:: MATRIX_2 semantic presentation
:: deftheorem Def1 defines --> MATRIX_2:def 1 :
theorem Th1: :: MATRIX_2:1
theorem Th2: :: MATRIX_2:2
definition
let c
1, c
2, c
3, c
4 be
set ;
func c
1,c
2 ][ c
3,c
4 -> tabular FinSequence equals :: MATRIX_2:def 2
<*<*a1,a2*>,<*a3,a4*>*>;
correctness
coherence
<*<*c1,c2*>,<*c3,c4*>*> is tabular FinSequence;
end;
:: deftheorem Def2 defines ][ MATRIX_2:def 2 :
theorem Th3: :: MATRIX_2:3
for b
1, b
2, b
3, b
4 being
set holds
(
len (b1,b2 ][ b3,b4) = 2 &
width (b1,b2 ][ b3,b4) = 2 &
Indices (b1,b2 ][ b3,b4) = [:(Seg 2),(Seg 2):] )
theorem Th4: :: MATRIX_2:4
for b
1, b
2, b
3, b
4 being
set holds
(
[1,1] in Indices (b1,b2 ][ b3,b4) &
[1,2] in Indices (b1,b2 ][ b3,b4) &
[2,1] in Indices (b1,b2 ][ b3,b4) &
[2,2] in Indices (b1,b2 ][ b3,b4) )
theorem Th5: :: MATRIX_2:5
theorem Th6: :: MATRIX_2:6
for b
1 being non
empty set for b
2, b
3, b
4, b
5 being
Element of b
1 holds
(
(b2,b3 ][ b4,b5) * 1,1
= b
2 &
(b2,b3 ][ b4,b5) * 1,2
= b
3 &
(b2,b3 ][ b4,b5) * 2,1
= b
4 &
(b2,b3 ][ b4,b5) * 2,2
= b
5 )
:: deftheorem Def3 defines Upper_Triangular_Matrix MATRIX_2:def 3 :
:: deftheorem Def4 defines Lower_Triangular_Matrix MATRIX_2:def 4 :
theorem Th7: :: MATRIX_2:7
theorem Th8: :: MATRIX_2:8
canceled;
theorem Th9: :: MATRIX_2:9
canceled;
theorem Th10: :: MATRIX_2:10
definition
let c
1 be
Nat;
let c
2 be
Field;
let c
3 be
Matrix of c
2;
canceled;assume E5:
c
1 in Seg (width c3)
;
func DelCol c
3,c
1 -> Matrix of a
2 means :: MATRIX_2:def 6
(
len a
4 = len a
3 & ( for b
1 being
Nat holds
( b
1 in dom a
3 implies a
4 . b
1 = Del (Line a3,b1),a
1 ) ) );
existence
ex b1 being Matrix of c2 st
( len b1 = len c3 & ( for b2 being Nat holds
( b2 in dom c3 implies b1 . b2 = Del (Line c3,b2),c1 ) ) )
uniqueness
for b1, b2 being Matrix of c2 holds
( len b1 = len c3 & ( for b3 being Nat holds
( b3 in dom c3 implies b1 . b3 = Del (Line c3,b3),c1 ) ) & len b2 = len c3 & ( for b3 being Nat holds
( b3 in dom c3 implies b2 . b3 = Del (Line c3,b3),c1 ) ) implies b1 = b2 )
end;
:: deftheorem Def5 MATRIX_2:def 5 :
canceled;
:: deftheorem Def6 defines DelCol MATRIX_2:def 6 :
theorem Th11: :: MATRIX_2:11
theorem Th12: :: MATRIX_2:12
theorem Th13: :: MATRIX_2:13
theorem Th14: :: MATRIX_2:14
theorem Th15: :: MATRIX_2:15
theorem Th16: :: MATRIX_2:16
theorem Th17: :: MATRIX_2:17
theorem Th18: :: MATRIX_2:18
definition
let c
1 be
Nat;
let c
2 be
Field;
let c
3 be
Matrix of c
2;
assume E12:
( c
1 in dom c
3 &
width c
3 > 0 )
;
func DelLine c
3,c
1 -> Matrix of a
2 means :: MATRIX_2:def 7
a
4 = {} if len a
3 = 1
otherwise (
width a
4 = width a
3 & ( for b
1 being
Nat holds
( b
1 in Seg (width a3) implies
Col a
4,b
1 = Del (Col a3,b1),a
1 ) ) );
existence
( not ( len c3 = 1 & ( for b1 being Matrix of c2 holds
not b1 = {} ) ) & not ( not len c3 = 1 & ( for b1 being Matrix of c2 holds
not ( width b1 = width c3 & ( for b2 being Nat holds
( b2 in Seg (width c3) implies Col b1,b2 = Del (Col c3,b2),c1 ) ) ) ) ) )
uniqueness
for b1, b2 being Matrix of c2 holds
( ( len c3 = 1 & b1 = {} & b2 = {} implies b1 = b2 ) & ( not len c3 = 1 & width b1 = width c3 & ( for b3 being Nat holds
( b3 in Seg (width c3) implies Col b1,b3 = Del (Col c3,b3),c1 ) ) & width b2 = width c3 & ( for b3 being Nat holds
( b3 in Seg (width c3) implies Col b2,b3 = Del (Col c3,b3),c1 ) ) implies b1 = b2 ) )
consistency
for b1 being Matrix of c2 holds
verum
;
end;
:: deftheorem Def7 defines DelLine MATRIX_2:def 7 :
:: deftheorem Def8 defines Deleting MATRIX_2:def 8 :
:: deftheorem Def9 defines permutational MATRIX_2:def 9 :
:: deftheorem Def10 defines len MATRIX_2:def 10 :
theorem Th19: :: MATRIX_2:19
:: deftheorem Def11 defines Permutations MATRIX_2:def 11 :
theorem Th20: :: MATRIX_2:20
theorem Th21: :: MATRIX_2:21
:: deftheorem Def12 defines len MATRIX_2:def 12 :
theorem Th22: :: MATRIX_2:22
:: deftheorem Def13 defines Group_of_Perm MATRIX_2:def 13 :
theorem Th23: :: MATRIX_2:23
theorem Th24: :: MATRIX_2:24
theorem Th25: :: MATRIX_2:25
theorem Th26: :: MATRIX_2:26
theorem Th27: :: MATRIX_2:27
canceled;
theorem Th28: :: MATRIX_2:28
:: deftheorem Def14 defines being_transposition MATRIX_2:def 14 :
:: deftheorem Def15 defines even MATRIX_2:def 15 :
theorem Th29: :: MATRIX_2:29
:: deftheorem Def16 defines - MATRIX_2:def 16 :
:: deftheorem Def17 defines FinOmega MATRIX_2:def 17 :
theorem Th30: :: MATRIX_2:30