let x1, x2, x3, x4, x5, x6, x7 be non pair set ; :: thesis: for s being State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7))
for a1, a2, a3, a4, a5, a6, a7 being Element of BOOLEAN st a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 holds
( (Following (s,6)) . (STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7)) = ((((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))) 'xor' (((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3))) & (Following (s,6)) . x1 = a1 & (Following (s,6)) . x2 = a2 & (Following (s,6)) . x3 = a3 & (Following (s,6)) . x4 = a4 & (Following (s,6)) . x5 = a5 & (Following (s,6)) . x6 = a6 & (Following (s,6)) . x7 = a7 )
set S = STC0Str (x1,x2,x3,x4,x5,x6,x7);
set C = STC0Circ (x1,x2,x3,x4,x5,x6,x7);
set A1out = GFA0AdderOutput (x1,x2,x3);
set A2out = GFA0AdderOutput (x5,x6,x7);
set C1out = GFA0CarryOutput (x1,x2,x3);
set C2out = GFA0CarryOutput (x5,x6,x7);
set C3out = GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4);
set C1C2x = [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2];
set S1out = STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7);
let s be State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7)); :: thesis: for a1, a2, a3, a4, a5, a6, a7 being Element of BOOLEAN st a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 holds
( (Following (s,6)) . (STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7)) = ((((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))) 'xor' (((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3))) & (Following (s,6)) . x1 = a1 & (Following (s,6)) . x2 = a2 & (Following (s,6)) . x3 = a3 & (Following (s,6)) . x4 = a4 & (Following (s,6)) . x5 = a5 & (Following (s,6)) . x6 = a6 & (Following (s,6)) . x7 = a7 )
let a1, a2, a3, a4, a5, a6, a7 be Element of BOOLEAN ; :: thesis: ( a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 implies ( (Following (s,6)) . (STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7)) = ((((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))) 'xor' (((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3))) & (Following (s,6)) . x1 = a1 & (Following (s,6)) . x2 = a2 & (Following (s,6)) . x3 = a3 & (Following (s,6)) . x4 = a4 & (Following (s,6)) . x5 = a5 & (Following (s,6)) . x6 = a6 & (Following (s,6)) . x7 = a7 ) )
assume A2: ( a1 = s . x1 & a2 = s . x2 & a3 = s . x3 & a4 = s . x4 & a5 = s . x5 & a6 = s . x6 & a7 = s . x7 ) ; :: thesis: ( (Following (s,6)) . (STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7)) = ((((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))) 'xor' (((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3))) & (Following (s,6)) . x1 = a1 & (Following (s,6)) . x2 = a2 & (Following (s,6)) . x3 = a3 & (Following (s,6)) . x4 = a4 & (Following (s,6)) . x5 = a5 & (Following (s,6)) . x6 = a6 & (Following (s,6)) . x7 = a7 )
A3: ( x1 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x2 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x3 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x4 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x5 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x6 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & x7 in InputVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) ) by ThSTC0S9;
A5: ( (Following (s,5)) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = (((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5)) & (Following (s,5)) . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = ((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3)) ) by A2, LmSTC0S15S1s6a, LmSTC0S15S1s6b;
Following (s,(5 + 1)) = Following (Following (s,5)) by FACIRC_1:12;
hence ( (Following (s,6)) . (STC0OutputS1 (x1,x2,x3,x4,x5,x6,x7)) = ((((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1)) 'xor' (((a5 '&' a6) 'or' (a6 '&' a7)) 'or' (a7 '&' a5))) 'xor' (((((a1 'xor' a2) 'xor' a3) '&' ((a5 'xor' a6) 'xor' a7)) 'or' (((a5 'xor' a6) 'xor' a7) '&' a4)) 'or' (a4 '&' ((a1 'xor' a2) 'xor' a3))) & (Following (s,6)) . x1 = a1 & (Following (s,6)) . x2 = a2 & (Following (s,6)) . x3 = a3 & (Following (s,6)) . x4 = a4 & (Following (s,6)) . x5 = a5 & (Following (s,6)) . x6 = a6 & (Following (s,6)) . x7 = a7 ) by A2, A3, CIRCCMB3:1, A5, LmSTC0S15S1s8; :: thesis: verum