:: FSM_1 semantic presentation
theorem Th1: :: FSM_1:1
for b
1, b
2 being
Nat holds
not ( b
1 < b
2 & ( for b
3 being
Nat holds
not ( b
2 = b
1 + b
3 & 1
<= b
3 ) ) )
theorem Th2: :: FSM_1:2
canceled;
theorem Th3: :: FSM_1:3
canceled;
theorem Th4: :: FSM_1:4
canceled;
theorem Th5: :: FSM_1:5
canceled;
theorem Th6: :: FSM_1:6
canceled;
theorem Th7: :: FSM_1:7
theorem Th8: :: FSM_1:8
theorem Th9: :: FSM_1:9
theorem Th10: :: FSM_1:10
theorem Th11: :: FSM_1:11
theorem Th12: :: FSM_1:12
theorem Th13: :: FSM_1:13
theorem Th14: :: FSM_1:14
theorem Th15: :: FSM_1:15
:: deftheorem Def1 defines -succ_of FSM_1:def 1 :
:: deftheorem Def2 defines -admissible FSM_1:def 2 :
theorem Th16: :: FSM_1:16
:: deftheorem Def3 defines -leads_to FSM_1:def 3 :
theorem Th17: :: FSM_1:17
:: deftheorem Def4 defines is_admissible_for FSM_1:def 4 :
theorem Th18: :: FSM_1:18
:: deftheorem Def5 defines leads_to_under FSM_1:def 5 :
theorem Th19: :: FSM_1:19
theorem Th20: :: FSM_1:20
theorem Th21: :: FSM_1:21
theorem Th22: :: FSM_1:22
theorem Th23: :: FSM_1:23
definition
let c
1 be
set ;
let c
2 be non
empty set ;
attr a
3 is
strict;
struct Mealy-FSM of c
1,c
2 -> FSM of a
1;
aggr Mealy-FSM(#
carrier,
Tran,
OFun,
InitS #)
-> Mealy-FSM of a
1,a
2;
sel OFun c
3 -> Function of
[:the carrier of a3,a1:],a
2;
attr a
3 is
strict;
struct Moore-FSM of c
1,c
2 -> FSM of a
1;
aggr Moore-FSM(#
carrier,
Tran,
OFun,
InitS #)
-> Moore-FSM of a
1,a
2;
sel OFun c
3 -> Function of the
carrier of a
3,a
2;
end;
registration
let c
1 be
set ;
let c
2 be non
empty set ;
let c
3 be non
empty finite set ;
let c
4 be
Function of
[:c3,c1:],c
3;
let c
5 be
Function of
[:c3,c1:],c
2;
let c
6 be
Element of c
3;
cluster Mealy-FSM(# a
3,a
4,a
5,a
6 #)
-> non
empty finite ;
coherence
( Mealy-FSM(# c3,c4,c5,c6 #) is finite & not Mealy-FSM(# c3,c4,c5,c6 #) is empty )
by GROUP_1:def 14, STRUCT_0:def 1;
end;
registration
let c
1 be
set ;
let c
2 be non
empty set ;
let c
3 be non
empty finite set ;
let c
4 be
Function of
[:c3,c1:],c
3;
let c
5 be
Function of c
3,c
2;
let c
6 be
Element of c
3;
cluster Moore-FSM(# a
3,a
4,a
5,a
6 #)
-> non
empty finite ;
coherence
( Moore-FSM(# c3,c4,c5,c6 #) is finite & not Moore-FSM(# c3,c4,c5,c6 #) is empty )
by GROUP_1:def 14, STRUCT_0:def 1;
end;
definition
let c
1, c
2 be non
empty set ;
let c
3 be non
empty Mealy-FSM of c
1,c
2;
let c
4 be
State of c
3;
let c
5 be
FinSequence of c
1;
func c
4,c
5 -response -> FinSequence of a
2 means :
Def6:
:: FSM_1:def 6
(
len a
6 = len a
5 & ( for b
1 being
Nat holds
( b
1 in dom a
5 implies a
6 . b
1 = the
OFun of a
3 . [((a4,a5 -admissible ) . b1),(a5 . b1)] ) ) );
existence
ex b1 being FinSequence of c2 st
( len b1 = len c5 & ( for b2 being Nat holds
( b2 in dom c5 implies b1 . b2 = the OFun of c3 . [((c4,c5 -admissible ) . b2),(c5 . b2)] ) ) )
uniqueness
for b1, b2 being FinSequence of c2 holds
( len b1 = len c5 & ( for b3 being Nat holds
( b3 in dom c5 implies b1 . b3 = the OFun of c3 . [((c4,c5 -admissible ) . b3),(c5 . b3)] ) ) & len b2 = len c5 & ( for b3 being Nat holds
( b3 in dom c5 implies b2 . b3 = the OFun of c3 . [((c4,c5 -admissible ) . b3),(c5 . b3)] ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines -response FSM_1:def 6 :
theorem Th24: :: FSM_1:24
:: deftheorem Def7 defines -response FSM_1:def 7 :
theorem Th25: :: FSM_1:25
theorem Th26: :: FSM_1:26
theorem Th27: :: FSM_1:27
for b
1, b
2 being non
empty set for b
3, b
4 being
FinSequence of b
1for b
5, b
6 being non
empty Mealy-FSM of b
1,b
2for b
7, b
8 being
State of b
5for b
9, b
10 being
State of b
6 holds
not ( b
7,b
3 -leads_to b
8 & b
9,b
3 -leads_to b
10 & b
8,b
4 -response <> b
10,b
4 -response & not b
7,
(b3 ^ b4) -response <> b
9,
(b3 ^ b4) -response )
:: deftheorem Def8 defines is_similar_to FSM_1:def 8 :
theorem Th28: :: FSM_1:28
theorem Th29: :: FSM_1:29
definition
let c
1, c
2 be non
empty set ;
let c
3, c
4 be non
empty Mealy-FSM of c
1,c
2;
pred c
3,c
4 -are_equivalent means :
Def9:
:: FSM_1:def 9
for b
1 being
FinSequence of a
1 holds the
InitS of a
3,b
1 -response = the
InitS of a
4,b
1 -response ;
reflexivity
for b1 being non empty Mealy-FSM of c1,c2
for b2 being FinSequence of c1 holds the InitS of b1,b2 -response = the InitS of b1,b2 -response
;
symmetry
for b1, b2 being non empty Mealy-FSM of c1,c2 holds
( ( for b3 being FinSequence of c1 holds the InitS of b1,b3 -response = the InitS of b2,b3 -response ) implies for b3 being FinSequence of c1 holds the InitS of b2,b3 -response = the InitS of b1,b3 -response )
;
end;
:: deftheorem Def9 defines -are_equivalent FSM_1:def 9 :
theorem Th30: :: FSM_1:30
definition
let c
1, c
2 be non
empty set ;
let c
3 be non
empty Mealy-FSM of c
1,c
2;
let c
4, c
5 be
State of c
3;
pred c
4,c
5 -are_equivalent means :
Def10:
:: FSM_1:def 10
for b
1 being
FinSequence of a
1 holds a
4,b
1 -response = a
5,b
1 -response ;
reflexivity
for b1 being State of c3
for b2 being FinSequence of c1 holds b1,b2 -response = b1,b2 -response
;
symmetry
for b1, b2 being State of c3 holds
( ( for b3 being FinSequence of c1 holds b1,b3 -response = b2,b3 -response ) implies for b3 being FinSequence of c1 holds b2,b3 -response = b1,b3 -response )
;
end;
:: deftheorem Def10 defines -are_equivalent FSM_1:def 10 :
theorem Th31: :: FSM_1:31
canceled;
theorem Th32: :: FSM_1:32
canceled;
theorem Th33: :: FSM_1:33
theorem Th34: :: FSM_1:34
theorem Th35: :: FSM_1:35
theorem Th36: :: FSM_1:36
theorem Th37: :: FSM_1:37
:: deftheorem Def11 defines -equivalent FSM_1:def 11 :
theorem Th38: :: FSM_1:38
theorem Th39: :: FSM_1:39
theorem Th40: :: FSM_1:40
theorem Th41: :: FSM_1:41
theorem Th42: :: FSM_1:42
theorem Th43: :: FSM_1:43
theorem Th44: :: FSM_1:44
definition
let c
1, c
2 be non
empty set ;
let c
3 be non
empty Mealy-FSM of c
1,c
2;
let c
4 be
Nat;
func c
4 -eq_states_EqR c
3 -> Equivalence_Relation of the
carrier of a
3 means :
Def12:
:: FSM_1:def 12
for b
1, b
2 being
State of a
3 holds
(
[b1,b2] in a
5 iff a
4 -equivalent b
1,b
2 );
existence
ex b1 being Equivalence_Relation of the carrier of c3 st
for b2, b3 being State of c3 holds
( [b2,b3] in b1 iff c4 -equivalent b2,b3 )
uniqueness
for b1, b2 being Equivalence_Relation of the carrier of c3 holds
( ( for b3, b4 being State of c3 holds
( [b3,b4] in b1 iff c4 -equivalent b3,b4 ) ) & ( for b3, b4 being State of c3 holds
( [b3,b4] in b2 iff c4 -equivalent b3,b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines -eq_states_EqR FSM_1:def 12 :
:: deftheorem Def13 defines -eq_states_partition FSM_1:def 13 :
theorem Th45: :: FSM_1:45
theorem Th46: :: FSM_1:46
theorem Th47: :: FSM_1:47
theorem Th48: :: FSM_1:48
theorem Th49: :: FSM_1:49
theorem Th50: :: FSM_1:50
theorem Th51: :: FSM_1:51
theorem Th52: :: FSM_1:52
:: deftheorem Def14 defines final FSM_1:def 14 :
theorem Th53: :: FSM_1:53
theorem Th54: :: FSM_1:54
theorem Th55: :: FSM_1:55
:: deftheorem Def15 defines final_states_partition FSM_1:def 15 :
theorem Th56: :: FSM_1:56
definition
let c
1, c
2 be non
empty set ;
let c
3 be non
empty finite Mealy-FSM of c
1,c
2;
let c
4 be
Element of
final_states_partition c
3;
let c
5 be
Element of c
1;
func c
5,c
4 -succ_class -> Element of
final_states_partition a
3 means :
Def16:
:: FSM_1:def 16
ex b
1 being
State of a
3ex b
2 being
Nat st
( b
1 in a
4 & b
2 + 1
= card the
carrier of a
3 & a
6 = Class (b2 -eq_states_EqR a3),
(the Tran of a3 . [b1,a5]) );
existence
ex b1 being Element of final_states_partition c3ex b2 being State of c3ex b3 being Nat st
( b2 in c4 & b3 + 1 = card the carrier of c3 & b1 = Class (b3 -eq_states_EqR c3),(the Tran of c3 . [b2,c5]) )
uniqueness
for b1, b2 being Element of final_states_partition c3 holds
( ex b3 being State of c3ex b4 being Nat st
( b3 in c4 & b4 + 1 = card the carrier of c3 & b1 = Class (b4 -eq_states_EqR c3),(the Tran of c3 . [b3,c5]) ) & ex b3 being State of c3ex b4 being Nat st
( b3 in c4 & b4 + 1 = card the carrier of c3 & b2 = Class (b4 -eq_states_EqR c3),(the Tran of c3 . [b3,c5]) ) implies b1 = b2 )
end;
:: deftheorem Def16 defines -succ_class FSM_1:def 16 :
:: deftheorem Def17 defines -class_response FSM_1:def 17 :
definition
let c
1, c
2 be non
empty set ;
let c
3 be non
empty finite Mealy-FSM of c
1,c
2;
func the_reduction_of c
3 -> strict Mealy-FSM of a
1,a
2 means :
Def18:
:: FSM_1:def 18
( the
carrier of a
4 = final_states_partition a
3 & ( for b
1 being
State of a
4for b
2 being
Element of a
1for b
3 being
State of a
3 holds
( b
3 in b
1 implies ( the
Tran of a
3 . b
3,b
2 in the
Tran of a
4 . b
1,b
2 & the
OFun of a
3 . b
3,b
2 = the
OFun of a
4 . b
1,b
2 ) ) ) & the
InitS of a
3 in the
InitS of a
4 );
existence
ex b1 being strict Mealy-FSM of c1,c2 st
( the carrier of b1 = final_states_partition c3 & ( for b2 being State of b1
for b3 being Element of c1
for b4 being State of c3 holds
( b4 in b2 implies ( the Tran of c3 . b4,b3 in the Tran of b1 . b2,b3 & the OFun of c3 . b4,b3 = the OFun of b1 . b2,b3 ) ) ) & the InitS of c3 in the InitS of b1 )
uniqueness
for b1, b2 being strict Mealy-FSM of c1,c2 holds
( the carrier of b1 = final_states_partition c3 & ( for b3 being State of b1
for b4 being Element of c1
for b5 being State of c3 holds
( b5 in b3 implies ( the Tran of c3 . b5,b4 in the Tran of b1 . b3,b4 & the OFun of c3 . b5,b4 = the OFun of b1 . b3,b4 ) ) ) & the InitS of c3 in the InitS of b1 & the carrier of b2 = final_states_partition c3 & ( for b3 being State of b2
for b4 being Element of c1
for b5 being State of c3 holds
( b5 in b3 implies ( the Tran of c3 . b5,b4 in the Tran of b2 . b3,b4 & the OFun of c3 . b5,b4 = the OFun of b2 . b3,b4 ) ) ) & the InitS of c3 in the InitS of b2 implies b1 = b2 )
end;
:: deftheorem Def18 defines the_reduction_of FSM_1:def 18 :
theorem Th57: :: FSM_1:57
theorem Th58: :: FSM_1:58
definition
let c
1, c
2 be non
empty set ;
let c
3, c
4 be non
empty Mealy-FSM of c
1,c
2;
pred c
3,c
4 -are_isomorphic means :
Def19:
:: FSM_1:def 19
ex b
1 being
Function of the
carrier of a
3,the
carrier of a
4 st
( b
1 is
bijective & b
1 . the
InitS of a
3 = the
InitS of a
4 & ( for b
2 being
State of a
3for b
3 being
Element of a
1 holds
( b
1 . (the Tran of a3 . b2,b3) = the
Tran of a
4 . (b1 . b2),b
3 & the
OFun of a
3 . b
2,b
3 = the
OFun of a
4 . (b1 . b2),b
3 ) ) );
reflexivity
for b1 being non empty Mealy-FSM of c1,c2 holds
ex b2 being Function of the carrier of b1,the carrier of b1 st
( b2 is bijective & b2 . the InitS of b1 = the InitS of b1 & ( for b3 being State of b1
for b4 being Element of c1 holds
( b2 . (the Tran of b1 . b3,b4) = the Tran of b1 . (b2 . b3),b4 & the OFun of b1 . b3,b4 = the OFun of b1 . (b2 . b3),b4 ) ) )
symmetry
for b1, b2 being non empty Mealy-FSM of c1,c2 holds
not ( ex b3 being Function of the carrier of b1,the carrier of b2 st
( b3 is bijective & b3 . the InitS of b1 = the InitS of b2 & ( for b4 being State of b1
for b5 being Element of c1 holds
( b3 . (the Tran of b1 . b4,b5) = the Tran of b2 . (b3 . b4),b5 & the OFun of b1 . b4,b5 = the OFun of b2 . (b3 . b4),b5 ) ) ) & ( for b3 being Function of the carrier of b2,the carrier of b1 holds
not ( b3 is bijective & b3 . the InitS of b2 = the InitS of b1 & ( for b4 being State of b2
for b5 being Element of c1 holds
( b3 . (the Tran of b2 . b4,b5) = the Tran of b1 . (b3 . b4),b5 & the OFun of b2 . b4,b5 = the OFun of b1 . (b3 . b4),b5 ) ) ) ) )
end;
:: deftheorem Def19 defines -are_isomorphic FSM_1:def 19 :
theorem Th59: :: FSM_1:59
theorem Th60: :: FSM_1:60
theorem Th61: :: FSM_1:61
for b
1, b
2 being non
empty set for b
3, b
4 being non
empty Mealy-FSM of b
2,b
1for b
5, b
6 being
State of b
3for b
7 being
Function of the
carrier of b
3,the
carrier of b
4 holds
( b
7 . the
InitS of b
3 = the
InitS of b
4 & ( for b
8 being
State of b
3for b
9 being
Element of b
2 holds
( b
7 . (the Tran of b3 . b8,b9) = the
Tran of b
4 . (b7 . b8),b
9 & the
OFun of b
3 . b
8,b
9 = the
OFun of b
4 . (b7 . b8),b
9 ) ) implies ( b
5,b
6 -are_equivalent iff b
7 . b
5,b
7 . b
6 -are_equivalent ) )
theorem Th62: :: FSM_1:62
:: deftheorem Def20 defines reduced FSM_1:def 20 :
theorem Th63: :: FSM_1:63
theorem Th64: :: FSM_1:64
theorem Th65: :: FSM_1:65
:: deftheorem Def21 defines accessible FSM_1:def 21 :
:: deftheorem Def22 defines connected FSM_1:def 22 :
theorem Th66: :: FSM_1:66
:: deftheorem Def23 defines accessibleStates FSM_1:def 23 :
theorem Th67: :: FSM_1:67
theorem Th68: :: FSM_1:68
theorem Th69: :: FSM_1:69
for b
1, b
2 being non
empty set for b
3 being non
empty finite Mealy-FSM of b
1,b
2for b
4 being
Function of
[:(accessibleStates b3),b1:],
accessibleStates b
3for b
5 being
Function of
[:(accessibleStates b3),b1:],b
2for b
6 being
Element of
accessibleStates b
3 holds
( b
4 = the
Tran of b
3 | [:(accessibleStates b3),b1:] & b
5 = the
OFun of b
3 | [:(accessibleStates b3),b1:] & b
6 = the
InitS of b
3 implies b
3,
Mealy-FSM(#
(accessibleStates b3),b
4,b
5,b
6 #)
-are_equivalent )
theorem Th70: :: FSM_1:70
:: deftheorem Def24 defines -Mealy_union FSM_1:def 24 :
theorem Th71: :: FSM_1:71
theorem Th72: :: FSM_1:72
theorem Th73: :: FSM_1:73
theorem Th74: :: FSM_1:74
theorem Th75: :: FSM_1:75
theorem Th76: :: FSM_1:76
theorem Th77: :: FSM_1:77
theorem Th78: :: FSM_1:78
theorem Th79: :: FSM_1:79
theorem Th80: :: FSM_1:80
theorem Th81: :: FSM_1:81
theorem Th82: :: FSM_1:82
theorem Th83: :: FSM_1:83
theorem Th84: :: FSM_1:84
theorem Th85: :: FSM_1:85