set f = 'xor' ;
let x, y, c be non pair set ; :: thesis: for s being State of (BitSubtracterWithBorrowCirc x,y,c)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following s,2) . (BitSubtracterOutput x,y,c) = (a1 'xor' a2) 'xor' a3 & (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) )
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); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following s,2) . (BitSubtracterOutput x,y,c) = (a1 'xor' a2) 'xor' a3 & (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) )
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies ( (Following s,2) . (BitSubtracterOutput x,y,c) = (a1 'xor' a2) 'xor' a3 & (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) ) )
assume that
A1: a1 = s . x and
A2: a2 = s . y and
A3: a3 = s . c ; :: thesis: ( (Following s,2) . (BitSubtracterOutput x,y,c) = (a1 'xor' a2) 'xor' a3 & (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) )
reconsider s1 = s | the carrier of (2GatesCircStr x,y,c,'xor' ) as State of (BitSubtracterCirc x,y,c) by FACIRC_1:26;
A4: dom s1 = the carrier of (2GatesCircStr x,y,c,'xor' ) by CIRCUIT1:4;
c in the carrier of (2GatesCircStr x,y,c,'xor' ) by FACIRC_1:60;
then A5: a3 = s1 . c by A3, A4, FUNCT_1:70;
y in the carrier of (2GatesCircStr x,y,c,'xor' ) by FACIRC_1:60;
then A6: a2 = s1 . y by A2, A4, FUNCT_1:70;
reconsider t = s as State of ((BitSubtracterCirc x,y,c) +* (BorrowCirc x,y,c)) ;
not InputVertices (2GatesCircStr x,y,c,'xor' ) is with_pair by FACIRC_1:59;
then InnerVertices (BorrowStr x,y,c) misses InputVertices (2GatesCircStr x,y,c,'xor' ) by Th1, FACIRC_1:5;
then A7: (Following t,2) . (2GatesCircOutput x,y,c,'xor' ) = (Following s1,2) . (2GatesCircOutput x,y,c,'xor' ) by FACIRC_1:32;
reconsider s2 = s | the carrier of (BorrowStr x,y,c) as State of (BorrowCirc x,y,c) by FACIRC_1:26;
A8: dom s2 = the carrier of (BorrowStr x,y,c) by CIRCUIT1:4;
x in the carrier of (2GatesCircStr x,y,c,'xor' ) by FACIRC_1:60;
then a1 = s1 . x by A1, A4, FUNCT_1:70;
hence (Following s,2) . (BitSubtracterOutput x,y,c) = (a1 'xor' a2) 'xor' a3 by A6, A5, A7, FACIRC_1:64; :: thesis: (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3)
not InputVertices (BorrowStr x,y,c) is with_pair by Th2;
then InnerVertices (2GatesCircStr x,y,c,'xor' ) misses InputVertices (BorrowStr x,y,c) by FACIRC_1:5, FACIRC_1:58;
then A9: (Following t,2) . (BorrowOutput x,y,c) = (Following s2,2) . (BorrowOutput x,y,c) by FACIRC_1:33;
c in the carrier of (BorrowStr x,y,c) by Th6;
then A10: a3 = s2 . c by A3, A8, FUNCT_1:70;
y in the carrier of (BorrowStr x,y,c) by Th6;
then A11: a2 = s2 . y by A2, A8, FUNCT_1:70;
x in the carrier of (BorrowStr x,y,c) by Th6;
then a1 = s2 . x by A1, A8, FUNCT_1:70;
hence (Following s,2) . (BorrowOutput x,y,c) = ((('not' a1) '&' a2) 'or' (a2 '&' a3)) 'or' (('not' a1) '&' a3) by A11, A10, A9, Lm2; :: thesis: verum