:: GOBOARD1 semantic presentation
theorem Th1: :: GOBOARD1:1
for b
1, b
2 being
Real holds
(
abs (b1 - b2) = 1 iff ( ( b
1 > b
2 & b
1 = b
2 + 1 ) or ( b
1 < b
2 & b
2 = b
1 + 1 ) ) )
theorem Th2: :: GOBOARD1:2
for b
1, b
2, b
3, b
4 being
Nat holds
(
(abs (b1 - b2)) + (abs (b3 - b4)) = 1 iff ( (
abs (b1 - b2) = 1 & b
3 = b
4 ) or (
abs (b3 - b4) = 1 & b
1 = b
2 ) ) )
theorem Th3: :: GOBOARD1:3
for b
1 being
Nat holds
( not ( b
1 > 1 & ( for b
2 being
Nat holds
not ( b
1 = b
2 + 1 & b
2 > 0 ) ) ) & not ( ex b
2 being
Nat st
( b
1 = b
2 + 1 & b
2 > 0 ) & not b
1 > 1 ) )
theorem Th4: :: GOBOARD1:4
canceled;
theorem Th5: :: GOBOARD1:5
canceled;
theorem Th6: :: GOBOARD1:6
canceled;
theorem Th7: :: GOBOARD1:7
theorem Th8: :: GOBOARD1:8
canceled;
theorem Th9: :: GOBOARD1:9
canceled;
:: deftheorem Def1 defines increasing GOBOARD1:def 1 :
:: deftheorem Def2 defines constant GOBOARD1:def 2 :
:: deftheorem Def3 defines X_axis GOBOARD1:def 3 :
:: deftheorem Def4 defines Y_axis GOBOARD1:def 4 :
theorem Th10: :: GOBOARD1:10
canceled;
theorem Th11: :: GOBOARD1:11
canceled;
theorem Th12: :: GOBOARD1:12
canceled;
theorem Th13: :: GOBOARD1:13
canceled;
theorem Th14: :: GOBOARD1:14
theorem Th15: :: GOBOARD1:15
for b
1 being
FinSequence of
REAL for b
2 being
Nat holds
( b
1 <> {} &
rng b
1 c= Seg b
2 & b
1 . 1
= 1 & b
1 . (len b1) = b
2 & ( for b
3 being
Nat holds
( 1
<= b
3 & b
3 <= (len b1) - 1 implies for b
4, b
5 being
Real holds
not ( b
4 = b
1 . b
3 & b
5 = b
1 . (b3 + 1) & not
abs (b4 - b5) = 1 & not b
4 = b
5 ) ) ) implies ( ( for b
3 being
Nat holds
not ( b
3 in Seg b
2 & ( for b
4 being
Nat holds
not ( b
4 in dom b
1 & b
1 . b
4 = b
3 ) ) ) ) & ( for b
3, b
4, b
5 being
Natfor b
6 being
Real holds
not ( b
3 in dom b
1 & b
1 . b
3 = b
5 & ( for b
7 being
Nat holds
( b
7 in dom b
1 & b
1 . b
7 = b
5 implies b
7 <= b
3 ) ) & b
3 < b
4 & b
4 in dom b
1 & b
6 = b
1 . b
4 & not b
5 < b
6 ) ) ) )
theorem Th16: :: GOBOARD1:16
theorem Th17: :: GOBOARD1:17
:: deftheorem Def5 defines empty-yielding GOBOARD1:def 5 :
:: deftheorem Def6 defines X_equal-in-line GOBOARD1:def 6 :
:: deftheorem Def7 defines Y_equal-in-column GOBOARD1:def 7 :
:: deftheorem Def8 defines Y_increasing-in-line GOBOARD1:def 8 :
:: deftheorem Def9 defines X_increasing-in-column GOBOARD1:def 9 :
Lemma17:
for b1 being non empty set
for b2 being Matrix of b1 holds
( not b2 is empty-yielding iff ( 0 < len b2 & 0 < width b2 ) )
theorem Th18: :: GOBOARD1:18
canceled;
theorem Th19: :: GOBOARD1:19
theorem Th20: :: GOBOARD1:20
theorem Th21: :: GOBOARD1:21
theorem Th22: :: GOBOARD1:22
theorem Th23: :: GOBOARD1:23
theorem Th24: :: GOBOARD1:24
definition
let c
1 be
Go-board;
let c
2 be
Nat;
assume E21:
( c
2 in Seg (width c1) &
width c
1 > 1 )
;
func DelCol c
1,c
2 -> Go-board means :
Def10:
:: GOBOARD1:def 10
(
len a
3 = len a
1 & ( for b
1 being
Nat holds
( b
1 in dom a
1 implies a
3 . b
1 = Del (Line a1,b1),a
2 ) ) );
existence
ex b1 being Go-board st
( len b1 = len c1 & ( for b2 being Nat holds
( b2 in dom c1 implies b1 . b2 = Del (Line c1,b2),c2 ) ) )
uniqueness
for b1, b2 being Go-board holds
( len b1 = len c1 & ( for b3 being Nat holds
( b3 in dom c1 implies b1 . b3 = Del (Line c1,b3),c2 ) ) & len b2 = len c1 & ( for b3 being Nat holds
( b3 in dom c1 implies b2 . b3 = Del (Line c1,b3),c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines DelCol GOBOARD1:def 10 :
theorem Th25: :: GOBOARD1:25
theorem Th26: :: GOBOARD1:26
theorem Th27: :: GOBOARD1:27
theorem Th28: :: GOBOARD1:28
theorem Th29: :: GOBOARD1:29
theorem Th30: :: GOBOARD1:30
theorem Th31: :: GOBOARD1:31
theorem Th32: :: GOBOARD1:32
theorem Th33: :: GOBOARD1:33
theorem Th34: :: GOBOARD1:34
theorem Th35: :: GOBOARD1:35
theorem Th36: :: GOBOARD1:36
theorem Th37: :: GOBOARD1:37
definition
let c
1 be
set ;
let c
2 be
FinSequence of c
1;
let c
3 be
Matrix of c
1;
pred c
2 is_sequence_on c
3 means :
Def11:
:: GOBOARD1:def 11
( ( for b
1 being
Nat holds
not ( b
1 in dom a
2 & ( for b
2, b
3 being
Nat holds
not (
[b2,b3] in Indices a
3 & a
2 /. b
1 = a
3 * b
2,b
3 ) ) ) ) & ( for b
1 being
Nat holds
( b
1 in dom a
2 & b
1 + 1
in dom a
2 implies for b
2, b
3, b
4, b
5 being
Nat holds
(
[b2,b3] in Indices a
3 &
[b4,b5] in Indices a
3 & a
2 /. b
1 = a
3 * b
2,b
3 & a
2 /. (b1 + 1) = a
3 * b
4,b
5 implies
(abs (b2 - b4)) + (abs (b3 - b5)) = 1 ) ) ) );
end;
:: deftheorem Def11 defines is_sequence_on GOBOARD1:def 11 :
for b
1 being
set for b
2 being
FinSequence of b
1for b
3 being
Matrix of b
1 holds
( b
2 is_sequence_on b
3 iff ( ( for b
4 being
Nat holds
not ( b
4 in dom b
2 & ( for b
5, b
6 being
Nat holds
not (
[b5,b6] in Indices b
3 & b
2 /. b
4 = b
3 * b
5,b
6 ) ) ) ) & ( for b
4 being
Nat holds
( b
4 in dom b
2 & b
4 + 1
in dom b
2 implies for b
5, b
6, b
7, b
8 being
Nat holds
(
[b5,b6] in Indices b
3 &
[b7,b8] in Indices b
3 & b
2 /. b
4 = b
3 * b
5,b
6 & b
2 /. (b4 + 1) = b
3 * b
7,b
8 implies
(abs (b5 - b7)) + (abs (b6 - b8)) = 1 ) ) ) ) );
Lemma31:
for b1 being set
for b2 being Matrix of b1 holds <*> b1 is_sequence_on b2
theorem Th38: :: GOBOARD1:38
theorem Th39: :: GOBOARD1:39
for b
1, b
2 being
FinSequence of
(TOP-REAL 2)for b
3 being
set for b
4 being
Matrix of b
3 holds
( ( for b
5 being
Nat holds
not ( b
5 in dom b
1 & ( for b
6, b
7 being
Nat holds
not (
[b6,b7] in Indices b
4 & b
1 /. b
5 = b
4 * b
6,b
7 ) ) ) ) & ( for b
5 being
Nat holds
not ( b
5 in dom b
2 & ( for b
6, b
7 being
Nat holds
not (
[b6,b7] in Indices b
4 & b
2 /. b
5 = b
4 * b
6,b
7 ) ) ) ) implies for b
5 being
Nat holds
not ( b
5 in dom (b1 ^ b2) & ( for b
6, b
7 being
Nat holds
not (
[b6,b7] in Indices b
4 &
(b1 ^ b2) /. b
5 = b
4 * b
6,b
7 ) ) ) )
theorem Th40: :: GOBOARD1:40
for b
1, b
2 being
FinSequence of
(TOP-REAL 2)for b
3 being
set for b
4 being
Matrix of b
3 holds
( ( for b
5 being
Nat holds
( b
5 in dom b
1 & b
5 + 1
in dom b
1 implies for b
6, b
7, b
8, b
9 being
Nat holds
(
[b6,b7] in Indices b
4 &
[b8,b9] in Indices b
4 & b
1 /. b
5 = b
4 * b
6,b
7 & b
1 /. (b5 + 1) = b
4 * b
8,b
9 implies
(abs (b6 - b8)) + (abs (b7 - b9)) = 1 ) ) ) & ( for b
5 being
Nat holds
( b
5 in dom b
2 & b
5 + 1
in dom b
2 implies for b
6, b
7, b
8, b
9 being
Nat holds
(
[b6,b7] in Indices b
4 &
[b8,b9] in Indices b
4 & b
2 /. b
5 = b
4 * b
6,b
7 & b
2 /. (b5 + 1) = b
4 * b
8,b
9 implies
(abs (b6 - b8)) + (abs (b7 - b9)) = 1 ) ) ) & ( for b
5, b
6, b
7, b
8 being
Nat holds
(
[b5,b6] in Indices b
4 &
[b7,b8] in Indices b
4 & b
1 /. (len b1) = b
4 * b
5,b
6 & b
2 /. 1
= b
4 * b
7,b
8 &
len b
1 in dom b
1 & 1
in dom b
2 implies
(abs (b5 - b7)) + (abs (b6 - b8)) = 1 ) ) implies for b
5 being
Nat holds
( b
5 in dom (b1 ^ b2) & b
5 + 1
in dom (b1 ^ b2) implies for b
6, b
7, b
8, b
9 being
Nat holds
(
[b6,b7] in Indices b
4 &
[b8,b9] in Indices b
4 &
(b1 ^ b2) /. b
5 = b
4 * b
6,b
7 &
(b1 ^ b2) /. (b5 + 1) = b
4 * b
8,b
9 implies
(abs (b6 - b8)) + (abs (b7 - b9)) = 1 ) ) )
theorem Th41: :: GOBOARD1:41
theorem Th42: :: GOBOARD1:42
theorem Th43: :: GOBOARD1:43
theorem Th44: :: GOBOARD1:44
theorem Th45: :: GOBOARD1:45
for b
1 being
Go-boardfor b
2 being
FinSequence of
(TOP-REAL 2) holds
( 1
<= len b
2 & b
2 /. 1
in rng (Line b1,1) & b
2 /. (len b2) in rng (Line b1,(len b1)) & b
2 is_sequence_on b
1 implies ( ( for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 <= len b
1 & ( for b
4 being
Nat holds
not ( b
4 in dom b
2 & b
2 /. b
4 in rng (Line b1,b3) ) ) ) ) & ( for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 <= len b
1 & 2
<= len b
2 & not
L~ b
2 meets rng (Line b1,b3) ) ) & ( for b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
<= b
3 & b
3 <= len b
1 & 1
<= b
4 & b
4 <= len b
1 & b
5 in dom b
2 & b
6 in dom b
2 & b
2 /. b
5 in rng (Line b1,b3) & ( for b
7 being
Nat holds
( b
7 in dom b
2 & b
2 /. b
7 in rng (Line b1,b3) implies b
7 <= b
5 ) ) & b
5 < b
6 & b
2 /. b
6 in rng (Line b1,b4) & not b
3 < b
4 ) ) ) )
theorem Th46: :: GOBOARD1:46
theorem Th47: :: GOBOARD1:47
theorem Th48: :: GOBOARD1:48
theorem Th49: :: GOBOARD1:49
for b
1 being
Go-boardfor b
2 being
FinSequence of
(TOP-REAL 2) holds
( 1
<= len b
2 & b
2 /. 1
in rng (Col b1,1) & b
2 /. (len b2) in rng (Col b1,(width b1)) & b
2 is_sequence_on b
1 implies ( ( for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 <= width b
1 & ( for b
4 being
Nat holds
not ( b
4 in dom b
2 & b
2 /. b
4 in rng (Col b1,b3) ) ) ) ) & ( for b
3 being
Nat holds
not ( 1
<= b
3 & b
3 <= width b
1 & 2
<= len b
2 & not
L~ b
2 meets rng (Col b1,b3) ) ) & ( for b
3, b
4, b
5, b
6 being
Nat holds
not ( 1
<= b
3 & b
3 <= width b
1 & 1
<= b
4 & b
4 <= width b
1 & b
5 in dom b
2 & b
6 in dom b
2 & b
2 /. b
5 in rng (Col b1,b3) & ( for b
7 being
Nat holds
( b
7 in dom b
2 & b
2 /. b
7 in rng (Col b1,b3) implies b
7 <= b
5 ) ) & b
5 < b
6 & b
2 /. b
6 in rng (Col b1,b4) & not b
3 < b
4 ) ) ) )
theorem Th50: :: GOBOARD1:50
theorem Th51: :: GOBOARD1:51
theorem Th52: :: GOBOARD1:52
theorem Th53: :: GOBOARD1:53