:: Rotating and reversing.(Finite sequences)
:: by Andrzej Trybulec
::
:: Received January 21, 1999
:: Copyright (c) 1999-2021 Association of Mizar Users

definition

end;

:: deftheorem REVROT_1:def 1 :
canceled;

:: deftheorem REVROT_1:def 2 :
canceled;

::$CD 2

theorem :: REVROT_1:1

canceled;

::$CD 2

theorem :: REVROT_1:2

canceled;

::$CD 2

theorem :: REVROT_1:3

canceled;

::$CD 2

theorem :: REVROT_1:4

canceled;

::$CD 2

theorem :: REVROT_1:5

canceled;

::$CD 2

theorem :: REVROT_1:6

canceled;

::$CD 2

theorem :: REVROT_1:7

canceled;

::$CD 2

theorem :: REVROT_1:8

canceled;

::$CD 2

theorem :: REVROT_1:9

canceled;

::$CD 2

theorem :: REVROT_1:10

canceled;

::$CD 2

theorem :: REVROT_1:11

canceled;

::$CD 2

theorem :: REVROT_1:12

canceled;

::$CD 2

theorem :: REVROT_1:13

canceled;

::$CD 2

theorem :: REVROT_1:14

canceled;

::$CD 2

theorem :: REVROT_1:15

canceled;

::$CD 2

theorem :: REVROT_1:16

canceled;

::$CD 2

theorem :: REVROT_1:17

canceled;

::$CD 2

theorem :: REVROT_1:18

canceled;

Th9: for i being 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))
by FINSEQ_6:174;

Th10: for i being 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))
by FINSEQ_6:175;

Th11: 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)
by FINSEQ_6:176;

Th14: 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
by FINSEQ_6:179;

Th15: 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
by FINSEQ_6:180;

Th16: for D being non empty set
for f being circular FinSequence of D
for p being Element of D st ( for i being Nat st 1 < i & i < len f holds
f /. i <> f /. 1 ) holds
Rotate ((Rotate (f,p)),(f /. 1)) = f
by FINSEQ_6:181;

Th17: for i being 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))
by FINSEQ_6:182;

Th18: for i being 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))
by FINSEQ_6:183;

theorem Th19: :: REVROT_1:19

for n being non zero Nat holds 0.REAL n <> 1.REAL n

proof end;

registration

let n be non zero 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) by Th20;

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) by Th22;

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 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 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;

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 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 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 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;

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;