:: Logical Correctness of Vector Calculation Programs
:: by Takaya Nishiyama , Hirofumi Fukura and Yatsuka Nakamura
::
:: Received July 13, 2004
:: Copyright (c) 2004-2011 Association of Mizar Users

begin

theorem :: PRGCOR_2:1

canceled;

theorem :: PRGCOR_2:2

canceled;

theorem :: PRGCOR_2:3

canceled;

theorem :: PRGCOR_2:4

canceled;

theorem :: PRGCOR_2:5

canceled;

definition

let D be non empty set ;
canceled;
canceled;
canceled;
canceled;
let p be XFinSequence of ;
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 PRGCOR_2:def 1 :
canceled;

:: deftheorem PRGCOR_2:def 2 :
canceled;

:: deftheorem PRGCOR_2:def 3 :
canceled;

:: deftheorem PRGCOR_2:def 4 :
canceled;

:: deftheorem Def5 defines is_an_xrep_of PRGCOR_2:def 5 :
for D being non empty set
for p being XFinSequence of
for q being FinSequence of D holds
( p is_an_xrep_of q iff ( 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 ) ) );

theorem :: PRGCOR_2:6

for D being non empty set
for p being XFinSequence of 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
coherence
( ( x in y implies a is set ) & ( x = y implies b is set ) & ( not x in y & not x = y implies c is set ) );
consistency
for b₁ being set st x in y & x = y holds
( b₁ = a iff b₁ = b );
;

end;

:: deftheorem Def6 defines IFLGT PRGCOR_2:def 6 :
for x, y, a, b, c being set holds
( ( x in y implies IFLGT (x,y,a,b,c) = a ) & ( x = y implies IFLGT (x,y,a,b,c) = b ) & ( not x in y & not x = y implies IFLGT (x,y,a,b,c) = c ) );

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 st
( len p = n & p is_an_xrep_of q )

proof end;

definition

let a, b be XFinSequence of ;
assume that
A1: b . 0 is Nat and
A2: b . 0 < len a ;
func inner_prd_prg (a,b) -> Real means :Def7: :: PRGCOR_2:def 7
ex s being XFinSequence of 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
ex b₁ being Real ex s being XFinSequence of 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))) ) & b₁ = s . n )

proof end;

uniqueness
for b₁, b₂ being Real st ex s being XFinSequence of 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))) ) & b₁ = s . n ) & ex s being XFinSequence of 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))) ) & b₂ = s . n ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines inner_prd_prg PRGCOR_2:def 7 :
for a, b being XFinSequence of st b . 0 is Nat & b . 0 < len a holds
for b₃ being Real holds
( b₃ = inner_prd_prg (a,b) iff ex s being XFinSequence of 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))) ) & b₃ = s . n ) );

theorem Th8: :: PRGCOR_2:8

for a being FinSequence of REAL
for s being XFinSequence of st 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 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 ;
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 :
for b, c being XFinSequence of
for a being Real
for m being Integer holds
( m scalar_prd_prg c,a,b iff ( 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) ) ) ) );

theorem :: PRGCOR_2:11

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

proof end;

definition

let b, c be XFinSequence of ;
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 :
for b, c being XFinSequence of
for m being Integer holds
( m vector_minus_prg c,b iff ( 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) ) ) ) );

theorem :: PRGCOR_2:12

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

proof end;

definition

let a, b, c be XFinSequence of ;
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 :
for a, b, c being XFinSequence of
for m being Integer holds
( m vector_add_prg c,a,b iff ( 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) ) ) ) );

theorem :: PRGCOR_2:13

for a, b being non empty XFinSequence of
for m being Nat st b . 0 is Nat & len a = m & len b = m & a . 0 = b . 0 & b . 0 < m holds
( ex c being XFinSequence of st m vector_add_prg c,a,b & ( for c being non empty XFinSequence of 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 ;
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 :
for a, b, c being XFinSequence of
for m being Integer holds
( m vector_sub_prg c,a,b iff ( 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) ) ) ) );

theorem :: PRGCOR_2:14

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

proof end;