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:: Consequences of the Sequent Calculus
:: by Patrick Braselmann and Peter Koepke
::
:: Received September 25, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

definition

let m, n be Nat;

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

definition

let m, n be Nat;
:: 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 a, b, c being Nat holds
( c in seq (a,b) iff ( 1 + a <= c & c <= b + a ) )

proof end;

theorem Th2: :: CALCUL_2:2

for a being Nat holds seq (a,0) = {}

proof end;

theorem Th3: :: CALCUL_2:3

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

proof end;

theorem Th4: :: CALCUL_2:4

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

proof end;

theorem Th5: :: CALCUL_2:5

for a, b being Nat 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 Al be QC-alphabet ;
let f be FinSequence of CQC-WFF Al;

cluster card f -> finite ;

coherence
len f is finite ;

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 m, n 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 Al being QC-alphabet
for f, g being FinSequence of CQC-WFF Al holds len (Sgm (seq ((len g),(len f)))) = len f

proof end;

theorem Th11: :: CALCUL_2:11

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

proof end;

theorem Th12: :: CALCUL_2:12

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

proof end;

theorem Th13: :: CALCUL_2:13

for Al being QC-alphabet
for i being Element of NAT
for f, g being FinSequence of CQC-WFF Al 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 Al being QC-alphabet
for f, g being FinSequence of CQC-WFF Al holds seq ((len g),(len f)) c= dom (g ^ f)

proof end;

theorem Th15: :: CALCUL_2:15

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

proof end;

theorem Th16: :: CALCUL_2:16

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

proof end;

theorem Th18: :: CALCUL_2:18

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

proof end;

theorem Th20: :: CALCUL_2:20

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

proof end;

definition

let Al be QC-alphabet ;
let f be FinSequence of CQC-WFF Al;

func Begin f -> Element of CQC-WFF Al means :Def3: :: CALCUL_2:def 3
it = f . 1 if 1 <= len f
otherwise it = VERUM Al;

existence
( ( 1 <= len f implies ex b₁ being Element of CQC-WFF Al st b₁ = f . 1 ) & ( not 1 <= len f implies ex b₁ being Element of CQC-WFF Al st b₁ = VERUM Al ) )

proof end;

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

end;

:: deftheorem Def3 defines Begin CALCUL_2:def 3 :
for Al being QC-alphabet
for f being FinSequence of CQC-WFF Al
for b₃ being Element of CQC-WFF Al holds
( ( 1 <= len f implies ( b₃ = Begin f iff b₃ = f . 1 ) ) & ( not 1 <= len f implies ( b₃ = Begin f iff b₃ = VERUM Al ) ) );

definition

let Al be QC-alphabet ;
let f be FinSequence of CQC-WFF Al;
assume A1: 1 <= len f ;

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

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

proof end;

uniqueness
for b₁, b₂ being Element of CQC-WFF Al st ex F being FinSequence of CQC-WFF Al st
( b₁ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Nat st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF Al st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) ) & ex F being FinSequence of CQC-WFF Al st
( b₂ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Nat st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF Al 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 Al being QC-alphabet
for f being FinSequence of CQC-WFF Al st 1 <= len f holds
for b₃ being Element of CQC-WFF Al holds
( b₃ = Impl f iff ex F being FinSequence of CQC-WFF Al st
( b₃ = F . (len f) & len F = len f & ( F . 1 = Begin f or len f = 0 ) & ( for n being Nat st 1 <= n & n < len f holds
ex p, q being Element of CQC-WFF Al st
( p = f . (n + 1) & q = F . n & F . (n + 1) = p => q ) ) ) );

:: Some details about the calculus in CALCUL_1

theorem Th21: :: CALCUL_2:21

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

proof end;

theorem Th22: :: CALCUL_2:22

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

proof end;

theorem Th23: :: CALCUL_2:23

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

proof end;

theorem Th24: :: CALCUL_2:24

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

proof end;

theorem Th25: :: CALCUL_2:25

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

proof end;

theorem Th26: :: CALCUL_2:26

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

proof end;

theorem Th27: :: CALCUL_2:27

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

proof end;

theorem Th28: :: CALCUL_2:28

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

proof end;

theorem Th29: :: CALCUL_2:29

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

proof end;

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

proof end;