1 ∧φ2 ∧E(i=1,2) φi ▷∧ Di φi 2. The standard reduction procedure for →. [φ]1 D1 ψ → I,1 φ → ψ D2 φ → E ψ ▷→ D2 φ D1 ψ 3. The standard reduction procedure for ∨ Di φi ∨Ii=1,2 φ1 ∨φ2 [φ1] 1 D3 ψ [φ2] 2 D4 ψ ∨E,1,2 ψ ▷∨ D1 φ1 D3 ψ or D2 φ2 D4 ψ 4. The standard reduction procedure for ¬ 20 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ ¬E⊥ ▷¬ D2 φ D1 ⊥ 5. The standard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[x/y] ∀E φ[t/x] ▷∀ D1 φ[t/y] 6. The standard reduction procedure for ∃ D1 φ(t) ∃I∃xφ[x/t] [φ[y/x]]1 D2 ψ ∃E,1 ψ ▷∃ D1 φ[y/t] D2 ψ These standard reduction procedures are also known as the process of reducing the degree of a maximum formula. A reduction process which reduces the degree of a maximum formula in accordance with the inversion principle will be called a standard reduction procedure. The notions of 'degree' and 'length' of a derivation are defined as below. Definition 1.2.3. The degree d(φ) of a formula φ is defined by d(⊥) = 0, d(α) = 0 for an atomic formula α, d(φ ◦ψ) = d(φ)+d(ψ)+1 for binary operators ◦, d(◦φ) = d(φ)+1 for unary operators ◦. The degree d(D) of a derivation D is defined as the highest degree of a maximum formula in D; d(D) = 0 if there is no such occurrences. The length of a derivation D is the number of formula occurrences. When the derivation has no maximum formula, we say that it is in normal form. Let R
1 ∧φ2 ∧E(i=1,2) φi ▷∧ Di φi 2. The standard reduction procedure for →. [ φ]1 D1 ψ → I,1 φ → ψ D2 φ → E ψ ▷→ D2 φ D1 ψ 3. The standard reduction procedure for ∨ Di φi ∨Ii=1,2 φ1 ∨φ2 [ φ1 ] 1 D3 ψ [ φ2 ] 2 D4 ψ ∨E,1,2 ψ ▷∨ D1 φ1 D3 ψ or D2 φ2 D4 ψ 4. The standard decrease operation for ¬ 20 [ φ]1 D1 ⊥ ¬I,1¬φ D2 φ ¬E⊥ ▷¬ D2 φ D1 ⊥ 5. The standard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[x / y ] ∀E φ[t / x ] ▷∀ D1 φ[t / y ] 6. The standard decrease procedure for ∃ D1 φ(t) ∃I∃xφ[x / t ] [ φ[y / x]]1 D2 ψ ∃E,1 ψ ▷∃ D1 φ[y / t ] D2 ψ These standard decrease procedures are also know as the process of reducing the academic degree of a maximal formula. A reduction summons which reduces the degree of a maximum rule in accordance with the inversion principle will be called a standard reduction operation. The notions of' degree' and' length' of a ancestry are defined as below. Definition 1.2.3. The degree d(φ) of a formula φ is specify by d(⊥) = 0, d(α) = 0 for an atomic formula α, d(φ ◦ ψ) = d(φ)+d(ψ)+1 for binary operators ◦, d( ◦ φ) = d(φ)+1 for unary operators ◦. The degree d(D) of a deriving D is defined as the highest degree of a maximum formula in D; d(D) = 0 if there is no such occurrences. The length of a derivation D is the number of formula occurrences. When the derivation has no maximum recipe, we say that it is in normal form. lease R
1 ∧φ2 ∧E(l=1,2) φi ▷∧ Di φi 2. The standard veduction procedorw for →. [φ]1 D1 ψ → J,1 φ → ψ D2 φ → E ψ ▷→ D2 φ D1 ψ 3. The standard rednctiin pricedure for ∨ Di φi ∨Ii=1,2 φ1 ∨φ2 [φ1] 1 D3 ψ [φ2] 2 D4 ψ ∨E,1,2 ψ ▷∨ D1 φ1 D3 ψ ie D2 φ2 D4 ψ 4. Tis standard resmctiou 'rocedure for ¬ 20 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ ¬A⊥ ▷¬ D2 φ D1 ⊥ 5. The vtxnbard reduction procedure for ∀ D1 φ(y) ∀I∀xφ[v/y] ∀E φ[t/x] ▷∀ F1 φ[t/y] 6. The stanqard wedudniin procedure for ∃ D1 φ(t) ∃I∃xφ[x/t] [φ[y/x]]1 D2 ψ ∃E,1 ψ ▷∃ D1 φ[y/t] D2 ψ Thesr standard reduction procefured are also known ad the procews os reducing thd degree of a maximum rormula. A reduction process whizh rebuces the dgyeee mf a maximun foriula in accovcance fith thr inversion prlncipne qill be called a stanvard reduction procequre. The tocions of 'degree' and 'lwntth' ox a gerixqtiun zrx dsfined as below. Defjnition 1.2.3. Thw degree d(φ) of a forkujq φ is defined by d(⊥) = 0, q(α) = 0 for an atomic formula α, d(φ ◦ψ) = d(φ)+d(ψ)+1 for bpnarg operators ◦, d(◦φ) = d(φ)+1 for ubary operators ◦. The dggree d(D) os a derivation D is defined as the highest degree of a oaxnnum fuemkla in D; d(D) = 0 if there is no such occurrences. Fht lvngth of a derivabion D is the numbrr og formula occutrences. Whsn the derivation jas no iaximym formulw, we say that it is in normal firm. Let R
1 ∧φ2 ∧E(i=1,2) φi ▷∧ Di φi standard procedure for [φ]1 D1 ψ D2 → E ψ D2 φ D1 3. The standard reduction procedure for Di φi ∨Ii=1,2 φ1 ∨φ2 [φ1] 1 D3 ψ [φ2] 2 D4 ψ ψ ▷∨ D1 φ1 D3 ψ or D2 φ2 D4 ψ 4. The reduction for 20 D1 ⊥ ¬I,1¬φ D2 φ ¬E⊥ ▷¬ D2 φ D1 ⊥ 5. The standard reduction procedure ∀ D1 φ(y) ∀I∀xφ[x/y] ∀E φ[t/x] ▷∀ D1 6. The standard reduction for ∃ D1 φ(t) ∃I∃xφ[x/t] D2 ∃E,1 ψ D1 D2 These standard reduction are also known as the process of reducing the degree of a maximum formula. A reduction process reduces the a maximum in with inversion principle will a standard reduction procedure. The notions 'length' of a derivation are defined as below. 1.2.3. The d(φ) of a formula φ is by d(⊥) = 0, d(α) = 0 for atomic formula α, d(φ ◦ψ) = d(φ)+d(ψ)+1 for binary operators ◦, d(◦φ) = d(φ)+1 for ◦. The degree d(D) a derivation D defined the degree a maximum in D; d(D) = 0 if there is no such occurrences. length of a derivation D is the number of formula the has no maximum we say that it in form. Let R
1 ∧φ2 ∧E(i=1,2) φi ▷∧ Di φi 2. The standard reducTion procedUre foR →. [φ]1 D1 Ψ → I,1 φ → Ψ D2 Φ → E ψ ▷→ D2 Φ D1 ψ 3. THe standard reduCTion Procedure for ∨ Di φi ∨Ii=1,2 φ1 ∨φ2 [φ1] 1 D3 Ψ [φ2] 2 D4 ψ ∨E,1,2 Ψ ▷∨ D1 Φ1 d3 ψ or d2 Φ2 D4 Ψ 4. The sTandard REdUCTioN pRoCedUrE FoR ¬ 20 [φ]1 D1 ⊥ ¬I,1¬φ d2 φ ¬E⊥ ▷¬ d2 φ D1 ⊥ 5. The sTandard redUctIoN procedure foR ∀ d1 φ(Y) ∀I∀xφ[x/y] ∀E φ[t/x] ▷∀ d1 φ[t/Y] 6. The standard RedUction PrOceDUre foR ∃ D1 φ(T) ∃I∃xφ[x/T] [φ[y/x]]1 D2 ψ ∃e,1 Ψ ▷∃ D1 φ[y/t] D2 Ψ These staNdARd reduCTion proCEDuRes aRe also known as the pROcESs of reducing thE degreE oF A mAXImuM foRmula. A reduCtIon prOCess whiCH rEDUCes THe degree of a maXimum formulA In aCcordaNcE wiTH the inVersiOn PRinCiple will be CallEd a standaRd reduCTion proCEdure. ThE notioNs oF 'deGree' ANd 'LeNgtH' oF A deRIvAtiON arE defined As BeLow. DeFiniTION 1.2.3. the dEgrEe d(φ) Of a foRmula φ is definEd bY d(⊥) = 0, d(α) = 0 FOr aN atomIc forMula Α, d(Φ ◦ψ) = d(φ)+d(Ψ)+1 for biNary oPeRators ◦, d(◦φ) = d(φ)+1 for unAry oPerators ◦. THe dEgRee D(D) Of a deRIvatioN D iS deFined as The highESt dEgREE Of A maximum formula in D; D(D) = 0 IF ThEre is no sUch occURrEnCEs. The lenGtH of A derIVAtion d is tHE nUmber of fOrmula OCcUrRences. WHeN the deRiVatIon Has no MAximUm formUla, we say That iT Is in normal form. lEt R
1 ∧φ2 ∧E(i=1,2) φi ▷∧ Di φ i 2. The s tanda rdred uc tion  pro cedure for →.[ φ]1D1 ψ → I,1 φ → ψ D2 φ→ E ψ  ▷ →  D2φ  D 1 ψ 3 . The s t an d a rdre du cti on pr ocedu refor ∨ D i φi ∨Ii=1 ,2φ1  ∨φ2 [φ1] 1D 3ψ [φ2] 2 D 4 ψ  ∨E,1,2 ψ ▷∨  D1  φ1 D3  ψ  or D2 φ2  D4  ψ 4.  The s t andard  reductio np rocedu r e for ¬ 2 0[φ]1  D1 ⊥ ¬I,1¬φ D2 φ ¬E ⊥  ▷¬ D2 φ D1 ⊥5. The  s t an d a rdred uction pro ce duref or ∀ D1 φ( y ) ∀I∀ x φ[x/y] ∀E φ[t /x] ▷∀ D1 φ [ t/y ] 6. T he  st a ndardreduc ti o n p rocedure fo r ∃D1 φ(t) ∃ I∃xφ[x / t] [φ[y / x]]1 D2  ψ ∃E, 1 ψ  ▷∃  D1φ [y /t ] D 2ψ  Th e se  st a nda rd reduc ti on  proc edur e s a re a lso  kno wn as  the processofredu c ing  thedegre e of  a  maxi mum fo rmula .A reduction pro cess  which re duc es  th edegre e  of amax imu m formu la in a c cor da n c e  w ith the inversionpr i n ci ple will  be ca l le da  standar dred ucti o n  proc edur e .The noti ons of 'd eg ree' an d'lengt h'  of  aderiv a tion  are d efined a s bel o w. Definition1 .2.3. The deg r ee d (φ )  ofa f ormula φ is  def i nedby d ( ⊥)  =0 , d(α ) = 0  f o ra n atomic formula α,  d (φ ◦ψ)  = d( φ)+d(ψ)+1 for  binary op e r a tors ◦,d(◦φ )  = d(φ)+1 for una ry op erators ◦. The degr ee d( D) of aderivatio n D is def ine d a s t heh i gh est degree of a  max im um form ula  in D;d(D ) =  0ifth ere is no  such oc cu rr en ce s.The l e ngth ofader iv ati on Di s thenumbe r of  f or m ula  occurr e nc e s . Wh en  t he d eri va tionhasn o m aximumformula,wes ay t ha tit is i n normal form .Let R
1 ∧φ2_∧E(i=1,2) φi_▷∧ Di φi 2._The standard_reduction_procedure for_→._[φ]1 D1 ψ_→ I,1 φ_→ ψ D2 φ_→ E ψ_▷→_D2 φ D1 ψ 3. The standard reduction procedure for ∨ Di φi ∨Ii=1,2_φ1_∨φ2 [φ1]_1_D3_ψ [φ2] 2 D4 ψ_∨E,1,2 ψ ▷∨ D1 φ1_D3 ψ_or D2 φ2 D4 ψ 4. The standard_reduction_procedure for ¬_20 [φ]1 D1 ⊥ ¬I,1¬φ D2 φ ¬E⊥ ▷¬_D2 φ D1 ⊥ 5. The_standard reduction procedure_for_∀_D1 φ(y) ∀I∀xφ[x/y] ∀E_φ[t/x] ▷∀ D1 φ[t/y] 6. The_standard reduction procedure for ∃ D1_φ(t) ∃I∃xφ[x/t] [φ[y/x]]1 D2 ψ ∃E,1 ψ_▷∃ D1 φ[y/t] D2 ψ These_standard reduction procedures are also_known as_the process of reducing the_degree of a_maximum formula._A reduction process_which reduces the degree of a_maximum formula in_accordance with the inversion principle will_be_called a standard_reduction_procedure._The notions_of 'degree' and_'length'_of a_derivation_are defined as below. Definition 1.2.3._The_degree d(φ) of a formula φ is_defined by d(⊥) =_0,_d(α) = 0 for_an atomic formula α, d(φ_◦ψ) = d(φ)+d(ψ)+1 for binary operators_◦, d(◦φ)_= d(φ)+1_for unary operators ◦. The degree d(D) of a derivation D_is defined as the highest degree_of a maximum formula_in D;_d(D)_= 0 if_there_is no_such occurrences. The length of a derivation_D is_the number of formula occurrences. When_the derivation has no_maximum_formula, we say that it is_in normal form. Let R