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 )] )
consider f, g, h being ManySortedSet of such that
A1: ( 0 -BitSubtracterStr a,b = f . 0 & 0 -BitSubtracterCirc a,b = g . 0 ) and
A2: f . 0 = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) and
A3: g . 0 = 1GateCircuit <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) and
h . 0 = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] and
for n being Nat
for S being non empty ManySortedSign
for A being non-empty MSAlgebra of 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;
thus 0 -BitSubtracterStr a,b = 1GateCircStr <*> ,((0 -tuples_on BOOLEAN ) --> TRUE ) by A1, A2; :: 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, A3; :: thesis: 0 -BitBorrowOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )]
InnerVertices (0 -BitSubtracterStr a,b) = {[<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )]} by A1, A2, CIRCCOMB:49;
hence 0 -BitBorrowOutput a,b = [<*> ,((0 -tuples_on BOOLEAN ) --> TRUE )] by TARSKI:def 1; :: thesis: verum