:: Rotating and reversing.(Finite sequences)
:: by Andrzej Trybulec
::
:: Received January 21, 1999
:: Copyright (c) 1999 Association of Mizar Users

begin

definition

let D be non empty set ;
let f be FinSequence of D;
:: original: constant
redefine attr f is constant means :Def1: :: REVROT_1:def 1
for n, m being Element of NAT st n in dom f & m in dom f holds
f /. n = f /. m;
compatibility
( f is constant iff for n, m being Element of NAT st n in dom f & m in dom f holds
f /. n = f /. m )

proof end;

end;

:: deftheorem Def1 defines constant REVROT_1:def 1 :
for D being non empty set
for f being FinSequence of D holds
( f is constant iff for n, m being Element of NAT st n in dom f & m in dom f holds
f /. n = f /. m );

theorem Th1: :: REVROT_1:1

for D being non empty set
for f being FinSequence of D st f just_once_values f /. (len f) holds
(f /. (len f)) .. f = len f

proof end;

theorem :: REVROT_1:2

for D being non empty set
for f being FinSequence of D holds f /^ (len f) = {} by RFINSEQ:40;

theorem Th3: :: REVROT_1:3

for D being non empty set
for f being non empty FinSequence of D holds f /. (len f) in rng f

proof end;

definition

let D be non empty set ;
let f be FinSequence of D;
let y be set ;
:: original: just_once_values
redefine pred f just_once_values y means :: REVROT_1:def 2
ex x being set st
( x in dom f & y = f /. x & ( for z being set st z in dom f & z <> x holds
f /. z <> y ) );
compatibility
( f just_once_values y iff ex x being set st
( x in dom f & y = f /. x & ( for z being set st z in dom f & z <> x holds
f /. z <> y ) ) )

proof end;

end;

:: deftheorem defines just_once_values REVROT_1:def 2 :
for D being non empty set
for f being FinSequence of D
for y being set holds
( f just_once_values y iff ex x being set st
( x in dom f & y = f /. x & ( for z being set st z in dom f & z <> x holds
f /. z <> y ) ) );

theorem Th4: :: REVROT_1:4

for D being non empty set
for f being FinSequence of D st f just_once_values f /. (len f) holds
f -: (f /. (len f)) = f

proof end;

theorem Th5: :: REVROT_1:5

for D being non empty set
for f being FinSequence of D st f just_once_values f /. (len f) holds
f :- (f /. (len f)) = <*(f /. (len f))*>

proof end;

theorem Th6: :: REVROT_1:6

for D being non empty set
for p being Element of D
for f being FinSequence of D holds 1 <= len (f :- p)

proof end;

theorem :: REVROT_1:7

for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f holds
len (f :- p) <= len f

proof end;

theorem :: REVROT_1:8

for D being non empty set
for f being non empty circular FinSequence of D holds Rev f is circular

proof end;

begin

theorem Th9: :: REVROT_1:9

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f & 1 <= i & i <= len (f :- p) holds
(Rotate (f,p)) /. i = f /. ((i -' 1) + (p .. f))

proof end;

theorem Th10: :: REVROT_1:10

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f & p .. f <= i & i <= len f holds
f /. i = (Rotate (f,p)) /. ((i + 1) -' (p .. f))

proof end;

theorem Th11: :: REVROT_1:11

for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f holds
(Rotate (f,p)) /. (len (f :- p)) = f /. (len f)

proof end;

theorem Th12: :: REVROT_1:12

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f & len (f :- p) < i & i <= len f holds
(Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))

proof end;

theorem :: REVROT_1:13

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being FinSequence of D st p in rng f & 1 < i & i <= p .. f holds
f /. i = (Rotate (f,p)) /. ((i + (len f)) -' (p .. f))

proof end;

theorem Th14: :: REVROT_1:14

for D being non empty set
for p being Element of D
for f being FinSequence of D holds len (Rotate (f,p)) = len f

proof end;

theorem Th15: :: REVROT_1:15

for D being non empty set
for p being Element of D
for f being FinSequence of D holds dom (Rotate (f,p)) = dom f

proof end;

theorem Th16: :: REVROT_1:16

for D being non empty set
for f being circular FinSequence of D
for p being Element of D st ( for i being Element of NAT st 1 < i & i < len f holds
f /. i <> f /. 1 ) holds
Rotate ((Rotate (f,p)),(f /. 1)) = f

proof end;

begin

theorem Th17: :: REVROT_1:17

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being circular FinSequence of D st p in rng f & len (f :- p) <= i & i <= len f holds
(Rotate (f,p)) /. i = f /. ((i + (p .. f)) -' (len f))

proof end;

theorem Th18: :: REVROT_1:18

for i being Element of NAT
for D being non empty set
for p being Element of D
for f being circular FinSequence of D st p in rng f & 1 <= i & i <= p .. f holds
f /. i = (Rotate (f,p)) /. ((i + (len f)) -' (p .. f))

proof end;

registration

let D be non trivial set ;
cluster V1() V5(D) Function-like V8() V30() FinSequence-like FinSubsequence-like circular FinSequence of D;
existence
ex b₁ being FinSequence of D st
( not b₁ is constant & b₁ is circular )

proof end;

end;

registration

let D be non trivial set ;
let p be Element of D;
let f be V8() circular FinSequence of D;
cluster Rotate (f,p) -> V8() ;
coherence
not Rotate (f,p) is constant

proof end;

end;

begin

theorem Th19: :: REVROT_1:19

for n being non empty Element of NAT holds 0.REAL n <> 1.REAL n

proof end;

registration

let n be non empty Element of NAT ;
cluster TOP-REAL n -> non trivial ;
coherence
not TOP-REAL n is trivial

proof end;

end;

theorem Th20: :: REVROT_1:20

for f, g being FinSequence of (TOP-REAL 2) st rng f c= rng g holds
rng (X_axis f) c= rng (X_axis g)

proof end;

theorem Th21: :: REVROT_1:21

for f, g being FinSequence of (TOP-REAL 2) st rng f = rng g holds
rng (X_axis f) = rng (X_axis g)

proof end;

theorem Th22: :: REVROT_1:22

for f, g being FinSequence of (TOP-REAL 2) st rng f c= rng g holds
rng (Y_axis f) c= rng (Y_axis g)

proof end;

theorem Th23: :: REVROT_1:23

for f, g being FinSequence of (TOP-REAL 2) st rng f = rng g holds
rng (Y_axis f) = rng (Y_axis g)

proof end;

begin

registration

let p be Point of (TOP-REAL 2);
let f be circular special FinSequence of (TOP-REAL 2);
cluster Rotate (f,p) -> special ;
coherence
Rotate (f,p) is special

proof end;

end;

theorem Th24: :: REVROT_1:24

for i being Element of NAT
for p being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p in rng f & 1 <= i & i < len (f :- p) holds
LSeg ((Rotate (f,p)),i) = LSeg (f,((i -' 1) + (p .. f)))

proof end;

theorem Th25: :: REVROT_1:25

for i being Element of NAT
for p being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p in rng f & p .. f <= i & i < len f holds
LSeg (f,i) = LSeg ((Rotate (f,p)),((i -' (p .. f)) + 1))

proof end;

theorem Th26: :: REVROT_1:26

for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2) holds Incr (X_axis f) = Incr (X_axis (Rotate (f,p)))

proof end;

theorem Th27: :: REVROT_1:27

for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2) holds Incr (Y_axis f) = Incr (Y_axis (Rotate (f,p)))

proof end;

theorem Th28: :: REVROT_1:28

for p being Point of (TOP-REAL 2)
for f being non empty circular FinSequence of (TOP-REAL 2) holds GoB (Rotate (f,p)) = GoB f

proof end;

theorem Th29: :: REVROT_1:29

for p being Point of (TOP-REAL 2)
for f being V8() standard special_circular_sequence holds Rev (Rotate (f,p)) = Rotate ((Rev f),p)

proof end;

begin

theorem Th30: :: REVROT_1:30

for f being circular s.c.c. FinSequence of (TOP-REAL 2) st len f > 4 holds
(LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) = {(f /. 1)}

proof end;

theorem Th31: :: REVROT_1:31

for i being Element of NAT
for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2) st p in rng f & len (f :- p) <= i & i < len f holds
LSeg ((Rotate (f,p)),i) = LSeg (f,((i + (p .. f)) -' (len f)))

proof end;

registration

let p be Point of (TOP-REAL 2);
let f be circular s.c.c. FinSequence of (TOP-REAL 2);
cluster Rotate (f,p) -> s.c.c. ;
coherence
Rotate (f,p) is s.c.c.

proof end;

end;

registration

let p be Point of (TOP-REAL 2);
let f be V8() standard special_circular_sequence;
cluster Rotate (f,p) -> unfolded ;
coherence
Rotate (f,p) is unfolded

proof end;

end;

theorem Th32: :: REVROT_1:32

for i being Element of NAT
for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2) st p in rng f & 1 <= i & i < p .. f holds
LSeg (f,i) = LSeg ((Rotate (f,p)),((i + (len f)) -' (p .. f)))

proof end;

theorem Th33: :: REVROT_1:33

for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2) holds L~ (Rotate (f,p)) = L~ f

proof end;

theorem Th34: :: REVROT_1:34

for p being Point of (TOP-REAL 2)
for f being circular FinSequence of (TOP-REAL 2)
for G being Go-board holds
( f is_sequence_on G iff Rotate (f,p) is_sequence_on G )

proof end;

registration

let p be Point of (TOP-REAL 2);
let f be non empty circular standard FinSequence of (TOP-REAL 2);
cluster Rotate (f,p) -> standard ;
coherence
Rotate (f,p) is standard

proof end;

end;

theorem Th35: :: REVROT_1:35

for f being V8() standard special_circular_sequence
for p being Point of (TOP-REAL 2)
for k being Element of NAT st p in rng f & 1 <= k & k < p .. f holds
left_cell (f,k) = left_cell ((Rotate (f,p)),((k + (len f)) -' (p .. f)))

proof end;

theorem Th36: :: REVROT_1:36

for p being Point of (TOP-REAL 2)
for f being V8() standard special_circular_sequence holds LeftComp (Rotate (f,p)) = LeftComp f

proof end;

theorem :: REVROT_1:37

for p being Point of (TOP-REAL 2)
for f being V8() standard special_circular_sequence holds RightComp (Rotate (f,p)) = RightComp f

proof end;

begin

Lm1: for f being V8() standard special_circular_sequence holds
( not f /. 1 = N-min (L~ f) or f is clockwise_oriented or Rev f is clockwise_oriented )

proof end;

registration

let p be Point of (TOP-REAL 2);
let f be V8() standard clockwise_oriented special_circular_sequence;
cluster Rotate (f,p) -> clockwise_oriented ;
coherence
Rotate (f,p) is clockwise_oriented

proof end;

end;

theorem :: REVROT_1:38

for f being V8() standard special_circular_sequence holds
( f is clockwise_oriented or Rev f is clockwise_oriented )

proof end;