strongly_P−valid relative_to R._(3) A closed non-canonical derivation D is strongly P−valid relative to_R iff every D′, such that_D▷D′, is strongly P−valid_relative to_R._(4) An open_derivation_φ1,...,φn D_ψ, where all open assumptions of D_are among_φ1,...,φn, is strongly P−valid relative to_R iff for every_P′_⊇ P and R′ ⊇ R,_and for every list of closed_derivation Di φi (1 ⩽_i_⩽_n), which are strongly P′−valid_relative to R′, D1 φ1,..., Dn_φn D ψ_is strongly P′−valid relative to R′. Theorem_4.2.4._Let S be a natural deduction_system_having rules for ∧, ∨, →,_∀,_and_∃. Let