:: GOBRD10 semantic presentation
:: deftheorem Def1 defines are_adjacent1 GOBRD10:def 1 :
theorem Th1: :: GOBRD10:1
theorem Th2: :: GOBRD10:2
:: deftheorem Def2 defines are_adjacent2 GOBRD10:def 2 :
theorem Th3: :: GOBRD10:3
theorem Th4: :: GOBRD10:4
:: deftheorem Def3 defines |-> GOBRD10:def 3 :
theorem Th5: :: GOBRD10:5
canceled;
theorem Th6: :: GOBRD10:6
for
b1,
b2,
b3 being
Nat st
b2 <= b1 &
b3 <= b1 holds
ex
b4 being
FinSequence of
NAT st
(
b4 . 1
= b2 &
b4 . (len b4) = b3 &
len b4 = ((b2 -' b3) + (b3 -' b2)) + 1 & ( for
b5,
b6 being
Nat st 1
<= b5 &
b5 <= len b4 &
b6 = b4 . b5 holds
b6 <= b1 ) & ( for
b5 being
Nat st 1
<= b5 &
b5 < len b4 & not
b4 . (b5 + 1) = (b4 /. b5) + 1 holds
b4 . b5 = (b4 /. (b5 + 1)) + 1 ) )
theorem Th7: :: GOBRD10:7
theorem Th8: :: GOBRD10:8
for
b1,
b2,
b3,
b4,
b5,
b6 being
Nat st
b3 <= b1 &
b4 <= b2 &
b5 <= b1 &
b6 <= b2 holds
ex
b7,
b8 being
FinSequence of
NAT st
( ( for
b9,
b10,
b11 being
Nat st
b9 in dom b7 &
b10 = b7 . b9 &
b11 = b8 . b9 holds
(
b10 <= b1 &
b11 <= b2 ) ) &
b7 . 1
= b3 &
b7 . (len b7) = b5 &
b8 . 1
= b4 &
b8 . (len b8) = b6 &
len b7 = len b8 &
len b7 = ((((b3 -' b5) + (b5 -' b3)) + (b4 -' b6)) + (b6 -' b4)) + 1 & ( for
b9 being
Nat st 1
<= b9 &
b9 < len b7 holds
b7 /. b9,
b8 /. b9,
b7 /. (b9 + 1),
b8 /. (b9 + 1) are_adjacent2 ) )
theorem Th9: :: GOBRD10:9
for
b1,
b2 being
Natfor
b3 being
set for
b4 being
Subset of
b3for
b5 being
Matrix of
b1,
b2,
bool b3 st ex
b6,
b7 being
Nat st
(
b6 in Seg b1 &
b7 in Seg b2 &
b5 * b6,
b7 c= b4 ) & ( for
b6,
b7,
b8,
b9 being
Nat st
b6 in Seg b1 &
b8 in Seg b1 &
b7 in Seg b2 &
b9 in Seg b2 &
b6,
b7,
b8,
b9 are_adjacent2 holds
(
b5 * b6,
b7 c= b4 iff
b5 * b8,
b9 c= b4 ) ) holds
for
b6,
b7 being
Nat st
b6 in Seg b1 &
b7 in Seg b2 holds
b5 * b6,
b7 c= b4