let c, x, y be set ; :: thesis:
let f be Function of (2 -tuples_on BOOLEAN ),BOOLEAN ; :: thesis:
let s be State of (2GatesCircuit x,y,c,f); :: thesis:
set p = <*[<*x,y*>,f],c*>;
set xyf = [<*x,y*>,f];
set S1 = 1GateCircStr <*x,y*>,f;
set A1 = 1GateCircuit x,y,f;
set S2 = 1GateCircStr <*[<*x,y*>,f],c*>,f;
set A2 = 1GateCircuit [<*x,y*>,f],c,f;
set S = 2GatesCircStr x,y,c,f;
assume c <> [<*x,y*>,f] ; :: thesis:
then A1: InputVertices (2GatesCircStr x,y,c,f) = {x,y,c} by Th57;
A2: InnerVertices (2GatesCircStr x,y,c,f) = {[<*x,y*>,f],(2GatesCircOutput x,y,c,f)} by Th56;
A3: ( x in {x,y} & y in {x,y} & c in {c} & c in {[<*x,y*>,f],c} ) by TARSKI:def 1, TARSKI:def 2;
A4: ( x in InputVertices (2GatesCircStr x,y,c,f) & y in InputVertices (2GatesCircStr x,y,c,f) & c in InputVertices (2GatesCircStr x,y,c,f) ) by A1, ENUMSET1:def 1;
reconsider xx = x, yy = y, cc = c as Vertex of (2GatesCircStr x,y,c,f) by Th60;
A5: ( (Following s) . xx = s . x & (Following s) . yy = s . y & (Following s) . cc = s . c ) by A4, CIRCUIT2:def 5;
then A6: ( (Following (Following s)) . xx = s . x & (Following (Following s)) . yy = s . y & (Following (Following s)) . cc = s . c ) by A4, CIRCUIT2:def 5;
reconsider xyf = [<*x,y*>,f] as Element of InnerVertices (2GatesCircStr x,y,c,f) by A2, TARSKI:def 2;
rng <*[<*x,y*>,f],c*> = {xyf,c} by FINSEQ_2:147;
then ( xyf in rng <*[<*x,y*>,f],c*> & c in rng <*[<*x,y*>,f],c*> ) by TARSKI:def 2;
then A7: ( xyf in InputVertices (1GateCircStr <*[<*x,y*>,f],c*>,f) & c in InputVertices (1GateCircStr <*[<*x,y*>,f],c*>,f) ) by CIRCCOMB:49;
reconsider s1 = s | the carrier of (1GateCircStr <*x,y*>,f) as State of (1GateCircuit x,y,f) by Th26;
set fs = Following s;
reconsider fs2 = (Following s) | the carrier of (1GateCircStr <*[<*x,y*>,f],c*>,f) as State of (1GateCircuit [<*x,y*>,f],c,f) by Th26;
reconsider fs1 = (Following s) | the carrier of (1GateCircStr <*x,y*>,f) as State of (1GateCircuit x,y,f) by Th26;
A8: dom fs2 = the carrier of (1GateCircStr <*[<*x,y*>,f],c*>,f) by CIRCUIT1:4;
A9: dom fs1 = the carrier of (1GateCircStr <*x,y*>,f) by CIRCUIT1:4;
reconsider vx = x, vy = y as Vertex of (1GateCircStr <*x,y*>,f) by Th43;
A10: xyf in InnerVertices (1GateCircStr <*x,y*>,f) by Th47;
reconsider xyf1 = xyf as Element of InnerVertices (1GateCircStr <*x,y*>,f) by Th47;
A11: ( dom s1 = the carrier of (1GateCircStr <*x,y*>,f) & InputVertices (1GateCircStr <*x,y*>,f) = rng <*x,y*> & rng <*x,y*> = {x,y} & InputVertices (1GateCircStr <*x,y*>,f) c= the carrier of (1GateCircStr <*x,y*>,f) ) by CIRCCOMB:49, CIRCUIT1:4, FINSEQ_2:147;
reconsider xyf' = xyf, c' = c as Vertex of (1GateCircStr <*[<*x,y*>,f],c*>,f) by A7;
reconsider v2 = [<*[<*x,y*>,f],c*>,f] as Element of InnerVertices (1GateCircStr <*[<*x,y*>,f],c*>,f) by Th47;
A12: Following s,(1 + 1) = Following (Following s) by Th15;
hence (Following s,2) . (2GatesCircOutput x,y,c,f) = (Following fs2) . v2 by CIRCCOMB:72
.= f . <*(fs2 . xyf'),(fs2 . c')*> by Th48
.= f . <*((Following s) . xyf),(fs2 . c)*> by A8, FUNCT_1:70
.= f . <*((Following s) . xyf),((Following s) . c')*> by A8, FUNCT_1:70
.= f . <*((Following s) . xyf),(s . cc)*> by A4, CIRCUIT2:def 5
.= f . <*((Following s1) . xyf),(s . cc)*> by A10, CIRCCOMB:72
.= f . <*(f . <*(s1 . x),(s1 . y)*>),(s . cc)*> by Th48
.= f . <*(f . <*(s . x),(s1 . y)*>),(s . cc)*> by A3, A11, FUNCT_1:70
.= f . <*(f . <*(s . x),(s . y)*>),(s . c)*> by A3, A11, FUNCT_1:70 ;
:: thesis:
thus (Following s,2) . [<*x,y*>,f] = (Following fs1) . xyf1 by A12, CIRCCOMB:72
.= f . <*(fs1 . vx),(fs1 . vy)*> by Th48
.= f . <*(s . x),(fs1 . vy)*> by A5, A9, FUNCT_1:70
.= f . <*(s . x),(s . y)*> by A5, A9, FUNCT_1:70 ; :: thesis:
thus ( (Following s,2) . x = s . x & (Following s,2) . y = s . y & (Following s,2) . c = s . c ) by A6, Th15; :: thesis: