:: Consequences of the Sequent Calculus
:: by Patrick Braselmann and Peter Koepke
::
:: Received September 25, 2004
:: Copyright (c) 2004 Association of Mizar Users

begin

definition

let m, n be natural number ;
func seq m,n -> set equals :: CALCUL_2:def 1
{ k where k is Element of NAT : ( 1 + m <= k & k <= n + m ) } ;
coherence
{ k where k is Element of NAT : ( 1 + m <= k & k <= n + m ) } is set ;

end;

:: deftheorem defines seq CALCUL_2:def 1 :

for m, n being natural number holds seq m,n = { k where k is Element of NAT : ( 1 + m <= k & k <= n + m ) } ;

definition

let m, n be natural number ;
:: original: seq
redefine func seq m,n -> Subset of NAT ;
coherence
seq m,n is Subset of NAT

proof end;

end;

theorem Th1: :: CALCUL_2:1

for c, a, b being natural number holds
( c in seq a,b iff ( 1 + a <= c & c <= b + a ) )

proof end;

theorem Th2: :: CALCUL_2:2

for a being natural number holds seq a,0 = {}

proof end;

theorem Th3: :: CALCUL_2:3

for b, a being natural number holds
( b = 0 or b + a in seq a,b )

proof end;

theorem Th4: :: CALCUL_2:4

for b1, b2, a being natural number holds
( b1 <= b2 iff seq a,b1 c= seq a,b2 )

proof end;

theorem Th5: :: CALCUL_2:5

for a, b being natural number holds (seq a,b) \/ {((a + b) + 1)} = seq a,(b + 1)

proof end;

theorem Th6: :: CALCUL_2:6

for m, n being Element of NAT holds seq m,n,n are_equipotent

proof end;

registration

let m, n be Element of NAT ;
cluster seq m,n -> finite ;
coherence
seq m,n is finite

proof end;

end;

registration

let f be FinSequence of CQC-WFF ;
cluster card f -> finite ;
coherence
len f is finite by CARD_1:69;

end;

theorem Th7: :: CALCUL_2:7

for m, n being Element of NAT holds seq m,n c= Seg (m + n)

proof end;

theorem :: CALCUL_2:8

for n, m being Element of NAT holds Seg n misses seq n,m

proof end;

theorem :: CALCUL_2:9

for f, g being FinSequence holds dom (f ^ g) = (dom f) \/ (seq (len f),(len g))

proof end;

theorem Th10: :: CALCUL_2:10

for g, f being FinSequence of CQC-WFF holds len (Sgm (seq (len g),(len f))) = len f

proof end;

theorem Th11: :: CALCUL_2:11

for g, f being FinSequence of CQC-WFF holds dom (Sgm (seq (len g),(len f))) = dom f

proof end;

theorem Th12: :: CALCUL_2:12

for g, f being FinSequence of CQC-WFF holds rng (Sgm (seq (len g),(len f))) = seq (len g),(len f)

proof end;

theorem Th13: :: CALCUL_2:13

for i being Element of NAT
for g, f being FinSequence of CQC-WFF st i in dom (Sgm (seq (len g),(len f))) holds
(Sgm (seq (len g),(len f))) . i = (len g) + i

proof end;

theorem Th14: :: CALCUL_2:14

for g, f being FinSequence of CQC-WFF holds seq (len g),(len f) c= dom (g ^ f)

proof end;

theorem Th15: :: CALCUL_2:15

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

proof end;

theorem Th16: :: CALCUL_2:16

for g, f being FinSequence of CQC-WFF holds Seq ((g ^ f) | (seq (len g),(len f))) = (Sgm (seq (len g),(len f))) * (g ^ f)

proof end;

theorem Th17: :: CALCUL_2:17

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

proof end;

theorem Th18: :: CALCUL_2:18

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

proof end;

definition

let D be non empty set ;
let f be FinSequence of D;
let P be Permutation of (dom f);
func Per f,P -> FinSequence of D equals :: CALCUL_2:def 2
P * f;
coherence
P * f is FinSequence of D

proof end;

end;

:: deftheorem defines Per CALCUL_2:def 2 :

for D being non empty set
for f being FinSequence of D
for P being Permutation of (dom f) holds Per f,P = P * f;

theorem Th19: :: CALCUL_2:19

for f being FinSequence of CQC-WFF
for P being Permutation of (dom f) holds dom (Per f,P) = dom f

proof end;

theorem Th20: :: CALCUL_2:20

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

proof end;

begin

definition

let f be FinSequence of CQC-WFF ;
func Begin f -> Element of CQC-WFF means :Def3: :: CALCUL_2:def 3
it = f . 1 if 1 <= len f
otherwise it = VERUM ;
existence
( ( 1 <= len f implies ex b₁ being Element of CQC-WFF st b₁ = f . 1 ) & ( not 1 <= len f implies ex b₁ being Element of CQC-WFF st b₁ = VERUM ) )

proof end;

uniqueness
for b₁, b₂ being Element of CQC-WFF holds
( ( 1 <= len f & b₁ = f . 1 & b₂ = f . 1 implies b₁ = b₂ ) & ( not 1 <= len f & b₁ = VERUM & b₂ = VERUM implies b₁ = b₂ ) ) ;
consistency
for b₁ being Element of CQC-WFF holds verum ;

end;

:: deftheorem Def3 defines Begin CALCUL_2:def 3 :

for f being FinSequence of CQC-WFF
for b₂ being Element of CQC-WFF holds
( ( 1 <= len f implies ( b₂ = Begin f iff b₂ = f . 1 ) ) & ( not 1 <= len f implies ( b₂ = Begin f iff b₂ = VERUM ) ) );

definition

let f be FinSequence of CQC-WFF ;
assume A1: 1 <= len f ;
func Impl f -> Element of CQC-WFF means :Def4: :: CALCUL_2:def 4
ex F being FinSequence of CQC-WFF st
( it = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Element of NAT st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) );
existence
ex b₁ being Element of CQC-WFF ex F being FinSequence of CQC-WFF st
( b₁ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Element of NAT st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) )

proof end;

uniqueness
for b₁, b₂ being Element of CQC-WFF st ex F being FinSequence of CQC-WFF st
( b₁ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Element of NAT st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) ) & ex F being FinSequence of CQC-WFF st
( b₂ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Element of NAT st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Impl CALCUL_2:def 4 :

for f being FinSequence of CQC-WFF st 1 <= len f holds
for b₂ being Element of CQC-WFF holds
( b₂ = Impl f iff ex F being FinSequence of CQC-WFF st
( b₂ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Element of NAT st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) ) );

theorem Th21: :: CALCUL_2:21

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF holds |- (f ^ <*p*>) ^ <*p*>

proof end;

theorem Th22: :: CALCUL_2:22

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

proof end;

theorem Th23: :: CALCUL_2:23

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

proof end;

theorem Th24: :: CALCUL_2:24

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

proof end;

theorem Th25: :: CALCUL_2:25

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

proof end;

theorem Th26: :: CALCUL_2:26

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

proof end;

theorem Th27: :: CALCUL_2:27

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

proof end;

theorem Th28: :: CALCUL_2:28

for g, f being FinSequence of CQC-WFF st 1 <= len g & |- f ^ g holds
|- f ^ <*(Impl (Rev g))*>

proof end;

theorem Th29: :: CALCUL_2:29

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF
for P being Permutation of (dom f) st |- (Per f,P) ^ <*(Impl (Rev (f ^ <*p*>)))*> holds
|- (Per f,P) ^ <*p*>

proof end;

theorem :: CALCUL_2:30

for p being Element of CQC-WFF
for f being FinSequence of CQC-WFF
for P being Permutation of (dom f) st |- f ^ <*p*> holds
|- (Per f,P) ^ <*p*>

proof end;

begin

notation

let n be Element of NAT ;
let c be set ;
synonym IdFinS c,n for n |-> c;

end;

theorem Th31: :: CALCUL_2:31

for n being Element of NAT
for c being set st 1 <= n holds
rng (IdFinS c,n) = rng <*c*>

proof end;

definition

let D be non empty set ;
let n be Element of NAT ;
let p be Element of D;
:: original: IdFinS
redefine func IdFinS p,n -> FinSequence of D;
coherence
IdFinS p,n is FinSequence of D

proof end;

end;

theorem :: CALCUL_2:32

for p, q being Element of CQC-WFF
for n being Element of NAT
for f being FinSequence of CQC-WFF st 1 <= n & |- (f ^ (IdFinS p,n)) ^ <*q*> holds
|- (f ^ <*p*>) ^ <*q*>

proof end;