�E⊥ ¬I,7¬a ∈ a [a ∈ a]2 ¬E⊥ → I,2a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ************************ de f a ∈ a ¬E⊥ Since Σ3 has a maximum formula ¬a ∈ a in ¬E−rule, the application of ¬−reduction 62 provides the derivation Σ4 below. [¬⊥]6 [¬⊥]6 [a ∈ a]2 ∆ ⊥ ¬E⊥ ¬I,2a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ∈ E a ∈ a →⊥ [¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a → E⊥ ¬E⊥ CR,6⊥ Σ4 still has maximum formulas, and so is reduced to Σ5 below by ▷∈− and ▷¬−reductions. [¬⊥]6 [¬⊥]6 [¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a ∆ ⊥ ¬E⊥ ¬E⊥ CR,6⊥ Σ5 includes the same derivation with Σ2. Again, Σ2 can be further reduced. Then, we have 63 the following infinite reduction sequence. [¬⊥]6 [¬⊥]6 [¬⊥]8 [¬⊥]8 [¬⊥]i [¬⊥]i... ⊥ ¬E⊥ ¬E⊥ CR,i⊥... ⊥ ¬E⊥ ¬E⊥ CR,8⊥ ¬E⊥ ¬E⊥ CR,6⊥ where i = 2 j + 4( j > 0). Therefore, Σ2 generates a non-terminating reduction sequence. Having the auxiliary reduction procedure ⊵CR(◦,⋆), Σ2 initiates a non-terminating reduction sequence which is not so much a looping reduction as what Tennant (2016) calls, a'sp
� E⊥ ¬I,7¬a ∈ a [ a ∈ a]2 ¬E⊥ → I,2a ∈ a →⊥ ∈ I a ∈ { x|x ∈ x →⊥ } * * * * * * * * * * * * * * * * * * * * * * * * de f a ∈ a ¬E⊥ Since Σ3 has a maximum formula ¬a ∈ a in ¬E−rule, the lotion of ¬−reduction 62 provide the derivation Σ4 below. [ ¬⊥]6 [ ¬⊥]6 [ a ∈ a]2 ∆ ⊥ ¬E⊥ ¬I,2a ∈ a →⊥ ∈ I a ∈ { x|x ∈ x →⊥ } ∈ east a ∈ a →⊥ [ ¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a → E⊥ ¬E⊥ CR,6⊥ Σ4 even has maximum formula, and indeed is reduced to Σ5 downstairs by ▷∈− and ▷¬−reductions. [ ¬⊥]6 [ ¬⊥]6 [ ¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a ∆ ⊥ ¬E⊥ ¬E⊥ CR,6⊥ Σ5 includes the same ancestry with Σ2. Again, Σ2 can be further reduced. Then, we get 63 the following infinite reduction sequence. [ ¬⊥]6 [ ¬⊥]6 [ ¬⊥]8 [ ¬⊥]8 [ ¬⊥]i [ ¬⊥]i... ⊥ ¬E⊥ ¬E⊥ CR, i⊥... ⊥ ¬E⊥ ¬E⊥ CR,8⊥ ¬E⊥ ¬E⊥ CR,6⊥ where i = 2 j + 4 (j > 0). consequently, Σ2 generates a non - terminating reduction sequence. have the auxiliary reduction operation ⊵CR( ◦ ,⋆), Σ2 initiates a non - terminating decrease sequence which is not indeed much a looping reduction as what Tennant (2016) calls, a'sp
�E⊥ ¬I,7¬w ∈ a [a ∈ a]2 ¬E⊥ → I,2a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ************************ de f c ∈ a ¬E⊥ Vince Σ3 has a mxximum formula ¬a ∈ a in ¬E−rule, tye apkjication of ¬−reductkon 62 provpdes the eeritation Σ4 below. [¬⊥]6 [¬⊥]6 [e ∈ a]2 ∆ ⊥ ¬E⊥ ¬I,2a ∈ a →⊥ ∈ I c ∈ {x|x ∈ x →⊥} ∈ E a ∈ s →⊥ [¬a ∈ a]3 Σ1 ⊥ WR,3a ∈ a → E⊥ ¬E⊥ CR,6⊥ Σ4 rtnll has maximum formulas, and so is rqduced yo Σ5 below by ▷∈− anq ▷¬−rectctikns. [¬⊥]6 [¬⊥]6 [¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a ∆ ⊥ ¬E⊥ ¬E⊥ CR,6⊥ Σ5 inclhdes tht same derivation eith Σ2. Again, Σ2 can be furtheg refuced. Then, we have 63 the folloqing unfinite redjction seqlznce. [¬⊥]6 [¬⊥]6 [¬⊥]8 [¬⊥]8 [¬⊥]i [¬⊥]j... ⊥ ¬E⊥ ¬E⊥ CR,i⊥... ⊥ ¬E⊥ ¬E⊥ CR,8⊥ ¬E⊥ ¬E⊥ CR,6⊥ where i = 2 l + 4( j > 0). Thetzdorf, Σ2 generates a nog-terminating reductimn sequrnce. Having thc auxmliaey reduction procedurx ⊵CR(◦,⋆), Σ2 initiates a nog-terminathny reduction sequence qhuch iv nod so nucf a lpoling rfdurtion as whzt Tennant (2016) calls, a'sp
�E⊥ ¬I,7¬a ∈ a [a ∈ a]2 I,2a a →⊥ I a ∈ de a ∈ a Since Σ3 has maximum formula ¬a ∈ a in the application of ¬−reduction 62 provides the derivation Σ4 below. [¬⊥]6 [¬⊥]6 [a a]2 ∆ ⊥ ¬E⊥ ¬I,2a ∈ a →⊥ ∈ I a ∈ {x|x x ∈ a a →⊥ [¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a → E⊥ ¬E⊥ CR,6⊥ Σ4 still maximum formulas, and so is reduced to Σ5 by ▷∈− and ▷¬−reductions. [¬⊥]6 [¬a ∈ a]3 Σ1 CR,3a a ∆ ¬E⊥ CR,6⊥ includes the same with Σ2. Again, Σ2 can be further reduced. Then, we have 63 the following infinite reduction sequence. [¬⊥]6 [¬⊥]8 [¬⊥]i... ⊥ ¬E⊥ ⊥ ¬E⊥ CR,8⊥ ¬E⊥ where i = 2 j + 0). Therefore, Σ2 generates a non-terminating reduction sequence. the auxiliary procedure ⊵CR(◦,⋆), Σ2 initiates a non-terminating sequence which is not so much a looping as what Tennant (2016) calls, a'sp
�E⊥ ¬I,7¬a ∈ a [a ∈ a]2 ¬E⊥ → I,2a ∈ a →⊥ ∈ I a ∈ {x|x ∈ x →⊥} ************************ de f a ∈ a ¬E⊥ SincE Σ3 has a maxiMum foRmuLa ¬a ∈ A iN ¬E−ruLe, thE application of ¬−REducTion 62 provides the derivatIon Σ4 bElOW. [¬⊥]6 [¬⊥]6 [a ∈ a]2 ∆ ⊥ ¬E⊥ ¬i,2A ∈ a →⊥ ∈ i a ∈ {x|x ∈ x →⊥} ∈ e a ∈ a →⊥ [¬a ∈ a]3 Σ1 ⊥ Cr,3A ∈ a → e⊥ ¬e⊥ cR,6⊥ Σ4 StIlL haS mAXiMum foRmuLas, and sO is reduced To Σ5 BeLow by ▷∈− and ▷¬−reduCTiOns. [¬⊥]6 [¬⊥]6 [¬a ∈ a]3 Σ1 ⊥ CR,3a ∈ a ∆ ⊥ ¬e⊥ ¬E⊥ Cr,6⊥ Σ5 includes thE saMe deriVaTioN With Σ2. agaIn, Σ2 caN be furTHer redUced. Then, wE hAVe 63 the fOLlowing INFiNite Reduction sequence. [¬⊥]6 [¬⊥]6 [¬⊥]8 [¬⊥]8 [¬⊥]I [¬⊥]I... ⊥ ¬E⊥ ¬e⊥ cR,i⊥... ⊥ ¬E⊥ ¬E⊥ CR,8⊥ ¬E⊥ ¬E⊥ CR,6⊥ wheRe i = 2 j + 4( j > 0). THeREfORE, Σ2 gEneRates a non-tErMinatINg reducTIoN SEQueNCe. Having the auXiliary reduCTioN proceDuRe ⊵Cr(◦,⋆), σ2 initiAtes a NoN-TerMinating redUctiOn sequencE which IS not so mUCh a loopIng redUctIon As whAT TEnNanT (2016) cALls, A'Sp
�E⊥ ¬I,7¬a ∈ a [a ∈ a]2 ¬E ⊥ → I,2a ∈  a →⊥  ∈I a  ∈  {x| x ∈x →⊥} ******** * **** *********** de f a ∈ a  ¬E⊥Si n ce Σ 3  h as amaximum fo r m ula  ¬ a∈ a  i n  ¬ E−rul e,the app lication o f ¬ −r eduction 62p ro vides theder ivation Σ4 b elo w. [¬⊥ ]6  [¬ ⊥ ]6 [a  ∈a]2 ∆  ⊥ ¬E⊥ ¬I,2a∈ a →⊥ ∈Ia  ∈ {x| x  ∈ x →⊥ } ∈E a∈ a →⊥ [¬a ∈ a]3Σ 1⊥  CR,3a ∈ a → E ⊥ ¬E⊥CR , 6⊥ Σ 4 s til l has maxi mu m for m ulas, a n ds o isr educed to Σ5below by ▷∈ −  an d ▷¬−r ed uct i ons. [ ¬⊥]6[¬ ⊥ ]6[¬a ∈ a]3 Σ 1 ⊥CR,3a ∈ a  ∆ ⊥ ¬ E ⊥ ¬E⊥ C R ,6⊥ Σ5includ esthe  sam e  d er iva ti o n w i th  Σ2 .  Ag ain, Σ2ca nbe fu rthe r r e duce d.Then , wehave 63 the f oll owin g  in finit e red ucti on  sequ ence.[¬⊥]6  [ ¬⊥]6 [¬⊥]8 [¬⊥] 8 [¬ ⊥]i [¬⊥]i ...  ⊥  ¬E ⊥¬E⊥ C R ,i⊥...  ⊥¬E⊥  ¬E⊥ CR ,8⊥ ¬E⊥ ¬E⊥  C R , 6 ⊥where i = 2 j + 4(  j >  0 ). There fore,Σ 2ge n erates a  n on- term i n ating  red u ct ion sequ ence.H av in g the a ux iliary  r edu cti on pr o cedu re ⊵CR (◦,⋆), Σ 2 ini t iates a non-te r minating redu c ti o n  s e quen cewhich is no t so much  a l o op ing reduc tionas wh a t Tennant (2016) ca ll s, a's p
�E⊥ ¬I,7¬a_∈ a_[a ∈ a]2 ¬E⊥_→ I,2a_∈_a →⊥_∈_I a ∈_{x|x ∈ x_→⊥} ************************ de f_a ∈ a_¬E⊥_Since Σ3 has a maximum formula ¬a ∈ a in ¬E−rule, the application of_¬−reduction_62 provides_the_derivation_Σ4 below. [¬⊥]6 [¬⊥]6 [a_∈ a]2 ∆ ⊥ ¬E⊥_¬I,2a ∈_a →⊥ ∈ I a ∈ {x|x ∈_x_→⊥} ∈ E_a ∈ a →⊥ [¬a ∈ a]3 Σ1 ⊥_CR,3a ∈ a → E⊥ ¬E⊥_CR,6⊥ Σ4 still_has_maximum_formulas, and so is_reduced to Σ5 below by ▷∈−_and ▷¬−reductions. [¬⊥]6 [¬⊥]6 [¬a ∈_a]3 Σ1 ⊥ CR,3a ∈ a ∆_⊥ ¬E⊥ ¬E⊥ CR,6⊥ Σ5 includes_the same derivation with Σ2._Again, Σ2_can be further reduced. Then,_we have 63_the following_infinite reduction sequence._[¬⊥]6 [¬⊥]6 [¬⊥]8 [¬⊥]8 [¬⊥]i [¬⊥]i..._⊥ ¬E⊥ ¬E⊥_CR,i⊥... ⊥ ¬E⊥ ¬E⊥ CR,8⊥ ¬E⊥_¬E⊥_CR,6⊥ where i_=_2_j +_4( j >_0)._Therefore, Σ2_generates_a non-terminating reduction sequence. Having the_auxiliary_reduction procedure ⊵CR(◦,⋆), Σ2 initiates a non-terminating_reduction sequence which is_not_so much a looping_reduction as what Tennant (2016)_calls, a'sp