let x, y, c be set ; ( x <> [<*y,c*>,and2] & y <> [<*x,c*>,and2a] & c <> [<*x,y*>,and2a] & c <> [<*x,y*>,'xor'] implies for s being State of (BitSubtracterWithBorrowCirc (x,y,c)) holds Following (s,2) is stable )
assume that
A1:
x <> [<*y,c*>,and2]
and
A2:
y <> [<*x,c*>,and2a]
and
A3:
c <> [<*x,y*>,and2a]
and
A4:
c <> [<*x,y*>,'xor']
; for s being State of (BitSubtracterWithBorrowCirc (x,y,c)) holds Following (s,2) is stable
set S = BitSubtracterWithBorrowStr (x,y,c);
set S1 = 2GatesCircStr (x,y,c,'xor');
set S2 = BorrowStr (x,y,c);
set A = BitSubtracterWithBorrowCirc (x,y,c);
set A1 = BitSubtracterCirc (x,y,c);
set A2 = BorrowCirc (x,y,c);
let s be State of (BitSubtracterWithBorrowCirc (x,y,c)); Following (s,2) is stable
reconsider s1 = s | the carrier of (2GatesCircStr (x,y,c,'xor')) as State of (BitSubtracterCirc (x,y,c)) by FACIRC_1:26;
reconsider s2 = s | the carrier of (BorrowStr (x,y,c)) as State of (BorrowCirc (x,y,c)) by FACIRC_1:26;
reconsider t = s as State of ((BitSubtracterCirc (x,y,c)) +* (BorrowCirc (x,y,c))) ;
A5:
InputVertices (BorrowStr (x,y,c)) = {x,y,c}
by A1, A2, A3, Th15;
A6:
InnerVertices (2GatesCircStr (x,y,c,'xor')) misses InputVertices (2GatesCircStr (x,y,c,'xor'))
by XBOOLE_1:79;
A7:
InnerVertices (BorrowStr (x,y,c)) misses InputVertices (BorrowStr (x,y,c))
by XBOOLE_1:79;
A8:
InnerVertices (2GatesCircStr (x,y,c,'xor')) misses InputVertices (BorrowStr (x,y,c))
by A4, A5, A6, FACIRC_1:57;
A9:
InnerVertices (BorrowStr (x,y,c)) misses InputVertices (2GatesCircStr (x,y,c,'xor'))
by A4, A5, A7, FACIRC_1:57;
then A10:
Following (s1,2) = (Following (t,2)) | the carrier of (2GatesCircStr (x,y,c,'xor'))
by FACIRC_1:30;
A11:
Following (s1,3) = (Following (t,3)) | the carrier of (2GatesCircStr (x,y,c,'xor'))
by A9, FACIRC_1:30;
A12:
Following (s2,2) = (Following (t,2)) | the carrier of (BorrowStr (x,y,c))
by A8, FACIRC_1:31;
A13:
Following (s2,3) = (Following (t,3)) | the carrier of (BorrowStr (x,y,c))
by A8, FACIRC_1:31;
Following (s1,2) is stable
by A4, FACIRC_1:63;
then A14: Following (s1,2) =
Following (Following (s1,2))
.=
Following (s1,(2 + 1))
by FACIRC_1:12
;
Following (s2,2) is stable
by A1, A2, A3, Th23;
then A15: Following (s2,2) =
Following (Following (s2,2))
.=
Following (s2,(2 + 1))
by FACIRC_1:12
;
A16:
Following (s,(2 + 1)) = Following (Following (s,2))
by FACIRC_1:12;
A17:
dom (Following (s,2)) = the carrier of (BitSubtracterWithBorrowStr (x,y,c))
by CIRCUIT1:3;
A18:
dom (Following (s,3)) = the carrier of (BitSubtracterWithBorrowStr (x,y,c))
by CIRCUIT1:3;
A19:
dom (Following (s1,2)) = the carrier of (2GatesCircStr (x,y,c,'xor'))
by CIRCUIT1:3;
A20:
dom (Following (s2,2)) = the carrier of (BorrowStr (x,y,c))
by CIRCUIT1:3;
A21:
the carrier of (BitSubtracterWithBorrowStr (x,y,c)) = the carrier of (2GatesCircStr (x,y,c,'xor')) \/ the carrier of (BorrowStr (x,y,c))
by CIRCCOMB:def 2;
now for a being object st a in the carrier of (BitSubtracterWithBorrowStr (x,y,c)) holds
(Following (s,2)) . a = (Following (Following (s,2))) . alet a be
object ;
( a in the carrier of (BitSubtracterWithBorrowStr (x,y,c)) implies (Following (s,2)) . a = (Following (Following (s,2))) . a )assume
a in the
carrier of
(BitSubtracterWithBorrowStr (x,y,c))
;
(Following (s,2)) . a = (Following (Following (s,2))) . athen
(
a in the
carrier of
(2GatesCircStr (x,y,c,'xor')) or
a in the
carrier of
(BorrowStr (x,y,c)) )
by A21, XBOOLE_0:def 3;
then
( (
(Following (s,2)) . a = (Following (s1,2)) . a &
(Following (s,3)) . a = (Following (s1,3)) . a ) or (
(Following (s,2)) . a = (Following (s2,2)) . a &
(Following (s,3)) . a = (Following (s2,3)) . a ) )
by A10, A11, A12, A13, A14, A15, A19, A20, FUNCT_1:47;
hence
(Following (s,2)) . a = (Following (Following (s,2))) . a
by A14, A15, FACIRC_1:12;
verum end;
hence
Following (s,2) = Following (Following (s,2))
by A16, A17, A18, FUNCT_1:2; CIRCUIT2:def 6 verum