:: CIRCCMB3 semantic presentation
theorem Th1: :: CIRCCMB3:1
:: deftheorem Def1 defines stabilizing CIRCCMB3:def 1 :
:: deftheorem Def2 defines stabilizing CIRCCMB3:def 2 :
:: deftheorem Def3 defines with_stabilization-limit CIRCCMB3:def 3 :
:: deftheorem Def4 defines Result CIRCCMB3:def 4 :
:: deftheorem Def5 defines stabilization-time CIRCCMB3:def 5 :
theorem Th2: :: CIRCCMB3:2
theorem Th3: :: CIRCCMB3:3
theorem Th4: :: CIRCCMB3:4
theorem Th5: :: CIRCCMB3:5
theorem Th6: :: CIRCCMB3:6
theorem Th7: :: CIRCCMB3:7
canceled;
theorem Th8: :: CIRCCMB3:8
theorem Th9: :: CIRCCMB3:9
theorem Th10: :: CIRCCMB3:10
theorem Th11: :: CIRCCMB3:11
theorem Th12: :: CIRCCMB3:12
theorem Th13: :: CIRCCMB3:13
theorem Th14: :: CIRCCMB3:14
for
x1,
x2,
x3,
x4 being
set holds
rng <*x1,x2,x3,x4*> = {x1,x2,x3,x4}
theorem Th15: :: CIRCCMB3:15
for
x1,
x2,
x3,
x4,
x5 being
set holds
rng <*x1,x2,x3,x4,x5*> = {x1,x2,x3,x4,x5}
definition
let x1 be
set ,
x2 be
set ,
x3 be
set ,
x4 be
set ;
redefine func <* as
<*c1,c2,c3,c4*> -> FinSeqLen of 4;
coherence
<*x1,x2,x3,x4*> is FinSeqLen of 4
let x5 be
set ;
redefine func <* as
<*c1,c2,c3,c4,c5*> -> FinSeqLen of 5;
coherence
<*x1,x2,x3,x4,x5*> is FinSeqLen of 5
end;
:: deftheorem Def6 defines one-gate CIRCCMB3:def 6 :
:: deftheorem Def7 defines one-gate CIRCCMB3:def 7 :
:: deftheorem Def8 defines Output CIRCCMB3:def 8 :
theorem Th16: :: CIRCCMB3:16
theorem Th17: :: CIRCCMB3:17
theorem Th18: :: CIRCCMB3:18
theorem Th19: :: CIRCCMB3:19
theorem Th20: :: CIRCCMB3:20
theorem Th21: :: CIRCCMB3:21
theorem Th22: :: CIRCCMB3:22
scheme :: CIRCCMB3:sch 102
s102{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> set ,
F5()
-> set ,
F6()
-> non
empty finite set ,
F7(
set ,
set ,
set ,
set ,
set )
-> Element of
F6() } :
theorem Th23: :: CIRCCMB3:23
theorem Th24: :: CIRCCMB3:24
theorem Th25: :: CIRCCMB3:25
theorem Th26: :: CIRCCMB3:26
:: deftheorem Def9 defines Signature CIRCCMB3:def 9 :
theorem Th27: :: CIRCCMB3:27
:: deftheorem Def10 defines Circuit CIRCCMB3:def 10 :
theorem Th28: :: CIRCCMB3:28
theorem Th29: :: CIRCCMB3:29
canceled;
theorem Th30: :: CIRCCMB3:30
theorem Th31: :: CIRCCMB3:31
theorem Th32: :: CIRCCMB3:32
theorem Th33: :: CIRCCMB3:33
theorem Th34: :: CIRCCMB3:34
theorem Th35: :: CIRCCMB3:35
scheme :: CIRCCMB3:sch 184
s184{
F1()
-> non
empty set ,
F2(
set ,
set )
-> Element of
F1() } :
( ex
f being
Function of 2
-tuples_on F1(),
F1() st
for
x,
y being
Element of
F1() holds
f . <*x,y*> = F2(
x,
y) & ( for
f1,
f2 being
Function of 2
-tuples_on F1(),
F1() st ( for
x,
y being
Element of
F1() holds
f1 . <*x,y*> = F2(
x,
y) ) & ( for
x,
y being
Element of
F1() holds
f2 . <*x,y*> = F2(
x,
y) ) holds
f1 = f2 ) )
scheme :: CIRCCMB3:sch 185
s185{
F1()
-> non
empty set ,
F2(
set ,
set ,
set )
-> Element of
F1() } :
( ex
f being
Function of 3
-tuples_on F1(),
F1() st
for
x,
y,
z being
Element of
F1() holds
f . <*x,y,z*> = F2(
x,
y,
z) & ( for
f1,
f2 being
Function of 3
-tuples_on F1(),
F1() st ( for
x,
y,
z being
Element of
F1() holds
f1 . <*x,y,z*> = F2(
x,
y,
z) ) & ( for
x,
y,
z being
Element of
F1() holds
f2 . <*x,y,z*> = F2(
x,
y,
z) ) holds
f1 = f2 ) )
theorem Th36: :: CIRCCMB3:36
theorem Th37: :: CIRCCMB3:37
theorem Th38: :: CIRCCMB3:38
for
f being
Function for
x1,
x2,
x3,
x4,
x5 being
set st
x1 in dom f &
x2 in dom f &
x3 in dom f &
x4 in dom f &
x5 in dom f holds
f * <*x1,x2,x3,x4,x5*> = <*(f . x1),(f . x2),(f . x3),(f . x4),(f . x5)*>
scheme :: CIRCCMB3:sch 193
s193{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> non
empty finite set ,
F5(
set ,
set ,
set )
-> Element of
F4(),
F6()
-> Function of 3
-tuples_on F4(),
F4() } :
for
s being
State of
(1GateCircuit <*F1(),F2(),F3()*>,F6()) for
a1,
a2,
a3 being
Element of
F4() st
a1 = s . F1() &
a2 = s . F2() &
a3 = s . F3() holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3()*>,F6())) = F5(
a1,
a2,
a3)
provided
scheme :: CIRCCMB3:sch 195
s195{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> set ,
F5()
-> non
empty finite set ,
F6(
set ,
set ,
set ,
set )
-> Element of
F5(),
F7()
-> Function of 4
-tuples_on F5(),
F5() } :
for
s being
State of
(1GateCircuit <*F1(),F2(),F3(),F4()*>,F7()) for
a1,
a2,
a3,
a4 being
Element of
F5() st
a1 = s . F1() &
a2 = s . F2() &
a3 = s . F3() &
a4 = s . F4() holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F7())) = F6(
a1,
a2,
a3,
a4)
provided
E34:
for
g being
Function of 4
-tuples_on F5(),
F5() holds
(
g = F7() iff for
a1,
a2,
a3,
a4 being
Element of
F5() holds
g . <*a1,a2,a3,a4*> = F6(
a1,
a2,
a3,
a4) )
scheme :: CIRCCMB3:sch 197
s197{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> set ,
F5()
-> set ,
F6()
-> non
empty finite set ,
F7(
set ,
set ,
set ,
set ,
set )
-> Element of
F6(),
F8()
-> Function of 5
-tuples_on F6(),
F6() } :
for
s being
State of
(1GateCircuit <*F1(),F2(),F3(),F4(),F5()*>,F8()) for
a1,
a2,
a3,
a4,
a5 being
Element of
F6() st
a1 = s . F1() &
a2 = s . F2() &
a3 = s . F3() &
a4 = s . F4() &
a5 = s . F5() holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F8())) = F7(
a1,
a2,
a3,
a4,
a5)
provided
E34:
for
g being
Function of 5
-tuples_on F6(),
F6() holds
(
g = F8() iff for
a1,
a2,
a3,
a4,
a5 being
Element of
F6() holds
g . <*a1,a2,a3,a4,a5*> = F7(
a1,
a2,
a3,
a4,
a5) )
theorem Th39: :: CIRCCMB3:39
theorem Th40: :: CIRCCMB3:40
theorem Th41: :: CIRCCMB3:41
theorem Th42: :: CIRCCMB3:42
theorem Th43: :: CIRCCMB3:43
theorem Th44: :: CIRCCMB3:44
theorem Th45: :: CIRCCMB3:45
for
x1,
x2,
x3 being
set for
X being non
empty finite set for
f being
Function of 3
-tuples_on X,
X for
S being
Signature of
X st
x1 in the
carrier of
S & not
x2 in InnerVertices S & not
x3 in InnerVertices S & not
Output (1GateCircStr <*x1,x2,x3*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2,x3*>,f)) = (InputVertices S) \/ {x2,x3}
theorem Th46: :: CIRCCMB3:46
for
x1,
x2,
x3 being
set for
X being non
empty finite set for
f being
Function of 3
-tuples_on X,
X for
S being
Signature of
X st
x2 in the
carrier of
S & not
x1 in InnerVertices S & not
x3 in InnerVertices S & not
Output (1GateCircStr <*x1,x2,x3*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2,x3*>,f)) = (InputVertices S) \/ {x1,x3}
theorem Th47: :: CIRCCMB3:47
for
x1,
x2,
x3 being
set for
X being non
empty finite set for
f being
Function of 3
-tuples_on X,
X for
S being
Signature of
X st
x3 in the
carrier of
S & not
x1 in InnerVertices S & not
x2 in InnerVertices S & not
Output (1GateCircStr <*x1,x2,x3*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2,x3*>,f)) = (InputVertices S) \/ {x1,x2}
theorem Th48: :: CIRCCMB3:48
theorem Th49: :: CIRCCMB3:49
theorem Th50: :: CIRCCMB3:50
theorem Th51: :: CIRCCMB3:51
theorem Th52: :: CIRCCMB3:52
scheme :: CIRCCMB3:sch 217
s217{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> non
empty finite set ,
F5(
set ,
set ,
set )
-> Element of
F4(),
F6()
-> finite Signature of
F4(),
F7()
-> Circuit of
F4(),
F6(),
F8()
-> Function of 3
-tuples_on F4(),
F4() } :
for
s being
State of
(F7() +* (1GateCircuit <*F1(),F2(),F3()*>,F8())) for
s' being
State of
F7() st
s' = s | the
carrier of
F6() holds
for
a1,
a2,
a3 being
Element of
F4() st (
F1()
in InnerVertices F6() implies
a1 = (Result s') . F1() ) & ( not
F1()
in InnerVertices F6() implies
a1 = s . F1() ) & (
F2()
in InnerVertices F6() implies
a2 = (Result s') . F2() ) & ( not
F2()
in InnerVertices F6() implies
a2 = s . F2() ) & (
F3()
in InnerVertices F6() implies
a3 = (Result s') . F3() ) & ( not
F3()
in InnerVertices F6() implies
a3 = s . F3() ) holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3()*>,F8())) = F5(
a1,
a2,
a3)
provided
E34:
for
g being
Function of 3
-tuples_on F4(),
F4() holds
(
g = F8() iff for
a1,
a2,
a3 being
Element of
F4() holds
g . <*a1,a2,a3*> = F5(
a1,
a2,
a3) )
and E36:
not
Output (1GateCircStr <*F1(),F2(),F3()*>,F8()) in InputVertices F6()
scheme :: CIRCCMB3:sch 218
s218{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> set ,
F5()
-> non
empty finite set ,
F6(
set ,
set ,
set ,
set )
-> Element of
F5(),
F7()
-> finite Signature of
F5(),
F8()
-> Circuit of
F5(),
F7(),
F9()
-> Function of 4
-tuples_on F5(),
F5() } :
for
s being
State of
(F8() +* (1GateCircuit <*F1(),F2(),F3(),F4()*>,F9())) for
s' being
State of
F8() st
s' = s | the
carrier of
F7() holds
for
a1,
a2,
a3,
a4 being
Element of
F5() st (
F1()
in InnerVertices F7() implies
a1 = (Result s') . F1() ) & ( not
F1()
in InnerVertices F7() implies
a1 = s . F1() ) & (
F2()
in InnerVertices F7() implies
a2 = (Result s') . F2() ) & ( not
F2()
in InnerVertices F7() implies
a2 = s . F2() ) & (
F3()
in InnerVertices F7() implies
a3 = (Result s') . F3() ) & ( not
F3()
in InnerVertices F7() implies
a3 = s . F3() ) & (
F4()
in InnerVertices F7() implies
a4 = (Result s') . F4() ) & ( not
F4()
in InnerVertices F7() implies
a4 = s . F4() ) holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F9())) = F6(
a1,
a2,
a3,
a4)
provided
E34:
for
g being
Function of 4
-tuples_on F5(),
F5() holds
(
g = F9() iff for
a1,
a2,
a3,
a4 being
Element of
F5() holds
g . <*a1,a2,a3,a4*> = F6(
a1,
a2,
a3,
a4) )
and E36:
not
Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F9()) in InputVertices F7()
scheme :: CIRCCMB3:sch 219
s219{
F1()
-> set ,
F2()
-> set ,
F3()
-> set ,
F4()
-> set ,
F5()
-> set ,
F6()
-> non
empty finite set ,
F7(
set ,
set ,
set ,
set ,
set )
-> Element of
F6(),
F8()
-> finite Signature of
F6(),
F9()
-> Circuit of
F6(),
F8(),
F10()
-> Function of 5
-tuples_on F6(),
F6() } :
for
s being
State of
(F9() +* (1GateCircuit <*F1(),F2(),F3(),F4(),F5()*>,F10())) for
s' being
State of
F9() st
s' = s | the
carrier of
F8() holds
for
a1,
a2,
a3,
a4,
a5 being
Element of
F6() st (
F1()
in InnerVertices F8() implies
a1 = (Result s') . F1() ) & ( not
F1()
in InnerVertices F8() implies
a1 = s . F1() ) & (
F2()
in InnerVertices F8() implies
a2 = (Result s') . F2() ) & ( not
F2()
in InnerVertices F8() implies
a2 = s . F2() ) & (
F3()
in InnerVertices F8() implies
a3 = (Result s') . F3() ) & ( not
F3()
in InnerVertices F8() implies
a3 = s . F3() ) & (
F4()
in InnerVertices F8() implies
a4 = (Result s') . F4() ) & ( not
F4()
in InnerVertices F8() implies
a4 = s . F4() ) & (
F5()
in InnerVertices F8() implies
a5 = (Result s') . F5() ) & ( not
F5()
in InnerVertices F8() implies
a5 = s . F5() ) holds
(Result s) . (Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F10())) = F7(
a1,
a2,
a3,
a4,
a5)
provided
E34:
for
g being
Function of 5
-tuples_on F6(),
F6() holds
(
g = F10() iff for
a1,
a2,
a3,
a4,
a5 being
Element of
F6() holds
g . <*a1,a2,a3,a4,a5*> = F7(
a1,
a2,
a3,
a4,
a5) )
and E36:
not
Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F10()) in InputVertices F8()
:: deftheorem Def11 defines with_nonpair_inputs CIRCCMB3:def 11 :
theorem Th53: :: CIRCCMB3:53
theorem Th54: :: CIRCCMB3:54
theorem Th55: :: CIRCCMB3:55
Lemma222:
for x being non pair set holds not {x} is with_pair
;
Lemma223:
for x, y being without_pairs set holds not x \/ y is with_pair
;
theorem Th56: :: CIRCCMB3:56
Lemma225:
for x, y being non pair set holds not {x,y} is with_pair
;
theorem Th57: :: CIRCCMB3:57
registration
let X be non
empty finite set ;
let S be
with_nonpair_inputs Signature of
X;
let x1 be
Vertex of
S,
x2 be
Vertex of
S;
let n be non
pair set ;
let f be
Function of 3
-tuples_on X,
X;
cluster a2 +* (1GateCircStr <*a3,a4,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,x2,n*>,f) is with_nonpair_inputs
by ;
cluster a2 +* (1GateCircStr <*a3,a5,a4*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,n,x2*>,f) is with_nonpair_inputs
by ;
cluster a2 +* (1GateCircStr <*a5,a3,a4*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n,x1,x2*>,f) is with_nonpair_inputs
by ;
end;
registration
let X be non
empty finite set ;
let S be
with_nonpair_inputs Signature of
X;
let x be
Vertex of
S;
let n1 be non
pair set ,
n2 be non
pair set ;
let f be
Function of 3
-tuples_on X,
X;
cluster a2 +* (1GateCircStr <*a3,a4,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x,n1,n2*>,f) is with_nonpair_inputs
by ;
cluster a2 +* (1GateCircStr <*a4,a3,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n1,x,n2*>,f) is with_nonpair_inputs
by ;
cluster a2 +* (1GateCircStr <*a4,a5,a3*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n1,n2,x*>,f) is with_nonpair_inputs
by ;
end;