:: CIRCCMB3 semantic presentation

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A
for x being set st x in InputVertices S holds
for n being Nat holds (Following s,n) . x = s . x

proof end;

definition

let S be non empty non void Circuit-like ManySortedSign ;
let A be non-empty Circuit of S;
let s be State of A;
attr s is stabilizing means :Def1: :: CIRCCMB3:def 1
ex n being Nat st Following s,n is stable;

end;

:: deftheorem Def1 defines stabilizing CIRCCMB3:def 1 :

definition

let S be non empty non void Circuit-like ManySortedSign ;
let A be non-empty Circuit of S;
attr A is stabilizing means :Def2: :: CIRCCMB3:def 2
for s being State of A holds s is stabilizing;
attr A is with_stabilization-limit means :: CIRCCMB3:def 3
ex n being Nat st
for s being State of A holds Following s,n is stable;

end;

:: deftheorem Def2 defines stabilizing CIRCCMB3:def 2 :

:: deftheorem defines with_stabilization-limit CIRCCMB3:def 3 :

registration

let S be non empty non void Circuit-like ManySortedSign ;
cluster non-empty with_stabilization-limit -> non-empty stabilizing MSAlgebra of S;
coherence

proof end;

end;

definition

let S be non empty non void Circuit-like ManySortedSign ;
let A be non-empty Circuit of S;
let s be State of A;
assume A1: s is stabilizing ;
func Result s -> State of A means :Def4: :: CIRCCMB3:def 4
( it is stable & ex n being Nat st it = Following s,n );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def4 defines Result CIRCCMB3:def 4 :

definition

let S be non empty non void Circuit-like ManySortedSign ;
let A be non-empty Circuit of S;
let s be State of A;
assume A1: s is stabilizing ;
func stabilization-time s -> Nat means :Def5: :: CIRCCMB3:def 5
( Following s,it is stable & ( for n being Nat st n < it holds
not Following s,n is stable ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines stabilization-time CIRCCMB3:def 5 :

theorem Th2: :: CIRCCMB3:2

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A st s is stabilizing holds
Result s = Following s,(stabilization-time s)

proof end;

theorem :: CIRCCMB3:3

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A
for n being Nat st Following s,n is stable holds
stabilization-time s <= n

proof end;

theorem Th4: :: CIRCCMB3:4

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A
for n being Nat st Following s,n is stable holds
Result s = Following s,n

proof end;

theorem Th5: :: CIRCCMB3:5

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A
for n being Nat st s is stabilizing & n >= stabilization-time s holds
Result s = Following s,n

proof end;

theorem :: CIRCCMB3:6

for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A st s is stabilizing holds
for x being set st x in InputVertices S holds
(Result s) . x = s . x

proof end;

theorem :: CIRCCMB3:7

canceled;

theorem Th8: :: CIRCCMB3:8

for S1, S2 being non empty non void Circuit-like ManySortedSign st InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 holds
for S being non empty non void Circuit-like ManySortedSign st S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st A = A1 +* A2 holds
for s being State of A
for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing holds
s is stabilizing

proof end;

theorem :: CIRCCMB3:9

for S1, S2 being non empty non void Circuit-like ManySortedSign st InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 holds
for S being non empty non void Circuit-like ManySortedSign st S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = s | the carrier of S2 & s2 is stabilizing holds
stabilization-time s = max (stabilization-time s1),(stabilization-time s2)

proof end;

theorem Th10: :: CIRCCMB3:10

for S1, S2 being non empty non void Circuit-like ManySortedSign st InputVertices S1 misses InnerVertices S2 holds
for S being non empty non void Circuit-like ManySortedSign st S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following s,(stabilization-time s1)) | the carrier of S2 & s2 is stabilizing holds
s is stabilizing

proof end;

theorem Th11: :: CIRCCMB3:11

proof end;

theorem :: CIRCCMB3:12

for S1, S2, S being non empty non void Circuit-like ManySortedSign st InputVertices S1 misses InnerVertices S2 & S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following s,(stabilization-time s1)) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1

proof end;

theorem Th13: :: CIRCCMB3:13

for x being set
for X being non empty finite set
for n being Nat
for p being FinSeqLen of n
for g being Function of n -tuples_on X,X
for s being State of (1GateCircuit p,g) holds s * p is Element of n -tuples_on X

proof end;

theorem Th14: :: CIRCCMB3:14

for x1, x2, x3, x4 being set holds rng <*x1,x2,x3,x4*> = {x1,x2,x3,x4}

proof end;

theorem Th15: :: CIRCCMB3:15

for x1, x2, x3, x4, x5 being set holds rng <*x1,x2,x3,x4,x5*> = {x1,x2,x3,x4,x5}

proof end;

definition

let x1, x2, x3, x4 be set ;
:: original: <*
redefine func <*x1,x2,x3,x4*> -> FinSeqLen of 4;
coherence

proof end;

let x5 be set ;
:: original: <*
redefine func <*x1,x2,x3,x4,x5*> -> FinSeqLen of 5;
coherence

proof end;

end;

definition

let S be ManySortedSign ;
attr S is one-gate means :Def6: :: CIRCCMB3:def 6
ex X being non empty finite set ex n being Nat ex p being FinSeqLen of n ex f being Function of n -tuples_on X,X st S = 1GateCircStr p,f;

end;

:: deftheorem Def6 defines one-gate CIRCCMB3:def 6 :

definition

let S be non empty ManySortedSign ;
let A be MSAlgebra of S;
attr A is one-gate means :Def7: :: CIRCCMB3:def 7
ex X being non empty finite set ex n being Nat ex p being FinSeqLen of n ex f being Function of n -tuples_on X,X st
( S = 1GateCircStr p,f & A = 1GateCircuit p,f );

end;

:: deftheorem Def7 defines one-gate CIRCCMB3:def 7 :

registration

let p be FinSequence;
let x be set ;
cluster 1GateCircStr p,x -> finite ;
coherence

proof end;

end;

registration

cluster one-gate -> non empty finite strict non void unsplit gate`1=arity ManySortedSign ;
coherence

proof end;

end;

registration

cluster non empty one-gate -> non empty gate`2=den ManySortedSign ;
coherence

proof end;

end;

registration

let X be non empty finite set ;
let n be Nat;
let p be FinSeqLen of n;
let f be Function of n -tuples_on X,X;
cluster 1GateCircStr p,f -> finite one-gate ;
coherence by Def6;

end;

registration

cluster non empty finite strict non void unsplit gate`1=arity gate`2=den one-gate ManySortedSign ;
existence

proof end;

end;

registration

let S be one-gate ManySortedSign ;
cluster one-gate -> strict non-empty MSAlgebra of S;
coherence

proof end;

end;

registration

let X be non empty finite set ;
let n be Nat;
let p be FinSeqLen of n;
let f be Function of n -tuples_on X,X;
cluster 1GateCircuit p,f -> one-gate ;
coherence by Def7;

end;

registration

let S be one-gate ManySortedSign ;
cluster strict non-empty one-gate MSAlgebra of S;
existence

proof end;

end;

definition

let S be one-gate ManySortedSign ;
func Output S -> Vertex of S equals :: CIRCCMB3:def 8
union the OperSymbols of S;
coherence

proof end;

end;

:: deftheorem defines Output CIRCCMB3:def 8 :

registration

let S be one-gate ManySortedSign ;
cluster Output S -> pair ;
coherence

proof end;

end;

theorem Th16: :: CIRCCMB3:16

for S being one-gate ManySortedSign
for p being FinSequence
for x being set st S = 1GateCircStr p,x holds
Output S = [p,x]

proof end;

theorem Th17: :: CIRCCMB3:17

for S being one-gate ManySortedSign holds InnerVertices S = {(Output S)}

proof end;

theorem :: CIRCCMB3:18

for S being one-gate ManySortedSign
for A being one-gate Circuit of S
for n being Nat
for X being non empty finite set
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n st A = 1GateCircuit p,f holds
S = 1GateCircStr p,f

proof end;

theorem :: CIRCCMB3:19

for n being Nat
for X being non empty finite set
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n
for s being State of (1GateCircuit p,f) holds (Following s) . (Output (1GateCircStr p,f)) = f . (s * p)

proof end;

theorem Th20: :: CIRCCMB3:20

for S being one-gate ManySortedSign
for A being one-gate Circuit of S
for s being State of A holds Following s is stable

proof end;

registration

let S be non empty non void Circuit-like ManySortedSign ;
cluster non-empty one-gate -> non-empty stabilizing with_stabilization-limit MSAlgebra of S;
coherence

proof end;

end;

theorem Th21: :: CIRCCMB3:21

for S being one-gate ManySortedSign
for A being one-gate Circuit of S
for s being State of A holds Result s = Following s

proof end;

theorem Th22: :: CIRCCMB3:22

for S being one-gate ManySortedSign
for A being one-gate Circuit of S
for s being State of A holds stabilization-time s <= 1

proof end;

scheme :: CIRCCMB3:sch 1
OneGate1Ex{ F₁() -> set , F₂() -> non empty finite set , F₃( set ) -> Element of F₂() } :

ex S being one-gate ManySortedSign ex A being one-gate Circuit of S st
( InputVertices S = {F₁()} & ( for s being State of A holds (Result s) . (Output S) = F₃((s . F₁())) ) )

proof end;

scheme :: CIRCCMB3:sch 2
OneGate2Ex{ F₁() -> set , F₂() -> set , F₃() -> non empty finite set , F₄( set , set ) -> Element of F₃() } :

ex S being one-gate ManySortedSign ex A being one-gate Circuit of S st
( InputVertices S = {F₁(),F₂()} & ( for s being State of A holds (Result s) . (Output S) = F₄((s . F₁()),(s . F₂())) ) )

proof end;

scheme :: CIRCCMB3:sch 3
OneGate3Ex{ F₁() -> set , F₂() -> set , F₃() -> set , F₄() -> non empty finite set , F₅( set , set , set ) -> Element of F₄() } :

ex S being one-gate ManySortedSign ex A being one-gate Circuit of S st
( InputVertices S = {F₁(),F₂(),F₃()} & ( for s being State of A holds (Result s) . (Output S) = F₅((s . F₁()),(s . F₂()),(s . F₃())) ) )

proof end;

scheme :: CIRCCMB3:sch 4
OneGate4Ex{ F₁() -> set , F₂() -> set , F₃() -> set , F₄() -> set , F₅() -> non empty finite set , F₆( set , set , set , set ) -> Element of F₅() } :

ex S being one-gate ManySortedSign ex A being one-gate Circuit of S st
( InputVertices S = {F₁(),F₂(),F₃(),F₄()} & ( for s being State of A holds (Result s) . (Output S) = F₆((s . F₁()),(s . F₂()),(s . F₃()),(s . F₄())) ) )

proof end;

scheme :: CIRCCMB3:sch 5
OneGate5Ex{ F₁() -> set , F₂() -> set , F₃() -> set , F₄() -> set , F₅() -> set , F₆() -> non empty finite set , F₇( set , set , set , set , set ) -> Element of F₆() } :

ex S being one-gate ManySortedSign ex A being one-gate Circuit of S st
( InputVertices S = {F₁(),F₂(),F₃(),F₄(),F₅()} & ( for s being State of A holds (Result s) . (Output S) = F₇((s . F₁()),(s . F₂()),(s . F₃()),(s . F₄()),(s . F₅())) ) )

proof end;

theorem Th23: :: CIRCCMB3:23

for f being constant Function holds f = (dom f) --> (the_value_of f)

proof end;

theorem Th24: :: CIRCCMB3:24

for X, Y being non empty set
for n, m being Nat st n <> 0 & n -tuples_on X = m -tuples_on Y holds
( X = Y & n = m )

proof end;

theorem :: CIRCCMB3:25

for S1, S2 being non empty ManySortedSign
for v being Vertex of S1 holds v is Vertex of (S1 +* S2)

proof end;

theorem :: CIRCCMB3:26

for S1, S2 being non empty ManySortedSign
for v being Vertex of S2 holds v is Vertex of (S1 +* S2)

proof end;

definition

let X be non empty finite set ;
mode Signature of X -> non empty non void unsplit gate`1=arity gate`2=den ManySortedSign means :Def9: :: CIRCCMB3:def 9
ex A being Circuit of it st
( the Sorts of A is constant & the_value_of the Sorts of A = X & A is gate`2=den );
existence

proof end;

end;

:: deftheorem Def9 defines Signature CIRCCMB3:def 9 :

theorem Th27: :: CIRCCMB3:27

for n being Nat
for X being non empty finite set
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n holds 1GateCircStr p,f is Signature of X

proof end;

registration

let X be non empty finite set ;
cluster non empty finite strict non void unsplit gate`1=arity gate`2=den one-gate Signature of X;
existence

proof end;

end;

definition

let n be Nat;
let X be non empty finite set ;
let f be Function of n -tuples_on X,X;
let p be FinSeqLen of n;
:: original: 1GateCircStr
redefine func 1GateCircStr p,f -> strict Signature of X;
coherence by Th27;

end;

definition

let X be non empty finite set ;
let S be Signature of X;
mode Circuit of X,S -> Circuit of S means :Def10: :: CIRCCMB3:def 10
( it is gate`2=den & the Sorts of it is constant & the_value_of the Sorts of it = X );
existence

proof end;

end;

:: deftheorem Def10 defines Circuit CIRCCMB3:def 10 :

registration

let X be non empty finite set ;
let S be Signature of X;
cluster -> non-empty gate`2=den Circuit of X,S;
coherence

proof end;

end;

theorem Th28: :: CIRCCMB3:28

for n being Nat
for X being non empty finite set
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n holds 1GateCircuit p,f is Circuit of X, 1GateCircStr p,f

proof end;

registration

let X be non empty finite set ;
let S be one-gate Signature of X;
cluster strict non-empty gate`2=den stabilizing with_stabilization-limit one-gate Circuit of X,S;
existence

proof end;

end;

registration

let X be non empty finite set ;
let S be Signature of X;
cluster strict non-empty gate`2=den Circuit of X,S;
existence

proof end;

end;

definition

let n be Nat;
let X be non empty finite set ;
let f be Function of n -tuples_on X,X;
let p be FinSeqLen of n;
:: original: 1GateCircuit
redefine func 1GateCircuit p,f -> strict Circuit of X, 1GateCircStr p,f;
coherence by Th28;

end;

theorem :: CIRCCMB3:29

canceled;

theorem Th30: :: CIRCCMB3:30

for X being non empty finite set
for S1, S2 being Signature of X
for A1 being Circuit of X,S1
for A2 being Circuit of X,S2 holds A1 tolerates A2

proof end;

theorem Th31: :: CIRCCMB3:31

for X being non empty finite set
for S1, S2 being Signature of X
for A1 being Circuit of X,S1
for A2 being Circuit of X,S2 holds A1 +* A2 is Circuit of S1 +* S2

proof end;

theorem Th32: :: CIRCCMB3:32

for X being non empty finite set
for S1, S2 being Signature of X
for A1 being Circuit of X,S1
for A2 being Circuit of X,S2 holds A1 +* A2 is gate`2=den

proof end;

theorem Th33: :: CIRCCMB3:33

for X being non empty finite set
for S1, S2 being Signature of X
for A1 being Circuit of X,S1
for A2 being Circuit of X,S2 holds
( the Sorts of (A1 +* A2) is constant & the_value_of the Sorts of (A1 +* A2) = X )

proof end;

registration

let S1, S2 be non empty finite ManySortedSign ;
cluster S1 +* S2 -> finite ;
coherence

proof end;

end;

registration

let X be non empty finite set ;
let S1, S2 be Signature of X;
cluster S1 +* S2 -> gate`2=den ;
coherence

proof end;

end;

definition

let X be non empty finite set ;
let S1, S2 be Signature of X;
:: original: +*
redefine func S1 +* S2 -> strict Signature of X;
coherence

proof end;

end;

definition

let X be non empty finite set ;
let S1, S2 be Signature of X;
let A1 be Circuit of X,S1;
let A2 be Circuit of X,S2;
:: original: +*
redefine func A1 +* A2 -> strict Circuit of X,S1 +* S2;
coherence

proof end;

end;

theorem Th34: :: CIRCCMB3:34

for x, y being set holds
( the_rank_of x in the_rank_of [x,y] & the_rank_of y in the_rank_of [x,y] )

proof end;

theorem Th35: :: CIRCCMB3:35

for S being non empty finite non void unsplit gate`1=arity gate`2=den ManySortedSign
for A being non-empty Circuit of S st A is gate`2=den holds
A is with_stabilization-limit

proof end;

registration

let X be non empty finite set ;
let S be finite Signature of X;
cluster -> non-empty gate`2=den with_stabilization-limit Circuit of X,S;
coherence by Th35;

end;

scheme :: CIRCCMB3:sch 6
1AryDef{ F₁() -> non empty set , F₂( set ) -> Element of F₁() } :

( ex f being Function of 1 -tuples_on F₁(),F₁() st
for x being Element of F₁() holds f . <*x*> = F₂(x) & ( for f1, f2 being Function of 1 -tuples_on F₁(),F₁() st ( for x being Element of F₁() holds f1 . <*x*> = F₂(x) ) & ( for x being Element of F₁() holds f2 . <*x*> = F₂(x) ) holds
f1 = f2 ) )

proof end;

scheme :: CIRCCMB3:sch 7
2AryDef{ 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 ) )

proof end;

scheme :: CIRCCMB3:sch 8
3AryDef{ 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 ) )

proof end;

theorem Th36: :: CIRCCMB3:36

for f being Function
for x being set st x in dom f holds
f * <*x*> = <*(f . x)*>

proof end;

theorem Th37: :: CIRCCMB3:37

for f being Function
for x1, x2, x3, x4 being set st x1 in dom f & x2 in dom f & x3 in dom f & x4 in dom f holds
f * <*x1,x2,x3,x4*> = <*(f . x1),(f . x2),(f . x3),(f . x4)*>

proof end;

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)*>

proof end;

scheme :: CIRCCMB3:sch 9
OneGate1Result{ F₁() -> set , F₂() -> non empty finite set , F₃( set ) -> Element of F₂(), F₄() -> Function of 1 -tuples_on F₂(),F₂() } :

for s being State of (1GateCircuit <*F₁()*>,F₄())
for a1 being Element of F₂() st a1 = s . F₁() holds
(Result s) . (Output (1GateCircStr <*F₁()*>,F₄())) = F₃(a1)

provided

A1: for g being Function of 1 -tuples_on F₂(),F₂() holds
( g = F₄() iff for a1 being Element of F₂() holds g . <*a1*> = F₃(a1) )

proof end;

scheme :: CIRCCMB3:sch 10
OneGate2Result{ F₁() -> set , F₂() -> set , F₃() -> non empty finite set , F₄( set , set ) -> Element of F₃(), F₅() -> Function of 2 -tuples_on F₃(),F₃() } :

for s being State of (1GateCircuit <*F₁(),F₂()*>,F₅())
for a1, a2 being Element of F₃() st a1 = s . F₁() & a2 = s . F₂() holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂()*>,F₅())) = F₄(a1,a2)

provided

A1: for g being Function of 2 -tuples_on F₃(),F₃() holds
( g = F₅() iff for a1, a2 being Element of F₃() holds g . <*a1,a2*> = F₄(a1,a2) )

proof end;

scheme :: CIRCCMB3:sch 11
OneGate3Result{ 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

A1: for g being Function of 3 -tuples_on F₄(),F₄() holds
( g = F₆() iff for a1, a2, a3 being Element of F₄() holds g . <*a1,a2,a3*> = F₅(a1,a2,a3) )

proof end;

scheme :: CIRCCMB3:sch 12
OneGate4Result{ 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

A1: 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) )

proof end;

scheme :: CIRCCMB3:sch 13
OneGate5Result{ 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

A1: 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) )

proof end;

theorem Th39: :: CIRCCMB3:39

for n being Nat
for X being non empty finite set
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n
for S being Signature of X st rng p c= the carrier of S & not Output (1GateCircStr p,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr p,f)) = InputVertices S

proof end;

theorem Th40: :: CIRCCMB3:40

for X1, X2 being set
for X being non empty finite set
for n being Nat
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n
for S being Signature of X st rng p = X1 \/ X2 & X1 c= the carrier of S & X2 misses InnerVertices S & not Output (1GateCircStr p,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr p,f)) = (InputVertices S) \/ X2

proof end;

theorem Th41: :: CIRCCMB3:41

for x1 being set
for X being non empty finite set
for f being Function of 1 -tuples_on X,X
for S being Signature of X st x1 in the carrier of S & not Output (1GateCircStr <*x1*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1*>,f)) = InputVertices S

proof end;

theorem Th42: :: CIRCCMB3:42

for x1, x2 being set
for X being non empty finite set
for f being Function of 2 -tuples_on X,X
for S being Signature of X st x1 in the carrier of S & not x2 in InnerVertices S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = (InputVertices S) \/ {x2}

proof end;

theorem Th43: :: CIRCCMB3:43

for x1, x2 being set
for X being non empty finite set
for f being Function of 2 -tuples_on X,X
for S being Signature of X st x2 in the carrier of S & not x1 in InnerVertices S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = (InputVertices S) \/ {x1}

proof end;

theorem Th44: :: CIRCCMB3:44

for x1, x2 being set
for X being non empty finite set
for f being Function of 2 -tuples_on X,X
for S being Signature of X st x1 in the carrier of S & x2 in the carrier of S & not Output (1GateCircStr <*x1,x2*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2*>,f)) = InputVertices S

proof end;

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}

proof end;

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}

proof end;

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}

proof end;

theorem Th48: :: CIRCCMB3:48

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 & x2 in the carrier of 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) \/ {x3}

proof end;

theorem Th49: :: CIRCCMB3:49

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 & x3 in the carrier of 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) \/ {x2}

proof end;

theorem Th50: :: CIRCCMB3:50

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 & x3 in the carrier of S & not x1 in InnerVertices S & not Output (1GateCircStr <*x1,x2,x3*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2,x3*>,f)) = (InputVertices S) \/ {x1}

proof end;

theorem Th51: :: CIRCCMB3:51

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 & x2 in the carrier of S & x3 in the carrier of S & not Output (1GateCircStr <*x1,x2,x3*>,f) in InputVertices S holds
InputVertices (S +* (1GateCircStr <*x1,x2,x3*>,f)) = InputVertices S

proof end;

theorem Th52: :: CIRCCMB3:52

for X being non empty finite set
for S being finite Signature of X
for A being Circuit of X,S
for n being Nat
for f being Function of n -tuples_on X,X
for p being FinSeqLen of n st not Output (1GateCircStr p,f) in InputVertices S holds
for s being State of (A +* (1GateCircuit p,f))
for s' being State of A st s' = s | the carrier of S holds
stabilization-time s <= 1 + (stabilization-time s')

proof end;

scheme :: CIRCCMB3:sch 14
Comb1CircResult{ F₁() -> set , F₂() -> non empty finite set , F₃( set ) -> Element of F₂(), F₄() -> finite Signature of F₂(), F₅() -> Circuit of F₂(),F₄(), F₆() -> Function of 1 -tuples_on F₂(),F₂() } :

for s being State of (F₅() +* (1GateCircuit <*F₁()*>,F₆()))
for s' being State of F₅() st s' = s | the carrier of F₄() holds
for a1 being Element of F₂() st ( F₁() in InnerVertices F₄() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₄() implies a1 = s . F₁() ) holds
(Result s) . (Output (1GateCircStr <*F₁()*>,F₆())) = F₃(a1)

provided

A1: for g being Function of 1 -tuples_on F₂(),F₂() holds
( g = F₆() iff for a1 being Element of F₂() holds g . <*a1*> = F₃(a1) ) and
A2: not Output (1GateCircStr <*F₁()*>,F₆()) in InputVertices F₄()

proof end;

scheme :: CIRCCMB3:sch 15
Comb2CircResult{ F₁() -> set , F₂() -> set , F₃() -> non empty finite set , F₄( set , set ) -> Element of F₃(), F₅() -> finite Signature of F₃(), F₆() -> Circuit of F₃(),F₅(), F₇() -> Function of 2 -tuples_on F₃(),F₃() } :

for s being State of (F₆() +* (1GateCircuit <*F₁(),F₂()*>,F₇()))
for s' being State of F₆() st s' = s | the carrier of F₅() holds
for a1, a2 being Element of F₃() st ( F₁() in InnerVertices F₅() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₅() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₅() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₅() implies a2 = s . F₂() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂()*>,F₇())) = F₄(a1,a2)

provided

A1: for g being Function of 2 -tuples_on F₃(),F₃() holds
( g = F₇() iff for a1, a2 being Element of F₃() holds g . <*a1,a2*> = F₄(a1,a2) ) and
A2: not Output (1GateCircStr <*F₁(),F₂()*>,F₇()) in InputVertices F₅()

proof end;

scheme :: CIRCCMB3:sch 16
Comb3CircResult{ 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

A1: 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
A2: not Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) in InputVertices F₆()

proof end;

scheme :: CIRCCMB3:sch 17
Comb4CircResult{ 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

A1: 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
A2: not Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄()*>,F₉()) in InputVertices F₇()

proof end;

scheme :: CIRCCMB3:sch 18
Comb5CircResult{ 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

A1: 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
A2: not Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) in InputVertices F₈()

proof end;

definition

let S be non empty ManySortedSign ;
attr S is with_nonpair_inputs means :Def11: :: CIRCCMB3:def 11
not InputVertices S is with_pair;

end;

:: deftheorem Def11 defines with_nonpair_inputs CIRCCMB3:def 11 :

registration

cluster NAT -> without_pairs ;
coherence

proof end;

let X be without_pairs set ;
cluster -> without_pairs Element of bool X;
coherence

proof end;

end;

registration

cluster natural-yielding -> nonpair-yielding set ;
coherence

proof end;

end;

registration

cluster -> natural-yielding FinSequence of NAT ;
coherence

proof end;

end;

registration

cluster one-to-one nonpair-yielding natural-yielding set ;
existence

proof end;

end;

registration

let n be Nat;
cluster one-to-one nonpair-yielding natural-yielding FinSeqLen of n;
existence

proof end;

end;

registration

let p be nonpair-yielding FinSequence;
let f be set ;
cluster 1GateCircStr p,f -> finite with_nonpair_inputs ;
coherence

proof end;

end;

registration

cluster non empty finite strict non void unsplit gate`1=arity gate`2=den one-gate with_nonpair_inputs ManySortedSign ;
existence

proof end;

let X be non empty finite set ;
cluster non empty finite strict non void unsplit gate`1=arity gate`2=den one-gate with_nonpair_inputs Signature of X;
existence

proof end;

end;

registration

let S be non empty with_nonpair_inputs ManySortedSign ;
cluster InputVertices S -> without_pairs ;
coherence by Def11;

end;

theorem :: CIRCCMB3:53

for S being non empty with_nonpair_inputs ManySortedSign
for x being Vertex of S st x is pair holds
x in InnerVertices S

proof end;

registration

let S be non empty unsplit gate`1=arity ManySortedSign ;
cluster InnerVertices S -> Relation-like ;
coherence

proof end;

end;

registration

let S be non empty non void unsplit gate`2=den ManySortedSign ;
cluster InnerVertices S -> Relation-like ;
coherence

proof end;

end;

registration

let S1, S2 be non empty unsplit gate`1=arity with_nonpair_inputs ManySortedSign ;
cluster S1 +* S2 -> with_nonpair_inputs ;
coherence

proof end;

end;

theorem :: CIRCCMB3:54

for x being non pair set
for R being Relation holds not x in R

proof end;

theorem Th55: :: CIRCCMB3:55

for x1 being set
for X being non empty finite set
for f being Function of 1 -tuples_on X,X
for S being with_nonpair_inputs Signature of X st ( x1 in the carrier of S or not x1 is pair ) holds
S +* (1GateCircStr <*x1*>,f) is with_nonpair_inputs

proof end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1 be Vertex of S;
let f be Function of 1 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1*>,f) -> with_nonpair_inputs ;
coherence by Th55;

end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1 be non pair set ;
let f be Function of 1 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1*>,f) -> gate`2=den with_nonpair_inputs ;
coherence ;

end;

Lm1: for x being non pair set holds not {x} is with_pair
;

Lm2: for x, y being without_pairs set holds not x \/ y is with_pair
;

theorem Th56: :: CIRCCMB3:56

for x1, x2 being set
for X being non empty finite set
for f being Function of 2 -tuples_on X,X
for S being with_nonpair_inputs Signature of X st ( x1 in the carrier of S or not x1 is pair ) & ( x2 in the carrier of S or not x2 is pair ) holds
S +* (1GateCircStr <*x1,x2*>,f) is with_nonpair_inputs

proof end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1 be Vertex of S;
let n2 be non pair set ;
let f be Function of 2 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1,n2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th56;
cluster S +* (1GateCircStr <*n2,x1*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th56;

end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1, x2 be Vertex of S;
let f be Function of 2 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1,x2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th56;

end;

Lm3: for x, y being non pair set holds not {x,y} is with_pair
;

theorem Th57: :: CIRCCMB3:57

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 with_nonpair_inputs Signature of X st ( x1 in the carrier of S or not x1 is pair ) & ( x2 in the carrier of S or not x2 is pair ) & ( x3 in the carrier of S or not x3 is pair ) holds
S +* (1GateCircStr <*x1,x2,x3*>,f) is with_nonpair_inputs

proof end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1, x2 be Vertex of S;
let n be non pair set ;
let f be Function of 3 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1,x2,n*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;
cluster S +* (1GateCircStr <*x1,n,x2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;
cluster S +* (1GateCircStr <*n,x1,x2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;

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, n2 be non pair set ;
let f be Function of 3 -tuples_on X,X;
cluster S +* (1GateCircStr <*x,n1,n2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;
cluster S +* (1GateCircStr <*n1,x,n2*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;
cluster S +* (1GateCircStr <*n1,n2,x*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;

end;

registration

let X be non empty finite set ;
let S be with_nonpair_inputs Signature of X;
let x1, x2, x3 be Vertex of S;
let f be Function of 3 -tuples_on X,X;
cluster S +* (1GateCircStr <*x1,x2,x3*>,f) -> gate`2=den with_nonpair_inputs ;
coherence by Th57;

end;