:: GOBOARD2 semantic presentation
defpred S1[ Nat] means for b1 being finite Subset of REAL holds
( b1 <> {} & card b1 = a1 implies ( b1 is bounded_above & upper_bound b1 in b1 & b1 is bounded_below & lower_bound b1 in b1 ) );
Lemma1:
S1[0]
by CARD_2:59;
Lemma2:
for b1 being Nat holds
( S1[b1] implies S1[b1 + 1] )
Lemma3:
for b1 being Nat holds
S1[b1]
from NAT_1:sch 1(Lemma1, Lemma2);
theorem Th1: :: GOBOARD2:1
theorem Th2: :: GOBOARD2:2
canceled;
theorem Th3: :: GOBOARD2:3
theorem Th4: :: GOBOARD2:4
theorem Th5: :: GOBOARD2:5
theorem Th6: :: GOBOARD2:6
theorem Th7: :: GOBOARD2:7
theorem Th8: :: GOBOARD2:8
theorem Th9: :: GOBOARD2:9
theorem Th10: :: GOBOARD2:10
theorem Th11: :: GOBOARD2:11
theorem Th12: :: GOBOARD2:12
theorem Th13: :: GOBOARD2:13
theorem Th14: :: GOBOARD2:14
theorem Th15: :: GOBOARD2:15
theorem Th16: :: GOBOARD2:16
theorem Th17: :: GOBOARD2:17
theorem Th18: :: GOBOARD2:18
theorem Th19: :: GOBOARD2:19
theorem Th20: :: GOBOARD2:20
defpred S2[ Nat] means for b1 being FinSequence of REAL holds
not ( card (rng b1) = a1 & ( for b2 being FinSequence of REAL holds
not ( rng b2 = rng b1 & len b2 = card (rng b1) & b2 is increasing ) ) );
Lemma15:
S2[0]
Lemma16:
for b1 being Nat holds
( S2[b1] implies S2[b1 + 1] )
Lemma17:
for b1 being Nat holds
S2[b1]
from NAT_1:sch 1(Lemma15, Lemma16);
theorem Th21: :: GOBOARD2:21
defpred S3[ Nat] means for b1, b2 being FinSequence of REAL holds
( len b1 = a1 & len b2 = a1 & rng b1 = rng b2 & b1 is increasing & b2 is increasing implies b1 = b2 );
Lemma18:
S3[0]
Lemma19:
for b1 being Nat holds
( S3[b1] implies S3[b1 + 1] )
Lemma20:
for b1 being Nat holds
S3[b1]
from NAT_1:sch 1(Lemma18, Lemma19);
theorem Th22: :: GOBOARD2:22
definition
let c
1, c
2 be
increasing FinSequence of
REAL ;
assume E21:
( c
1 <> {} & c
2 <> {} )
;
func GoB c
1,c
2 -> Matrix of
(TOP-REAL 2) means :
Def1:
:: GOBOARD2:def 1
(
len a
3 = len a
1 &
width a
3 = len a
2 & ( for b
1, b
2 being
Nat holds
(
[b1,b2] in Indices a
3 implies a
3 * b
1,b
2 = |[(a1 . b1),(a2 . b2)]| ) ) );
existence
ex b1 being Matrix of (TOP-REAL 2) st
( len b1 = len c1 & width b1 = len c2 & ( for b2, b3 being Nat holds
( [b2,b3] in Indices b1 implies b1 * b2,b3 = |[(c1 . b2),(c2 . b3)]| ) ) )
uniqueness
for b1, b2 being Matrix of (TOP-REAL 2) holds
( len b1 = len c1 & width b1 = len c2 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices b1 implies b1 * b3,b4 = |[(c1 . b3),(c2 . b4)]| ) ) & len b2 = len c1 & width b2 = len c2 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices b2 implies b2 * b3,b4 = |[(c1 . b3),(c2 . b4)]| ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines GoB GOBOARD2:def 1 :
:: deftheorem Def2 defines Incr GOBOARD2:def 2 :
:: deftheorem Def3 defines GoB GOBOARD2:def 3 :
theorem Th23: :: GOBOARD2:23
theorem Th24: :: GOBOARD2:24
theorem Th25: :: GOBOARD2:25
theorem Th26: :: GOBOARD2:26
theorem Th27: :: GOBOARD2:27
theorem Th28: :: GOBOARD2:28
theorem Th29: :: GOBOARD2:29