 which generates a non-terminating reduction sequence and so is not fully normalizable. Proof. Two claims establish the result. Claim 1. There is a closed derivation Σ9 of ⊥. 83 First, we have an open derivation Σ7 of ⊥ from [¬Ψ]. [¬Ψ]2 [T (⌜Ψ⌝)]4 [Ψ]1 [¬Ψ]2 [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [⊥]5 ¬E,5⊥ With Σ7, there is a derivation Σ8 of ⊥ from [Ψ]. [Ψ]9 [¬T (⌜Ψ⌝)]6 [¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ ΨE,6⊥ Then, we have a closed derivation Σ9 of ⊥. [¬Ψ]2 Σ7 ⊥ CR,2 Ψ [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]11 CRE,11⊥ Claim 2. Σ9 initiates a non-terminating reduction sequence and so is not fully normalizable. Since Ψ in CRE−rule of Σ9 is the major premise which is not an assumption, we apply 84 ▷CRE to Σ9 and have a derivation below. [Ψ]12 Σ8 ⊥ ¬I,12¬Ψ [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [⊥]5 ¬E,5⊥ It still has a major premise ¬Ψ derived by ¬I−rule. By applying ▷¬ and ▷Ψ, the following derivation is achieved. [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]3 CRE,3�
which generates a non - terminating reduction sequence and so is not amply normalizable. validation. Two claims establish the resultant role. call 1. There is a closed ancestry Σ9 of ⊥. 83 foremost, we have an open derivation Σ7 of ⊥ from [ ¬Ψ ]. [ ¬Ψ]2 [ T (⌜ Ψ ⌝ )]4 [ Ψ]1 [ ¬Ψ]2 [ ⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [ ⊥]5 ¬E,5⊥ With Σ7, there be a derivation Σ8 of ⊥ from [ Ψ ]. [ Ψ]9 [ ¬T (⌜ Ψ ⌝ )]6 [ ¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I T (⌜ Ψ ⌝) [ ⊥]10 ¬E,10⊥ ΨE,6⊥ Then, we get a closed derivation Σ9 of ⊥. [ ¬Ψ]2 Σ7 ⊥ CR,2 Ψ [ Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [ ⊥]11 CRE,11⊥ Claim 2. Σ9 initiates a non - terminating reduction sequence and so is not amply normalizable. Since Ψ in CRE−rule of Σ9 is the major premise which is not an assumption, we apply 84 ▷CRE to Σ9 and have a ancestry below. [ Ψ]12 Σ8 ⊥ ¬I,12¬Ψ [ T (⌜ Ψ ⌝ )]4 [ Ψ]1 [ Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [ ⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [ ⊥]5 ¬E,5⊥ It still have a major premise ¬Ψ derived by ¬I−rule. By applying ▷¬ and ▷Ψ, the following derivation is achieved. [ deoxythymidine monophosphate (⌜ Ψ ⌝ )]4 [ Ψ]1 [ Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [ ⊥]3 CRE,3 �
 whlch generates a non-termikating reduction sequenre and ao is nog fully normalizable. Proof. Txo coaims establish the result. Zlaim 1. Thvre is a xlostd derivation Σ9 of ⊥. 83 First, we habc an m'en derivation Σ7 of ⊥ from [¬Ψ]. [¬Ψ]2 [T (⌜Ψ⌝)]4 [Ψ]1 [¬Ψ]2 [⊥]3 CRE,3⊥ T A,1⊥ ΨK,4 Ψ [⊥]5 ¬E,5⊥ With Σ7, there is a derivation Σ8 of ⊥ from [Ψ]. [Ψ]9 [¬H (⌜Ψ⌝)]6 [¬Ψ]2 Σ7 ⊥ CR,2 Ψ T I J (⌜Ψ⌝) [⊥]10 ¬E,10⊥ ΨQ,6⊥ Thsn, we have a closed derivation Σ9 of ⊥. [¬Ψ]2 Σ7 ⊥ CR,2 Ψ [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]11 CRE,11⊥ Ckaim 2. Σ9 initiates a non-termlnatlng reduction sequfnce and so is git fully noroalizable. Since Ψ in CRG−rule of Σ9 is the major premise wfich ns not an awsymphhon, we applb 84 ▷CRE to Σ9 and havc a derhvation below. [Ψ]12 Σ8 ⊥ ¬I,12¬Ψ [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [⊥]5 ¬E,5⊥ It stmll has a major premyse ¬Ψ derieeb by ¬I−rule. By applyint ▷¬ and ▷Ψ, the foluiwivg secivztion ls echieved. [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]3 CRE,3�
which generates a non-terminating reduction sequence and not normalizable. Proof. claims establish the a derivation Σ9 of 83 First, we an open derivation Σ7 of ⊥ [¬Ψ]. [¬Ψ]2 [T (⌜Ψ⌝)]4 [Ψ]1 [¬Ψ]2 [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [⊥]5 With Σ7, there is a derivation Σ8 of ⊥ from [Ψ]. [Ψ]9 [¬T [¬Ψ]2 ⊥ Ψ I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ ΨE,6⊥ Then, we have a closed derivation Σ9 of ⊥. [¬Ψ]2 ⊥ CR,2 Ψ [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]11 Claim 2. Σ9 initiates non-terminating reduction sequence and so not normalizable. Since in of is the major which is not an assumption, we apply 84 ▷CRE to Σ9 and have a derivation below. [Ψ]12 ⊥ ¬I,12¬Ψ [Ψ]1 [Ψ]9 ⊥ [⊥]3 T E,1⊥ ΨI,4 ¬E,5⊥ It still has a major by ¬I−rule. By applying ▷¬ and ▷Ψ, the derivation is [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ [⊥]3 CRE,3�
 which generates a non-terminaTing reductIon seQueNce AnD so iS not Fully normalizaBLe. PrOof. Two claims establish tHe resUlT. claiM 1. thEre is A closed DErIVAtiOn σ9 oF ⊥. 83 FiRsT, We Have aN opEn derivAtion Σ7 of ⊥ frOm [¬Ψ]. [¬ψ]2 [T (⌜ψ⌝)]4 [Ψ]1 [¬Ψ]2 [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 Ψ [⊥]5 ¬E,5⊥ wItH Σ7, there is a DerIvation Σ8 of ⊥ frOm [Ψ]. [ψ]9 [¬T (⌜Ψ⌝)]6 [¬Ψ]2 Σ7 ⊥ Cr,2 Ψ t I T (⌜ψ⌝) [⊥]10 ¬e,10⊥ ΨE,6⊥ ThEn, wE have A closeD DerivaTion Σ9 of ⊥. [¬Ψ]2 Σ7 ⊥ cR,2 ψ [ψ]9 Σ8 ⊥ ¬I,9¬Ψ [⊥]11 CRe,11⊥ claim 2. Σ9 iNITiAtes A non-terminating reDUcTIon sequence and So is noT fULlY NOrmAliZable. Since ψ iN CRE−rULe of Σ9 is THe MAJOr pREmise which is nOt an assumptIOn, wE apply 84 ▷cRe to σ9 And havE a derIvATioN below. [Ψ]12 Σ8 ⊥ ¬I,12¬Ψ [T (⌜ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬i,9¬Ψ [⊥]3 CRE,3⊥ T E,1⊥ ΨI,4 ψ [⊥]5 ¬E,5⊥ It stILl has a mAJor premIse ¬Ψ deRivEd bY ¬I−ruLE. BY aPplYiNG ▷¬ anD ▷ψ, tHe fOLloWing deriVaTiOn is aChieVED. [t (⌜ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8 ⊥ ¬i,9¬Ψ [⊥]3 CrE,3�
 which generates a non-ter minating r educt ion  se qu ence  and  so is not ful l y no rmalizable. Proof. Two  clai ms esta b li sh th e resul t .C l aim  1 .The re is  a cl ose d deriv ation Σ9 o f ⊥ .83 First, we ha ve an open  de rivation Σ7of⊥ from  [ ¬Ψ] .  [¬Ψ] 2 [ T (⌜Ψ ⌝)]4 [ Ψ ]1 [¬Ψ ]2 [⊥]3 C RE , 3⊥ T E , 1⊥ ΨI,4 Ψ  [ ⊥]5¬E,5⊥ With Σ7, th e re is a derivatio n Σ8 o f⊥  f r o m [ Ψ].  [Ψ]9 [¬T(⌜ Ψ⌝)]6 [¬Ψ]2 Σ 7  ⊥ C R ,2Ψ  T I T (⌜Ψ⌝)[⊥]10 ¬E,10 ⊥  ΨE ,6⊥ Th en , w e  havea clo se d  de rivation Σ9  of⊥. [¬Ψ]2Σ7 ⊥ C R ,2 Ψ [Ψ ] 9 Σ8 ⊥¬I,9¬Ψ  [⊥ ]11  CRE , 11 ⊥Cla im 2.Σ 9ini t iat es a non -t er minat ingr e d u ctio n s eque nce a nd so is notful ly n o rma lizab le. S ince  Ψ  in C RE−rul e ofΣ9  is the major p remi se whichisno t a nassum p tion,weapp ly 84 ▷ CRE toΣ 9 a nd h a ve  a derivation belo w. [ Ψ] 12 Σ8 ⊥¬I,12¬ Ψ  [ T( ⌜Ψ⌝)]4 [ Ψ] 1 [ Ψ]9Σ 8  ⊥ ¬I ,9¬Ψ [⊥ ]3 CRE,3 ⊥ T E, 1 ⊥ΨI ,4 Ψ [⊥ ]5  ¬E,5⊥  I t s til l has a ma jor pr emise ¬Ψ  deri v ed by ¬I−rule. By applying ▷ ¬  a n d  ▷ Ψ , th e f ollowing de riva t ionis a c hi eve d . [T(⌜Ψ⌝) ]4 [Ψ ] 1 [Ψ]9 Σ8 ⊥ ¬I,9¬Ψ[⊥ ]3 CRE ,3�
 which_generates a_non-terminating reduction sequence and_so is_not_fully normalizable._Proof._Two claims establish_the result. Claim_1. There is a_closed derivation Σ9_of_⊥. 83 First, we have an open derivation Σ7 of ⊥ from [¬Ψ]. [¬Ψ]2_[T_(⌜Ψ⌝)]4 [Ψ]1_[¬Ψ]2_[⊥]3_CRE,3⊥ T E,1⊥ ΨI,4 Ψ_[⊥]5 ¬E,5⊥ With Σ7, there_is a_derivation Σ8 of ⊥ from [Ψ]. [Ψ]9 [¬T_(⌜Ψ⌝)]6_[¬Ψ]2 Σ7 ⊥_CR,2 Ψ T I T (⌜Ψ⌝) [⊥]10 ¬E,10⊥ ΨE,6⊥_Then, we have a closed derivation_Σ9 of ⊥._[¬Ψ]2_Σ7_⊥ CR,2 Ψ [Ψ]9_Σ8 ⊥ ¬I,9¬Ψ [⊥]11 CRE,11⊥ Claim_2. Σ9 initiates a non-terminating reduction_sequence and so is not fully normalizable._Since Ψ in CRE−rule of Σ9_is the major premise which_is not_an assumption, we apply 84_▷CRE to Σ9_and have_a derivation below._[Ψ]12 Σ8 ⊥ ¬I,12¬Ψ [T (⌜Ψ⌝)]4_[Ψ]1 [Ψ]9 Σ8_⊥ ¬I,9¬Ψ [⊥]3 CRE,3⊥ T E,1⊥_ΨI,4_Ψ [⊥]5 ¬E,5⊥_It_still_has a_major premise ¬Ψ_derived_by ¬I−rule._By_applying ▷¬ and ▷Ψ, the following_derivation_is achieved. [T (⌜Ψ⌝)]4 [Ψ]1 [Ψ]9 Σ8_⊥ ¬I,9¬Ψ [⊥]3 CRE,3�