:: PRGCOR_2 semantic presentation

Lm1: for x being set
for n being Nat st x in n holds
x is Nat

proof end;

theorem Th1: :: PRGCOR_2:1

for n, m being Nat holds
( n in m iff n < m )

proof end;

Lm2: for p being XFinSequence
for D being set holds
( rng p c= D iff p is XFinSequence of D )
by ORDINAL1:def 8;

registration

let D be non empty set ;
cluster non empty T-Sequence of D;
existence

proof end;

end;

theorem Th2: :: PRGCOR_2:2

for D being non empty set
for f being non empty XFinSequence of D holds len f > 0

proof end;

definition

let D be set ;
let q be FinSequence of D;
func FS2XFS q -> XFinSequence of D means :Def1: :: PRGCOR_2:def 1
( len it = len q & ( for i being Nat st i < len q holds
q . (i + 1) = it . i ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines FS2XFS PRGCOR_2:def 1 :

definition

let D be set ;
let q be XFinSequence of D;
func XFS2FS q -> FinSequence of D means :: PRGCOR_2:def 2
( len it = len q & ( for i being Nat st 1 <= i & i <= len q holds
q . (i -' 1) = it . i ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines XFS2FS PRGCOR_2:def 2 :

theorem Th3: :: PRGCOR_2:3

for k being Nat
for a being set holds k --> a is XFinSequence

proof end;

theorem Th4: :: PRGCOR_2:4

for D being set
for n being Nat
for r being set st r in D holds
( n --> r is XFinSequence of D & ( for q2 being XFinSequence st q2 = n --> r holds
len q2 = n ) )

proof end;

definition

let D be non empty set ;
let q be FinSequence of D;
let n be Nat;
assume A1: ( n > len q & NAT c= D ) ;
func FS2XFS* q,n -> non empty XFinSequence of D means :Def3: :: PRGCOR_2:def 3
( len q = it . 0 & len it = n & ( for i being Nat st 1 <= i & i <= len q holds
it . i = q . i ) & ( for j being Nat st len q < j & j < n holds
it . j = 0 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines FS2XFS* PRGCOR_2:def 3 :

definition

let D be non empty set ;
let p be non empty XFinSequence of D;
assume A1: ( NAT c= D & p . 0 is Nat & p . 0 in len p ) ;
func XFS2FS* p -> FinSequence of D means :Def4: :: PRGCOR_2:def 4
for m being Nat st m = p . 0 holds
( len it = m & ( for i being Nat st 1 <= i & i <= m holds
it . i = p . i ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines XFS2FS* PRGCOR_2:def 4 :

theorem Th5: :: PRGCOR_2:5

for D being non empty set
for p being non empty XFinSequence of D st NAT c= D & p . 0 = 0 & 0 < len p holds
XFS2FS* p = {}

proof end;

definition

let D be non empty set ;
let p be XFinSequence of D;
let q be FinSequence of D;
pred p is_an_xrep_of q means :Def5: :: PRGCOR_2:def 5
( NAT c= D & p . 0 = len q & len q < len p & ( for i being Nat st 1 <= i & i <= len q holds
p . i = q . i ) );

end;

:: deftheorem Def5 defines is_an_xrep_of PRGCOR_2:def 5 :

theorem :: PRGCOR_2:6

for D being non empty set
for p being non empty XFinSequence of D st NAT c= D & p . 0 is Nat & p . 0 in len p holds
p is_an_xrep_of XFS2FS* p

proof end;

definition

let x, y, a, b, c be set ;
func IFLGT x,y,a,b,c -> set equals :Def6: :: PRGCOR_2:def 6
a if x in y
b if x = y
otherwise c;
correctness ;

end;

:: deftheorem Def6 defines IFLGT PRGCOR_2:def 6 :

theorem Th7: :: PRGCOR_2:7

for D being non empty set
for q being FinSequence of D
for n being Nat st NAT c= D & n > len q holds
ex p being XFinSequence of D st
( len p = n & p is_an_xrep_of q )

proof end;

definition

let b be XFinSequence of REAL ;
let n be Nat;
:: original: .
redefine func b . n -> Real;
coherence

proof end;

end;

definition

let a, b be XFinSequence of REAL ;
assume A1: ( b . 0 is Nat & 0 <= b . 0 & b . 0 < len a ) ;
func inner_prd_prg a,b -> Real means :Def7: :: PRGCOR_2:def 7
ex s being XFinSequence of REAL ex n being Integer st
( len s = len a & s . 0 = 0 & n = b . 0 & ( n <> 0 implies for i being Nat st i < n holds
s . (i + 1) = (s . i) + ((a . (i + 1)) * (b . (i + 1))) ) & it = s . n );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines inner_prd_prg PRGCOR_2:def 7 :

theorem Th8: :: PRGCOR_2:8

for a being FinSequence of REAL
for s being XFinSequence of REAL st len s > len a & s . 0 = 0 & ( for i being Nat st i < len a holds
s . (i + 1) = (s . i) + (a . (i + 1)) ) holds
Sum a = s . (len a)

proof end;

theorem :: PRGCOR_2:9

for a being FinSequence of REAL ex s being XFinSequence of REAL st
( len s = (len a) + 1 & s . 0 = 0 & ( for i being Nat st i < len a holds
s . (i + 1) = (s . i) + (a . (i + 1)) ) & Sum a = s . (len a) )

proof end;

theorem :: PRGCOR_2:10

for a, b being FinSequence of REAL
for n being Nat st len a = len b & n > len a holds
|(a,b)| = inner_prd_prg (FS2XFS* a,n),(FS2XFS* b,n)

proof end;

definition

let b, c be XFinSequence of REAL ;
let a be Real;
let m be Integer;
pred m scalar_prd_prg c,a,b means :Def8: :: PRGCOR_2:def 8
( len c = m & len b = m & ex n being Integer st
( c . 0 = b . 0 & n = b . 0 & ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = a * (b . i) ) ) );

end;

:: deftheorem Def8 defines scalar_prd_prg PRGCOR_2:def 8 :

theorem :: PRGCOR_2:11

for b being non empty XFinSequence of REAL
for a being Real
for m being Nat st b . 0 is Nat & len b = m & 0 <= b . 0 & b . 0 < m holds
( ex c being XFinSequence of REAL st m scalar_prd_prg c,a,b & ( for c being non empty XFinSequence of REAL st m scalar_prd_prg c,a,b holds
XFS2FS* c = a * (XFS2FS* b) ) )

proof end;

definition

let b, c be XFinSequence of REAL ;
let m be Integer;
pred m vector_minus_prg c,b means :Def9: :: PRGCOR_2:def 9
( len c = m & len b = m & ex n being Integer st
( c . 0 = b . 0 & n = b . 0 & ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = - (b . i) ) ) );

end;

:: deftheorem Def9 defines vector_minus_prg PRGCOR_2:def 9 :

theorem :: PRGCOR_2:12

for b being non empty XFinSequence of REAL
for m being Nat st b . 0 is Nat & len b = m & 0 <= b . 0 & b . 0 < m holds
( ex c being XFinSequence of REAL st m vector_minus_prg c,b & ( for c being non empty XFinSequence of REAL st m vector_minus_prg c,b holds
XFS2FS* c = - (XFS2FS* b) ) )

proof end;

definition

let a, b, c be XFinSequence of REAL ;
let m be Integer;
pred m vector_add_prg c,a,b means :Def10: :: PRGCOR_2:def 10
( len c = m & len a = m & len b = m & ex n being Integer st
( c . 0 = b . 0 & n = b . 0 & ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = (a . i) + (b . i) ) ) );

end;

:: deftheorem Def10 defines vector_add_prg PRGCOR_2:def 10 :

theorem :: PRGCOR_2:13

for a, b being non empty XFinSequence of REAL
for m being Nat st b . 0 is Nat & len a = m & len b = m & a . 0 = b . 0 & 0 <= b . 0 & b . 0 < m holds
( ex c being XFinSequence of REAL st m vector_add_prg c,a,b & ( for c being non empty XFinSequence of REAL st m vector_add_prg c,a,b holds
XFS2FS* c = (XFS2FS* a) + (XFS2FS* b) ) )

proof end;

definition

let a, b, c be XFinSequence of REAL ;
let m be Integer;
pred m vector_sub_prg c,a,b means :Def11: :: PRGCOR_2:def 11
( len c = m & len a = m & len b = m & ex n being Integer st
( c . 0 = b . 0 & n = b . 0 & ( n <> 0 implies for i being Nat st 1 <= i & i <= n holds
c . i = (a . i) - (b . i) ) ) );

end;

:: deftheorem Def11 defines vector_sub_prg PRGCOR_2:def 11 :

theorem :: PRGCOR_2:14

for a, b being non empty XFinSequence of REAL
for m being Nat st b . 0 is Nat & len a = m & len b = m & a . 0 = b . 0 & 0 <= b . 0 & b . 0 < m holds
( ex c being XFinSequence of REAL st m vector_sub_prg c,a,b & ( for c being non empty XFinSequence of REAL st m vector_sub_prg c,a,b holds
XFS2FS* c = (XFS2FS* a) - (XFS2FS* b) ) )

proof end;