let a, b be FinSequence; :: thesis: ( 0 -BitSubtracterStr (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitSubtracterCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitBorrowOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] )
A1: ex f, g, h being ManySortedSet of NAT st
( 0 -BitSubtracterStr (a,b) = f . 0 & 0 -BitSubtracterCirc (a,b) = g . 0 & f . 0 = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & g . 0 = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & h . 0 = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] & ( for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra over S
for z being set st S = f . n & A = g . n & z = h . n holds
( f . (n + 1) = S +* (BitSubtracterWithBorrowStr ((a . (n + 1)),(b . (n + 1)),z)) & g . (n + 1) = A +* (BitSubtracterWithBorrowCirc ((a . (n + 1)),(b . (n + 1)),z)) & h . (n + 1) = BorrowOutput ((a . (n + 1)),(b . (n + 1)),z) ) ) ) by Def2;
hence 0 -BitSubtracterStr (a,b) = 1GateCircStr (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) ; :: thesis: ( 0 -BitSubtracterCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) & 0 -BitBorrowOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] )
thus 0 -BitSubtracterCirc (a,b) = 1GateCircuit (<*>,((0 -tuples_on BOOLEAN) --> TRUE)) by A1; :: thesis: 0 -BitBorrowOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)]
InnerVertices (0 -BitSubtracterStr (a,b)) = {[<*>,((0 -tuples_on BOOLEAN) --> TRUE)]} by A1, CIRCCOMB:42;
hence 0 -BitBorrowOutput (a,b) = [<*>,((0 -tuples_on BOOLEAN) --> TRUE)] by TARSKI:def 1; :: thesis: verum