Article FSM_1, MML version 4.99.1005
:: FSM_1:th 1
theorem
for b1, b2 being natural set
st b1 < b2
holds ex b3 being Element of NAT st
b2 = b1 + b3 & 1 <= b3;
:: FSM_1:th 7
theorem
for b1 being set
for b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b2,b3
st b4 is bijective(b1, b2) & b5 is bijective(b2, b3)
holds b5 * b4 is bijective(b1, b3);
:: FSM_1:th 8
theorem
for b1 being set
for b2, b3 being symmetric transitive total Relation of b1,b1
st Class b2 = Class b3
holds b2 = b3;
:: FSM_1:th 9
theorem
for b1 being non empty set
for b2 being a_partition of b1 holds
b2 is not empty;
:: FSM_1:th 10
theorem
for b1 being finite set
for b2 being a_partition of b1 holds
b2 is finite;
:: FSM_1:condreg 1
registration
let a1 be finite set;
cluster -> finite (a_partition of a1);
end;
:: FSM_1:exreg 1
registration
let a1 be non empty finite set;
cluster non empty with_non-empty_elements finite a_partition of a1;
end;
:: FSM_1:th 11
theorem
for b1 being non empty set
for b2 being a_partition of b1
for b3 being set
st b3 in b2
holds ex b4 being Element of b1 st
b4 in b3;
:: FSM_1:th 12
theorem
for b1 being non empty finite set
for b2 being a_partition of b1 holds
card b2 <= card b1;
:: FSM_1:th 13
theorem
for b1 being non empty finite set
for b2, b3 being a_partition of b1
st b2 is_finer_than b3
holds card b3 <= card b2;
:: FSM_1:th 14
theorem
for b1 being non empty finite set
for b2, b3 being a_partition of b1
st b2 is_finer_than b3
for b4 being Element of b3 holds
ex b5 being Element of b2 st
b5 c= b4;
:: FSM_1:th 15
theorem
for b1 being non empty finite set
for b2, b3 being a_partition of b1
st b2 is_finer_than b3 & card b2 = card b3
holds b2 = b3;
:: FSM_1:structnot 1 => FSM_1:struct 1
definition
let a1 be set;
struct(1-sorted) FSM(#
carrier -> set,
Tran -> Function-like quasi_total Relation of [:the carrier of it,A1:],the carrier of it,
InitS -> Element of the carrier of it
#);
end;
:: FSM_1:attrnot 1 => FSM_1:attr 1
definition
let a1 be set;
let a2 be FSM over a1;
attr a2 is strict;
end;
:: FSM_1:exreg 2
registration
let a1 be set;
cluster strict FSM over a1;
end;
:: FSM_1:aggrnot 1 => FSM_1:aggr 1
definition
let a1, a2 be set;
let a3 be Function-like quasi_total Relation of [:a2,a1:],a2;
let a4 be Element of a2;
aggr FSM(#a2,a3,a4#) -> strict FSM over a1;
end;
:: FSM_1:selnot 1 => FSM_1:sel 1
definition
let a1 be set;
let a2 be FSM over a1;
sel the Tran of a2 -> Function-like quasi_total Relation of [:the carrier of a2,a1:],the carrier of a2;
end;
:: FSM_1:selnot 2 => FSM_1:sel 2
definition
let a1 be set;
let a2 be FSM over a1;
sel the InitS of a2 -> Element of the carrier of a2;
end;
:: FSM_1:modenot 1
definition
let a1 be set;
let a2 be FSM over a1;
mode State of a2 is Element of the carrier of a2;
end;
:: FSM_1:exreg 3
registration
let a1 be set;
cluster non empty finite FSM over a1;
end;
:: FSM_1:funcnot 1 => FSM_1:func 1
definition
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be Element of a1;
let a4 be Element of the carrier of a2;
func A3 -succ_of A4 -> Element of the carrier of a2 equals
(the Tran of a2) . [a4,a3];
end;
:: FSM_1:def 1
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 holds
b3 -succ_of b4 = (the Tran of b2) . [b4,b3];
:: FSM_1:funcnot 2 => FSM_1:func 2
definition
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be Element of the carrier of a2;
let a4 be FinSequence of a1;
func (A3,A4)-admissible -> FinSequence of the carrier of a2 means
it . 1 = a3 &
len it = (len a4) + 1 &
(for b1 being natural set
st 1 <= b1 & b1 <= len a4
holds ex b2 being Element of a1 st
ex b3, b4 being Element of the carrier of a2 st
b2 = a4 . b1 & b3 = it . b1 & b4 = it . (b1 + 1) & b2 -succ_of b3 = b4);
end;
:: FSM_1:def 2
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of the carrier of b2
for b4 being FinSequence of b1
for b5 being FinSequence of the carrier of b2 holds
b5 = (b3,b4)-admissible
iff
b5 . 1 = b3 &
len b5 = (len b4) + 1 &
(for b6 being natural set
st 1 <= b6 & b6 <= len b4
holds ex b7 being Element of b1 st
ex b8, b9 being Element of the carrier of b2 st
b7 = b4 . b6 & b8 = b5 . b6 & b9 = b5 . (b6 + 1) & b7 -succ_of b8 = b9);
:: FSM_1:th 16
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of the carrier of b2 holds
(b3,<*> b1)-admissible = <*b3*>;
:: FSM_1:prednot 1 => FSM_1:pred 1
definition
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be FinSequence of a1;
let a4, a5 be Element of the carrier of a2;
pred A4,A3 -leads_to A5 means
(a4,a3)-admissible . ((len a3) + 1) = a5;
end;
:: FSM_1:dfs 3
definiens
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be FinSequence of a1;
let a4, a5 be Element of the carrier of a2;
To prove
a4,a3 -leads_to a5
it is sufficient to prove
thus (a4,a3)-admissible . ((len a3) + 1) = a5;
:: FSM_1:def 3
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being FinSequence of b1
for b4, b5 being Element of the carrier of b2 holds
b4,b3 -leads_to b5
iff
(b4,b3)-admissible . ((len b3) + 1) = b5;
:: FSM_1:th 17
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of the carrier of b2 holds
b3,<*> b1 -leads_to b3;
:: FSM_1:prednot 2 => FSM_1:pred 2
definition
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be FinSequence of a1;
let a4 be FinSequence of the carrier of a2;
pred A4 is_admissible_for A3 means
ex b1 being Element of the carrier of a2 st
b1 = a4 . 1 & (b1,a3)-admissible = a4;
end;
:: FSM_1:dfs 4
definiens
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be FinSequence of a1;
let a4 be FinSequence of the carrier of a2;
To prove
a4 is_admissible_for a3
it is sufficient to prove
thus ex b1 being Element of the carrier of a2 st
b1 = a4 . 1 & (b1,a3)-admissible = a4;
:: FSM_1:def 4
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being FinSequence of b1
for b4 being FinSequence of the carrier of b2 holds
b4 is_admissible_for b3
iff
ex b5 being Element of the carrier of b2 st
b5 = b4 . 1 & (b5,b3)-admissible = b4;
:: FSM_1:th 18
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of the carrier of b2 holds
<*b3*> is_admissible_for <*> b1;
:: FSM_1:funcnot 3 => FSM_1:func 3
definition
let a1 be non empty set;
let a2 be non empty FSM over a1;
let a3 be Element of the carrier of a2;
let a4 be FinSequence of a1;
func A3 leads_to_under A4 -> Element of the carrier of a2 means
a3,a4 -leads_to it;
end;
:: FSM_1:def 5
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being Element of the carrier of b2
for b4 being FinSequence of b1
for b5 being Element of the carrier of b2 holds
b5 = b3 leads_to_under b4
iff
b3,b4 -leads_to b5;
:: FSM_1:th 19
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3 being FinSequence of b1
for b4, b5 being Element of the carrier of b2 holds
(b4,b3)-admissible . len ((b4,b3)-admissible) = b5
iff
b4,b3 -leads_to b5;
:: FSM_1:th 20
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3, b4 being FinSequence of b1
for b5 being Element of the carrier of b2
for b6 being Element of NAT
st 1 <= b6 & b6 <= len b3
holds (b5,b3 ^ b4)-admissible . b6 = (b5,b3)-admissible . b6;
:: FSM_1:th 21
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3, b4 being FinSequence of b1
for b5, b6 being Element of the carrier of b2
st b5,b3 -leads_to b6
holds (b5,b3 ^ b4)-admissible . ((len b3) + 1) = b6;
:: FSM_1:th 22
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3, b4 being FinSequence of b1
for b5, b6 being Element of the carrier of b2
st b5,b3 -leads_to b6
for b7 being Element of NAT
st 1 <= b7 & b7 <= (len b4) + 1
holds (b5,b3 ^ b4)-admissible . ((len b3) + b7) = (b6,b4)-admissible . b7;
:: FSM_1:th 23
theorem
for b1 being non empty set
for b2 being non empty FSM over b1
for b3, b4 being FinSequence of b1
for b5, b6 being Element of the carrier of b2
st b5,b3 -leads_to b6
holds (b5,b3 ^ b4)-admissible = (Del((b5,b3)-admissible,(len b3) + 1)) ^ ((b6,b4)-admissible);
:: FSM_1:structnot 2 => FSM_1:struct 2
definition
let a1 be set;
let a2 be non empty set;
struct(FSM over a1) Mealy-FSM(#
carrier -> set,
Tran -> Function-like quasi_total Relation of [:the carrier of it,A1:],the carrier of it,
OFun -> Function-like quasi_total Relation of [:the carrier of it,A1:],A2,
InitS -> Element of the carrier of it
#);
end;
:: FSM_1:attrnot 2 => FSM_1:attr 2
definition
let a1 be set;
let a2 be non empty set;
let a3 be Mealy-FSM over a1,a2;
attr a3 is strict;
end;
:: FSM_1:exreg 4
registration
let a1 be set;
let a2 be non empty set;
cluster strict Mealy-FSM over a1,a2;
end;
:: FSM_1:aggrnot 2 => FSM_1:aggr 2
definition
let a1 be set;
let a2 be non empty set;
let a3 be set;
let a4 be Function-like quasi_total Relation of [:a3,a1:],a3;
let a5 be Function-like quasi_total Relation of [:a3,a1:],a2;
let a6 be Element of a3;
aggr Mealy-FSM(#a3,a4,a5,a6#) -> strict Mealy-FSM over a1,a2;
end;
:: FSM_1:selnot 3 => FSM_1:sel 3
definition
let a1 be set;
let a2 be non empty set;
let a3 be Mealy-FSM over a1,a2;
sel the OFun of a3 -> Function-like quasi_total Relation of [:the carrier of a3,a1:],a2;
end;
:: FSM_1:structnot 3 => FSM_1:struct 3
definition
let a1 be set;
let a2 be non empty set;
struct(FSM over a1) Moore-FSM(#
carrier -> set,
Tran -> Function-like quasi_total Relation of [:the carrier of it,A1:],the carrier of it,
OFun -> Function-like quasi_total Relation of the carrier of it,A2,
InitS -> Element of the carrier of it
#);
end;
:: FSM_1:attrnot 3 => FSM_1:attr 3
definition
let a1 be set;
let a2 be non empty set;
let a3 be Moore-FSM over a1,a2;
attr a3 is strict;
end;
:: FSM_1:exreg 5
registration
let a1 be set;
let a2 be non empty set;
cluster strict Moore-FSM over a1,a2;
end;
:: FSM_1:aggrnot 3 => FSM_1:aggr 3
definition
let a1 be set;
let a2 be non empty set;
let a3 be set;
let a4 be Function-like quasi_total Relation of [:a3,a1:],a3;
let a5 be Function-like quasi_total Relation of a3,a2;
let a6 be Element of a3;
aggr Moore-FSM(#a3,a4,a5,a6#) -> strict Moore-FSM over a1,a2;
end;
:: FSM_1:selnot 4 => FSM_1:sel 4
definition
let a1 be set;
let a2 be non empty set;
let a3 be Moore-FSM over a1,a2;
sel the OFun of a3 -> Function-like quasi_total Relation of the carrier of a3,a2;
end;
:: FSM_1:funcreg 1
registration
let a1 be set;
let a2 be non empty finite set;
let a3 be Function-like quasi_total Relation of [:a2,a1:],a2;
let a4 be Element of a2;
cluster FSM(#a2,a3,a4#) -> non empty finite strict;
end;
:: FSM_1:funcreg 2
registration
let a1 be set;
let a2 be non empty set;
let a3 be non empty finite set;
let a4 be Function-like quasi_total Relation of [:a3,a1:],a3;
let a5 be Function-like quasi_total Relation of [:a3,a1:],a2;
let a6 be Element of a3;
cluster Mealy-FSM(#a3,a4,a5,a6#) -> non empty finite strict;
end;
:: FSM_1:funcreg 3
registration
let a1 be set;
let a2 be non empty set;
let a3 be non empty finite set;
let a4 be Function-like quasi_total Relation of [:a3,a1:],a3;
let a5 be Function-like quasi_total Relation of a3,a2;
let a6 be Element of a3;
cluster Moore-FSM(#a3,a4,a5,a6#) -> non empty finite strict;
end;
:: FSM_1:exreg 6
registration
let a1 be set;
let a2 be non empty set;
cluster non empty finite Mealy-FSM over a1,a2;
end;
:: FSM_1:exreg 7
registration
let a1 be set;
let a2 be non empty set;
cluster non empty finite Moore-FSM over a1,a2;
end;
:: FSM_1:funcnot 4 => FSM_1:func 4
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be Element of the carrier of a3;
let a5 be FinSequence of a1;
func (A4,A5)-response -> FinSequence of a2 means
len it = len a5 &
(for b1 being Element of NAT
st b1 in dom a5
holds it . b1 = (the OFun of a3) . [(a4,a5)-admissible . b1,a5 . b1]);
end;
:: FSM_1:def 6
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being Element of the carrier of b3
for b5 being FinSequence of b1
for b6 being FinSequence of b2 holds
b6 = (b4,b5)-response
iff
len b6 = len b5 &
(for b7 being Element of NAT
st b7 in dom b5
holds b6 . b7 = (the OFun of b3) . [(b4,b5)-admissible . b7,b5 . b7]);
:: FSM_1:th 24
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being Element of the carrier of b3 holds
(b4,<*> b1)-response = <*> b2;
:: FSM_1:funcnot 5 => FSM_1:func 5
definition
let a1, a2 be non empty set;
let a3 be non empty Moore-FSM over a1,a2;
let a4 be Element of the carrier of a3;
let a5 be FinSequence of a1;
func (A4,A5)-response -> FinSequence of a2 means
len it = (len a5) + 1 &
(for b1 being Element of NAT
st b1 in Seg ((len a5) + 1)
holds it . b1 = (the OFun of a3) . ((a4,a5)-admissible . b1));
end;
:: FSM_1:def 7
theorem
for b1, b2 being non empty set
for b3 being non empty Moore-FSM over b1,b2
for b4 being Element of the carrier of b3
for b5 being FinSequence of b1
for b6 being FinSequence of b2 holds
b6 = (b4,b5)-response
iff
len b6 = (len b5) + 1 &
(for b7 being Element of NAT
st b7 in Seg ((len b5) + 1)
holds b6 . b7 = (the OFun of b3) . ((b4,b5)-admissible . b7));
:: FSM_1:th 25
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4 being non empty Moore-FSM over b1,b2
for b5 being Element of the carrier of b4 holds
(b5,b3)-response . 1 = (the OFun of b4) . b5;
:: FSM_1:th 26
theorem
for b1, b2 being non empty set
for b3, b4 being FinSequence of b1
for b5 being non empty Mealy-FSM over b1,b2
for b6, b7 being Element of the carrier of b5
st b6,b3 -leads_to b7
holds (b6,b3 ^ b4)-response = (b6,b3)-response ^ ((b7,b4)-response);
:: FSM_1:th 27
theorem
for b1, b2 being non empty set
for b3, b4 being FinSequence of b1
for b5, b6 being non empty Mealy-FSM over b1,b2
for b7, b8 being Element of the carrier of b5
for b9, b10 being Element of the carrier of b6
st b7,b3 -leads_to b8 & b9,b3 -leads_to b10 & (b8,b4)-response <> (b10,b4)-response
holds (b7,b3 ^ b4)-response <> (b9,b3 ^ b4)-response;
:: FSM_1:prednot 3 => FSM_1:pred 3
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be non empty Moore-FSM over a1,a2;
pred A3 is_similar_to A4 means
for b1 being FinSequence of a1 holds
<*(the OFun of a4) . the InitS of a4*> ^ ((the InitS of a3,b1)-response) = (the InitS of a4,b1)-response;
end;
:: FSM_1:dfs 8
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be non empty Moore-FSM over a1,a2;
To prove
a3 is_similar_to a4
it is sufficient to prove
thus for b1 being FinSequence of a1 holds
<*(the OFun of a4) . the InitS of a4*> ^ ((the InitS of a3,b1)-response) = (the InitS of a4,b1)-response;
:: FSM_1:def 8
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being non empty Moore-FSM over b1,b2 holds
b3 is_similar_to b4
iff
for b5 being FinSequence of b1 holds
<*(the OFun of b4) . the InitS of b4*> ^ ((the InitS of b3,b5)-response) = (the InitS of b4,b5)-response;
:: FSM_1:th 28
theorem
for b1, b2 being non empty set
for b3 being non empty finite Moore-FSM over b1,b2 holds
ex b4 being non empty finite Mealy-FSM over b1,b2 st
b4 is_similar_to b3;
:: FSM_1:th 29
theorem
for b1 being non empty set
for b2 being non empty finite set
for b3 being non empty finite Mealy-FSM over b1,b2 holds
ex b4 being non empty finite Moore-FSM over b1,b2 st
b3 is_similar_to b4;
:: FSM_1:prednot 4 => FSM_1:pred 4
definition
let a1, a2 be non empty set;
let a3, a4 be non empty Mealy-FSM over a1,a2;
pred A3,A4 -are_equivalent means
for b1 being FinSequence of a1 holds
(the InitS of a3,b1)-response = (the InitS of a4,b1)-response;
symmetry;
:: for a1, a2 being non empty set
:: for a3, a4 being non empty Mealy-FSM over a1,a2
:: st a3,a4 -are_equivalent
:: holds a4,a3 -are_equivalent;
reflexivity;
:: for a1, a2 being non empty set
:: for a3 being non empty Mealy-FSM over a1,a2 holds
:: a3,a3 -are_equivalent;
end;
:: FSM_1:dfs 9
definiens
let a1, a2 be non empty set;
let a3, a4 be non empty Mealy-FSM over a1,a2;
To prove
a3,a4 -are_equivalent
it is sufficient to prove
thus for b1 being FinSequence of a1 holds
(the InitS of a3,b1)-response = (the InitS of a4,b1)-response;
:: FSM_1:def 9
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Mealy-FSM over b1,b2 holds
b3,b4 -are_equivalent
iff
for b5 being FinSequence of b1 holds
(the InitS of b3,b5)-response = (the InitS of b4,b5)-response;
:: FSM_1:th 30
theorem
for b1, b2 being non empty set
for b3, b4, b5 being non empty Mealy-FSM over b1,b2
st b3,b4 -are_equivalent & b4,b5 -are_equivalent
holds b3,b5 -are_equivalent;
:: FSM_1:prednot 5 => FSM_1:pred 5
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4, a5 be Element of the carrier of a3;
pred A4,A5 -are_equivalent means
for b1 being FinSequence of a1 holds
(a4,b1)-response = (a5,b1)-response;
symmetry;
:: for a1, a2 being non empty set
:: for a3 being non empty Mealy-FSM over a1,a2
:: for a4, a5 being Element of the carrier of a3
:: st a4,a5 -are_equivalent
:: holds a5,a4 -are_equivalent;
reflexivity;
:: for a1, a2 being non empty set
:: for a3 being non empty Mealy-FSM over a1,a2
:: for a4 being Element of the carrier of a3 holds
:: a4,a4 -are_equivalent;
end;
:: FSM_1:dfs 10
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4, a5 be Element of the carrier of a3;
To prove
a4,a5 -are_equivalent
it is sufficient to prove
thus for b1 being FinSequence of a1 holds
(a4,b1)-response = (a5,b1)-response;
:: FSM_1:def 10
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5 being Element of the carrier of b3 holds
b4,b5 -are_equivalent
iff
for b6 being FinSequence of b1 holds
(b4,b6)-response = (b5,b6)-response;
:: FSM_1:th 33
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5, b6 being Element of the carrier of b3
st b4,b5 -are_equivalent & b5,b6 -are_equivalent
holds b4,b6 -are_equivalent;
:: FSM_1:th 34
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being FinSequence of b1
for b5 being non empty Mealy-FSM over b1,b2
for b6, b7 being Element of the carrier of b5
st b6 = (the Tran of b5) . [b7,b3]
for b8 being Element of NAT
st b8 in Seg ((len b4) + 1)
holds (b7,<*b3*> ^ b4)-admissible . (b8 + 1) = (b6,b4)-admissible . b8;
:: FSM_1:th 35
theorem
for b1, b2 being non empty set
for b3 being Element of b1
for b4 being FinSequence of b1
for b5 being non empty Mealy-FSM over b1,b2
for b6, b7 being Element of the carrier of b5
st b6 = (the Tran of b5) . [b7,b3]
holds (b7,<*b3*> ^ b4)-response = <*(the OFun of b5) . [b7,b3]*> ^ ((b6,b4)-response);
:: FSM_1:th 36
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b2,b1
for b4, b5 being Element of the carrier of b3 holds
b4,b5 -are_equivalent
iff
for b6 being Element of b2 holds
(the OFun of b3) . [b4,b6] = (the OFun of b3) . [b5,b6] &
(the Tran of b3) . [b4,b6],(the Tran of b3) . [b5,b6] -are_equivalent;
:: FSM_1:th 37
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b2,b1
for b4, b5 being Element of the carrier of b3
st b4,b5 -are_equivalent
for b6 being FinSequence of b2
for b7 being Element of NAT
st b7 in dom b6
holds ex b8, b9 being Element of the carrier of b3 st
b8 = (b4,b6)-admissible . b7 & b9 = (b5,b6)-admissible . b7 & b8,b9 -are_equivalent;
:: FSM_1:prednot 6 => FSM_1:pred 6
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4, a5 be Element of the carrier of a3;
let a6 be natural set;
pred A6 -equivalent A4,A5 means
for b1 being FinSequence of a1
st len b1 <= a6
holds (a4,b1)-response = (a5,b1)-response;
end;
:: FSM_1:dfs 11
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4, a5 be Element of the carrier of a3;
let a6 be natural set;
To prove
a6 -equivalent a4,a5
it is sufficient to prove
thus for b1 being FinSequence of a1
st len b1 <= a6
holds (a4,b1)-response = (a5,b1)-response;
:: FSM_1:def 11
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5 being Element of the carrier of b3
for b6 being natural set holds
b6 -equivalent b4,b5
iff
for b7 being FinSequence of b1
st len b7 <= b6
holds (b4,b7)-response = (b5,b7)-response;
:: FSM_1:th 38
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being Element of the carrier of b3
for b5 being natural set holds
b5 -equivalent b4,b4;
:: FSM_1:th 39
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5 being Element of the carrier of b3
for b6 being natural set
st b6 -equivalent b4,b5
holds b6 -equivalent b5,b4;
:: FSM_1:th 40
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5, b6 being Element of the carrier of b3
for b7 being natural set
st b7 -equivalent b4,b5 & b7 -equivalent b5,b6
holds b7 -equivalent b4,b6;
:: FSM_1:th 41
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3
for b5, b6 being Element of the carrier of b4
st b5,b6 -are_equivalent
holds b1 -equivalent b5,b6;
:: FSM_1:th 42
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5 being Element of the carrier of b3 holds
0 -equivalent b4,b5;
:: FSM_1:th 43
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4, b5 being Element of the carrier of b3
for b6, b7 being natural set
st b6 + b7 -equivalent b4,b5
holds b6 -equivalent b4,b5;
:: FSM_1:th 44
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b2,b1
for b4, b5 being Element of the carrier of b3
for b6 being natural set
st 1 <= b6
holds b6 -equivalent b4,b5
iff
1 -equivalent b4,b5 &
(for b7 being Element of b2
for b8 being Element of NAT
st b8 = b6 - 1
holds b8 -equivalent (the Tran of b3) . [b4,b7],(the Tran of b3) . [b5,b7]);
:: FSM_1:funcnot 6 => FSM_1:func 6
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be natural set;
func A4 -eq_states_EqR A3 -> symmetric transitive total Relation of the carrier of a3,the carrier of a3 means
for b1, b2 being Element of the carrier of a3 holds
[b1,b2] in it
iff
a4 -equivalent b1,b2;
end;
:: FSM_1:def 12
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being natural set
for b5 being symmetric transitive total Relation of the carrier of b3,the carrier of b3 holds
b5 = b4 -eq_states_EqR b3
iff
for b6, b7 being Element of the carrier of b3 holds
[b6,b7] in b5
iff
b4 -equivalent b6,b7;
:: FSM_1:funcnot 7 => FSM_1:func 7
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be natural set;
func A4 -eq_states_partition A3 -> non empty a_partition of the carrier of a3 equals
Class (a4 -eq_states_EqR a3);
end;
:: FSM_1:def 13
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being natural set holds
b4 -eq_states_partition b3 = Class (b4 -eq_states_EqR b3);
:: FSM_1:th 45
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3 holds
(b1 + 1) -eq_states_partition b4 is_finer_than b1 -eq_states_partition b4;
:: FSM_1:th 46
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being natural set
st Class (b4 -eq_states_EqR b3) = Class ((b4 + 1) -eq_states_EqR b3)
for b5 being natural set holds
Class ((b4 + b5) -eq_states_EqR b3) = Class (b4 -eq_states_EqR b3);
:: FSM_1:th 47
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3
st b1 -eq_states_partition b4 = (b1 + 1) -eq_states_partition b4
for b5 being Element of NAT holds
(b1 + b5) -eq_states_partition b4 = b1 -eq_states_partition b4;
:: FSM_1:th 48
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3
st (b1 + 1) -eq_states_partition b4 <> b1 -eq_states_partition b4
for b5 being Element of NAT
st b5 <= b1
holds (b5 + 1) -eq_states_partition b4 <> b5 -eq_states_partition b4;
:: FSM_1:th 49
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty finite Mealy-FSM over b2,b3
st b1 -eq_states_partition b4 <> (b1 + 1) -eq_states_partition b4
holds card (b1 -eq_states_partition b4) < card ((b1 + 1) -eq_states_partition b4);
:: FSM_1:th 50
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being Element of the carrier of b3 holds
Class(0 -eq_states_EqR b3,b4) = the carrier of b3;
:: FSM_1:th 51
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2 holds
0 -eq_states_partition b3 = {the carrier of b3};
:: FSM_1:th 52
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty finite Mealy-FSM over b2,b3
st b1 + 1 = card the carrier of b4
holds (b1 + 1) -eq_states_partition b4 = b1 -eq_states_partition b4;
:: FSM_1:attrnot 4 => FSM_1:attr 4
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be a_partition of the carrier of a3;
attr a4 is final means
for b1, b2 being Element of the carrier of a3 holds
b1,b2 -are_equivalent
iff
ex b3 being Element of a4 st
b1 in b3 & b2 in b3;
end;
:: FSM_1:dfs 14
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be a_partition of the carrier of a3;
To prove
a4 is final
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a3 holds
b1,b2 -are_equivalent
iff
ex b3 being Element of a4 st
b1 in b3 & b2 in b3;
:: FSM_1:def 14
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being a_partition of the carrier of b3 holds
b4 is final(b1, b2, b3)
iff
for b5, b6 being Element of the carrier of b3 holds
b5,b6 -are_equivalent
iff
ex b7 being Element of b4 st
b5 in b7 & b6 in b7;
:: FSM_1:th 53
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3
st b1 -eq_states_partition b4 is final(b2, b3, b4)
holds (b1 + 1) -eq_states_EqR b4 = b1 -eq_states_EqR b4;
:: FSM_1:th 54
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty Mealy-FSM over b2,b3
st b1 -eq_states_partition b4 = (b1 + 1) -eq_states_partition b4
holds b1 -eq_states_partition b4 is final(b2, b3, b4);
:: FSM_1:th 55
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty finite Mealy-FSM over b2,b3
st b1 + 1 = card the carrier of b4
holds ex b5 being Element of NAT st
b5 <= b1 & b5 -eq_states_partition b4 is final(b2, b3, b4);
:: FSM_1:funcnot 8 => FSM_1:func 8
definition
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
func final_states_partition A3 -> a_partition of the carrier of a3 means
it is final(a1, a2, a3);
end;
:: FSM_1:def 15
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4 being a_partition of the carrier of b3 holds
b4 = final_states_partition b3
iff
b4 is final(b1, b2, b3);
:: FSM_1:th 56
theorem
for b1 being Element of NAT
for b2, b3 being non empty set
for b4 being non empty finite Mealy-FSM over b2,b3
st b1 + 1 = card the carrier of b4
holds final_states_partition b4 = b1 -eq_states_partition b4;
:: FSM_1:funcnot 9 => FSM_1:func 9
definition
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
let a4 be Element of final_states_partition a3;
let a5 be Element of a1;
func (A5,A4)-succ_class -> Element of final_states_partition a3 means
ex b1 being Element of the carrier of a3 st
ex b2 being Element of NAT st
b1 in a4 &
b2 + 1 = card the carrier of a3 &
it = Class(b2 -eq_states_EqR a3,(the Tran of a3) . [b1,a5]);
end;
:: FSM_1:def 16
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4 being Element of final_states_partition b3
for b5 being Element of b1
for b6 being Element of final_states_partition b3 holds
b6 = (b5,b4)-succ_class
iff
ex b7 being Element of the carrier of b3 st
ex b8 being Element of NAT st
b7 in b4 &
b8 + 1 = card the carrier of b3 &
b6 = Class(b8 -eq_states_EqR b3,(the Tran of b3) . [b7,b5]);
:: FSM_1:funcnot 10 => FSM_1:func 10
definition
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
let a4 be Element of final_states_partition a3;
let a5 be Element of a1;
func (A4,A5)-class_response -> Element of a2 means
ex b1 being Element of the carrier of a3 st
b1 in a4 & it = (the OFun of a3) . [b1,a5];
end;
:: FSM_1:def 17
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4 being Element of final_states_partition b3
for b5 being Element of b1
for b6 being Element of b2 holds
b6 = (b4,b5)-class_response
iff
ex b7 being Element of the carrier of b3 st
b7 in b4 & b6 = (the OFun of b3) . [b7,b5];
:: FSM_1:funcnot 11 => FSM_1:func 11
definition
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
func the_reduction_of A3 -> strict Mealy-FSM over a1,a2 means
the carrier of it = final_states_partition a3 &
(for b1 being Element of the carrier of it
for b2 being Element of a1
for b3 being Element of the carrier of a3
st b3 in b1
holds (the Tran of a3) .(b3,b2) in (the Tran of it) .(b1,b2) &
(the OFun of a3) .(b3,b2) = (the OFun of it) .(b1,b2)) &
the InitS of a3 in the InitS of it;
end;
:: FSM_1:def 18
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4 being strict Mealy-FSM over b1,b2 holds
b4 = the_reduction_of b3
iff
the carrier of b4 = final_states_partition b3 &
(for b5 being Element of the carrier of b4
for b6 being Element of b1
for b7 being Element of the carrier of b3
st b7 in b5
holds (the Tran of b3) .(b7,b6) in (the Tran of b4) .(b5,b6) &
(the OFun of b3) .(b7,b6) = (the OFun of b4) .(b5,b6)) &
the InitS of b3 in the InitS of b4;
:: FSM_1:funcreg 4
registration
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
cluster the_reduction_of a3 -> non empty finite strict;
end;
:: FSM_1:th 57
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4, b5 being non empty finite Mealy-FSM over b1,b2
for b6 being Element of the carrier of b5
for b7 being Element of the carrier of b4
st b4 = the_reduction_of b5 & b6 in b7
for b8 being Element of NAT
st b8 in Seg ((len b3) + 1)
holds (b6,b3)-admissible . b8 in (b7,b3)-admissible . b8;
:: FSM_1:th 58
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2 holds
b3,the_reduction_of b3 -are_equivalent;
:: FSM_1:prednot 7 => FSM_1:pred 7
definition
let a1, a2 be non empty set;
let a3, a4 be non empty Mealy-FSM over a1,a2;
pred A3,A4 -are_isomorphic means
ex b1 being Function-like quasi_total Relation of the carrier of a3,the carrier of a4 st
b1 is bijective(the carrier of a3, the carrier of a4) &
b1 . the InitS of a3 = the InitS of a4 &
(for b2 being Element of the carrier of a3
for b3 being Element of a1 holds
b1 . ((the Tran of a3) .(b2,b3)) = (the Tran of a4) .(b1 . b2,b3) &
(the OFun of a3) .(b2,b3) = (the OFun of a4) .(b1 . b2,b3));
symmetry;
:: for a1, a2 being non empty set
:: for a3, a4 being non empty Mealy-FSM over a1,a2
:: st a3,a4 -are_isomorphic
:: holds a4,a3 -are_isomorphic;
reflexivity;
:: for a1, a2 being non empty set
:: for a3 being non empty Mealy-FSM over a1,a2 holds
:: a3,a3 -are_isomorphic;
end;
:: FSM_1:dfs 19
definiens
let a1, a2 be non empty set;
let a3, a4 be non empty Mealy-FSM over a1,a2;
To prove
a3,a4 -are_isomorphic
it is sufficient to prove
thus ex b1 being Function-like quasi_total Relation of the carrier of a3,the carrier of a4 st
b1 is bijective(the carrier of a3, the carrier of a4) &
b1 . the InitS of a3 = the InitS of a4 &
(for b2 being Element of the carrier of a3
for b3 being Element of a1 holds
b1 . ((the Tran of a3) .(b2,b3)) = (the Tran of a4) .(b1 . b2,b3) &
(the OFun of a3) .(b2,b3) = (the OFun of a4) .(b1 . b2,b3));
:: FSM_1:def 19
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Mealy-FSM over b1,b2 holds
b3,b4 -are_isomorphic
iff
ex b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4 st
b5 is bijective(the carrier of b3, the carrier of b4) &
b5 . the InitS of b3 = the InitS of b4 &
(for b6 being Element of the carrier of b3
for b7 being Element of b1 holds
b5 . ((the Tran of b3) .(b6,b7)) = (the Tran of b4) .(b5 . b6,b7) &
(the OFun of b3) .(b6,b7) = (the OFun of b4) .(b5 . b6,b7));
:: FSM_1:th 59
theorem
for b1, b2 being non empty set
for b3, b4, b5 being non empty Mealy-FSM over b1,b2
st b3,b4 -are_isomorphic & b4,b5 -are_isomorphic
holds b3,b5 -are_isomorphic;
:: FSM_1:th 60
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b2
for b4, b5 being non empty Mealy-FSM over b2,b1
for b6 being Element of the carrier of b4
for b7 being Function-like quasi_total Relation of the carrier of b4,the carrier of b5
st for b8 being Element of the carrier of b4
for b9 being Element of b2 holds
b7 . ((the Tran of b4) .(b8,b9)) = (the Tran of b5) .(b7 . b8,b9)
for b8 being Element of NAT
st 1 <= b8 & b8 <= (len b3) + 1
holds b7 . ((b6,b3)-admissible . b8) = (b7 . b6,b3)-admissible . b8;
:: FSM_1:th 61
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Mealy-FSM over b2,b1
for b5, b6 being Element of the carrier of b3
for b7 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st b7 . the InitS of b3 = the InitS of b4 &
(for b8 being Element of the carrier of b3
for b9 being Element of b2 holds
b7 . ((the Tran of b3) .(b8,b9)) = (the Tran of b4) .(b7 . b8,b9) &
(the OFun of b3) .(b8,b9) = (the OFun of b4) .(b7 . b8,b9))
holds b5,b6 -are_equivalent
iff
b7 . b5,b7 . b6 -are_equivalent;
:: FSM_1:th 62
theorem
for b1, b2 being non empty set
for b3, b4 being non empty finite Mealy-FSM over b1,b2
for b5, b6 being Element of the carrier of b3
st b3 = the_reduction_of b4 & b5 <> b6
holds not b5,b6 -are_equivalent;
:: FSM_1:attrnot 5 => FSM_1:attr 5
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
attr a3 is reduced means
for b1, b2 being Element of the carrier of a3
st b1 <> b2
holds not b1,b2 -are_equivalent;
end;
:: FSM_1:dfs 20
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
To prove
a3 is reduced
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a3
st b1 <> b2
holds not b1,b2 -are_equivalent;
:: FSM_1:def 20
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2 holds
b3 is reduced(b1, b2)
iff
for b4, b5 being Element of the carrier of b3
st b4 <> b5
holds not b4,b5 -are_equivalent;
:: FSM_1:th 63
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2 holds
the_reduction_of b3 is reduced(b1, b2);
:: FSM_1:exreg 8
registration
let a1, a2 be non empty set;
cluster non empty finite reduced Mealy-FSM over a1,a2;
end;
:: FSM_1:th 64
theorem
for b1, b2 being non empty set
for b3 being non empty finite reduced Mealy-FSM over b1,b2 holds
b3,the_reduction_of b3 -are_isomorphic;
:: FSM_1:th 65
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2 holds
b3 is reduced(b1, b2)
iff
ex b4 being non empty finite Mealy-FSM over b1,b2 st
b3,the_reduction_of b4 -are_isomorphic;
:: FSM_1:attrnot 6 => FSM_1:attr 6
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be Element of the carrier of a3;
attr a4 is accessible means
ex b1 being FinSequence of a1 st
the InitS of a3,b1 -leads_to a4;
end;
:: FSM_1:dfs 21
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
let a4 be Element of the carrier of a3;
To prove
a4 is accessible
it is sufficient to prove
thus ex b1 being FinSequence of a1 st
the InitS of a3,b1 -leads_to a4;
:: FSM_1:def 21
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2
for b4 being Element of the carrier of b3 holds
b4 is accessible(b1, b2, b3)
iff
ex b5 being FinSequence of b1 st
the InitS of b3,b5 -leads_to b4;
:: FSM_1:attrnot 7 => FSM_1:attr 7
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
attr a3 is connected means
for b1 being Element of the carrier of a3 holds
b1 is accessible(a1, a2, a3);
end;
:: FSM_1:dfs 22
definiens
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
To prove
a3 is connected
it is sufficient to prove
thus for b1 being Element of the carrier of a3 holds
b1 is accessible(a1, a2, a3);
:: FSM_1:def 22
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2 holds
b3 is connected(b1, b2)
iff
for b4 being Element of the carrier of b3 holds
b4 is accessible(b1, b2, b3);
:: FSM_1:exreg 9
registration
let a1, a2 be non empty set;
cluster non empty finite connected Mealy-FSM over a1,a2;
end;
:: FSM_1:th 66
theorem
for b1, b2 being non empty set
for b3 being non empty finite connected Mealy-FSM over b1,b2 holds
the_reduction_of b3 is connected(b1, b2);
:: FSM_1:exreg 10
registration
let a1, a2 be non empty set;
cluster non empty finite reduced connected Mealy-FSM over a1,a2;
end;
:: FSM_1:funcnot 12 => FSM_1:func 12
definition
let a1, a2 be non empty set;
let a3 be non empty Mealy-FSM over a1,a2;
func accessibleStates A3 -> set equals
{b1 where b1 is Element of the carrier of a3: b1 is accessible(a1, a2, a3)};
end;
:: FSM_1:def 23
theorem
for b1, b2 being non empty set
for b3 being non empty Mealy-FSM over b1,b2 holds
accessibleStates b3 = {b4 where b4 is Element of the carrier of b3: b4 is accessible(b1, b2, b3)};
:: FSM_1:funcreg 5
registration
let a1, a2 be non empty set;
let a3 be non empty finite Mealy-FSM over a1,a2;
cluster accessibleStates a3 -> non empty finite;
end;
:: FSM_1:th 67
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b2,b1 holds
accessibleStates b3 c= the carrier of b3 &
(for b4 being Element of the carrier of b3 holds
b4 in accessibleStates b3
iff
b4 is accessible(b2, b1, b3));
:: FSM_1:th 68
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b2,b1 holds
(the Tran of b3) | [:accessibleStates b3,b2:] is Function-like quasi_total Relation of [:accessibleStates b3,b2:],accessibleStates b3;
:: FSM_1:th 69
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4 being Function-like quasi_total Relation of [:accessibleStates b3,b1:],accessibleStates b3
for b5 being Function-like quasi_total Relation of [:accessibleStates b3,b1:],b2
for b6 being Element of accessibleStates b3
st b4 = (the Tran of b3) | [:accessibleStates b3,b1:] &
b5 = (the OFun of b3) | [:accessibleStates b3,b1:] &
b6 = the InitS of b3
holds b3,Mealy-FSM(#accessibleStates b3,b4,b5,b6#) -are_equivalent;
:: FSM_1:th 70
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b2,b1 holds
ex b4 being non empty finite connected Mealy-FSM over b2,b1 st
the Tran of b4 = (the Tran of b3) | [:accessibleStates b3,b2:] &
the OFun of b4 = (the OFun of b3) | [:accessibleStates b3,b2:] &
the InitS of b4 = the InitS of b3 &
b3,b4 -are_equivalent;
:: FSM_1:funcnot 13 => FSM_1:func 13
definition
let a1 be set;
let a2 be non empty set;
let a3, a4 be non empty Mealy-FSM over a1,a2;
func A3 -Mealy_union A4 -> strict Mealy-FSM over a1,a2 means
the carrier of it = (the carrier of a3) \/ the carrier of a4 &
the Tran of it = (the Tran of a3) +* the Tran of a4 &
the OFun of it = (the OFun of a3) +* the OFun of a4 &
the InitS of it = the InitS of a3;
end;
:: FSM_1:def 24
theorem
for b1 being set
for b2 being non empty set
for b3, b4 being non empty Mealy-FSM over b1,b2
for b5 being strict Mealy-FSM over b1,b2 holds
b5 = b3 -Mealy_union b4
iff
the carrier of b5 = (the carrier of b3) \/ the carrier of b4 &
the Tran of b5 = (the Tran of b3) +* the Tran of b4 &
the OFun of b5 = (the OFun of b3) +* the OFun of b4 &
the InitS of b5 = the InitS of b3;
:: FSM_1:funcreg 6
registration
let a1 be set;
let a2 be non empty set;
let a3, a4 be non empty finite Mealy-FSM over a1,a2;
cluster a3 -Mealy_union a4 -> non empty finite strict;
end;
:: FSM_1:th 71
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4, b5 being non empty Mealy-FSM over b1,b2
for b6 being Element of the carrier of b4
for b7 being non empty finite Mealy-FSM over b1,b2
for b8 being Element of the carrier of b7
st b7 = b4 -Mealy_union b5 & the carrier of b4 misses the carrier of b5 & b6 = b8
holds (b6,b3)-admissible = (b8,b3)-admissible;
:: FSM_1:th 72
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4, b5 being non empty Mealy-FSM over b1,b2
for b6 being Element of the carrier of b4
for b7 being non empty finite Mealy-FSM over b1,b2
for b8 being Element of the carrier of b7
st b7 = b4 -Mealy_union b5 & the carrier of b4 misses the carrier of b5 & b6 = b8
holds (b6,b3)-response = (b8,b3)-response;
:: FSM_1:th 73
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4, b5 being non empty Mealy-FSM over b1,b2
for b6 being Element of the carrier of b5
for b7 being non empty finite Mealy-FSM over b1,b2
for b8 being Element of the carrier of b7
st b7 = b4 -Mealy_union b5 & the carrier of b4 misses the carrier of b5 & b6 = b8
holds (b6,b3)-admissible = (b8,b3)-admissible;
:: FSM_1:th 74
theorem
for b1, b2 being non empty set
for b3 being FinSequence of b1
for b4, b5 being non empty Mealy-FSM over b1,b2
for b6 being Element of the carrier of b5
for b7 being non empty finite Mealy-FSM over b1,b2
for b8 being Element of the carrier of b7
st b7 = b4 -Mealy_union b5 & the carrier of b4 misses the carrier of b5 & b6 = b8
holds (b6,b3)-response = (b8,b3)-response;
:: FSM_1:th 75
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty reduced Mealy-FSM over b1,b2
st b3 = b4 -Mealy_union b5 & the carrier of b4 misses the carrier of b5 & b4,b5 -are_equivalent
holds ex b6 being Element of the carrier of the_reduction_of b3 st
the InitS of b4 in b6 & the InitS of b5 in b6 & b6 = the InitS of the_reduction_of b3;
:: FSM_1:th 76
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty reduced connected Mealy-FSM over b1,b2
for b6, b7 being Element of the carrier of b3
for b8, b9 being Element of the carrier of b4
st b8 = b6 & b9 = b7 & the carrier of b4 misses the carrier of b5 & b4,b5 -are_equivalent & b3 = b4 -Mealy_union b5 & not b8,b9 -are_equivalent
holds not b6,b7 -are_equivalent;
:: FSM_1:th 77
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty reduced connected Mealy-FSM over b1,b2
for b6, b7 being Element of the carrier of b3
for b8, b9 being Element of the carrier of b4
st b8 = b6 & b9 = b7 & the carrier of b5 misses the carrier of b4 & b5,b4 -are_equivalent & b3 = b5 -Mealy_union b4 & not b8,b9 -are_equivalent
holds not b6,b7 -are_equivalent;
:: FSM_1:th 78
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty finite reduced connected Mealy-FSM over b1,b2
st the carrier of b4 misses the carrier of b5 & b4,b5 -are_equivalent & b3 = b4 -Mealy_union b5
for b6 being Element of the carrier of the_reduction_of b3
for b7, b8 being Element of b6
st b7 in the carrier of b4 & b8 in the carrier of b4
holds b7 = b8;
:: FSM_1:th 79
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty finite reduced connected Mealy-FSM over b1,b2
st the carrier of b4 misses the carrier of b5 & b4,b5 -are_equivalent & b3 = b4 -Mealy_union b5
for b6 being Element of the carrier of the_reduction_of b3
for b7, b8 being Element of b6
st b7 in the carrier of b5 & b8 in the carrier of b5
holds b7 = b8;
:: FSM_1:th 80
theorem
for b1, b2 being non empty set
for b3 being non empty finite Mealy-FSM over b1,b2
for b4, b5 being non empty finite reduced connected Mealy-FSM over b1,b2
st the carrier of b4 misses the carrier of b5 & b4,b5 -are_equivalent & b3 = b4 -Mealy_union b5
for b6 being Element of the carrier of the_reduction_of b3 holds
ex b7, b8 being Element of b6 st
b7 in the carrier of b4 &
b8 in the carrier of b5 &
(for b9 being Element of b6
st b9 <> b7
holds b9 = b8);
:: FSM_1:th 81
theorem
for b1, b2 being non empty set
for b3, b4 being non empty finite Mealy-FSM over b1,b2 holds
ex b5, b6 being non empty finite Mealy-FSM over b1,b2 st
the carrier of b5 misses the carrier of b6 & b5,b3 -are_isomorphic & b6,b4 -are_isomorphic;
:: FSM_1:th 82
theorem
for b1, b2 being non empty set
for b3, b4 being non empty Mealy-FSM over b1,b2
st b3,b4 -are_isomorphic
holds b3,b4 -are_equivalent;
:: FSM_1:th 83
theorem
for b1, b2 being non empty set
for b3, b4 being non empty finite reduced connected Mealy-FSM over b1,b2
st the carrier of b3 misses the carrier of b4 & b3,b4 -are_equivalent
holds b3,b4 -are_isomorphic;
:: FSM_1:th 84
theorem
for b1, b2 being non empty set
for b3, b4 being non empty finite connected Mealy-FSM over b1,b2
st b3,b4 -are_equivalent
holds the_reduction_of b3,the_reduction_of b4 -are_isomorphic;
:: FSM_1:th 85
theorem
for b1, b2 being non empty set
for b3, b4 being non empty finite reduced connected Mealy-FSM over b1,b2 holds
b3,b4 -are_isomorphic
iff
b3,b4 -are_equivalent;