:: NORMSP_1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
attr c1 is strict;
struct NORMSTR -> RLSStruct ;
aggr NORMSTR(# carrier, Zero, add, Mult, norm #) -> NORMSTR ;
sel norm c1 -> Function of the carrier of c1, REAL ;
end;

registration
cluster non empty NORMSTR ;
existence
not for b1 being NORMSTR holds b1 is empty
proof end;
end;

deffunc H1( NORMSTR ) -> Element of the carrier of $1 = 0. $1;

definition
let X be non empty NORMSTR ;
let x be Point of X;
func ||.x.|| -> Real equals :: NORMSP_1:def 1
the norm of X . x;
coherence
the norm of X . x is Real
;
end;

:: deftheorem defines ||. NORMSP_1:def 1 :
for X being non empty NORMSTR
for x being Point of X holds ||.x.|| = the norm of X . x;

consider V being RealLinearSpace;

Lm1: the carrier of ((0). V) = {(0. V)}
by RLSUB_1:def 3;

reconsider niltonil = the carrier of ((0). V) --> 0 as Function of the carrier of ((0). V), REAL by FUNCOP_1:57;

Lm2: for u being VECTOR of ((0). V) holds niltonil . u = 0
by FUNCOP_1:13;

0. V is VECTOR of ((0). V)
by Lm1, TARSKI:def 1;

then Lm3: niltonil . (0. V) = 0
by Lm2;

Lm4: for u being VECTOR of ((0). V)
for a being Real holds niltonil . (a * u) = (abs a) * (niltonil . u)
proof end;

Lm5: for u, v being VECTOR of ((0). V) holds niltonil . (u + v) <= (niltonil . u) + (niltonil . v)
proof end;

reconsider X = NORMSTR(# the carrier of ((0). V),the Zero of ((0). V),the add of ((0). V),the Mult of ((0). V),niltonil #) as non empty NORMSTR by STRUCT_0:def 1;

Lm6: now
let x, y be Point of X; :: thesis: for a being Real holds
( ( ||.x.|| = 0 implies x = H1(X) ) & ( x = H1(X) implies ||.x.|| = 0 ) & ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )

let a be Real; :: thesis: ( ( ||.x.|| = 0 implies x = H1(X) ) & ( x = H1(X) implies ||.x.|| = 0 ) & ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
reconsider u = x, w = y as VECTOR of ((0). V) ;
thus ( ||.x.|| = 0 iff x = H1(X) ) :: thesis: ( ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| )
proof
H1(X) = the Zero of X
.= 0. ((0). V)
.= 0. V by RLSUB_1:19 ;
hence ( ||.x.|| = 0 iff x = H1(X) ) by Lm1, Lm3, TARSKI:def 1; :: thesis: verum
end;
a * x = the Mult of X . [a,u]
.= a * u ;
hence ||.(a * x).|| = (abs a) * ||.x.|| by Lm4; :: thesis: ||.(x + y).|| <= ||.x.|| + ||.y.||
x + y = the add of X . [x,y]
.= u + w ;
hence ||.(x + y).|| <= ||.x.|| + ||.y.|| by Lm5; :: thesis: verum
end;

definition
let IT be non empty NORMSTR ;
attr IT is RealNormSpace-like means :Def2: :: NORMSP_1:def 2
for x, y being Point of IT
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. IT ) & ( x = 0. IT implies ||.x.|| = 0 ) & ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| );
end;

:: deftheorem Def2 defines RealNormSpace-like NORMSP_1:def 2 :
for IT being non empty NORMSTR holds
( IT is RealNormSpace-like iff for x, y being Point of IT
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. IT ) & ( x = 0. IT implies ||.x.|| = 0 ) & ||.(a * x).|| = (abs a) * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| ) );

registration
cluster non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like strict RealNormSpace-like NORMSTR ;
existence
ex b1 being non empty NORMSTR st
( b1 is RealNormSpace-like & b1 is RealLinearSpace-like & b1 is Abelian & b1 is add-associative & b1 is right_zeroed & b1 is right_complementable & b1 is strict )
proof end;
end;

definition
mode RealNormSpace is non empty Abelian add-associative right_zeroed right_complementable RealLinearSpace-like RealNormSpace-like NORMSTR ;
end;

theorem :: NORMSP_1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace holds ||.(0. RNS).|| = 0 by Def2;

theorem Th6: :: NORMSP_1:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x being Point of RNS holds ||.(- x).|| = ||.x.||
proof end;

theorem Th7: :: NORMSP_1:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| <= ||.x.|| + ||.y.||
proof end;

theorem Th8: :: NORMSP_1:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x being Point of RNS holds 0 <= ||.x.||
proof end;

theorem :: NORMSP_1:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for a, b being Real
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.((a * x) + (b * y)).|| <= ((abs a) * ||.x.||) + ((abs b) * ||.y.||)
proof end;

theorem Th10: :: NORMSP_1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS holds
( ||.(x - y).|| = 0 iff x = y )
proof end;

theorem Th11: :: NORMSP_1:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.(x - y).|| = ||.(y - x).||
proof end;

theorem Th12: :: NORMSP_1:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS holds ||.x.|| - ||.y.|| <= ||.(x - y).||
proof end;

theorem Th13: :: NORMSP_1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS holds abs (||.x.|| - ||.y.||) <= ||.(x - y).||
proof end;

theorem Th14: :: NORMSP_1:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, z, y being Point of RNS holds ||.(x - z).|| <= ||.(x - y).|| + ||.(y - z).||
proof end;

theorem :: NORMSP_1:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x, y being Point of RNS st x <> y holds
||.(x - y).|| <> 0 by Th10;

definition
let RNS be 1-sorted ;
mode sequence of RNS is Function of NAT ,the carrier of RNS;
end;

theorem :: NORMSP_1:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for f being Function
for RNS being non empty 1-sorted
for x being Element of RNS holds
( f is sequence of RNS iff ( dom f = NAT & ( for d being set st d in NAT holds
f . d is Element of RNS ) ) )
proof end;

theorem :: NORMSP_1:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th19: :: NORMSP_1:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for x being Element of RNS ex S being sequence of RNS st rng S = {x}
proof end;

theorem Th20: :: NORMSP_1:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS st ex x being Element of RNS st
for n being Nat holds S . n = x holds
ex x being Element of RNS st rng S = {x}
proof end;

theorem Th21: :: NORMSP_1:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS st ex x being Element of RNS st rng S = {x} holds
for n being Nat holds S . n = S . (n + 1)
proof end;

theorem Th22: :: NORMSP_1:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n being Nat holds S . n = S . (n + 1) ) holds
for n, k being Nat holds S . n = S . (n + k)
proof end;

theorem Th23: :: NORMSP_1:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, k being Nat holds S . n = S . (n + k) ) holds
for n, m being Nat holds S . n = S . m
proof end;

theorem Th24: :: NORMSP_1:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS st ( for n, m being Nat holds S . n = S . m ) holds
ex x being Element of RNS st
for n being Nat holds S . n = x
proof end;

theorem :: NORMSP_1:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace ex S being sequence of RNS st rng S = {(0. RNS)} by Th19;

definition
let RNS be non empty 1-sorted ;
let S be sequence of RNS;
canceled;
:: original: constant
redefine attr S is constant means :Def4: :: NORMSP_1:def 4
ex x being Element of RNS st
for n being Nat holds S . n = x;
compatibility
( S is constant iff ex x being Element of RNS st
for n being Nat holds S . n = x )
proof end;
end;

:: deftheorem NORMSP_1:def 3 :
canceled;

:: deftheorem Def4 defines constant NORMSP_1:def 4 :
for RNS being non empty 1-sorted
for S being sequence of RNS holds
( S is constant iff ex x being Element of RNS st
for n being Nat holds S . n = x );

theorem :: NORMSP_1:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being non empty 1-sorted
for S being sequence of RNS holds
( S is constant iff ex x being Element of RNS st rng S = {x} )
proof end;

Lm7: for RNS being non empty 1-sorted
for S being sequence of RNS
for n being Nat holds S . n is Element of RNS
;

definition
let RNS be non empty 1-sorted ;
let S be sequence of RNS;
let n be Nat;
:: original: .
redefine func S . n -> Element of RNS;
coherence
S . n is Element of RNS
by Lm7;
end;

definition
let RNS be RealLinearSpace;
let S1, S2 be sequence of RNS;
func S1 + S2 -> sequence of RNS means :Def5: :: NORMSP_1:def 5
for n being Nat holds it . n = (S1 . n) + (S2 . n);
existence
ex b1 being sequence of RNS st
for n being Nat holds b1 . n = (S1 . n) + (S2 . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Nat holds b1 . n = (S1 . n) + (S2 . n) ) & ( for n being Nat holds b2 . n = (S1 . n) + (S2 . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines + NORMSP_1:def 5 :
for RNS being RealLinearSpace
for S1, S2, b4 being sequence of RNS holds
( b4 = S1 + S2 iff for n being Nat holds b4 . n = (S1 . n) + (S2 . n) );

definition
let RNS be RealLinearSpace;
let S1, S2 be sequence of RNS;
func S1 - S2 -> sequence of RNS means :Def6: :: NORMSP_1:def 6
for n being Nat holds it . n = (S1 . n) - (S2 . n);
existence
ex b1 being sequence of RNS st
for n being Nat holds b1 . n = (S1 . n) - (S2 . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Nat holds b1 . n = (S1 . n) - (S2 . n) ) & ( for n being Nat holds b2 . n = (S1 . n) - (S2 . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines - NORMSP_1:def 6 :
for RNS being RealLinearSpace
for S1, S2, b4 being sequence of RNS holds
( b4 = S1 - S2 iff for n being Nat holds b4 . n = (S1 . n) - (S2 . n) );

definition
let RNS be RealLinearSpace;
let S be sequence of RNS;
let x be Element of RNS;
func S - x -> sequence of RNS means :Def7: :: NORMSP_1:def 7
for n being Nat holds it . n = (S . n) - x;
existence
ex b1 being sequence of RNS st
for n being Nat holds b1 . n = (S . n) - x
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Nat holds b1 . n = (S . n) - x ) & ( for n being Nat holds b2 . n = (S . n) - x ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines - NORMSP_1:def 7 :
for RNS being RealLinearSpace
for S being sequence of RNS
for x being Element of RNS
for b4 being sequence of RNS holds
( b4 = S - x iff for n being Nat holds b4 . n = (S . n) - x );

definition
let RNS be RealLinearSpace;
let S be sequence of RNS;
let a be Real;
func a * S -> sequence of RNS means :Def8: :: NORMSP_1:def 8
for n being Nat holds it . n = a * (S . n);
existence
ex b1 being sequence of RNS st
for n being Nat holds b1 . n = a * (S . n)
proof end;
uniqueness
for b1, b2 being sequence of RNS st ( for n being Nat holds b1 . n = a * (S . n) ) & ( for n being Nat holds b2 . n = a * (S . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines * NORMSP_1:def 8 :
for RNS being RealLinearSpace
for S being sequence of RNS
for a being Real
for b4 being sequence of RNS holds
( b4 = a * S iff for n being Nat holds b4 . n = a * (S . n) );

definition
let RNS be RealNormSpace;
let S be sequence of RNS;
attr S is convergent means :Def9: :: NORMSP_1:def 9
ex g being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r;
end;

:: deftheorem Def9 defines convergent NORMSP_1:def 9 :
for RNS being RealNormSpace
for S being sequence of RNS holds
( S is convergent iff ex g being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - g).|| < r );

theorem :: NORMSP_1:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th34: :: NORMSP_1:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
S1 + S2 is convergent
proof end;

theorem Th35: :: NORMSP_1:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
S1 - S2 is convergent
proof end;

theorem Th36: :: NORMSP_1:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x being Point of RNS
for S being sequence of RNS st S is convergent holds
S - x is convergent
proof end;

theorem Th37: :: NORMSP_1:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for a being Real
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
a * S is convergent
proof end;

definition
let RNS be RealNormSpace;
let S be sequence of RNS;
func ||.S.|| -> Real_Sequence means :Def10: :: NORMSP_1:def 10
for n being Nat holds it . n = ||.(S . n).||;
existence
ex b1 being Real_Sequence st
for n being Nat holds b1 . n = ||.(S . n).||
proof end;
uniqueness
for b1, b2 being Real_Sequence st ( for n being Nat holds b1 . n = ||.(S . n).|| ) & ( for n being Nat holds b2 . n = ||.(S . n).|| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines ||. NORMSP_1:def 10 :
for RNS being RealNormSpace
for S being sequence of RNS
for b3 being Real_Sequence holds
( b3 = ||.S.|| iff for n being Nat holds b3 . n = ||.(S . n).|| );

theorem :: NORMSP_1:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem Th39: :: NORMSP_1:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
||.S.|| is convergent
proof end;

definition
let RNS be RealNormSpace;
let S be sequence of RNS;
assume A1: S is convergent ;
func lim S -> Point of RNS means :Def11: :: NORMSP_1:def 11
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - it).|| < r;
existence
ex b1 being Point of RNS st
for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b1).|| < r
by A1, Def9;
uniqueness
for b1, b2 being Point of RNS st ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b1).|| < r ) & ( for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b2).|| < r ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines lim NORMSP_1:def 11 :
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
for b3 being Point of RNS holds
( b3 = lim S iff for r being Real st 0 < r holds
ex m being Nat st
for n being Nat st m <= n holds
||.((S . n) - b3).|| < r );

theorem :: NORMSP_1:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
canceled;

theorem :: NORMSP_1:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for g being Point of RNS
for S being sequence of RNS st S is convergent & lim S = g holds
( ||.(S - g).|| is convergent & lim ||.(S - g).|| = 0 )
proof end;

theorem :: NORMSP_1:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
lim (S1 + S2) = (lim S1) + (lim S2)
proof end;

theorem :: NORMSP_1:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for S1, S2 being sequence of RNS st S1 is convergent & S2 is convergent holds
lim (S1 - S2) = (lim S1) - (lim S2)
proof end;

theorem :: NORMSP_1:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for RNS being RealNormSpace
for x being Point of RNS
for S being sequence of RNS st S is convergent holds
lim (S - x) = (lim S) - x
proof end;

theorem :: NORMSP_1:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for a being Real
for RNS being RealNormSpace
for S being sequence of RNS st S is convergent holds
lim (a * S) = a * (lim S)
proof end;