:: CIRCCMB2 semantic presentation
theorem Th1: :: CIRCCMB2:1
theorem Th2: :: CIRCCMB2:2
theorem Th3: :: CIRCCMB2:3
theorem Th4: :: CIRCCMB2:4
scheme :: CIRCCMB2:sch 63
s63{
F1(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F2(
set ,
set )
-> set ,
P1[
set ,
set ,
set ],
F3()
-> ManySortedSet of
NAT ,
F4()
-> ManySortedSet of
NAT } :
provided
E39:
ex
S being non
empty ManySortedSign ex
x being
set st
(
S = F3()
. 0 &
x = F4()
. 0 &
P1[
S,
x,0] )
and E40:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = F3()
. n &
x = F4()
. n holds
(
F3()
. (n + 1) = F1(
S,
x,
n) &
F4()
. (n + 1) = F2(
x,
n) )
and E41:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = F3()
. n &
x = F4()
. n &
P1[
S,
x,
n] holds
P1[
F1(
S,
x,
n),
F2(
x,
n),
n + 1]
defpred S1[ set , set , set ] means verum;
scheme :: CIRCCMB2:sch 92
s92{
F1()
-> non
empty ManySortedSign ,
F2()
-> set ,
F3(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F4(
set ,
set )
-> set ,
F5()
-> Element of
NAT } :
( ex
S being non
empty ManySortedSign ex
f,
h being
ManySortedSet of
NAT st
(
S = f . F5() &
f . 0
= F1() &
h . 0
= F2() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = f . n &
x = h . n holds
(
f . (n + 1) = F3(
S,
x,
n) &
h . (n + 1) = F4(
x,
n) ) ) ) & ( for
S1,
S2 being non
empty ManySortedSign st ex
f,
h being
ManySortedSet of
NAT st
(
S1 = f . F5() &
f . 0
= F1() &
h . 0
= F2() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = f . n &
x = h . n holds
(
f . (n + 1) = F3(
S,
x,
n) &
h . (n + 1) = F4(
x,
n) ) ) ) & ex
f,
h being
ManySortedSet of
NAT st
(
S2 = f . F5() &
f . 0
= F1() &
h . 0
= F2() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = f . n &
x = h . n holds
(
f . (n + 1) = F3(
S,
x,
n) &
h . (n + 1) = F4(
x,
n) ) ) ) holds
S1 = S2 ) )
scheme :: CIRCCMB2:sch 93
s93{
F1()
-> non
empty ManySortedSign ,
F2(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F3()
-> set ,
F4(
set ,
set )
-> set ,
F5()
-> Element of
NAT } :
provided
E39:
(
F1() is
unsplit &
F1() is
gate`1=arity &
F1() is
gate`2isBoolean & not
F1() is
void & not
F1() is
empty &
F1() is
strict )
and E40:
for
S being non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign for
x being
set for
n being
Element of
NAT holds
(
F2(
S,
x,
n) is
unsplit &
F2(
S,
x,
n) is
gate`1=arity &
F2(
S,
x,
n) is
gate`2isBoolean & not
F2(
S,
x,
n) is
void & not
F2(
S,
x,
n) is
empty &
F2(
S,
x,
n) is
strict )
theorem Th5: :: CIRCCMB2:5
theorem Th6: :: CIRCCMB2:6
theorem Th7: :: CIRCCMB2:7
theorem Th8: :: CIRCCMB2:8
theorem Th9: :: CIRCCMB2:9
scheme :: CIRCCMB2:sch 120
s120{
F1()
-> non
empty ManySortedSign ,
F2()
-> non-empty MSAlgebra of
F1(),
F3()
-> set ,
F4(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F5(
set ,
set ,
set ,
set )
-> set ,
F6(
set ,
set )
-> set } :
scheme :: CIRCCMB2:sch 122
s122{
F1(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F2(
set ,
set ,
set ,
set )
-> set ,
F3(
set ,
set )
-> set ,
P1[
set ,
set ,
set ,
set ],
F4()
-> ManySortedSet of
NAT ,
F5()
-> ManySortedSet of
NAT ,
F6()
-> ManySortedSet of
NAT } :
provided
E39:
ex
S being non
empty ManySortedSign ex
A being
non-empty MSAlgebra of
S ex
x being
set st
(
S = F4()
. 0 &
A = F5()
. 0 &
x = F6()
. 0 &
P1[
S,
A,
x,0] )
and E40:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = F4()
. n &
A = F5()
. n &
x = F6()
. n holds
(
F4()
. (n + 1) = F1(
S,
x,
n) &
F5()
. (n + 1) = F2(
S,
A,
x,
n) &
F6()
. (n + 1) = F3(
x,
n) )
and E41:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = F4()
. n &
A = F5()
. n &
x = F6()
. n &
P1[
S,
A,
x,
n] holds
P1[
F1(
S,
x,
n),
F2(
S,
A,
x,
n),
F3(
x,
n),
n + 1]
and E44:
for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set for
n being
Element of
NAT holds
F2(
S,
A,
x,
n) is
non-empty MSAlgebra of
F1(
S,
x,
n)
defpred S2[ set , set , set , set ] means verum;
scheme :: CIRCCMB2:sch 126
s126{
F1(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F2(
set ,
set ,
set ,
set )
-> set ,
F3(
set ,
set )
-> set ,
F4()
-> ManySortedSet of
NAT ,
F5()
-> ManySortedSet of
NAT ,
F6()
-> ManySortedSet of
NAT ,
F7()
-> ManySortedSet of
NAT ,
F8()
-> ManySortedSet of
NAT ,
F9()
-> ManySortedSet of
NAT } :
(
F4()
= F5() &
F6()
= F7() &
F8()
= F9() )
provided
E39:
ex
S being non
empty ManySortedSign ex
A being
non-empty MSAlgebra of
S st
(
S = F4()
. 0 &
A = F6()
. 0 )
and E40:
(
F4()
. 0
= F5()
. 0 &
F6()
. 0
= F7()
. 0 &
F8()
. 0
= F9()
. 0 )
and E41:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = F4()
. n &
A = F6()
. n &
x = F8()
. n holds
(
F4()
. (n + 1) = F1(
S,
x,
n) &
F6()
. (n + 1) = F2(
S,
A,
x,
n) &
F8()
. (n + 1) = F3(
x,
n) )
and E44:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = F5()
. n &
A = F7()
. n &
x = F9()
. n holds
(
F5()
. (n + 1) = F1(
S,
x,
n) &
F7()
. (n + 1) = F2(
S,
A,
x,
n) &
F9()
. (n + 1) = F3(
x,
n) )
and E45:
for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set for
n being
Element of
NAT holds
F2(
S,
A,
x,
n) is
non-empty MSAlgebra of
F1(
S,
x,
n)
scheme :: CIRCCMB2:sch 129
s129{
F1()
-> non
empty ManySortedSign ,
F2()
-> non-empty MSAlgebra of
F1(),
F3(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F4(
set ,
set ,
set ,
set )
-> set ,
F5(
set ,
set )
-> set ,
F6()
-> ManySortedSet of
NAT ,
F7()
-> ManySortedSet of
NAT ,
F8()
-> ManySortedSet of
NAT } :
provided
E39:
(
F6()
. 0
= F1() &
F7()
. 0
= F2() )
and E40:
for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = F6()
. n &
A = F7()
. n &
x = F8()
. n holds
(
F6()
. (n + 1) = F3(
S,
x,
n) &
F7()
. (n + 1) = F4(
S,
A,
x,
n) &
F8()
. (n + 1) = F5(
x,
n) )
and E41:
for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set for
n being
Element of
NAT holds
F4(
S,
A,
x,
n) is
non-empty MSAlgebra of
F3(
S,
x,
n)
scheme :: CIRCCMB2:sch 130
s130{
F1()
-> non
empty ManySortedSign ,
F2()
-> non-empty MSAlgebra of
F1(),
F3()
-> set ,
F4(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F5(
set ,
set ,
set ,
set )
-> set ,
F6(
set ,
set )
-> set ,
F7()
-> Element of
NAT } :
provided
scheme :: CIRCCMB2:sch 131
s131{
F1()
-> non
empty ManySortedSign ,
F2()
-> non
empty ManySortedSign ,
F3()
-> non-empty MSAlgebra of
F1(),
F4()
-> set ,
F5(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F6(
set ,
set ,
set ,
set )
-> set ,
F7(
set ,
set )
-> set ,
F8()
-> Element of
NAT } :
provided
scheme :: CIRCCMB2:sch 132
s132{
F1()
-> non
empty ManySortedSign ,
F2()
-> non
empty ManySortedSign ,
F3()
-> non-empty MSAlgebra of
F1(),
F4()
-> set ,
F5(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F6(
set ,
set ,
set ,
set )
-> set ,
F7(
set ,
set )
-> set ,
F8()
-> Element of
NAT } :
for
A1,
A2 being
non-empty MSAlgebra of
F2() st ex
f,
g,
h being
ManySortedSet of
NAT st
(
F2()
= f . F8() &
A1 = g . F8() &
f . 0
= F1() &
g . 0
= F3() &
h . 0
= F4() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = f . n &
A = g . n &
x = h . n holds
(
f . (n + 1) = F5(
S,
x,
n) &
g . (n + 1) = F6(
S,
A,
x,
n) &
h . (n + 1) = F7(
x,
n) ) ) ) & ex
f,
g,
h being
ManySortedSet of
NAT st
(
F2()
= f . F8() &
A2 = g . F8() &
f . 0
= F1() &
g . 0
= F3() &
h . 0
= F4() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = f . n &
A = g . n &
x = h . n holds
(
f . (n + 1) = F5(
S,
x,
n) &
g . (n + 1) = F6(
S,
A,
x,
n) &
h . (n + 1) = F7(
x,
n) ) ) ) holds
A1 = A2
provided
scheme :: CIRCCMB2:sch 133
s133{
F1()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F2()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F3()
-> strict gate`2=den Boolean Circuit of
F1(),
F4(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F5(
set ,
set ,
set ,
set )
-> set ,
F6()
-> set ,
F7(
set ,
set )
-> set ,
F8()
-> Element of
NAT } :
provided
E39:
for
S being non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign for
x being
set for
n being
Element of
NAT holds
(
F4(
S,
x,
n) is
unsplit &
F4(
S,
x,
n) is
gate`1=arity &
F4(
S,
x,
n) is
gate`2isBoolean & not
F4(
S,
x,
n) is
void &
F4(
S,
x,
n) is
strict )
and E40:
ex
f,
h being
ManySortedSet of
NAT st
(
F2()
= f . F8() &
f . 0
= F1() &
h . 0
= F6() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
x being
set st
S = f . n &
x = h . n holds
(
f . (n + 1) = F4(
S,
x,
n) &
h . (n + 1) = F7(
x,
n) ) ) )
and E41:
for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set for
n being
Element of
NAT holds
F5(
S,
A,
x,
n) is
non-empty MSAlgebra of
F4(
S,
x,
n)
and E44:
for
S,
S1 being non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign for
A being
strict gate`2=den Boolean Circuit of
S for
x being
set for
n being
Element of
NAT st
S1 = F4(
S,
x,
n) holds
F5(
S,
A,
x,
n) is
strict gate`2=den Boolean Circuit of
S1
:: deftheorem Def1 defines MSAlg CIRCCMB2:def 1 :
scheme :: CIRCCMB2:sch 138
s138{
F1()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F2()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F3()
-> strict gate`2=den Boolean Circuit of
F1(),
F4(
set ,
set )
-> non
empty non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F5(
set ,
set )
-> set ,
F6()
-> set ,
F7(
set ,
set )
-> set ,
F8()
-> Element of
NAT } :
provided
scheme :: CIRCCMB2:sch 141
s141{
F1()
-> non
empty ManySortedSign ,
F2()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F3()
-> non-empty MSAlgebra of
F1(),
F4()
-> set ,
F5(
set ,
set ,
set )
-> non
empty ManySortedSign ,
F6(
set ,
set ,
set ,
set )
-> set ,
F7(
set ,
set )
-> set ,
F8()
-> Element of
NAT } :
for
A1,
A2 being
strict gate`2=den Boolean Circuit of
F2() st ex
f,
g,
h being
ManySortedSet of
NAT st
(
F2()
= f . F8() &
A1 = g . F8() &
f . 0
= F1() &
g . 0
= F3() &
h . 0
= F4() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = f . n &
A = g . n &
x = h . n holds
(
f . (n + 1) = F5(
S,
x,
n) &
g . (n + 1) = F6(
S,
A,
x,
n) &
h . (n + 1) = F7(
x,
n) ) ) ) & ex
f,
g,
h being
ManySortedSet of
NAT st
(
F2()
= f . F8() &
A2 = g . F8() &
f . 0
= F1() &
g . 0
= F3() &
h . 0
= F4() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A being
non-empty MSAlgebra of
S for
x being
set st
S = f . n &
A = g . n &
x = h . n holds
(
f . (n + 1) = F5(
S,
x,
n) &
g . (n + 1) = F6(
S,
A,
x,
n) &
h . (n + 1) = F7(
x,
n) ) ) ) holds
A1 = A2
provided
theorem Th10: :: CIRCCMB2:10
theorem Th11: :: CIRCCMB2:11
theorem Th12: :: CIRCCMB2:12
theorem Th13: :: CIRCCMB2:13
theorem Th14: :: CIRCCMB2:14
theorem Th15: :: CIRCCMB2:15
theorem Th16: :: CIRCCMB2:16
theorem Th17: :: CIRCCMB2:17
theorem Th18: :: CIRCCMB2:18
theorem Th19: :: CIRCCMB2:19
theorem Th20: :: CIRCCMB2:20
theorem Th21: :: CIRCCMB2:21
theorem Th22: :: CIRCCMB2:22
theorem Th23: :: CIRCCMB2:23
theorem Th24: :: CIRCCMB2:24
theorem Th25: :: CIRCCMB2:25
scheme :: CIRCCMB2:sch 163
s163{
F1()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F2()
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F3()
-> strict gate`2=den Boolean Circuit of
F1(),
F4()
-> strict gate`2=den Boolean Circuit of
F2(),
F5(
set ,
set )
-> non
empty strict non
void unsplit gate`1=arity gate`2isBoolean ManySortedSign ,
F6(
set ,
set )
-> set ,
F7()
-> ManySortedSet of
NAT ,
F8()
-> set ,
F9(
set ,
set )
-> set ,
F10(
Element of
NAT )
-> Element of
NAT } :
provided
E39:
for
x being
set for
n being
Element of
NAT holds
F6(
x,
n) is
strict gate`2=den Boolean Circuit of
F5(
x,
n)
and E40:
for
s being
State of
F3() holds
Following s,
F10(0) is
stable
and E41:
for
n being
Element of
NAT for
x being
set for
A being
non-empty Circuit of
F5(
x,
n) st
x = F7()
. n &
A = F6(
x,
n) holds
for
s being
State of
A holds
Following s,
F10(1) is
stable
and E44:
ex
f,
g being
ManySortedSet of
NAT st
(
F2()
= f . F10(2) &
F4()
= g . F10(2) &
f . 0
= F1() &
g . 0
= F3() &
F7()
. 0
= F8() & ( for
n being
Element of
NAT for
S being non
empty ManySortedSign for
A1 being
non-empty MSAlgebra of
S for
x being
set for
A2 being
non-empty MSAlgebra of
F5(
x,
n) st
S = f . n &
A1 = g . n &
x = F7()
. n &
A2 = F6(
x,
n) holds
(
f . (n + 1) = S +* F5(
x,
n) &
g . (n + 1) = A1 +* A2 &
F7()
. (n + 1) = F9(
x,
n) ) ) )
and E45:
(
InnerVertices F1() is
Relation & not
InputVertices F1() is
with_pair )
and E46:
(
F7()
. 0
= F8() &
F8()
in InnerVertices F1() )
and E48:
for
n being
Element of
NAT for
x being
set holds
InnerVertices F5(
x,
n) is
Relation
and E49:
for
n being
Element of
NAT for
x being
set st
x = F7()
. n holds
not
(InputVertices F5(x,n)) \ {x} is
with_pair
and E76:
for
n being
Element of
NAT for
x being
set st
x = F7()
. n holds
(
F7()
. (n + 1) = F9(
x,
n) &
x in InputVertices F5(
x,
n) &
F9(
x,
n)
in InnerVertices F5(
x,
n) )