let f be 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)))
let p be 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)))
let k be Nat; ( p in rng f & 1 <= k & k < p .. f implies left_cell (f,k) = left_cell ((Rotate (f,p)),((k + (len f)) -' (p .. f))) )
assume that
A1:
p in rng f
and
A2:
1 <= k
and
A3:
k < p .. f
; left_cell (f,k) = left_cell ((Rotate (f,p)),((k + (len f)) -' (p .. f)))
set n = (k + (len f)) -' (p .. f);
A4:
p .. f <= len f
by A1, FINSEQ_4:21;
then
k < len f
by A3, XXREAL_0:2;
then A5:
k + 1 <= len f
by NAT_1:13;
0 < k
by A2;
then A6:
0 + 1 < k + 1
by XREAL_1:6;
len f <= k + (len f)
by NAT_1:11;
then A7:
((k + (len f)) -' (p .. f)) + 1 = ((k + (len f)) + 1) -' (p .. f)
by A4, NAT_D:38, XXREAL_0:2;
A8:
k + 1 <= p .. f
by A3, NAT_1:13;
then
(k + 1) + (len f) <= (len f) + (p .. f)
by XREAL_1:6;
then
((k + (len f)) -' (p .. f)) + 1 <= len f
by A7, NAT_D:53;
then A9:
((k + (len f)) -' (p .. f)) + 1 <= len (Rotate (f,p))
by Th14;
A10:
((k + (len f)) -' (p .. f)) + 1 = ((k + 1) + (len f)) -' (p .. f)
by A7;
A11:
for i1, j1, i2, j2 being Nat st [i1,j1] in Indices (GoB (Rotate (f,p))) & [i2,j2] in Indices (GoB (Rotate (f,p))) & (Rotate (f,p)) /. ((k + (len f)) -' (p .. f)) = (GoB (Rotate (f,p))) * (i1,j1) & (Rotate (f,p)) /. (((k + (len f)) -' (p .. f)) + 1) = (GoB (Rotate (f,p))) * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i2,(j2 -' 1)) ) holds
( i1 = i2 & j1 = j2 + 1 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j2) )
proof
A12:
left_cell (
f,
k)
= left_cell (
f,
k)
;
let i1,
j1,
i2,
j2 be
Nat;
( [i1,j1] in Indices (GoB (Rotate (f,p))) & [i2,j2] in Indices (GoB (Rotate (f,p))) & (Rotate (f,p)) /. ((k + (len f)) -' (p .. f)) = (GoB (Rotate (f,p))) * (i1,j1) & (Rotate (f,p)) /. (((k + (len f)) -' (p .. f)) + 1) = (GoB (Rotate (f,p))) * (i2,j2) & not ( i1 = i2 & j1 + 1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),(i1 -' 1),j1) ) & not ( i1 + 1 = i2 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j1) ) & not ( i1 = i2 + 1 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i2,(j2 -' 1)) ) implies ( i1 = i2 & j1 = j2 + 1 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j2) ) )
assume that A13:
(
[i1,j1] in Indices (GoB (Rotate (f,p))) &
[i2,j2] in Indices (GoB (Rotate (f,p))) )
and A14:
(
(Rotate (f,p)) /. ((k + (len f)) -' (p .. f)) = (GoB (Rotate (f,p))) * (
i1,
j1) &
(Rotate (f,p)) /. (((k + (len f)) -' (p .. f)) + 1) = (GoB (Rotate (f,p))) * (
i2,
j2) )
;
( ( i1 = i2 & j1 + 1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),(i1 -' 1),j1) ) or ( i1 + 1 = i2 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j1) ) or ( i1 = i2 + 1 & j1 = j2 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i2,(j2 -' 1)) ) or ( i1 = i2 & j1 = j2 + 1 & left_cell (f,k) = cell ((GoB (Rotate (f,p))),i1,j2) ) )
A15:
GoB (Rotate (f,p)) = GoB f
by Th28;
then A16:
(
f /. k = (GoB f) * (
i1,
j1) &
f /. (k + 1) = (GoB f) * (
i2,
j2) )
by A1, A2, A3, A6, A8, A10, A14, Th18;
then A17:
( (
i1 = i2 &
j1 + 1
= j2 &
left_cell (
f,
k)
= cell (
(GoB f),
(i1 -' 1),
j1) ) or (
i1 + 1
= i2 &
j1 = j2 &
left_cell (
f,
k)
= cell (
(GoB f),
i1,
j1) ) or (
i1 = i2 + 1 &
j1 = j2 &
left_cell (
f,
k)
= cell (
(GoB f),
i2,
(j2 -' 1)) ) or (
i1 = i2 &
j1 = j2 + 1 &
left_cell (
f,
k)
= cell (
(GoB f),
i1,
j2) ) )
by A2, A5, A13, A15, A12, GOBOARD5:def 7;
end;
1 + (p .. f) <= k + (len f)
by A2, A4, XREAL_1:7;
then
1 <= (k + (len f)) -' (p .. f)
by NAT_D:55;
hence
left_cell (f,k) = left_cell ((Rotate (f,p)),((k + (len f)) -' (p .. f)))
by A9, A11, GOBOARD5:def 7; verum