:: 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 :
for b
1, b
2 being
Natfor b
3 being
FinSequence of
NAT holds
( b
3 = b
1 |-> b
2 iff (
len b
3 = b
1 & ( for b
4 being
Nat holds
( 1
<= b
4 & b
4 <= b
1 implies b
3 . b
4 = b
2 ) ) ) );
theorem Th5: :: GOBRD10:5
canceled;
theorem Th6: :: GOBRD10:6
for b
1, b
2, b
3 being
Nat holds
not ( b
2 <= b
1 & b
3 <= b
1 & ( for b
4 being
FinSequence of
NAT holds
not ( b
4 . 1
= b
2 & b
4 . (len b4) = b
3 &
len b
4 = ((b2 -' b3) + (b3 -' b2)) + 1 & ( for b
5, b
6 being
Nat holds
( 1
<= b
5 & b
5 <= len b
4 & b
6 = b
4 . b
5 implies b
6 <= b
1 ) ) & ( for b
5 being
Nat holds
not ( 1
<= b
5 & b
5 < len b
4 & not b
4 . (b5 + 1) = (b4 /. b5) + 1 & not b
4 . b
5 = (b4 /. (b5 + 1)) + 1 ) ) ) ) )
theorem Th7: :: GOBRD10:7
theorem Th8: :: GOBRD10:8
for b
1, b
2, b
3, b
4, b
5, b
6 being
Nat holds
not ( b
3 <= b
1 & b
4 <= b
2 & b
5 <= b
1 & b
6 <= b
2 & ( for b
7, b
8 being
FinSequence of
NAT holds
not ( ( for b
9, b
10, b
11 being
Nat holds
( b
9 in dom b
7 & b
10 = b
7 . b
9 & b
11 = b
8 . b
9 implies ( b
10 <= b
1 & b
11 <= b
2 ) ) ) & b
7 . 1
= b
3 & b
7 . (len b7) = b
5 & b
8 . 1
= b
4 & b
8 . (len b8) = b
6 &
len b
7 = len b
8 &
len b
7 = ((((b3 -' b5) + (b5 -' b3)) + (b4 -' b6)) + (b6 -' b4)) + 1 & ( for b
9 being
Nat holds
( 1
<= b
9 & b
9 < len b
7 implies b
7 /. b
9,b
8 /. b
9,b
7 /. (b9 + 1),b
8 /. (b9 + 1) are_adjacent2 ) ) ) ) )
theorem Th9: :: GOBRD10:9
for b
1, b
2 being
Natfor b
3 being
set for b
4 being
Subset of b
3for b
5 being
Matrix of b
1,b
2,
bool b
3 holds
( ex b
6, b
7 being
Nat st
( b
6 in Seg b
1 & b
7 in Seg b
2 & b
5 * b
6,b
7 c= b
4 ) & ( for b
6, b
7, b
8, b
9 being
Nat holds
( b
6 in Seg b
1 & b
8 in Seg b
1 & b
7 in Seg b
2 & b
9 in Seg b
2 & b
6,b
7,b
8,b
9 are_adjacent2 implies ( b
5 * b
6,b
7 c= b
4 iff b
5 * b
8,b
9 c= b
4 ) ) ) implies for b
6, b
7 being
Nat holds
( b
6 in Seg b
1 & b
7 in Seg b
2 implies b
5 * b
6,b
7 c= b
4 ) )