:: Real Sequences and Basic Operations on Them
:: by Jaros{\l}aw Kotowicz
::
:: Copyright (c) 1990-2021 Association of Mizar Users

definition end;

theorem Th1: :: SEQ_1:1
for f being Function holds
( f is Real_Sequence iff ( dom f = NAT & ( for x being object st x in NAT holds
f . x is real ) ) )
proof end;

theorem Th2: :: SEQ_1:2
for f being Function holds
( f is Real_Sequence iff ( dom f = NAT & ( for n being Nat holds f . n is real ) ) )
proof end;

registration
existence
ex b1 being PartFunc of NAT,REAL st b1 is non-zero
proof end;
end;

theorem :: SEQ_1:3
for f being non-zero PartFunc of NAT,REAL holds rng f c= REAL \ by ;

theorem Th4: :: SEQ_1:4
for seq being Real_Sequence holds
( seq is non-zero iff for x being object st x in NAT holds
seq . x <> 0 )
proof end;

theorem Th5: :: SEQ_1:5
for seq being Real_Sequence holds
( seq is non-zero iff for n being Nat holds seq . n <> 0 )
proof end;

theorem :: SEQ_1:6
for r being Real ex seq being Real_Sequence st rng seq = {r}
proof end;

scheme :: SEQ_1:sch 1
ExRealSeq{ F1( object ) -> Real } :
ex seq being Real_Sequence st
for n being Nat holds seq . n = F1(n)
proof end;

::
:: Basic Operations on Sequences
::
scheme :: SEQ_1:sch 2
PartFuncExD9{ F1() -> non empty set , F2() -> non empty set , P1[ object , object ] } :
ex f being PartFunc of F1(),F2() st
( ( for d being Element of F1() holds
( d in dom f iff ex c being Element of F2() st P1[d,c] ) ) & ( for d being Element of F1() st d in dom f holds
P1[d,f . d] ) )
proof end;

scheme :: SEQ_1:sch 3
LambdaPFD9{ F1() -> non empty set , F2() -> non empty set , F3( object ) -> Element of F2(), P1[ object ] } :
ex f being PartFunc of F1(),F2() st
( ( for d being Element of F1() holds
( d in dom f iff P1[d] ) ) & ( for d being Element of F1() st d in dom f holds
f . d = F3(d) ) )
proof end;

scheme :: SEQ_1:sch 4
UnPartFuncD9{ F1() -> set , F2() -> set , F3() -> set , F4( object ) -> object } :
for f, g being PartFunc of F1(),F2() st dom f = F3() & ( for c being Element of F1() st c in dom f holds
f . c = F4(c) ) & dom g = F3() & ( for c being Element of F1() st c in dom g holds
g . c = F4(c) ) holds
f = g
proof end;

theorem Th7: :: SEQ_1:7
for seq, seq1, seq2 being Real_Sequence holds
( seq = seq1 + seq2 iff for n being Nat holds seq . n = (seq1 . n) + (seq2 . n) )
proof end;

theorem Th8: :: SEQ_1:8
for seq, seq1, seq2 being Real_Sequence holds
( seq = seq1 (#) seq2 iff for n being Nat holds seq . n = (seq1 . n) * (seq2 . n) )
proof end;

theorem Th9: :: SEQ_1:9
for r being Real
for seq1, seq2 being Real_Sequence holds
( seq1 = r (#) seq2 iff for n being Nat holds seq1 . n = r * (seq2 . n) )
proof end;

theorem :: SEQ_1:10
for seq1, seq2 being Real_Sequence holds
( seq1 = - seq2 iff for n being Nat holds seq1 . n = - (seq2 . n) )
proof end;

theorem :: SEQ_1:11
for seq1, seq2 being Real_Sequence holds seq1 - seq2 = seq1 + (- seq2) ;

theorem Th12: :: SEQ_1:12
for seq, seq1 being Real_Sequence holds
( seq1 = abs seq iff for n being Nat holds seq1 . n = |.(seq . n).| )
proof end;

theorem Th13: :: SEQ_1:13
for seq1, seq2, seq3 being Real_Sequence holds (seq1 + seq2) + seq3 = seq1 + (seq2 + seq3)
proof end;

theorem Th14: :: SEQ_1:14
for seq1, seq2, seq3 being Real_Sequence holds (seq1 (#) seq2) (#) seq3 = seq1 (#) (seq2 (#) seq3)
proof end;

theorem Th15: :: SEQ_1:15
for seq1, seq2, seq3 being Real_Sequence holds (seq1 + seq2) (#) seq3 = (seq1 (#) seq3) + (seq2 (#) seq3)
proof end;

theorem :: SEQ_1:16
for seq1, seq2, seq3 being Real_Sequence holds seq3 (#) (seq1 + seq2) = (seq3 (#) seq1) + (seq3 (#) seq2) by Th15;

theorem :: SEQ_1:17
for seq being Real_Sequence holds - seq = (- 1) (#) seq ;

theorem Th18: :: SEQ_1:18
for r being Real
for seq1, seq2 being Real_Sequence holds r (#) (seq1 (#) seq2) = (r (#) seq1) (#) seq2
proof end;

theorem Th19: :: SEQ_1:19
for r being Real
for seq1, seq2 being Real_Sequence holds r (#) (seq1 (#) seq2) = seq1 (#) (r (#) seq2)
proof end;

theorem Th20: :: SEQ_1:20
for seq1, seq2, seq3 being Real_Sequence holds (seq1 - seq2) (#) seq3 = (seq1 (#) seq3) - (seq2 (#) seq3)
proof end;

theorem :: SEQ_1:21
for seq1, seq2, seq3 being Real_Sequence holds (seq3 (#) seq1) - (seq3 (#) seq2) = seq3 (#) (seq1 - seq2) by Th20;

theorem Th22: :: SEQ_1:22
for r being Real
for seq1, seq2 being Real_Sequence holds r (#) (seq1 + seq2) = (r (#) seq1) + (r (#) seq2)
proof end;

theorem Th23: :: SEQ_1:23
for r, p being Real
for seq being Real_Sequence holds (r * p) (#) seq = r (#) (p (#) seq)
proof end;

theorem Th24: :: SEQ_1:24
for r being Real
for seq1, seq2 being Real_Sequence holds r (#) (seq1 - seq2) = (r (#) seq1) - (r (#) seq2)
proof end;

theorem :: SEQ_1:25
for r being Real
for seq, seq1 being Real_Sequence holds r (#) (seq1 /" seq) = (r (#) seq1) /" seq
proof end;

theorem :: SEQ_1:26
for seq1, seq2, seq3 being Real_Sequence holds seq1 - (seq2 + seq3) = (seq1 - seq2) - seq3
proof end;

theorem :: SEQ_1:27
for seq being Real_Sequence holds 1 (#) seq = seq
proof end;

theorem :: SEQ_1:28
canceled;

::$CT theorem :: SEQ_1:29 for seq1, seq2 being Real_Sequence holds seq1 - (- seq2) = seq1 + seq2 ; theorem :: SEQ_1:30 for seq1, seq2, seq3 being Real_Sequence holds seq1 - (seq2 - seq3) = (seq1 - seq2) + seq3 proof end; theorem :: SEQ_1:31 for seq1, seq2, seq3 being Real_Sequence holds seq1 + (seq2 - seq3) = (seq1 + seq2) - seq3 proof end; theorem :: SEQ_1:32 for seq1, seq2 being Real_Sequence holds ( (- seq1) (#) seq2 = - (seq1 (#) seq2) & seq1 (#) (- seq2) = - (seq1 (#) seq2) ) by Th18; theorem Th32: :: SEQ_1:33 for seq being Real_Sequence st seq is non-zero holds seq " is non-zero proof end; theorem :: SEQ_1:34 canceled; ::$CT
theorem Th33: :: SEQ_1:35
for seq, seq1 being Real_Sequence holds
( ( seq is non-zero & seq1 is non-zero ) iff seq (#) seq1 is non-zero )
proof end;

theorem Th34: :: SEQ_1:36
for seq, seq1 being Real_Sequence holds (seq ") (#) (seq1 ") = (seq (#) seq1) "
proof end;

theorem :: SEQ_1:37
for seq, seq1 being Real_Sequence st seq is non-zero holds
(seq1 /" seq) (#) seq = seq1
proof end;

theorem :: SEQ_1:38
for seq, seq1, seq9, seq19 being Real_Sequence holds (seq9 /" seq) (#) (seq19 /" seq1) = (seq9 (#) seq19) /" (seq (#) seq1)
proof end;

theorem :: SEQ_1:39
for seq, seq1 being Real_Sequence st seq is non-zero & seq1 is non-zero holds
seq /" seq1 is non-zero
proof end;

theorem Th38: :: SEQ_1:40
for seq, seq1 being Real_Sequence holds (seq /" seq1) " = seq1 /" seq
proof end;

theorem :: SEQ_1:41
for seq, seq1, seq2 being Real_Sequence holds seq2 (#) (seq1 /" seq) = (seq2 (#) seq1) /" seq
proof end;

theorem :: SEQ_1:42
for seq, seq1, seq2 being Real_Sequence holds seq2 /" (seq /" seq1) = (seq2 (#) seq1) /" seq
proof end;

theorem Th41: :: SEQ_1:43
for seq, seq1, seq2 being Real_Sequence st seq1 is non-zero holds
seq2 /" seq = (seq2 (#) seq1) /" (seq (#) seq1)
proof end;

theorem Th42: :: SEQ_1:44
for r being Real
for seq being Real_Sequence st r <> 0 & seq is non-zero holds
r (#) seq is non-zero
proof end;

theorem :: SEQ_1:45
for seq being Real_Sequence st seq is non-zero holds
- seq is non-zero by Th42;

theorem Th44: :: SEQ_1:46
for r being Real
for seq being Real_Sequence holds (r (#) seq) " = (r ") (#) (seq ")
proof end;

Lm1: (- 1) " = - 1
;

theorem :: SEQ_1:47
for seq being Real_Sequence holds (- seq) " = (- 1) (#) (seq ") by ;

theorem :: SEQ_1:48
for seq, seq1 being Real_Sequence holds
( - (seq1 /" seq) = (- seq1) /" seq & seq1 /" (- seq) = - (seq1 /" seq) )
proof end;

theorem :: SEQ_1:49
for seq, seq1, seq19 being Real_Sequence holds
( (seq1 /" seq) + (seq19 /" seq) = (seq1 + seq19) /" seq & (seq1 /" seq) - (seq19 /" seq) = (seq1 - seq19) /" seq )
proof end;

theorem :: SEQ_1:50
for seq, seq1, seq9, seq19 being Real_Sequence st seq is non-zero & seq9 is non-zero holds
( (seq1 /" seq) + (seq19 /" seq9) = ((seq1 (#) seq9) + (seq19 (#) seq)) /" (seq (#) seq9) & (seq1 /" seq) - (seq19 /" seq9) = ((seq1 (#) seq9) - (seq19 (#) seq)) /" (seq (#) seq9) )
proof end;

theorem :: SEQ_1:51
for seq, seq1, seq9, seq19 being Real_Sequence holds (seq19 /" seq) /" (seq9 /" seq1) = (seq19 (#) seq1) /" (seq (#) seq9)
proof end;

theorem Th50: :: SEQ_1:52
for seq, seq9 being Real_Sequence holds abs (seq (#) seq9) = (abs seq) (#) (abs seq9)
proof end;

theorem :: SEQ_1:53
for seq being Real_Sequence st seq is non-zero holds
abs seq is non-zero
proof end;

theorem Th52: :: SEQ_1:54
for seq being Real_Sequence holds (abs seq) " = abs (seq ")
proof end;

theorem :: SEQ_1:55
for seq, seq9 being Real_Sequence holds abs (seq9 /" seq) = (abs seq9) /" (abs seq)
proof end;

theorem :: SEQ_1:56
for r being Real
for seq being Real_Sequence holds abs (r (#) seq) = |.r.| (#) (abs seq)
proof end;

definition
let b be Real;
func seq_const b -> Real_Sequence equals :Def1: :: SEQ_1:def 1
NAT --> b;
coherence
proof end;
end;

:: deftheorem Def1 defines seq_const SEQ_1:def 1 :
for b being Real holds seq_const b = NAT --> b;

registration
let b be Real;
coherence ;
end;

registration
let b be non zero Real;
coherence
proof end;
end;

registration
existence
ex b1 being Real_Sequence st b1 is constant
proof end;
end;

theorem :: SEQ_1:57
for a being Real
for k being Nat holds () . k = a by ;

registration
let k be object ;
reduce () . k to 0 ;
reducibility
() . k = 0
proof end;
end;

registration
let k be object ;
reduce () . k to 0 ;
reducibility
() . k = 0
;
end;