:: Preliminaries to Automatic Generation of Mizar Documentation for Circuits
:: by Grzegorz Bancerek and Adam Naumowicz
::
:: Received July 26, 2002
:: Copyright (c) 2002 Association of Mizar Users


theorem Th1: :: CIRCCMB3:1
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 Element of 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 Element of NAT st Following s,n is stable;
end;

:: deftheorem Def1 defines stabilizing CIRCCMB3:def 1 :
for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S
for s being State of A holds
( s is stabilizing iff ex n being Element of NAT st Following s,n is stable );

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 Element of NAT st
for s being State of A holds Following s,n is stable;
end;

:: deftheorem Def2 defines stabilizing CIRCCMB3:def 2 :
for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S holds
( A is stabilizing iff for s being State of A holds s is stabilizing );

:: deftheorem defines with_stabilization-limit CIRCCMB3:def 3 :
for S being non empty non void Circuit-like ManySortedSign
for A being non-empty Circuit of S holds
( A is with_stabilization-limit iff ex n being Element of NAT st
for s being State of A holds Following s,n is stable );

registration
let S be non empty non void Circuit-like ManySortedSign ;
cluster non-empty with_stabilization-limit -> non-empty stabilizing MSAlgebra of S;
coherence
for b1 being non-empty Circuit of S st b1 is with_stabilization-limit holds
b1 is stabilizing
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 Element of NAT st it = Following s,n );
existence
ex b1 being State of A st
( b1 is stable & ex n being Element of NAT st b1 = Following s,n )
proof end;
uniqueness
for b1, b2 being State of A st b1 is stable & ex n being Element of NAT st b1 = Following s,n & b2 is stable & ex n being Element of NAT st b2 = Following s,n holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines Result CIRCCMB3:def 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 st s is stabilizing holds
for b4 being State of A holds
( b4 = Result s iff ( b4 is stable & ex n being Element of NAT st b4 = Following s,n ) );

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 -> Element of NAT means :Def5: :: CIRCCMB3:def 5
( Following s,it is stable & ( for n being Element of NAT st n < it holds
not Following s,n is stable ) );
existence
ex b1 being Element of NAT st
( Following s,b1 is stable & ( for n being Element of NAT st n < b1 holds
not Following s,n is stable ) )
proof end;
uniqueness
for b1, b2 being Element of NAT st Following s,b1 is stable & ( for n being Element of NAT st n < b1 holds
not Following s,n is stable ) & Following s,b2 is stable & ( for n being Element of NAT st n < b2 holds
not Following s,n is stable ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines stabilization-time CIRCCMB3:def 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 st s is stabilizing holds
for b4 being Element of NAT holds
( b4 = stabilization-time s iff ( Following s,b4 is stable & ( for n being Element of NAT st n < b4 holds
not Following s,n is stable ) ) );

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 Element of 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 Element of 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 Element of 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
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
stabilization-time s = (stabilization-time s1) + (stabilization-time s2)
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 Element of 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
<*x1,x2,x3,x4*> is FinSeqLen of 4
proof end;
let x5 be set ;
:: original: <*
redefine func <*x1,x2,x3,x4,x5*> -> FinSeqLen of 5;
coherence
<*x1,x2,x3,x4,x5*> is FinSeqLen of 5
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 Element of 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 :
for S being ManySortedSign holds
( S is one-gate iff ex X being non empty finite set ex n being Element of NAT ex p being FinSeqLen of n ex f being Function of n -tuples_on X,X st S = 1GateCircStr p,f );

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 Element of 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 :
for S being non empty ManySortedSign
for A being MSAlgebra of S holds
( A is one-gate iff ex X being non empty finite set ex n being Element of 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 ) );

registration
let p be FinSequence;
let x be set ;
cluster 1GateCircStr p,x -> finite ;
coherence
1GateCircStr p,x is finite
proof end;
end;

registration
cluster one-gate -> non empty finite non void strict unsplit gate`1=arity ManySortedSign ;
coherence
for b1 being ManySortedSign st b1 is one-gate holds
( b1 is strict & not b1 is void & not b1 is empty & b1 is unsplit & b1 is gate`1=arity & b1 is finite )
proof end;
end;

registration
cluster non empty one-gate -> non empty gate`2=den ManySortedSign ;
coherence
for b1 being non empty ManySortedSign st b1 is one-gate holds
b1 is gate`2=den
proof end;
end;

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

registration
cluster one-gate ManySortedSign ;
existence
ex b1 being ManySortedSign st b1 is one-gate
proof end;
end;

registration
let S be one-gate ManySortedSign ;
cluster one-gate -> strict non-empty MSAlgebra of S;
coherence
for b1 being Circuit of S st b1 is one-gate holds
( b1 is strict & b1 is non-empty )
proof end;
end;

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

registration
let S be one-gate ManySortedSign ;
cluster non-empty one-gate MSAlgebra of S;
existence
ex b1 being Circuit of S st
( b1 is one-gate & b1 is non-empty )
proof end;
end;

definition
let S be one-gate ManySortedSign ;
func Output S -> Vertex of S equals :: CIRCCMB3:def 8
 union the carrier' of S;
coherence
union the carrier' of S is Vertex of S
proof end;
end;

:: deftheorem defines Output CIRCCMB3:def 8 :
for S being one-gate ManySortedSign holds Output S = union the carrier' of S;

registration
let S be one-gate ManySortedSign ;
cluster Output S -> pair ;
coherence
Output S is pair
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 Element of 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 Element of 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 with_stabilization-limit MSAlgebra of S;
coherence
for b1 being non-empty Circuit of S st b1 is one-gate holds
b1 is with_stabilization-limit
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 Element of 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
ex b1 being non empty non void unsplit gate`1=arity gate`2=den ManySortedSign ex A being Circuit of b1 st
( the Sorts of A is constant & the_value_of the Sorts of A = X & A is gate`2=den )
proof end;
end;

:: deftheorem Def9 defines Signature CIRCCMB3:def 9 :
for X being non empty finite set
for b2 being non empty non void unsplit gate`1=arity gate`2=den ManySortedSign holds
( b2 is Signature of X iff ex A being Circuit of b2 st
( the Sorts of A is constant & the_value_of the Sorts of A = X & A is gate`2=den ) );

theorem Th27: :: CIRCCMB3:27
for n being Element of 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 non void strict unsplit gate`1=arity gate`2=den one-gate Signature of X;
existence
ex b1 being Signature of X st
( b1 is strict & b1 is one-gate )
proof end;
end;

definition
let n be Element of 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
1GateCircStr p,f is strict Signature of X 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
ex b1 being Circuit of S st
( b1 is gate`2=den & the Sorts of b1 is constant & the_value_of the Sorts of b1 = X )
proof end;
end;

:: deftheorem Def10 defines Circuit CIRCCMB3:def 10 :
for X being non empty finite set
for S being Signature of X
for b3 being Circuit of S holds
( b3 is Circuit of X,S iff ( b3 is gate`2=den & the Sorts of b3 is constant & the_value_of the Sorts of b3 = X ) );

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
for b1 being Circuit of X,S holds
( b1 is gate`2=den & b1 is non-empty )
proof end;
end;

theorem Th28: :: CIRCCMB3:28
for n being Element of 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 one-gate Circuit of X,S;
existence
ex b1 being Circuit of X,S st
( b1 is strict & b1 is one-gate )
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
ex b1 being Circuit of X,S st b1 is strict
proof end;
end;

definition
let n be Element of 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
1GateCircuit p,f is strict Circuit of X, 1GateCircStr p,f 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
S1 +* S2 is finite
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
S1 +* S2 is gate`2=den
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
S1 +* S2 is strict Signature of X
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
A1 +* A2 is strict Circuit of X,S1 +* S2
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 -> with_stabilization-limit Circuit of X,S;
coherence
for b1 being Circuit of X,S holds b1 is with_stabilization-limit 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 Element of 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 Element of 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 Element of 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 :
for S being non empty ManySortedSign holds
( S is with_nonpair_inputs iff not InputVertices S is with_pair );

registration
cluster NAT -> without_pairs ;
coherence
not NAT is with_pair
proof end;
let X be without_pairs set ;
cluster -> without_pairs Element of bool X;
coherence
for b1 being Subset of X holds not b1 is with_pair
proof end;
end;

registration
cluster natural-valued -> nonpair-yielding set ;
coherence
for b1 being Function st b1 is natural-valued holds
b1 is nonpair-yielding
proof end;
end;

registration
cluster one-to-one natural-valued set ;
existence
ex b1 being FinSequence st
( b1 is one-to-one & b1 is natural-valued )
proof end;
end;

registration
let n be Element of NAT ;
cluster one-to-one natural-valued FinSeqLen of n;
existence
ex b1 being FinSeqLen of n st
( b1 is one-to-one & b1 is natural-valued )
proof end;
end;

registration
let p be nonpair-yielding FinSequence;
let f be set ;
cluster 1GateCircStr p,f -> with_nonpair_inputs ;
coherence
1GateCircStr p,f is with_nonpair_inputs
proof end;
end;

registration
cluster non empty finite non void strict unsplit gate`1=arity gate`2=den one-gate with_nonpair_inputs ManySortedSign ;
existence
ex b1 being one-gate ManySortedSign st b1 is with_nonpair_inputs
proof end;
let X be non empty finite set ;
cluster non empty finite non void strict unsplit gate`1=arity gate`2=den one-gate with_nonpair_inputs Signature of X;
existence
ex b1 being one-gate Signature of X st b1 is with_nonpair_inputs
proof end;
end;

registration
let S be non empty with_nonpair_inputs ManySortedSign ;
cluster InputVertices S -> without_pairs ;
coherence
not InputVertices S is with_pair 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
InnerVertices S is Relation-like
proof end;
end;

registration
let S be non empty non void unsplit gate`2=den ManySortedSign ;
cluster InnerVertices S -> Relation-like ;
coherence
InnerVertices S is Relation-like
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
S1 +* S2 is with_nonpair_inputs
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
S +* (1GateCircStr <*x1*>,f) is with_nonpair_inputs 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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1*>,f) is with_nonpair_inputs ;
end;

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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,n2*>,f) is with_nonpair_inputs by Th56;
cluster S +* (1GateCircStr <*n2,x1*>,f) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n2,x1*>,f) is with_nonpair_inputs 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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,x2*>,f) is with_nonpair_inputs by Th56;
end;

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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,x2,n*>,f) is with_nonpair_inputs by Th57;
cluster S +* (1GateCircStr <*x1,n,x2*>,f) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,n,x2*>,f) is with_nonpair_inputs by Th57;
cluster S +* (1GateCircStr <*n,x1,x2*>,f) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n,x1,x2*>,f) is with_nonpair_inputs 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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x,n1,n2*>,f) is with_nonpair_inputs by Th57;
cluster S +* (1GateCircStr <*n1,x,n2*>,f) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n1,x,n2*>,f) is with_nonpair_inputs by Th57;
cluster S +* (1GateCircStr <*n1,n2,x*>,f) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*n1,n2,x*>,f) is with_nonpair_inputs 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) -> with_nonpair_inputs ;
coherence
S +* (1GateCircStr <*x1,x2,x3*>,f) is with_nonpair_inputs by Th57;
end;