:: CIRCCOMB semantic presentation
theorem Th1: :: CIRCCOMB:1
canceled;
theorem Th2: :: CIRCCOMB:2
theorem Th3: :: CIRCCOMB:3
theorem Th4: :: CIRCCOMB:4
theorem Th5: :: CIRCCOMB:5
theorem Th6: :: CIRCCOMB:6
:: deftheorem Def1 defines tolerates CIRCCOMB:def 1 :
:: deftheorem Def2 defines +* CIRCCOMB:def 2 :
theorem Th7: :: CIRCCOMB:7
theorem Th8: :: CIRCCOMB:8
theorem Th9: :: CIRCCOMB:9
theorem Th10: :: CIRCCOMB:10
theorem Th11: :: CIRCCOMB:11
theorem Th12: :: CIRCCOMB:12
theorem Th13: :: CIRCCOMB:13
theorem Th14: :: CIRCCOMB:14
theorem Th15: :: CIRCCOMB:15
theorem Th16: :: CIRCCOMB:16
theorem Th17: :: CIRCCOMB:17
theorem Th18: :: CIRCCOMB:18
theorem Th19: :: CIRCCOMB:19
theorem Th20: :: CIRCCOMB:20
theorem Th21: :: CIRCCOMB:21
:: deftheorem Def3 defines tolerates CIRCCOMB:def 3 :
:: deftheorem Def4 defines +* CIRCCOMB:def 4 :
theorem Th22: :: CIRCCOMB:22
theorem Th23: :: CIRCCOMB:23
theorem Th24: :: CIRCCOMB:24
theorem Th25: :: CIRCCOMB:25
theorem Th26: :: CIRCCOMB:26
theorem Th27: :: CIRCCOMB:27
theorem Th28: :: CIRCCOMB:28
theorem Th29: :: CIRCCOMB:29
theorem Th30: :: CIRCCOMB:30
theorem Th31: :: CIRCCOMB:31
theorem Th32: :: CIRCCOMB:32
theorem Th33: :: CIRCCOMB:33
theorem Th34: :: CIRCCOMB:34
theorem Th35: :: CIRCCOMB:35
theorem Th36: :: CIRCCOMB:36
theorem Th37: :: CIRCCOMB:37
theorem Th38: :: CIRCCOMB:38
theorem Th39: :: CIRCCOMB:39
theorem Th40: :: CIRCCOMB:40
theorem Th41: :: CIRCCOMB:41
theorem Th42: :: CIRCCOMB:42
definition
let c
1 be
set ;
let c
2 be
FinSequence;
let c
3 be
set ;
func 1GateCircStr c
2,c
1,c
3 -> strict non
void ManySortedSign means :
Def5:
:: CIRCCOMB:def 5
( the
carrier of a
4 = (rng a2) \/ {a3} & the
OperSymbols of a
4 = {[a2,a1]} & the
Arity of a
4 . [a2,a1] = a
2 & the
ResultSort of a
4 . [a2,a1] = a
3 );
existence
ex b1 being strict non void ManySortedSign st
( the carrier of b1 = (rng c2) \/ {c3} & the OperSymbols of b1 = {[c2,c1]} & the Arity of b1 . [c2,c1] = c2 & the ResultSort of b1 . [c2,c1] = c3 )
uniqueness
for b1, b2 being strict non void ManySortedSign holds
( the carrier of b1 = (rng c2) \/ {c3} & the OperSymbols of b1 = {[c2,c1]} & the Arity of b1 . [c2,c1] = c2 & the ResultSort of b1 . [c2,c1] = c3 & the carrier of b2 = (rng c2) \/ {c3} & the OperSymbols of b2 = {[c2,c1]} & the Arity of b2 . [c2,c1] = c2 & the ResultSort of b2 . [c2,c1] = c3 implies b1 = b2 )
end;
:: deftheorem Def5 defines 1GateCircStr CIRCCOMB:def 5 :
theorem Th43: :: CIRCCOMB:43
theorem Th44: :: CIRCCOMB:44
theorem Th45: :: CIRCCOMB:45
definition
let c
1 be
set ;
let c
2 be
FinSequence;
func 1GateCircStr c
2,c
1 -> strict non
void ManySortedSign means :
Def6:
:: CIRCCOMB:def 6
( the
carrier of a
3 = (rng a2) \/ {[a2,a1]} & the
OperSymbols of a
3 = {[a2,a1]} & the
Arity of a
3 . [a2,a1] = a
2 & the
ResultSort of a
3 . [a2,a1] = [a2,a1] );
existence
ex b1 being strict non void ManySortedSign st
( the carrier of b1 = (rng c2) \/ {[c2,c1]} & the OperSymbols of b1 = {[c2,c1]} & the Arity of b1 . [c2,c1] = c2 & the ResultSort of b1 . [c2,c1] = [c2,c1] )
uniqueness
for b1, b2 being strict non void ManySortedSign holds
( the carrier of b1 = (rng c2) \/ {[c2,c1]} & the OperSymbols of b1 = {[c2,c1]} & the Arity of b1 . [c2,c1] = c2 & the ResultSort of b1 . [c2,c1] = [c2,c1] & the carrier of b2 = (rng c2) \/ {[c2,c1]} & the OperSymbols of b2 = {[c2,c1]} & the Arity of b2 . [c2,c1] = c2 & the ResultSort of b2 . [c2,c1] = [c2,c1] implies b1 = b2 )
end;
:: deftheorem Def6 defines 1GateCircStr CIRCCOMB:def 6 :
theorem Th46: :: CIRCCOMB:46
theorem Th47: :: CIRCCOMB:47
theorem Th48: :: CIRCCOMB:48
theorem Th49: :: CIRCCOMB:49
theorem Th50: :: CIRCCOMB:50
theorem Th51: :: CIRCCOMB:51
:: deftheorem Def7 defines unsplit CIRCCOMB:def 7 :
:: deftheorem Def8 defines gate`1=arity CIRCCOMB:def 8 :
:: deftheorem Def9 defines gate`2isBoolean CIRCCOMB:def 9 :
:: deftheorem Def10 defines gate`2=den CIRCCOMB:def 10 :
:: deftheorem Def11 defines gate`2=den CIRCCOMB:def 11 :
theorem Th52: :: CIRCCOMB:52
theorem Th53: :: CIRCCOMB:53
theorem Th54: :: CIRCCOMB:54
theorem Th55: :: CIRCCOMB:55
theorem Th56: :: CIRCCOMB:56
theorem Th57: :: CIRCCOMB:57
theorem Th58: :: CIRCCOMB:58
theorem Th59: :: CIRCCOMB:59
:: deftheorem Def12 defines FinSeqLen CIRCCOMB:def 12 :
definition
let c
1 be
Nat;
let c
2, c
3 be non
empty set ;
let c
4 be
Function of c
1 -tuples_on c
2,c
3;
let c
5 be
FinSeqLen of c
1;
let c
6 be
set ;
assume E53:
( c
6 in rng c
5 implies c
2 = c
3 )
;
func 1GateCircuit c
5,c
4,c
6 -> strict non-empty MSAlgebra of
1GateCircStr a
5,a
4,a
6 means :: CIRCCOMB:def 13
( the
Sorts of a
7 = ((rng a5) --> a2) +* ({a6} --> a3) & the
Charact of a
7 . [a5,a4] = a
4 );
existence
ex b1 being strict non-empty MSAlgebra of 1GateCircStr c5,c4,c6 st
( the Sorts of b1 = ((rng c5) --> c2) +* ({c6} --> c3) & the Charact of b1 . [c5,c4] = c4 )
uniqueness
for b1, b2 being strict non-empty MSAlgebra of 1GateCircStr c5,c4,c6 holds
( the Sorts of b1 = ((rng c5) --> c2) +* ({c6} --> c3) & the Charact of b1 . [c5,c4] = c4 & the Sorts of b2 = ((rng c5) --> c2) +* ({c6} --> c3) & the Charact of b2 . [c5,c4] = c4 implies b1 = b2 )
end;
:: deftheorem Def13 defines 1GateCircuit CIRCCOMB:def 13 :
definition
let c
1 be
Nat;
let c
2 be non
empty set ;
let c
3 be
Function of c
1 -tuples_on c
2,c
2;
let c
4 be
FinSeqLen of c
1;
func 1GateCircuit c
4,c
3 -> strict non-empty MSAlgebra of
1GateCircStr a
4,a
3 means :
Def14:
:: CIRCCOMB:def 14
( the
Sorts of a
5 = the
carrier of
(1GateCircStr a4,a3) --> a
2 & the
Charact of a
5 . [a4,a3] = a
3 );
existence
ex b1 being strict non-empty MSAlgebra of 1GateCircStr c4,c3 st
( the Sorts of b1 = the carrier of (1GateCircStr c4,c3) --> c2 & the Charact of b1 . [c4,c3] = c3 )
uniqueness
for b1, b2 being strict non-empty MSAlgebra of 1GateCircStr c4,c3 holds
( the Sorts of b1 = the carrier of (1GateCircStr c4,c3) --> c2 & the Charact of b1 . [c4,c3] = c3 & the Sorts of b2 = the carrier of (1GateCircStr c4,c3) --> c2 & the Charact of b2 . [c4,c3] = c3 implies b1 = b2 )
end;
:: deftheorem Def14 defines 1GateCircuit CIRCCOMB:def 14 :
theorem Th60: :: CIRCCOMB:60
theorem Th61: :: CIRCCOMB:61
theorem Th62: :: CIRCCOMB:62
theorem Th63: :: CIRCCOMB:63
theorem Th64: :: CIRCCOMB:64
:: deftheorem Def15 defines Boolean CIRCCOMB:def 15 :
theorem Th65: :: CIRCCOMB:65
theorem Th66: :: CIRCCOMB:66
theorem Th67: :: CIRCCOMB:67
theorem Th68: :: CIRCCOMB:68
theorem Th69: :: CIRCCOMB:69
theorem Th70: :: CIRCCOMB:70
theorem Th71: :: CIRCCOMB:71
theorem Th72: :: CIRCCOMB:72