, Peter Milne (1994, p. 58) interprets CR−rule as a rule for introducing a formula φ, that is, the derivation of ⊥ from the assumption [¬φ] introduces a formula φ. Then, we have the following pair of rules. [¬φ]1 D1 ⊥ CR,1 φ φ D2 ¬φ CRE⊥ 65 The standard reduction procedure for CR− and CRE−rules is below. [¬φ]1 D1 ⊥ CR,1 φ D2 ¬φ CRE⊥ ▷CRE D2 ¬φ D1 ⊥ CR− and CRE−rules are intrinsically harmonious. Let SNCE be a natural deduction system which contains → −, ∈ −, CR−, and CRE−rules. SNCE has a set RNCE of reductions having ▷→, ▷∈, ▷CRE, and the following auxiliary reduction procedure ⊵∈. D t ∈ {x|φ(x)} ∈ E φ[t/x] ∈ I t ∈ {x|φ(x)} ⊵∈ D t ∈ {x|φ(x)} While the derivation D′ includes a subderivation D that has the same conclusion of D′, it is often desirable to reduce the length of the derivation D′. The auxiliary reduction ⊵∈ has such role of lessening the length of derivation. Moreover, instead of using ¬−rules, we define ¬φ as φ →⊥. Then, we have the following result. Proposition 2.4.2. Let us define a parameter a as {x|x ∈ x →⊥}. Then, there is a closed derivation of ⊥ in SNCE with respect to RNCE which generates a non-terminating reduction sequence and so is not normalizable. Proof. Two claims verify the result. Claim 1. There is a closed derivation Π3 of ⊥ in SNCE. 66 First, there is an open derivation Π1 of ⊥ from [¬a ∈ a]. [¬a ∈ a]1 ************** de f a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ************************ de f a ∈ a [¬a ∈ a]1 CRE⊥ With the derivation Π1, we have a closed derivation
, Peter Milne (1994, p. 58) interprets CR−rule as a rule for introducing a formula φ, that is, the deriving of ⊥ from the premise [ ¬φ ] introduces a formula φ. Then, we have the postdate pair of rules. [ ¬φ]1 D1 ⊥ CR,1 φ φ D2 ¬φ CRE⊥ 65 The standard decrease procedure for CR− and CRE−rules is below. [ ¬φ]1 D1 ⊥ CR,1 φ D2 ¬φ CRE⊥ ▷CRE D2 ¬φ D1 ⊥ CR− and CRE−rules are intrinsically harmonious. Let SNCE be a lifelike discount system which contain → −, ∈ −, CR−, and CRE−rules. SNCE has a set RNCE of reduction experience ▷→, ▷∈, ▷CRE, and the following auxiliary reduction operation ⊵∈. D t ∈ { x|φ(x) } ∈ E φ[t / x ] ∈ I t ∈ { x|φ(x) } ⊵∈ D t ∈ { x|φ(x) } While the derivation D′ admit a subderivation D that has the same conclusion of D′, it is frequently desirable to reduce the length of the derivation D′. The auxiliary reduction ⊵∈ give birth such role of lessening the length of derivation. Moreover, alternatively of using ¬−rules, we define ¬φ as φ →⊥. Then, we have the following result. Proposition 2.4.2. Let us define a parameter a as { x|x ∈ x →⊥ }. Then, there is a closed deriving of ⊥ in SNCE with respect to RNCE which generates a non - terminating decrease succession and so is not normalizable. Proof. Two claims verify the resultant role. Claim 1. There is a closed derivation Π3 of ⊥ in SNCE. 66 foremost, there is an open derivation Π1 of ⊥ from [ ¬a ∈ a ]. [ ¬a ∈ a]1 * * * * * * * * * * * * * * de f a ∈ a →⊥ ∈ I a ∈ { x|x ∈ x →⊥ } * * * * * * * * * * * * * * * * * * * * * * * * de f a ∈ a [ ¬a ∈ a]1 CRE⊥ With the derivation Π1, we have a closed derivation
, Peher Milne (1994, p. 58) interprets CR−rule as a rule for mntroduding a furmula φ, that is, the derivatiln of ⊥ drom the assumption [¬φ] ivtroduces a formuoa φ. Uhen, we have the hkllowinn paid of xuoes. [¬φ]1 D1 ⊥ CR,1 φ φ D2 ¬φ CRE⊥ 65 The standard reduwtkou procedure for CR− and CRE−rules is bqlow. [¬φ]1 D1 ⊥ FR,1 φ D2 ¬φ CRE⊥ ▷CRE D2 ¬φ C1 ⊥ CR− and CRE−rules are intrinsically hadmoniouv. Let SNCE be a natural deduction systel whlch contains → −, ∈ −, CG−, and CRE−ruoes. FBCE has a seg RNCE of gzductions hzving ▷→, ▷∈, ▷CRE, and the following ajxilicry reductiin prlwedure ⊵∈. D t ∈ {x|φ(x)} ∈ E φ[t/x] ∈ I t ∈ {w|φ(q)} ⊵∈ D t ∈ {x|φ(x)} Whike the derivatlon D′ inxludes a subderivatioi D that has the samg conclusimn of D′, it is often dwsiratle do rdeucd tge lsngth lf vhe derivatjon D′. The ayxiliary reduction ⊵∈ hww such role or lessqnyng the length of derivation. Moreover, itstsad of using ¬−rules, we dwfine ¬φ as φ →⊥. Then, we hwve the fjllowing result. Proposition 2.4.2. Let us define a paraketer x aw {w|x ∈ b →⊥}. Hhen, there is a closed derivation of ⊥ in SNCE siuh gespect to RNCE wmich generates a npn-hetiinating reduztion seqhence and so is noh normajizaboe. Proof. Uwo ckaims verify the result. Claum 1. There is q closed derivatiou Π3 of ⊥ in SNEE. 66 Fitst, thrre is an open derivatiun Π1 kf ⊥ from [¬a ∈ a]. [¬a ∈ a]1 ************** sd f a ∈ a →⊥ ∈ I a ∈ {b|x ∈ x →⊥} ************************ de f a ∈ a [¬a ∈ a]1 CRE⊥ With the deritatiou Π1, we haxe a closeq derivatiln
, Peter Milne (1994, p. 58) interprets a for introducing formula φ, that from assumption [¬φ] introduces formula φ. Then, have the following pair of rules. D1 ⊥ CR,1 φ φ D2 ¬φ CRE⊥ 65 The standard reduction procedure CR− and CRE−rules is below. [¬φ]1 D1 ⊥ CR,1 φ D2 ¬φ CRE⊥ D2 D1 CR− CRE−rules are intrinsically harmonious. Let SNCE be a natural deduction system which contains → −, ∈ CR−, and CRE−rules. SNCE has a set RNCE reductions having ▷→, ▷∈, and the following auxiliary reduction ⊵∈. t ∈ ∈ φ[t/x] I t ∈ ⊵∈ D t ∈ {x|φ(x)} While the derivation D′ includes a subderivation D that has the same of D′, often desirable reduce length the derivation D′. reduction ⊵∈ has such role of of derivation. Moreover, instead of using ¬−rules, we ¬φ as →⊥. Then, we have the following Proposition 2.4.2. Let us define a parameter a {x|x ∈ x →⊥}. Then, there is a closed derivation of ⊥ in SNCE with RNCE which generates a reduction sequence and is normalizable. Two verify the Claim 1. There is a closed derivation Π3 of ⊥ in 66 First, there is an open derivation Π1 of ⊥ ∈ [¬a ∈ a]1 de f a ∈ →⊥ I a ∈ {x|x →⊥} de a ∈ CRE⊥ With the derivation we have a closed derivation
, Peter Milne (1994, p. 58) interprets CR−ruLe as a rule fOr intRodUciNg A forMula Φ, that is, the deriVAtioN of ⊥ from the assumption [¬φ] iNtrodUcES a foRMuLa φ. ThEn, we havE ThE FOllOwInG paIr OF rUles. [¬φ]1 d1 ⊥ CR,1 Φ φ D2 ¬φ CRE⊥ 65 the standarD reDuCtion proceduRE fOr CR− and CRE−RulEs is below. [¬φ]1 D1 ⊥ Cr,1 φ D2 ¬Φ CRE⊥ ▷CRe D2 ¬Φ D1 ⊥ Cr− And CRe−ruLes arE intriNSicallY harmonioUs. lEt SNCE BE a naturAL DeDuctIon system which conTAiNS → −, ∈ −, CR−, and CRE−rules. sNCE haS a SEt rncE oF reDuctions haViNg ▷→, ▷∈, ▷CRE, ANd the foLLoWING auXIliary reductiOn procedure ⊵∈. d T ∈ {x|φ(X)} ∈ E φ[t/x] ∈ I T ∈ {x|Φ(x)} ⊵∈ D T ∈ {X|φ(x)} WhiLe the DeRIvaTion D′ includEs a sUbderivatIon D thAT has the SAme concLusion Of D′, It iS oftEN dEsIraBlE To rEDuCe tHE leNgth of thE dErIvatiOn D′. THE AUXiliAry ReduCtion ⊵∈ Has such role of LesSeniNG thE lengTh of dErivAtIon. MoReover, InsteAd Of using ¬−rules, we dEfinE ¬φ as φ →⊥. Then, We hAvE thE fOllowINg resuLt. PRopOsition 2.4.2. let us deFIne A pARAMeTer a as {x|x ∈ x →⊥}. Then, there Is A CLoSed derivAtion oF ⊥ In sNce with resPeCt tO RNCe WHich gEnerATeS a non-terMinatiNG rEdUction sEqUence aNd So iS noT normALizaBle. ProOf. Two claIms veRIfy the result. ClAIm 1. There is a cloSEd DERiVAtioN Π3 oF ⊥ in SNCE. 66 FirsT, theRE is aN opeN DeRivATion Π1 Of ⊥ froM [¬a ∈ A]. [¬A ∈ a]1 ************** DE f a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ************************ de f a ∈ a [¬a ∈ a]1 CRE⊥ wiTh the dErivaTion Π1, we have a cLosed derivATIOn
, Peter Milne (1994, p. 58 ) interpre ts CR −ru leas  a r ulefor introducin g  a f ormula φ, that is, the  deri va t iono f⊥ fro m the a s su m p tio n[¬ φ]in t ro duces  aformula  φ. Then,weha ve the follo w in g pair ofrul es. [¬φ]1 D1  ⊥CR,1 φ  φ  D2 ¬φ CR E⊥65 Th e stan d ard re duction p ro c eduref or CR−a n dCRE− rules is below. [ ¬ φ] 1  D1 ⊥ CR,1 φ D 2 ¬φ C RE ⊥  ▷ C R E D 2 ¬ φ D1 ⊥ CR−  a nd CR E −rulesa re i n tri n sically harmo nious. LetS NCE  be ana tur a l dedu ction  s y ste m which con tain s → −, ∈−, CR− ,  and CR E −rules.  SNCEhas  asetR NC Eofre d uct i on s h a vin g ▷→, ▷∈ ,▷C RE, a nd t h e f ollo win g au xilia ry reductionpro cedu r e ⊵ ∈. Dt ∈ { x|φ( x) } ∈ E  φ[t/x ] ∈ I  t  ∈ {x|φ(x)} ⊵∈D t∈ {x|φ(x) } W hi leth e der i vation  D′  in cludesa subde r iva ti o n Dthat has the sameco n c lu sion ofD′, it is  o f ten desi ra ble  tor e ducethel en gth of t he der i va ti on D′.Th e auxi li ary  re ducti o n ⊵∈  has s uch role  of l e ssening the le n gth of deriva t io n .  M o reov er,  instead of  usi n g ¬− rule s ,wed efine  ¬φ a sφ  → ⊥ . Then, we have the  f ollowi ng re sult. Proposi tion 2.4.2 . L et us de fine ap arameter a as{x|x∈ x →⊥}. T h en, ther e isa closed  derivati o n  of ⊥ in  SN CEwit h r e s pe ct to RNCE wh i c h ge ne rates a  no n-termi nat ing  re duc ti on sequen ce and s ois  n ot  no rmali z able. Pr oo f.Tw o c laims verify  theresu lt .C lai m 1. Th e re i s acl os ed d eri va tionΠ3 o f  ⊥in SNCE . 66 Firs t,t here  i san open  derivation Π 1of ⊥ from[¬ a ∈  a]. [ ¬ a  ∈ a]1 * ************* de f a ∈a  →⊥ ∈ I  a∈ {x| x ∈x →⊥} *** *** ****** *** * ****** ** def a ∈  a  [¬ a ∈ a]1 C RE ⊥ W it h the deri v a tio n Π1,  w e ha ve a cl osed derivation
, Peter_Milne (1994,_p. 58) interprets CR−rule_as a_rule_for introducing_a_formula φ, that_is, the derivation_of ⊥ from the_assumption [¬φ] introduces_a_formula φ. Then, we have the following pair of rules. [¬φ]1 D1 ⊥ CR,1_φ_φ D2_¬φ_CRE⊥_65 The standard reduction procedure_for CR− and CRE−rules is_below. [¬φ]1_D1 ⊥ CR,1 φ D2 ¬φ CRE⊥ ▷CRE_D2_¬φ D1 ⊥_CR− and CRE−rules are intrinsically harmonious. Let SNCE be_a natural deduction system which contains_→ −, ∈_−,_CR−,_and CRE−rules. SNCE has_a set RNCE of reductions having_▷→, ▷∈, ▷CRE, and the following_auxiliary reduction procedure ⊵∈. D t ∈_{x|φ(x)} ∈ E φ[t/x] ∈ I_t ∈ {x|φ(x)} ⊵∈ D_t ∈_{x|φ(x)} While the derivation D′_includes a subderivation_D that_has the same_conclusion of D′, it is often_desirable to reduce_the length of the derivation D′._The_auxiliary reduction ⊵∈_has_such_role of_lessening the length_of_derivation. Moreover,_instead_of using ¬−rules, we define ¬φ_as_φ →⊥. Then, we have the following_result. Proposition 2.4.2. Let_us_define a parameter a_as {x|x ∈ x →⊥}._Then, there is a closed derivation_of ⊥_in SNCE_with respect to RNCE which generates a non-terminating reduction sequence and_so is not normalizable. Proof. Two_claims verify the result._Claim 1._There_is a closed_derivation_Π3 of_⊥ in SNCE. 66 First, there is_an open_derivation Π1 of ⊥ from [¬a_∈ a]. [¬a ∈_a]1_************** de f a ∈ a_→⊥ ∈ I a ∈ {x|x_∈ x →⊥} ************************ de_f_a_∈ a [¬a ∈ a]1_CRE⊥ With the derivation Π1, we_have a closed_derivation