:: A Sequent Calculus for First-Order Logic
:: by Patrick Braselmann and Peter Koepke
::
:: Received September 25, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

registration

let g be FinSequence;
let N be set ;
cluster g | N -> FinSubsequence-like ;
coherence
g | N is FinSubsequence-like ;

end;

definition

let D be non empty set ;
let f be FinSequence of D;
func Ant f -> FinSequence of D means :Def1: :: CALCUL_1:def 1
for i being Element of NAT st len f = i + 1 holds
it = f | (Seg i) if len f > 0
otherwise it = {} ;
existence
( ( len f > 0 implies ex b₁ being FinSequence of D st
for i being Element of NAT st len f = i + 1 holds
b₁ = f | (Seg i) ) & ( not len f > 0 implies ex b₁ being FinSequence of D st b₁ = {} ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of D holds
( ( len f > 0 & ( for i being Element of NAT st len f = i + 1 holds
b₁ = f | (Seg i) ) & ( for i being Element of NAT st len f = i + 1 holds
b₂ = f | (Seg i) ) implies b₁ = b₂ ) & ( not len f > 0 & b₁ = {} & b₂ = {} implies b₁ = b₂ ) )

proof end;

consistency
for b₁ being FinSequence of D holds verum ;

end;

:: deftheorem Def1 defines Ant CALCUL_1:def 1 :
for D being non empty set
for f, b₃ being FinSequence of D holds
( ( len f > 0 implies ( b₃ = Ant f iff for i being Element of NAT st len f = i + 1 holds
b₃ = f | (Seg i) ) ) & ( not len f > 0 implies ( b₃ = Ant f iff b₃ = {} ) ) );

definition

let D be non empty set ;
let f be FinSequence of D;
assume A1: len f > 0 ;
func Suc f -> Element of D equals :Def2: :: CALCUL_1:def 2
f . (len f);
coherence
f . (len f) is Element of D

proof end;

end;

:: deftheorem Def2 defines Suc CALCUL_1:def 2 :
for D being non empty set
for f being FinSequence of D st len f > 0 holds
Suc f = f . (len f);

definition

let f be Relation;
let p be set ;
pred p is_tail_of f means :Def3: :: CALCUL_1:def 3
p in rng f;

end;

:: deftheorem Def3 defines is_tail_of CALCUL_1:def 3 :
for f being Relation
for p being set holds
( p is_tail_of f iff p in rng f );

Lm1: now

let f be FinSequence; :: thesis: for p being set st p is_tail_of f holds
ex i being Element of NAT st
( i in dom f & f . i = p )
let p be set ; :: thesis: ( p is_tail_of f implies ex i being Element of NAT st
( i in dom f & f . i = p ) )
assume p is_tail_of f ; :: thesis: ex i being Element of NAT st
( i in dom f & f . i = p )
then p in rng f by Def3;
then consider i being set such that
A1: i in dom f and
A2: f . i = p by FUNCT_1:def 5;
ex n being Nat st dom f = Seg n by FINSEQ_1:def 2;
then reconsider i = i as Element of NAT by A1;
take i = i; :: thesis: ( i in dom f & f . i = p )
thus ( i in dom f & f . i = p ) by A1, A2; :: thesis: verum

end;

Lm2: now

let f be FinSequence; :: thesis: for p being set st ex i being Element of NAT st
( i in dom f & f . i = p ) holds
p is_tail_of f
let p be set ; :: thesis: ( ex i being Element of NAT st
( i in dom f & f . i = p ) implies p is_tail_of f )
assume ex i being Element of NAT st
( i in dom f & f . i = p ) ; :: thesis: p is_tail_of f
then p in rng f by FUNCT_1:def 5;
hence p is_tail_of f by Def3; :: thesis: verum

end;

definition

let f, g be FinSequence of CQC-WFF ;
pred f is_Subsequence_of g means :Def4: :: CALCUL_1:def 4
ex N being Subset of NAT st f c= Seq (g | N);

end;

:: deftheorem Def4 defines is_Subsequence_of CALCUL_1:def 4 :
for f, g being FinSequence of CQC-WFF holds
( f is_Subsequence_of g iff ex N being Subset of NAT st f c= Seq (g | N) );

theorem Th1: :: CALCUL_1:1

for f, g being FinSequence of CQC-WFF st f is_Subsequence_of g holds
( rng f c= rng g & ex N being Subset of NAT st rng f c= rng (g | N) )

proof end;

theorem Th2: :: CALCUL_1:2

for f being FinSequence of CQC-WFF st len f > 0 holds
( (len (Ant f)) + 1 = len f & len (Ant f) < len f )

proof end;

theorem Th3: :: CALCUL_1:3

for f being FinSequence of CQC-WFF st len f > 0 holds
( f = (Ant f) ^ <*(Suc f)*> & rng f = (rng (Ant f)) \/ {(Suc f)} )

proof end;

theorem Th4: :: CALCUL_1:4

for f being FinSequence of CQC-WFF st len f > 1 holds
len (Ant f) > 0

proof end;

theorem Th5: :: CALCUL_1:5

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF holds
( Suc (f ^ <*p*>) = p & Ant (f ^ <*p*>) = f )

proof end;

theorem Th6: :: CALCUL_1:6

for fin, fin1 being FinSequence holds
( len fin <= len (fin ^ fin1) & len fin1 <= len (fin ^ fin1) & ( fin <> {} implies ( 1 <= len fin & len fin1 < len (fin1 ^ fin) ) ) )

proof end;

theorem Th7: :: CALCUL_1:7

for f, g being FinSequence of CQC-WFF holds Seq ((f ^ g) | (dom f)) = (f ^ g) | (dom f)

proof end;

theorem Th8: :: CALCUL_1:8

for f, g being FinSequence of CQC-WFF holds f is_Subsequence_of f ^ g

proof end;

theorem Th9: :: CALCUL_1:9

for b, c being set
for fin being FinSequence holds 1 < len ((fin ^ <*b*>) ^ <*c*>)

proof end;

theorem Th10: :: CALCUL_1:10

for b being set
for fin being FinSequence holds
( 1 <= len (fin ^ <*b*>) & len (fin ^ <*b*>) in dom (fin ^ <*b*>) )

proof end;

theorem Th11: :: CALCUL_1:11

for m, n being Element of NAT st 0 < m holds
len (Sgm ((Seg n) \/ {(n + m)})) = n + 1

proof end;

theorem Th12: :: CALCUL_1:12

for m, n being Element of NAT st 0 < m holds
dom (Sgm ((Seg n) \/ {(n + m)})) = Seg (n + 1)

proof end;

theorem Th13: :: CALCUL_1:13

for f, g being FinSequence of CQC-WFF st 0 < len f holds
f is_Subsequence_of ((Ant f) ^ g) ^ <*(Suc f)*>

proof end;

theorem Th14: :: CALCUL_1:14

for c, d being set
for f being FinSequence of CQC-WFF holds
( 1 in dom <*c,d*> & 2 in dom <*c,d*> & (f ^ <*c,d*>) . ((len f) + 1) = c & (f ^ <*c,d*>) . ((len f) + 2) = d )

proof end;

begin

definition

let f be FinSequence of CQC-WFF ;
func still_not-bound_in f -> Subset of bound_QC-variables means :Def5: :: CALCUL_1:def 5
for a being set holds
( a in it iff ex i being Element of NAT ex p being Element of CQC-WFF st
( i in dom f & p = f . i & a in still_not-bound_in p ) );
existence
ex b₁ being Subset of bound_QC-variables st
for a being set holds
( a in b₁ iff ex i being Element of NAT ex p being Element of CQC-WFF st
( i in dom f & p = f . i & a in still_not-bound_in p ) )

proof end;

uniqueness
for b₁, b₂ being Subset of bound_QC-variables st ( for a being set holds
( a in b₁ iff ex i being Element of NAT ex p being Element of CQC-WFF st
( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) & ( for a being set holds
( a in b₂ iff ex i being Element of NAT ex p being Element of CQC-WFF st
( i in dom f & p = f . i & a in still_not-bound_in p ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines still_not-bound_in CALCUL_1:def 5 :
for f being FinSequence of CQC-WFF
for b₂ being Subset of bound_QC-variables holds
( b₂ = still_not-bound_in f iff for a being set holds
( a in b₂ iff ex i being Element of NAT ex p being Element of CQC-WFF st
( i in dom f & p = f . i & a in still_not-bound_in p ) ) );

definition

func set_of_CQC-WFF-seq -> set means :Def6: :: CALCUL_1:def 6
for a being set holds
( a in it iff a is FinSequence of CQC-WFF );
existence
ex b₁ being set st
for a being set holds
( a in b₁ iff a is FinSequence of CQC-WFF )

proof end;

uniqueness
for b₁, b₂ being set st ( for a being set holds
( a in b₁ iff a is FinSequence of CQC-WFF ) ) & ( for a being set holds
( a in b₂ iff a is FinSequence of CQC-WFF ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines set_of_CQC-WFF-seq CALCUL_1:def 6 :
for b₁ being set holds
( b₁ = set_of_CQC-WFF-seq iff for a being set holds
( a in b₁ iff a is FinSequence of CQC-WFF ) );

definition

let PR be FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:];
let n be Nat;
pred PR,n is_a_correct_step means :Def7: :: CALCUL_1:def 7
ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) if (PR . n) `2 = 0
ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> if (PR . n) `2 = 1
ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) if (PR . n) `2 = 2
ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) if (PR . n) `2 = 3
ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) if (PR . n) `2 = 4
ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) if (PR . n) `2 = 5
ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) if (PR . n) `2 = 6
ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) if (PR . n) `2 = 7
ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) if (PR . n) `2 = 8
ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) if (PR . n) `2 = 9
;
consistency
( ( (PR . n) `2 = 0 & (PR . n) `2 = 1 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 2 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 3 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 4 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 5 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 6 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 7 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 8 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 0 & (PR . n) `2 = 9 implies ( ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 2 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 3 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 4 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 5 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 6 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 7 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 8 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 1 & (PR . n) `2 = 9 implies ( ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 3 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 4 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 5 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 6 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 7 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 2 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 4 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 5 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 3 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 5 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 6 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 7 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 8 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 & (PR . n) `2 = 9 implies ( ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 7 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 & (PR . n) `2 = 8 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 8 & (PR . n) `2 = 9 implies ( ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) ) ;

end;

:: deftheorem Def7 defines is_a_correct_step CALCUL_1:def 7 :
for PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:]
for n being Nat holds
( ( (PR . n) `2 = 0 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF st
( Suc f is_tail_of Ant f & (PR . n) `1 = f ) ) ) & ( (PR . n) `2 = 1 implies ( PR,n is_a_correct_step iff ex f being FinSequence of CQC-WFF st (PR . n) `1 = f ^ <*VERUM*> ) ) & ( (PR . n) `2 = 2 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & Ant f is_Subsequence_of Ant g & Suc f = Suc g & (PR . i) `1 = f & (PR . n) `1 = g ) ) ) & ( (PR . n) `2 = 3 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*(Suc f)*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 4 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF ex p being Element of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant (Ant f)) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 5 implies ( PR,n is_a_correct_step iff ex i, j being Element of NAT ex f, g being FinSequence of CQC-WFF st
( 1 <= i & i < n & 1 <= j & j < i & Ant f = Ant g & f = (PR . j) `1 & g = (PR . i) `1 & (Ant f) ^ <*((Suc f) '&' (Suc g))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 6 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*p*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 7 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p, q being Element of CQC-WFF st
( 1 <= i & i < n & p '&' q = Suc f & f = (PR . i) `1 & (Ant f) ^ <*q*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 8 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = All (x,p) & f = (PR . i) `1 & (Ant f) ^ <*(p . (x,y))*> = (PR . n) `1 ) ) ) & ( (PR . n) `2 = 9 implies ( PR,n is_a_correct_step iff ex i being Element of NAT ex f being FinSequence of CQC-WFF ex p being Element of CQC-WFF ex x, y being bound_QC-variable st
( 1 <= i & i < n & Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & f = (PR . i) `1 & (Ant f) ^ <*(All (x,p))*> = (PR . n) `1 ) ) ) );

definition

let PR be FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:];
attr PR is a_proof means :Def8: :: CALCUL_1:def 8
( PR <> {} & ( for n being Nat st 1 <= n & n <= len PR holds
PR,n is_a_correct_step ) );

end;

:: deftheorem Def8 defines a_proof CALCUL_1:def 8 :
for PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:] holds
( PR is a_proof iff ( PR <> {} & ( for n being Nat st 1 <= n & n <= len PR holds
PR,n is_a_correct_step ) ) );

definition

let f be FinSequence of CQC-WFF ;
pred |- f means :Def9: :: CALCUL_1:def 9
ex PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:] st
( PR is a_proof & f = (PR . (len PR)) `1 );

end;

:: deftheorem Def9 defines |- CALCUL_1:def 9 :
for f being FinSequence of CQC-WFF holds
( |- f iff ex PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:] st
( PR is a_proof & f = (PR . (len PR)) `1 ) );

definition

let p be Element of CQC-WFF ;
let X be Subset of CQC-WFF;
pred p is_formal_provable_from X means :Def10: :: CALCUL_1:def 10
ex f being FinSequence of CQC-WFF st
( rng (Ant f) c= X & Suc f = p & |- f );

end;

:: deftheorem Def10 defines is_formal_provable_from CALCUL_1:def 10 :
for p being Element of CQC-WFF
for X being Subset of CQC-WFF holds
( p is_formal_provable_from X iff ex f being FinSequence of CQC-WFF st
( rng (Ant f) c= X & Suc f = p & |- f ) );

definition

let X be Subset of CQC-WFF;
let A be non empty set ;
let J be interpretation of A;
let v be Element of Valuations_in A;
pred J,v |= X means :Def11: :: CALCUL_1:def 11
for p being Element of CQC-WFF st p in X holds
J,v |= p;

end;

:: deftheorem Def11 defines |= CALCUL_1:def 11 :
for X being Subset of CQC-WFF
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= X iff for p being Element of CQC-WFF st p in X holds
J,v |= p );

definition

let X be Subset of CQC-WFF;
let p be Element of CQC-WFF ;
pred X |= p means :: CALCUL_1:def 12
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A st J,v |= X holds
J,v |= p;

end;

:: deftheorem defines |= CALCUL_1:def 12 :
for X being Subset of CQC-WFF
for p being Element of CQC-WFF holds
( X |= p iff for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A st J,v |= X holds
J,v |= p );

definition

let p be Element of CQC-WFF ;
pred |= p means :: CALCUL_1:def 13
{} CQC-WFF |= p;

end;

:: deftheorem defines |= CALCUL_1:def 13 :
for p being Element of CQC-WFF holds
( |= p iff {} CQC-WFF |= p );

definition

let f be FinSequence of CQC-WFF ;
let A be non empty set ;
let J be interpretation of A;
let v be Element of Valuations_in A;
pred J,v |= f means :Def14: :: CALCUL_1:def 14
J,v |= rng f;

end;

:: deftheorem Def14 defines |= CALCUL_1:def 14 :
for f being FinSequence of CQC-WFF
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= f iff J,v |= rng f );

definition

let f be FinSequence of CQC-WFF ;
let p be Element of CQC-WFF ;
pred f |= p means :Def15: :: CALCUL_1:def 15
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A st J,v |= f holds
J,v |= p;

end;

:: deftheorem Def15 defines |= CALCUL_1:def 15 :
for f being FinSequence of CQC-WFF
for p being Element of CQC-WFF holds
( f |= p iff for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A st J,v |= f holds
J,v |= p );

theorem Th15: :: CALCUL_1:15

for f being FinSequence of CQC-WFF st Suc f is_tail_of Ant f holds
Ant f |= Suc f

proof end;

theorem Th16: :: CALCUL_1:16

for f, g being FinSequence of CQC-WFF st Ant f is_Subsequence_of Ant g & Suc f = Suc g & Ant f |= Suc f holds
Ant g |= Suc g

proof end;

theorem Th17: :: CALCUL_1:17

for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A
for f being FinSequence of CQC-WFF st len f > 0 holds
( ( J,v |= Ant f & J,v |= Suc f ) iff J,v |= f )

proof end;

theorem Th18: :: CALCUL_1:18

for f, g being FinSequence of CQC-WFF st len f > 1 & len g > 1 & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & Ant f |= Suc f & Ant g |= Suc g holds
Ant (Ant f) |= Suc f

proof end;

theorem Th19: :: CALCUL_1:19

for p being Element of CQC-WFF
for f, g being FinSequence of CQC-WFF st len f > 1 & Ant f = Ant g & 'not' p = Suc (Ant f) & 'not' (Suc f) = Suc g & Ant f |= Suc f & Ant g |= Suc g holds
Ant (Ant f) |= p

proof end;

theorem Th20: :: CALCUL_1:20

for f, g being FinSequence of CQC-WFF st Ant f = Ant g & Ant f |= Suc f & Ant g |= Suc g holds
Ant f |= (Suc f) '&' (Suc g)

proof end;

theorem Th21: :: CALCUL_1:21

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st Ant f |= p '&' q holds
Ant f |= p

proof end;

theorem Th22: :: CALCUL_1:22

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st Ant f |= p '&' q holds
Ant f |= q

proof end;

theorem Th23: :: CALCUL_1:23

for p being Element of CQC-WFF
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A
for Sub being CQC_Substitution holds
( J,v |= [p,Sub] iff J,v |= p )

proof end;

theorem Th24: :: CALCUL_1:24

for p being Element of CQC-WFF
for x, y being bound_QC-variable
for A being non empty set
for J being interpretation of A
for v being Element of Valuations_in A holds
( J,v |= p . (x,y) iff ex a being Element of A st
( v . y = a & J,v . (x | a) |= p ) )

proof end;

theorem Th25: :: CALCUL_1:25

for p being Element of CQC-WFF
for x being bound_QC-variable
for f being FinSequence of CQC-WFF st Suc f = All (x,p) & Ant f |= Suc f holds
for y being bound_QC-variable holds Ant f |= p . (x,y)

proof end;

theorem Th26: :: CALCUL_1:26

for x being bound_QC-variable
for A being non empty set
for v being Element of Valuations_in A
for a being Element of A
for X being set st X c= bound_QC-variables & not x in X holds
(v . (x | a)) | X = v | X

proof end;

theorem Th27: :: CALCUL_1:27

for A being non empty set
for J being interpretation of A
for f being FinSequence of CQC-WFF
for v, w being Element of Valuations_in A st v | (still_not-bound_in f) = w | (still_not-bound_in f) & J,v |= f holds
J,w |= f

proof end;

theorem Th28: :: CALCUL_1:28

for p being Element of CQC-WFF
for y, x being bound_QC-variable
for A being non empty set
for v being Element of Valuations_in A
for a being Element of A st not y in still_not-bound_in (All (x,p)) holds
((v . (y | a)) . (x | a)) | (still_not-bound_in p) = (v . (x | a)) | (still_not-bound_in p)

proof end;

theorem Th29: :: CALCUL_1:29

for p being Element of CQC-WFF
for x, y being bound_QC-variable
for f being FinSequence of CQC-WFF st Suc f = p . (x,y) & Ant f |= Suc f & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) holds
Ant f |= All (x,p)

proof end;

theorem Th30: :: CALCUL_1:30

for f being FinSequence of CQC-WFF holds Ant (f ^ <*VERUM*>) |= Suc (f ^ <*VERUM*>)

proof end;

theorem Th31: :: CALCUL_1:31

for PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:]
for n being Nat st 1 <= n & n <= len PR & not (PR . n) `2 = 0 & not (PR . n) `2 = 1 & not (PR . n) `2 = 2 & not (PR . n) `2 = 3 & not (PR . n) `2 = 4 & not (PR . n) `2 = 5 & not (PR . n) `2 = 6 & not (PR . n) `2 = 7 & not (PR . n) `2 = 8 holds
(PR . n) `2 = 9

proof end;

Lm3: for n being Element of NAT
for PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:] st 1 <= n & n <= len PR holds
(PR . n) `1 is FinSequence of CQC-WFF

proof end;

theorem :: CALCUL_1:32

for p being Element of CQC-WFF
for X being Subset of CQC-WFF st p is_formal_provable_from X holds
X |= p

proof end;

begin

theorem Th33: :: CALCUL_1:33

for f being FinSequence of CQC-WFF st Suc f is_tail_of Ant f holds
|- f

proof end;

theorem Th34: :: CALCUL_1:34

for PR, PR1 being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:]
for n being Nat st 1 <= n & n <= len PR holds
( PR,n is_a_correct_step iff PR ^ PR1,n is_a_correct_step )

proof end;

theorem Th35: :: CALCUL_1:35

for n being Element of NAT
for PR1, PR being FinSequence of [:set_of_CQC-WFF-seq,Proof_Step_Kinds:] st 1 <= n & n <= len PR1 & PR1,n is_a_correct_step holds
PR ^ PR1,n + (len PR) is_a_correct_step

proof end;

theorem Th36: :: CALCUL_1:36

for f, g being FinSequence of CQC-WFF st Ant f is_Subsequence_of Ant g & Suc f = Suc g & |- f holds
|- g

proof end;

theorem Th37: :: CALCUL_1:37

for f, g being FinSequence of CQC-WFF st 1 < len f & 1 < len g & Ant (Ant f) = Ant (Ant g) & 'not' (Suc (Ant f)) = Suc (Ant g) & Suc f = Suc g & |- f & |- g holds
|- (Ant (Ant f)) ^ <*(Suc f)*>

proof end;

theorem Th38: :: CALCUL_1:38

for p being Element of CQC-WFF
for f, g being FinSequence of CQC-WFF st len f > 1 & Ant f = Ant g & Suc (Ant f) = 'not' p & 'not' (Suc f) = Suc g & |- f & |- g holds
|- (Ant (Ant f)) ^ <*p*>

proof end;

theorem Th39: :: CALCUL_1:39

for f, g being FinSequence of CQC-WFF st Ant f = Ant g & |- f & |- g holds
|- (Ant f) ^ <*((Suc f) '&' (Suc g))*>

proof end;

theorem Th40: :: CALCUL_1:40

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st p '&' q = Suc f & |- f holds
|- (Ant f) ^ <*p*>

proof end;

theorem Th41: :: CALCUL_1:41

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st p '&' q = Suc f & |- f holds
|- (Ant f) ^ <*q*>

proof end;

theorem Th42: :: CALCUL_1:42

for p being Element of CQC-WFF
for x, y being bound_QC-variable
for f being FinSequence of CQC-WFF st Suc f = All (x,p) & |- f holds
|- (Ant f) ^ <*(p . (x,y))*>

proof end;

theorem Th43: :: CALCUL_1:43

for p being Element of CQC-WFF
for x, y being bound_QC-variable
for f being FinSequence of CQC-WFF st Suc f = p . (x,y) & not y in still_not-bound_in (Ant f) & not y in still_not-bound_in (All (x,p)) & |- f holds
|- (Ant f) ^ <*(All (x,p))*>

proof end;

theorem Th44: :: CALCUL_1:44

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f & |- (Ant f) ^ <*('not' (Suc f))*> holds
|- (Ant f) ^ <*p*>

proof end;

theorem Th45: :: CALCUL_1:45

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF st 1 <= len f & |- f & |- f ^ <*p*> holds
|- (Ant f) ^ <*p*>

proof end;

theorem Th46: :: CALCUL_1:46

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*p*>) ^ <*q*> holds
|- (f ^ <*('not' q)*>) ^ <*('not' p)*>

proof end;

theorem :: CALCUL_1:47

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*('not' p)*>) ^ <*('not' q)*> holds
|- (f ^ <*q*>) ^ <*p*>

proof end;

theorem Th48: :: CALCUL_1:48

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*('not' p)*>) ^ <*q*> holds
|- (f ^ <*('not' q)*>) ^ <*p*>

proof end;

theorem Th49: :: CALCUL_1:49

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*p*>) ^ <*('not' q)*> holds
|- (f ^ <*q*>) ^ <*('not' p)*>

proof end;

theorem :: CALCUL_1:50

for p, r, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds
|- (f ^ <*(p 'or' q)*>) ^ <*r*>

proof end;

theorem :: CALCUL_1:51

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f ^ <*p*> holds
|- f ^ <*(p 'or' q)*>

proof end;

theorem :: CALCUL_1:52

for q, p being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f ^ <*q*> holds
|- f ^ <*(p 'or' q)*>

proof end;

theorem Th53: :: CALCUL_1:53

for p, r, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- (f ^ <*p*>) ^ <*r*> & |- (f ^ <*q*>) ^ <*r*> holds
|- (f ^ <*(p 'or' q)*>) ^ <*r*>

proof end;

theorem Th54: :: CALCUL_1:54

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f ^ <*p*> holds
|- f ^ <*('not' ('not' p))*>

proof end;

theorem Th55: :: CALCUL_1:55

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f ^ <*('not' ('not' p))*> holds
|- f ^ <*p*>

proof end;

theorem :: CALCUL_1:56

for p, q being Element of CQC-WFF
for f being FinSequence of CQC-WFF st |- f ^ <*(p => q)*> & |- f ^ <*p*> holds
|- f ^ <*q*>

proof end;

theorem Th57: :: CALCUL_1:57

for p being Element of CQC-WFF
for x, y being bound_QC-variable holds ('not' p) . (x,y) = 'not' (p . (x,y))

proof end;

theorem :: CALCUL_1:58

for p being Element of CQC-WFF
for x being bound_QC-variable
for f being FinSequence of CQC-WFF st ex y being bound_QC-variable st |- f ^ <*(p . (x,y))*> holds
|- f ^ <*(Ex (x,p))*>

proof end;

theorem Th59: :: CALCUL_1:59

for f, g being FinSequence of CQC-WFF holds still_not-bound_in (f ^ g) = (still_not-bound_in f) \/ (still_not-bound_in g)

proof end;

theorem Th60: :: CALCUL_1:60

for p being Element of CQC-WFF holds still_not-bound_in <*p*> = still_not-bound_in p

proof end;

theorem :: CALCUL_1:61

for p, q being Element of CQC-WFF
for x, y being bound_QC-variable
for f being FinSequence of CQC-WFF st |- (f ^ <*(p . (x,y))*>) ^ <*q*> & not y in still_not-bound_in ((f ^ <*(Ex (x,p))*>) ^ <*q*>) holds
|- (f ^ <*(Ex (x,p))*>) ^ <*q*>

proof end;

theorem Th62: :: CALCUL_1:62

for f being FinSequence of CQC-WFF holds still_not-bound_in f = union { (still_not-bound_in p) where p is Element of CQC-WFF : ex i being Element of NAT st
( i in dom f & p = f . i ) }

proof end;

theorem Th63: :: CALCUL_1:63

for f being FinSequence of CQC-WFF holds still_not-bound_in f is finite

proof end;

theorem Th64: :: CALCUL_1:64

( card bound_QC-variables = omega & not bound_QC-variables is finite )

proof end;

theorem Th65: :: CALCUL_1:65

for f being FinSequence of CQC-WFF holds
not for x being bound_QC-variable holds x in still_not-bound_in f

proof end;

theorem Th66: :: CALCUL_1:66

for p being Element of CQC-WFF
for x being bound_QC-variable
for f being FinSequence of CQC-WFF st |- f ^ <*(All (x,p))*> holds
|- f ^ <*(All (x,('not' ('not' p))))*>

proof end;

theorem Th67: :: CALCUL_1:67

for p being Element of CQC-WFF
for x being bound_QC-variable
for f being FinSequence of CQC-WFF st |- f ^ <*(All (x,('not' ('not' p))))*> holds
|- f ^ <*(All (x,p))*>

proof end;

theorem :: CALCUL_1:68

for p being Element of CQC-WFF
for x being bound_QC-variable
for f being FinSequence of CQC-WFF holds
( |- f ^ <*(All (x,p))*> iff |- f ^ <*('not' (Ex (x,('not' p))))*> )

proof end;

definition

let f be FinSequence;
let p be set ;
redefine pred p is_tail_of f means :: CALCUL_1:def 16
ex i being Element of NAT st
( i in dom f & f . i = p );
compatibility
( p is_tail_of f iff ex i being Element of NAT st
( i in dom f & f . i = p ) ) by Lm1, Lm2;

end;

:: deftheorem defines is_tail_of CALCUL_1:def 16 :
for f being FinSequence
for p being set holds
( p is_tail_of f iff ex i being Element of NAT st
( i in dom f & f . i = p ) );