:: 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 b
1, b
2, b
3, b
4 being
set holds
rng <*b1,b2,b3,b4*> = {b1,b2,b3,b4}
theorem Th15: :: CIRCCMB3:15
for b
1, b
2, b
3, b
4, b
5 being
set holds
rng <*b1,b2,b3,b4,b5*> = {b1,b2,b3,b4,b5}
definition
let c
1, c
2, c
3, c
4 be
set ;
redefine func <* as
<*c1,c2,c3,c4*> -> FinSeqLen of 4;
coherence
<*c1,c2,c3,c4*> is FinSeqLen of 4
let c
5 be
set ;
redefine func <* as
<*c1,c2,c3,c4,c5*> -> FinSeqLen of 5;
coherence
<*c1,c2,c3,c4,c5*> 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 5
s5{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> set , F
6()
-> non
empty finite set , F
7(
set ,
set ,
set ,
set ,
set )
-> Element of F
6() } :
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 7
s7{ F
1()
-> non
empty set , F
2(
set ,
set )
-> Element of F
1() } :
( ex b
1 being
Function of 2
-tuples_on F
1(),F
1() st
for b
2, b
3 being
Element of F
1() holds b
1 . <*b2,b3*> = F
2(b
2,b
3) & ( for b
1, b
2 being
Function of 2
-tuples_on F
1(),F
1() holds
( ( for b
3, b
4 being
Element of F
1() holds b
1 . <*b3,b4*> = F
2(b
3,b
4) ) & ( for b
3, b
4 being
Element of F
1() holds b
2 . <*b3,b4*> = F
2(b
3,b
4) ) implies b
1 = b
2 ) ) )
scheme :: CIRCCMB3:sch 8
s8{ F
1()
-> non
empty set , F
2(
set ,
set ,
set )
-> Element of F
1() } :
( ex b
1 being
Function of 3
-tuples_on F
1(),F
1() st
for b
2, b
3, b
4 being
Element of F
1() holds b
1 . <*b2,b3,b4*> = F
2(b
2,b
3,b
4) & ( for b
1, b
2 being
Function of 3
-tuples_on F
1(),F
1() holds
( ( for b
3, b
4, b
5 being
Element of F
1() holds b
1 . <*b3,b4,b5*> = F
2(b
3,b
4,b
5) ) & ( for b
3, b
4, b
5 being
Element of F
1() holds b
2 . <*b3,b4,b5*> = F
2(b
3,b
4,b
5) ) implies b
1 = b
2 ) ) )
theorem Th36: :: CIRCCMB3:36
theorem Th37: :: CIRCCMB3:37
theorem Th38: :: CIRCCMB3:38
for b
1 being
Functionfor b
2, b
3, b
4, b
5, b
6 being
set holds
( b
2 in dom b
1 & b
3 in dom b
1 & b
4 in dom b
1 & b
5 in dom b
1 & b
6 in dom b
1 implies b
1 * <*b2,b3,b4,b5,b6*> = <*(b1 . b2),(b1 . b3),(b1 . b4),(b1 . b5),(b1 . b6)*> )
scheme :: CIRCCMB3:sch 11
s11{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> non
empty finite set , F
5(
set ,
set ,
set )
-> Element of F
4(), F
6()
-> Function of 3
-tuples_on F
4(),F
4() } :
for b
1 being
State of
(1GateCircuit <*F1(),F2(),F3()*>,F6())for b
2, b
3, b
4 being
Element of F
4() holds
( b
2 = b
1 . F
1() & b
3 = b
1 . F
2() & b
4 = b
1 . F
3() implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3()*>,F6())) = F
5(b
2,b
3,b
4) )
provided
scheme :: CIRCCMB3:sch 12
s12{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> non
empty finite set , F
6(
set ,
set ,
set ,
set )
-> Element of F
5(), F
7()
-> Function of 4
-tuples_on F
5(),F
5() } :
for b
1 being
State of
(1GateCircuit <*F1(),F2(),F3(),F4()*>,F7())for b
2, b
3, b
4, b
5 being
Element of F
5() holds
( b
2 = b
1 . F
1() & b
3 = b
1 . F
2() & b
4 = b
1 . F
3() & b
5 = b
1 . F
4() implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F7())) = F
6(b
2,b
3,b
4,b
5) )
provided
E37:
for b
1 being
Function of 4
-tuples_on F
5(),F
5() holds
( b
1 = F
7() iff for b
2, b
3, b
4, b
5 being
Element of F
5() holds b
1 . <*b2,b3,b4,b5*> = F
6(b
2,b
3,b
4,b
5) )
scheme :: CIRCCMB3:sch 13
s13{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> set , F
6()
-> non
empty finite set , F
7(
set ,
set ,
set ,
set ,
set )
-> Element of F
6(), F
8()
-> Function of 5
-tuples_on F
6(),F
6() } :
for b
1 being
State of
(1GateCircuit <*F1(),F2(),F3(),F4(),F5()*>,F8())for b
2, b
3, b
4, b
5, b
6 being
Element of F
6() holds
( b
2 = b
1 . F
1() & b
3 = b
1 . F
2() & b
4 = b
1 . F
3() & b
5 = b
1 . F
4() & b
6 = b
1 . F
5() implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F8())) = F
7(b
2,b
3,b
4,b
5,b
6) )
provided
E37:
for b
1 being
Function of 5
-tuples_on F
6(),F
6() holds
( b
1 = F
8() iff for b
2, b
3, b
4, b
5, b
6 being
Element of F
6() holds b
1 . <*b2,b3,b4,b5,b6*> = F
7(b
2,b
3,b
4,b
5,b
6) )
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 b
1, b
2, b
3 being
set for b
4 being non
empty finite set for b
5 being
Function of 3
-tuples_on b
4,b
4for b
6 being
Signature of b
4 holds
( b
1 in the
carrier of b
6 & not b
2 in InnerVertices b
6 & not b
3 in InnerVertices b
6 & not
Output (1GateCircStr <*b1,b2,b3*>,b5) in InputVertices b
6 implies
InputVertices (b6 +* (1GateCircStr <*b1,b2,b3*>,b5)) = (InputVertices b6) \/ {b2,b3} )
theorem Th46: :: CIRCCMB3:46
for b
1, b
2, b
3 being
set for b
4 being non
empty finite set for b
5 being
Function of 3
-tuples_on b
4,b
4for b
6 being
Signature of b
4 holds
( b
2 in the
carrier of b
6 & not b
1 in InnerVertices b
6 & not b
3 in InnerVertices b
6 & not
Output (1GateCircStr <*b1,b2,b3*>,b5) in InputVertices b
6 implies
InputVertices (b6 +* (1GateCircStr <*b1,b2,b3*>,b5)) = (InputVertices b6) \/ {b1,b3} )
theorem Th47: :: CIRCCMB3:47
for b
1, b
2, b
3 being
set for b
4 being non
empty finite set for b
5 being
Function of 3
-tuples_on b
4,b
4for b
6 being
Signature of b
4 holds
( b
3 in the
carrier of b
6 & not b
1 in InnerVertices b
6 & not b
2 in InnerVertices b
6 & not
Output (1GateCircStr <*b1,b2,b3*>,b5) in InputVertices b
6 implies
InputVertices (b6 +* (1GateCircStr <*b1,b2,b3*>,b5)) = (InputVertices b6) \/ {b1,b2} )
theorem Th48: :: CIRCCMB3:48
theorem Th49: :: CIRCCMB3:49
theorem Th50: :: CIRCCMB3:50
theorem Th51: :: CIRCCMB3:51
theorem Th52: :: CIRCCMB3:52
scheme :: CIRCCMB3:sch 16
s16{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> non
empty finite set , F
5(
set ,
set ,
set )
-> Element of F
4(), F
6()
-> finite Signature of F
4(), F
7()
-> Circuit of F
4(),F
6(), F
8()
-> Function of 3
-tuples_on F
4(),F
4() } :
for b
1 being
State of
(F7() +* (1GateCircuit <*F1(),F2(),F3()*>,F8()))for b
2 being
State of F
7() holds
( b
2 = b
1 | the
carrier of F
6() implies for b
3, b
4, b
5 being
Element of F
4() holds
( ( F
1()
in InnerVertices F
6() implies b
3 = (Result b2) . F
1() ) & ( not F
1()
in InnerVertices F
6() implies b
3 = b
1 . F
1() ) & ( F
2()
in InnerVertices F
6() implies b
4 = (Result b2) . F
2() ) & ( not F
2()
in InnerVertices F
6() implies b
4 = b
1 . F
2() ) & ( F
3()
in InnerVertices F
6() implies b
5 = (Result b2) . F
3() ) & ( not F
3()
in InnerVertices F
6() implies b
5 = b
1 . F
3() ) implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3()*>,F8())) = F
5(b
3,b
4,b
5) ) )
provided
E51:
for b
1 being
Function of 3
-tuples_on F
4(),F
4() holds
( b
1 = F
8() iff for b
2, b
3, b
4 being
Element of F
4() holds b
1 . <*b2,b3,b4*> = F
5(b
2,b
3,b
4) )
and
E52:
not
Output (1GateCircStr <*F1(),F2(),F3()*>,F8()) in InputVertices F
6()
scheme :: CIRCCMB3:sch 17
s17{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> non
empty finite set , F
6(
set ,
set ,
set ,
set )
-> Element of F
5(), F
7()
-> finite Signature of F
5(), F
8()
-> Circuit of F
5(),F
7(), F
9()
-> Function of 4
-tuples_on F
5(),F
5() } :
for b
1 being
State of
(F8() +* (1GateCircuit <*F1(),F2(),F3(),F4()*>,F9()))for b
2 being
State of F
8() holds
( b
2 = b
1 | the
carrier of F
7() implies for b
3, b
4, b
5, b
6 being
Element of F
5() holds
( ( F
1()
in InnerVertices F
7() implies b
3 = (Result b2) . F
1() ) & ( not F
1()
in InnerVertices F
7() implies b
3 = b
1 . F
1() ) & ( F
2()
in InnerVertices F
7() implies b
4 = (Result b2) . F
2() ) & ( not F
2()
in InnerVertices F
7() implies b
4 = b
1 . F
2() ) & ( F
3()
in InnerVertices F
7() implies b
5 = (Result b2) . F
3() ) & ( not F
3()
in InnerVertices F
7() implies b
5 = b
1 . F
3() ) & ( F
4()
in InnerVertices F
7() implies b
6 = (Result b2) . F
4() ) & ( not F
4()
in InnerVertices F
7() implies b
6 = b
1 . F
4() ) implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F9())) = F
6(b
3,b
4,b
5,b
6) ) )
provided
E51:
for b
1 being
Function of 4
-tuples_on F
5(),F
5() holds
( b
1 = F
9() iff for b
2, b
3, b
4, b
5 being
Element of F
5() holds b
1 . <*b2,b3,b4,b5*> = F
6(b
2,b
3,b
4,b
5) )
and
E52:
not
Output (1GateCircStr <*F1(),F2(),F3(),F4()*>,F9()) in InputVertices F
7()
scheme :: CIRCCMB3:sch 18
s18{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5()
-> set , F
6()
-> non
empty finite set , F
7(
set ,
set ,
set ,
set ,
set )
-> Element of F
6(), F
8()
-> finite Signature of F
6(), F
9()
-> Circuit of F
6(),F
8(), F
10()
-> Function of 5
-tuples_on F
6(),F
6() } :
for b
1 being
State of
(F9() +* (1GateCircuit <*F1(),F2(),F3(),F4(),F5()*>,F10()))for b
2 being
State of F
9() holds
( b
2 = b
1 | the
carrier of F
8() implies for b
3, b
4, b
5, b
6, b
7 being
Element of F
6() holds
( ( F
1()
in InnerVertices F
8() implies b
3 = (Result b2) . F
1() ) & ( not F
1()
in InnerVertices F
8() implies b
3 = b
1 . F
1() ) & ( F
2()
in InnerVertices F
8() implies b
4 = (Result b2) . F
2() ) & ( not F
2()
in InnerVertices F
8() implies b
4 = b
1 . F
2() ) & ( F
3()
in InnerVertices F
8() implies b
5 = (Result b2) . F
3() ) & ( not F
3()
in InnerVertices F
8() implies b
5 = b
1 . F
3() ) & ( F
4()
in InnerVertices F
8() implies b
6 = (Result b2) . F
4() ) & ( not F
4()
in InnerVertices F
8() implies b
6 = b
1 . F
4() ) & ( F
5()
in InnerVertices F
8() implies b
7 = (Result b2) . F
5() ) & ( not F
5()
in InnerVertices F
8() implies b
7 = b
1 . F
5() ) implies
(Result b1) . (Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F10())) = F
7(b
3,b
4,b
5,b
6,b
7) ) )
provided
E51:
for b
1 being
Function of 5
-tuples_on F
6(),F
6() holds
( b
1 = F
10() iff for b
2, b
3, b
4, b
5, b
6 being
Element of F
6() holds b
1 . <*b2,b3,b4,b5,b6*> = F
7(b
2,b
3,b
4,b
5,b
6) )
and
E52:
not
Output (1GateCircStr <*F1(),F2(),F3(),F4(),F5()*>,F10()) in InputVertices F
8()
:: deftheorem Def11 defines with_nonpair_inputs CIRCCMB3:def 11 :
theorem Th53: :: CIRCCMB3:53
theorem Th54: :: CIRCCMB3:54
theorem Th55: :: CIRCCMB3:55
Lemma53:
for b1 being non pair set holds
not {b1} is with_pair
;
Lemma54:
for b1, b2 being without_pairs set holds
not b1 \/ b2 is with_pair
;
theorem Th56: :: CIRCCMB3:56
Lemma56:
for b1, b2 being non pair set holds
not {b1,b2} is with_pair
;
theorem Th57: :: CIRCCMB3:57
registration
let c
1 be non
empty finite set ;
let c
2 be
with_nonpair_inputs Signature of c
1;
let c
3, c
4 be
Vertex of c
2;
let c
5 be non
pair set ;
let c
6 be
Function of 3
-tuples_on c
1,c
1;
cluster a
2 +* (1GateCircStr <*a3,a4,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c3,c4,c5*>,c6) is with_nonpair_inputs
by Th57;
cluster a
2 +* (1GateCircStr <*a3,a5,a4*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c3,c5,c4*>,c6) is with_nonpair_inputs
by Th57;
cluster a
2 +* (1GateCircStr <*a5,a3,a4*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c5,c3,c4*>,c6) is with_nonpair_inputs
by Th57;
end;
registration
let c
1 be non
empty finite set ;
let c
2 be
with_nonpair_inputs Signature of c
1;
let c
3 be
Vertex of c
2;
let c
4, c
5 be non
pair set ;
let c
6 be
Function of 3
-tuples_on c
1,c
1;
cluster a
2 +* (1GateCircStr <*a3,a4,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c3,c4,c5*>,c6) is with_nonpair_inputs
by Th57;
cluster a
2 +* (1GateCircStr <*a4,a3,a5*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c4,c3,c5*>,c6) is with_nonpair_inputs
by Th57;
cluster a
2 +* (1GateCircStr <*a4,a5,a3*>,a6) -> gate`2=den with_nonpair_inputs ;
coherence
c2 +* (1GateCircStr <*c4,c5,c3*>,c6) is with_nonpair_inputs
by Th57;
end;