:: TOLER_1 semantic presentation

theorem Th1: :: TOLER_1:1
canceled;

theorem Th2: :: TOLER_1:2
{} is reflexive
proof end;

theorem Th3: :: TOLER_1:3
{} is symmetric
proof end;

theorem Th4: :: TOLER_1:4
{} is irreflexive
proof end;

theorem Th5: :: TOLER_1:5
{} is antisymmetric
proof end;

theorem Th6: :: TOLER_1:6
{} is asymmetric
proof end;

theorem Th7: :: TOLER_1:7
{} is connected
proof end;

theorem Th8: :: TOLER_1:8
{} is strongly_connected
proof end;

theorem Th9: :: TOLER_1:9
{} is transitive
proof end;

registration
cluster {} -> reflexive irreflexive symmetric antisymmetric asymmetric connected strongly_connected transitive ;
coherence
( {} is reflexive & {} is irreflexive & {} is symmetric & {} is antisymmetric & {} is asymmetric & {} is connected & {} is strongly_connected & {} is transitive )
by Th2, Th3, Th4, Th5, Th6, Th7, Th8, Th9;
end;

notation
let c1 be set ;
synonym Total c1 for nabla c1;
end;

definition
let c1 be Relation;
let c2 be set ;
redefine func |_2 as c1 |_2 c2 -> Relation of a2,a2;
coherence
c1 |_2 c2 is Relation of c2,c2
proof end;
end;

theorem Th10: :: TOLER_1:10
canceled;

theorem Th11: :: TOLER_1:11
canceled;

theorem Th12: :: TOLER_1:12
canceled;

theorem Th13: :: TOLER_1:13
for b1 being set holds rng (Total b1) = b1
proof end;

theorem Th14: :: TOLER_1:14
canceled;

theorem Th15: :: TOLER_1:15
for b1, b2, b3 being set holds
( b2 in b1 & b3 in b1 implies [b2,b3] in Total b1 )
proof end;

theorem Th16: :: TOLER_1:16
for b1, b2, b3 being set holds
( b2 in field (Total b1) & b3 in field (Total b1) implies [b2,b3] in Total b1 )
proof end;

theorem Th17: :: TOLER_1:17
canceled;

theorem Th18: :: TOLER_1:18
canceled;

theorem Th19: :: TOLER_1:19
for b1 being set holds Total b1 is strongly_connected
proof end;

theorem Th20: :: TOLER_1:20
canceled;

theorem Th21: :: TOLER_1:21
for b1 being set holds Total b1 is connected
proof end;

theorem Th22: :: TOLER_1:22
canceled;

theorem Th23: :: TOLER_1:23
canceled;

theorem Th24: :: TOLER_1:24
canceled;

theorem Th25: :: TOLER_1:25
for b1 being set
for b2 being Tolerance of b1 holds rng b2 = b1
proof end;

theorem Th26: :: TOLER_1:26
canceled;

theorem Th27: :: TOLER_1:27
for b1, b2 being set
for b3 being reflexive total Relation of b1 holds
( b2 in b1 iff [b2,b2] in b3 )
proof end;

theorem Th28: :: TOLER_1:28
for b1 being set
for b2 being Tolerance of b1 holds b2 is_reflexive_in b1
proof end;

theorem Th29: :: TOLER_1:29
for b1 being set
for b2 being Tolerance of b1 holds b2 is_symmetric_in b1
proof end;

theorem Th30: :: TOLER_1:30
canceled;

theorem Th31: :: TOLER_1:31
canceled;

theorem Th32: :: TOLER_1:32
for b1, b2, b3 being set
for b4 being Relation of b1,b2 holds
( b4 is symmetric implies b4 |_2 b3 is symmetric )
proof end;

definition
let c1 be set ;
let c2 be Tolerance of c1;
let c3 be Subset of c1;
redefine func |_2 as c2 |_2 c3 -> Tolerance of a3;
coherence
c2 |_2 c3 is Tolerance of c3
proof end;
end;

theorem Th33: :: TOLER_1:33
for b1, b2 being set
for b3 being Tolerance of b2 holds
( b1 c= b2 implies b3 |_2 b1 is Tolerance of b1 )
proof end;

definition
let c1 be set ;
let c2 be Tolerance of c1;
canceled;
canceled;
mode TolSet of c2 -> set means :Def3: :: TOLER_1:def 3
for b1, b2 being set holds
( b1 in a3 & b2 in a3 implies [b1,b2] in a2 );
existence
ex b1 being set st
for b2, b3 being set holds
( b2 in b1 & b3 in b1 implies [b2,b3] in c2 )
proof end;
end;

:: deftheorem Def1 TOLER_1:def 1 :
canceled;

:: deftheorem Def2 TOLER_1:def 2 :
canceled;

:: deftheorem Def3 defines TolSet TOLER_1:def 3 :
for b1 being set
for b2 being Tolerance of b1
for b3 being set holds
( b3 is TolSet of b2 iff for b4, b5 being set holds
( b4 in b3 & b5 in b3 implies [b4,b5] in b2 ) );

theorem Th34: :: TOLER_1:34
for b1 being set
for b2 being Tolerance of b1 holds
{} is TolSet of b2
proof end;

definition
let c1 be set ;
let c2 be Tolerance of c1;
let c3 be TolSet of c2;
attr a3 is TolClass-like means :Def4: :: TOLER_1:def 4
for b1 being set holds
not ( not b1 in a3 & b1 in a1 & ( for b2 being set holds
not ( b2 in a3 & not [b1,b2] in a2 ) ) );
end;

:: deftheorem Def4 defines TolClass-like TOLER_1:def 4 :
for b1 being set
for b2 being Tolerance of b1
for b3 being TolSet of b2 holds
( b3 is TolClass-like iff for b4 being set holds
not ( not b4 in b3 & b4 in b1 & ( for b5 being set holds
not ( b5 in b3 & not [b4,b5] in b2 ) ) ) );

registration
let c1 be set ;
let c2 be Tolerance of c1;
cluster TolClass-like TolSet of a2;
existence
ex b1 being TolSet of c2 st b1 is TolClass-like
proof end;
end;

definition
let c1 be set ;
let c2 be Tolerance of c1;
mode TolClass is TolClass-like TolSet of a2;
end;

theorem Th35: :: TOLER_1:35
canceled;

theorem Th36: :: TOLER_1:36
canceled;

theorem Th37: :: TOLER_1:37
canceled;

theorem Th38: :: TOLER_1:38
for b1 being set
for b2 being Tolerance of b1 holds
( {} is TolClass of b2 implies b2 = {} )
proof end;

theorem Th39: :: TOLER_1:39
{} is Tolerance of {} by PARTFUN1:def 4, RELAT_1:60, RELSET_1:25;

theorem Th40: :: TOLER_1:40
for b1 being set
for b2 being Tolerance of b1
for b3, b4 being set holds
( [b3,b4] in b2 implies {b3,b4} is TolSet of b2 )
proof end;

theorem Th41: :: TOLER_1:41
for b1 being set
for b2 being Tolerance of b1
for b3 being set holds
( b3 in b1 implies {b3} is TolSet of b2 )
proof end;

theorem Th42: :: TOLER_1:42
for b1 being set
for b2 being Tolerance of b1
for b3, b4 being set holds
( b3 is TolSet of b2 & b4 is TolSet of b2 implies b3 /\ b4 is TolSet of b2 )
proof end;

theorem Th43: :: TOLER_1:43
for b1, b2 being set
for b3 being Tolerance of b2 holds
( b1 is TolSet of b3 implies b1 c= b2 )
proof end;

theorem Th44: :: TOLER_1:44
canceled;

theorem Th45: :: TOLER_1:45
for b1 being set
for b2 being Tolerance of b1
for b3 being TolSet of b2 holds
ex b4 being TolClass of b2 st b3 c= b4
proof end;

theorem Th46: :: TOLER_1:46
for b1 being set
for b2 being Tolerance of b1
for b3, b4 being set holds
not ( [b3,b4] in b2 & ( for b5 being TolClass of b2 holds
not ( b3 in b5 & b4 in b5 ) ) )
proof end;

theorem Th47: :: TOLER_1:47
for b1 being set
for b2 being Tolerance of b1
for b3 being set holds
not ( b3 in b1 & ( for b4 being TolClass of b2 holds
not b3 in b4 ) )
proof end;

theorem Th48: :: TOLER_1:48
canceled;

theorem Th49: :: TOLER_1:49
for b1 being set
for b2 being Tolerance of b1 holds b2 c= Total b1
proof end;

theorem Th50: :: TOLER_1:50
for b1 being set
for b2 being Tolerance of b1 holds id b1 c= b2
proof end;

scheme :: TOLER_1:sch 1
s1{ F1() -> set , P1[ set , set ] } :
ex b1 being Tolerance of F1() st
for b2, b3 being set holds
( b2 in F1() & b3 in F1() implies ( [b2,b3] in b1 iff P1[b2,b3] ) )
provided
E20: for b1 being set holds
( b1 in F1() implies P1[b1,b1] ) and E21: for b1, b2 being set holds
( b1 in F1() & b2 in F1() & P1[b1,b2] implies P1[b2,b1] )
proof end;

theorem Th51: :: TOLER_1:51
for b1 being set holds
ex b2 being Tolerance of union b1 st
for b3 being set holds
( b3 in b1 implies b3 is TolSet of b2 )
proof end;

theorem Th52: :: TOLER_1:52
for b1 being set
for b2, b3 being Tolerance of union b1 holds
( ( for b4, b5 being set holds
( [b4,b5] in b2 iff ex b6 being set st
( b6 in b1 & b4 in b6 & b5 in b6 ) ) ) & ( for b4, b5 being set holds
( [b4,b5] in b3 iff ex b6 being set st
( b6 in b1 & b4 in b6 & b5 in b6 ) ) ) implies b2 = b3 )
proof end;

theorem Th53: :: TOLER_1:53
for b1 being set
for b2, b3 being Tolerance of b1 holds
( ( for b4 being set holds
( b4 is TolClass of b2 iff b4 is TolClass of b3 ) ) implies b2 = b3 )
proof end;

notation
let c1, c2 be set ;
let c3 be Relation of c1,c2;
let c4 be set ;
synonym neighbourhood c4,c3 for Class c1,c2;
end;

theorem Th54: :: TOLER_1:54
for b1, b2 being set
for b3 being Tolerance of b1
for b4 being set holds
( b4 in neighbourhood , iff [b2,b4] in b3 )
proof end;

theorem Th55: :: TOLER_1:55
canceled;

theorem Th56: :: TOLER_1:56
canceled;

theorem Th57: :: TOLER_1:57
canceled;

theorem Th58: :: TOLER_1:58
for b1, b2 being set
for b3 being Tolerance of b1
for b4 being set holds
( ( for b5 being set holds
( b5 in b4 iff ( b2 in b5 & b5 is TolClass of b3 ) ) ) implies neighbourhood , = union b4 )
proof end;

theorem Th59: :: TOLER_1:59
for b1, b2 being set
for b3 being Tolerance of b1
for b4 being set holds
( ( for b5 being set holds
( b5 in b4 iff ( b2 in b5 & b5 is TolSet of b3 ) ) ) implies neighbourhood , = union b4 )
proof end;

definition
let c1 be set ;
let c2 be Tolerance of c1;
canceled;
func TolSets c2 -> set means :Def6: :: TOLER_1:def 6
for b1 being set holds
( b1 in a3 iff b1 is TolSet of a2 );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff b2 is TolSet of c2 )
proof end;
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff b3 is TolSet of c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is TolSet of c2 ) ) implies b1 = b2 )
proof end;
func TolClasses c2 -> set means :Def7: :: TOLER_1:def 7
for b1 being set holds
( b1 in a3 iff b1 is TolClass of a2 );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff b2 is TolClass of c2 )
proof end;
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff b3 is TolClass of c2 ) ) & ( for b3 being set holds
( b3 in b2 iff b3 is TolClass of c2 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def5 TOLER_1:def 5 :
canceled;

:: deftheorem Def6 defines TolSets TOLER_1:def 6 :
for b1 being set
for b2 being Tolerance of b1
for b3 being set holds
( b3 = TolSets b2 iff for b4 being set holds
( b4 in b3 iff b4 is TolSet of b2 ) );

:: deftheorem Def7 defines TolClasses TOLER_1:def 7 :
for b1 being set
for b2 being Tolerance of b1
for b3 being set holds
( b3 = TolClasses b2 iff for b4 being set holds
( b4 in b3 iff b4 is TolClass of b2 ) );

theorem Th60: :: TOLER_1:60
canceled;

theorem Th61: :: TOLER_1:61
canceled;

theorem Th62: :: TOLER_1:62
canceled;

theorem Th63: :: TOLER_1:63
canceled;

theorem Th64: :: TOLER_1:64
for b1 being set
for b2, b3 being Tolerance of b1 holds
( TolClasses b2 c= TolClasses b3 implies b2 c= b3 )
proof end;

theorem Th65: :: TOLER_1:65
for b1 being set
for b2, b3 being Tolerance of b1 holds
( TolClasses b2 = TolClasses b3 implies b2 = b3 )
proof end;

theorem Th66: :: TOLER_1:66
for b1 being set
for b2 being Tolerance of b1 holds union (TolClasses b2) = b1
proof end;

theorem Th67: :: TOLER_1:67
for b1 being set
for b2 being Tolerance of b1 holds union (TolSets b2) = b1
proof end;

theorem Th68: :: TOLER_1:68
for b1 being set
for b2 being Tolerance of b1 holds
( ( for b3 being set holds
( b3 in b1 implies neighbourhood , is TolSet of b2 ) ) implies b2 is transitive )
proof end;

theorem Th69: :: TOLER_1:69
for b1 being set
for b2 being Tolerance of b1 holds
( b2 is transitive implies for b3 being set holds
( b3 in b1 implies neighbourhood , is TolClass of b2 ) )
proof end;

theorem Th70: :: TOLER_1:70
for b1 being set
for b2 being Tolerance of b1
for b3 being set
for b4 being TolClass of b2 holds
( b3 in b4 implies b4 c= neighbourhood , )
proof end;

theorem Th71: :: TOLER_1:71
for b1 being set
for b2, b3 being Tolerance of b1 holds
( TolSets b2 c= TolSets b3 iff b2 c= b3 )
proof end;

theorem Th72: :: TOLER_1:72
for b1 being set
for b2 being Tolerance of b1 holds TolClasses b2 c= TolSets b2
proof end;

theorem Th73: :: TOLER_1:73
for b1 being set
for b2, b3 being Tolerance of b1 holds
( ( for b4 being set holds
( b4 in b1 implies neighbourhood , c= neighbourhood , ) ) implies b2 c= b3 )
proof end;

theorem Th74: :: TOLER_1:74
for b1 being set
for b2 being Tolerance of b1 holds b2 c= b2 * b2
proof end;