 For the reduction procedures for generalized elimination rules for ∀ and ∃, we apply the same restrictions. The all free variables in D1 of both standard reduction procedures for ∀ and ∃ are different from both y and t. Since there can be no free occurrence of x in φ, φ[x/y][t/x] is the same as φ[t/y]. The assumptions in D2 of both stadard reduction procedures for ∀ and ∃ that ψ depends do not contain any occurrence of y, and y does not occur in ψ. For the process for ∀, the assumptions in D1 on which ∀xφ[x/y] depends do not contain any occurrence of y. 38 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ [⊥]2 D3 ψ ¬E,2 ψ ▷¬ D2 φ D1 ⊥ D3 ψ 5. The standard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[x/y] [φ[x/y][t/x]]1 D1 ψ ∀E,1 ψ ▷∀ D1 φ[t/y] D2 ψ 6. The standard reduction procedure for ∃ D1 φ(t) ∃I∃xφ[x/t] [φ[y/x]]1 D2 ψ ∃E,1 ψ ▷∃ D1 φ(t) D2[y/t] ψ When all elimination rules are formulated in generalized form, all normal derivations are not in full normal form. For instance, there is a derivation which has no maximum formula and so is in normal form, but is not in full normal form. We consider the following two derivations. [φ ∧ (ψ ∧σ)] ∧E φ ∧ψ ∧E σ [φ ∧ (ψ ∧σ)] [ψ ∧σ ]1 ∧E,1 ψ ∧σ [σ ]2 ∧E,2 σ The elimination rules for ∧ in the left side derivation are stated in standard form. On the other hand, the right side derivation consists of generalized elimination rules for ∧. The right side derivation is in normal form but is not in full normal form. To reduce the degree 39 of the major premise in the last ∧
For the reduction procedures for generalized elimination rules for ∀ and ∃, we apply the like restriction. The all free variables in D1 of both standard decrease procedures for ∀ and ∃ are unlike from both y and t. Since there can be no free occurrence of x in φ, φ[x / y][t / x ] is the like as φ[t / y ]. The assumptions in D2 of both stadard reduction procedures for ∀ and ∃ that ψ depend do not contain any occurrence of y, and y does not happen in ψ. For the process for ∀, the assumptions in D1 on which ∀xφ[x / y ] depends do not contain any happening of y. 38 [ φ]1 D1 ⊥ ¬I,1¬φ D2 φ [ ⊥]2 D3 ψ ¬E,2 ψ ▷¬ D2 φ D1 ⊥ D3 ψ 5. The standard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[x / y ] [ φ[x / y][t / x]]1 D1 ψ ∀E,1 ψ ▷∀ D1 φ[t / y ] D2 ψ 6. The standard decrease procedure for ∃ D1 φ(t) ∃I∃xφ[x / t ] [ φ[y / x]]1 D2 ψ ∃E,1 ψ ▷∃ D1 φ(t) D2[y / t ] ψ When all elimination dominion are explicate in generalized form, all normal derivations are not in full normal shape. For instance, there is a derivation which has no maximal formula and indeed is in normal form, but is not in full normal form. We study the following two derivations. [ φ ∧ (ψ ∧σ) ] ∧E φ ∧ψ ∧E σ [ φ ∧ (ψ ∧σ) ] [ ψ ∧σ ] 1 ∧E,1 ψ ∧σ [ σ ] 2 ∧E,2 σ The elimination rules for ∧ in the left side deriving are stated in standard form. On the other hand, the right side derivation consists of generalized elimination rules for ∧. The right slope derivation is in normal form but is not in wide normal form. To reduce the degree 39 of the major premise in the last ∧
 Fog the reduction procedurts for generalizeb elimiiation dules fof ∀ and ∃, we apply the same redteictiins. The all free variacles in D1 of both staidard reduction 'docedurcf fod ∀ anb ∃ are different from both y and t. Since tfexe can be no free occurrence of x in φ, φ[x/y][t/x] os the same as φ[t/i]. The wssujitlons in D2 of both stadard reductjon probedures for ∀ and ∃ that ψ depends do not contwin wny occurrence of j, and y doew noe occur in ψ. Fur the probzss for ∀, thg assumptions in D1 on which ∀xφ[x/y] aepenbs do not cinraij any occurrxnce os y. 38 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ [⊥]2 D3 ψ ¬E,2 ψ ▷¬ D2 φ D1 ⊥ D3 ψ 5. The standsrd reeuction procedure for ∀ D1 φ(y) ∀I∀xφ[x/y] [φ[x/y][t/x]]1 D1 ψ ∀E,1 ψ ▷∀ D1 φ[t/y] D2 ψ 6. The standard rwdyctiot prmcedjee wor ∃ V1 φ(t) ∃I∃xφ[x/t] [φ[j/x]]1 V2 ψ ∃E,1 ψ ▷∃ D1 φ(t) S2[y/t] ψ When aol elimination rulex wgr formulated in gegewalized form, all normal derivations are nof in full normal form. Fir instance, there is w derivatyon which has no maximum formula and so is in norkal fkfm, ymb is boh in full normal form. We consider the followigf uwo derivations. [φ ∧ (ψ ∧σ)] ∧E φ ∧ψ ∧E σ [φ ∧ (ψ ∧σ)] [ψ ∧σ ]1 ∧E,1 ψ ∧σ [σ ]2 ∧E,2 σ The elimiuztjon rules for ∧ in hhe lefj side derivatijn ate stated in standard form. In the other yand, the right sidz derivation conxists of generalized eliminacion rhles for ∧. Tje right akde derivation ir ik nmrmal form but is not in ftll normao foxm. To reauce the dqgree 39 of hhe major premise in the last ∧
For the reduction procedures for generalized elimination ∀ ∃, we the same restrictions. D1 both standard reduction for ∀ and are different from both y and Since there can be no free occurrence of x in φ, φ[x/y][t/x] is same as φ[t/y]. The assumptions in D2 of both stadard reduction procedures for and that depends not contain any occurrence of y, and y does not occur in ψ. For the process ∀, the assumptions in D1 on which ∀xφ[x/y] do not contain any of y. 38 [φ]1 D1 ¬I,1¬φ φ [⊥]2 ψ ψ D2 φ D1 D3 ψ 5. The standard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[x/y] [φ[x/y][t/x]]1 D1 ψ ∀E,1 ψ D1 φ[t/y] 6. The reduction for D1 φ(t) ∃I∃xφ[x/t] ψ ∃E,1 ψ ▷∃ D1 φ(t) all elimination rules are formulated in generalized form, normal derivations not in full normal form. For there is a derivation which has no maximum and so is in normal form, but is not in full normal form. We consider two derivations. [φ ∧ ∧σ)] ∧E φ ∧E [φ (ψ [ψ ∧σ ∧E,1 ψ ∧σ [σ ]2 ∧E,2 σ The elimination rules for in the left side derivation are stated in standard form. other the right side consists of generalized elimination for The right side derivation normal but full form. reduce the degree 39 the major premise in the ∧
 For the reduction procedures For generalIzed eLimInaTiOn ruLes fOr ∀ and ∃, we apply thE Same Restrictions. The all free VariaBlES in D1 OF bOth stAndard rEDuCTIon PrOcEduReS FoR ∀ and ∃ aRe dIfferenT from both y And T. SInce there can BE nO free occurRenCe of x in φ, φ[x/y][t/X] is The samE aS φ[t/Y]. the asSumPtionS in D2 of BOth staDard reducTiON proceDUres for ∀ AND ∃ tHat ψ Depends do not contaIN aNY occurrence of y, And y doEs NOt OCCur In ψ. for the procEsS for ∀, tHE assumpTIoNS IN D1 oN Which ∀xφ[x/y] depeNds do not conTAin Any occUrRenCE of y. 38 [φ]1 D1 ⊥ ¬i,1¬φ D2 φ [⊥]2 D3 Ψ ¬E,2 Ψ ▷¬ d2 φ D1 ⊥ d3 ψ 5. The standaRd reDuction prOcedurE For ∀ D1 φ(y) ∀I∀XΦ[x/y] [φ[x/y][t/X]]1 D1 ψ ∀E,1 ψ ▷∀ D1 Φ[t/y] d2 ψ 6. THe stANdArD reDuCTioN PrOceDUre For ∃ D1 φ(t) ∃I∃xΦ[x/T] [φ[Y/x]]1 D2 ψ ∃E,1 Ψ ▷∃ D1 φ(t) d2[Y/T] Ψ when All ElimInatiOn rules are forMulAted IN geNeralIzed fOrm, aLl NormaL derivAtionS aRe not in full normAl foRm. For instAncE, tHerE iS a derIVation WhiCh hAs no maxImum forMUla AnD SO Is In normal form, but is nOt IN FuLl normal Form. We COnSiDEr the folLoWinG two DERivatIons. [Φ ∧ (Ψ ∧σ)] ∧e φ ∧ψ ∧E σ [φ ∧ (ψ ∧σ)] [Ψ ∧σ ]1 ∧E,1 ψ ∧σ [σ ]2 ∧e,2 Σ THe EliminaTiOn ruleS fOr ∧ iN thE left SIde dErivatIon are stAted iN Standard form. On THe other hand, thE RiGHT sIDe deRivAtion consisTs of GEnerAlizED eLimINatioN ruleS fOR ∧. THE right side derivatioN iS in norMal foRm but is not in fUll normal fORM. to reduce The dEGrEE 39 of the major preMise iN the last ∧
 For the reduction procedu res for ge neral ize d e li mina tion  rules for ∀ a n d ∃,  we apply the same res trict io n s. T h eall f ree var i ab l e s i nD1  of  b o th  stan dar d reduc tion proce dur es  for ∀ and ∃ ar e differen t f rom both y a ndt. Sin ce  th e re ca n b e nofree o c curren ce of x i nφ , φ[x/ y ][t/x]i s  t he s ame as φ[t/y]. Th e  a s sumptions in D 2 of b ot h  s t a dar d r eduction p ro cedur e s for ∀ an d ∃  th a t ψ depends d o not conta i n a ny occ ur ren c e of y , and  y doe s not occur  inψ. For th e proc e ss for∀ , the a ssumpt ion s i n D1 on  w hic h∀ xφ[ x /y ] d e pen ds do no tco ntain  any o c c urre nce  ofy. 38  [φ]1 D1 ⊥ ¬I ,1¬ φ D2 φ [ ⊥]2 D 3 ψ ¬ E,2ψ▷¬ D2  φ D1⊥ D3ψ5. The standard  red uction pr oce du refo r ∀ D 1  φ(y)∀I∀ xφ[ x/y] [φ [x/y][t / x]] 1D 1 ψ∀E,1 ψ ▷∀ D1 φ[t/y ]D 2  ψ  6. Thestanda r dre d uction p ro ced uref o r ∃ D 1 φ( t )∃I∃xφ[x/ t] [φ[ y /x ]] 1 D2 ψ∃E ,1 ψ ▷ ∃D1φ(t ) D2[ y /t]ψ When  all eli minat i on rules are f o rmulated in g e ne r a li z ed f orm , all norma l de r ivat ions ar e n o t infullno r ma l  form. For instance ,thereis aderivation wh ich has no m a ximum fo rmul a  a n d so is in nor mal f orm, but i s  not infullnormal f orm. We c o n sider th e f oll owi ngt w oderivations.[ φ  ∧ ( ψ∧σ)] ∧E  φ∧ψ ∧E σ  [φ  ∧(ψ∧σ) ][ψ ∧σ ]1∧E,1 ψ ∧ σ[σ  ] 2∧E, 2 σ T h e elimin at ion  r ule s for ∧ in t he le ft s id ed eri vationa re s tate din  sta nda rd  form . On the  otherhand, the  ri g ht s id ederivat ion consistsof  generaliz ed  el iminat i o n rulesfor ∧. The right side d e rivatio n i s innorm al form b utis not  in full n ormalform.  T o r e d uce t h e  d egr ee  39 of the m ajo r pre mi se i n the l ast ∧
 For_the reduction_procedures for generalized elimination_rules for_∀_and ∃,_we_apply the same_restrictions. The all_free variables in D1_of both standard_reduction_procedures for ∀ and ∃ are different from both y and t. Since there_can_be no_free_occurrence_of x in φ, φ[x/y][t/x]_is the same as φ[t/y]._The assumptions_in D2 of both stadard reduction procedures for_∀_and ∃ that_ψ depends do not contain any occurrence of y,_and y does not occur in_ψ. For the_process_for_∀, the assumptions in_D1 on which ∀xφ[x/y] depends do_not contain any occurrence of y._38 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ_[⊥]2 D3 ψ ¬E,2 ψ ▷¬_D2 φ D1 ⊥ D3_ψ 5._The standard reduction procedure for_∀ D1 φ(y)_∀I∀xφ[x/y] [φ[x/y][t/x]]1_D1 ψ ∀E,1_ψ ▷∀ D1 φ[t/y] D2 ψ_6. The standard_reduction procedure for ∃ D1 φ(t)_∃I∃xφ[x/t]_[φ[y/x]]1 D2 ψ_∃E,1_ψ_▷∃ D1_φ(t) D2[y/t] ψ_When_all elimination_rules_are formulated in generalized form, all_normal_derivations are not in full normal form._For instance, there is_a_derivation which has no_maximum formula and so is_in normal form, but is not_in full_normal form._We consider the following two derivations. [φ ∧ (ψ ∧σ)] ∧E_φ ∧ψ ∧E σ [φ ∧_(ψ ∧σ)] [ψ ∧σ_]1 ∧E,1_ψ_∧σ [σ ]2_∧E,2_σ The_elimination rules for ∧ in the left_side derivation_are stated in standard form. On_the other hand, the_right_side derivation consists of generalized elimination_rules for ∧. The right side_derivation is in normal form_but_is_not in full normal form._To reduce the degree 39 of_the major premise_in the last ∧