YES Problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) mark(f(X)) -> active(f(mark(X))) mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3)) mark(c()) -> active(c()) mark(true()) -> active(true()) mark(false()) -> active(false()) f(mark(X)) -> f(X) f(active(X)) -> f(X) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Proof: Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [1], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [1 0 0] [1 0 1] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [0 0 1] , [0] [true] = [0] [0], [0] [c] = [0] [0], [1 0 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [f](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 1] [1 0 1] active(f(X)) = [0 0 0]X >= [0 0 0]X = mark(if(X,c(),f(true()))) [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] active(if(true(),X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X = mark(X) [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 0 1] [1] [1 0 0] active(if(false(),X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y = mark(Y) [0 0 1] [0 0 1] [0] [0 0 1] [1 0 1] [1 0 1] mark(f(X)) = [0 0 0]X >= [0 0 0]X = active(f(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(if(mark(X1),mark(X2),X3)) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0] [0] mark(c()) = [0] >= [0] = active(c()) [0] [0] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [0] [0] mark(false()) = [0] >= [0] = active(false()) [1] [1] [1 0 1] [1 0 1] f(mark(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] f(active(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] if(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [0 0 1] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(if(true(),X,Y)) -> mark(X) mark(f(X)) -> active(f(mark(X))) mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3)) mark(c()) -> active(c()) mark(true()) -> active(true()) mark(false()) -> active(false()) f(mark(X)) -> f(X) f(active(X)) -> f(X) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [1 1 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 1]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [1 0 0] , [0] [true] = [0] [0], [0] [c] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] , [1 1 0] [0] [f](x0) = [0 0 1]x0 + [1] [0 0 0] [0] orientation: [1 1 1] [1] [1 1 0] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X = mark(if(X,c(),f(true()))) [0 0 0] [0] [0 0 0] [1 1 1] [1 0 0] [1 1 0] active(if(true(),X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X = mark(X) [0 0 1] [1 0 0] [0 0 1] [1 1 1] [1] [1 1 1] [1] mark(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(f(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 1] [1 0 0] [1 1 0] [1 1 1] [1 0 0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(if(mark(X1),mark(X2),X3)) [0 0 0] [0 0 1] [1 0 0] [0 0 0] [0 0 1] [1 0 0] [0] [0] mark(c()) = [0] >= [0] = active(c()) [0] [0] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [1 1 0] [0] [1 1 0] [0] f(mark(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = f(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] f(active(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = f(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 0 0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 0 0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 1 0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 0 0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(X1,active(X2),X3) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 0 0] [0 0 0] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] if(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 1]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [1 1 0] [0 0 0] [0 0 1] [1 0 0] problem: active(if(true(),X,Y)) -> mark(X) mark(f(X)) -> active(f(mark(X))) mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3)) mark(c()) -> active(c()) mark(true()) -> active(true()) mark(false()) -> active(false()) f(mark(X)) -> f(X) f(active(X)) -> f(X) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [0], [1 0 1] [mark](x0) = [0 1 0]x0 [0 1 0] , [1 0 1] [1 0 1] [1 1 0] [1] [if](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [0] [0 1 0] [0 1 0] [0 0 0] [0], [0] [true] = [0] [0], [0] [c] = [0] [0], [active](x0) = x0 , [1 0 1] [0] [f](x0) = [0 1 0]x0 + [1] [0 1 0] [1] orientation: [1 0 1] [1 1 0] [1] [1 0 1] active(if(true(),X,Y)) = [0 1 0]X + [0 0 0]Y + [0] >= [0 1 0]X = mark(X) [0 1 0] [0 0 0] [0] [0 1 0] [1 1 1] [1] [1 1 1] [0] mark(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = active(f(mark(X))) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1 1 1] [1 1 0] [1] [1 1 1] [1 1 1] [1 1 0] [1] mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = active(if(mark(X1),mark(X2),X3)) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [0] [0] mark(c()) = [0] >= [0] = active(c()) [0] [0] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [1 1 1] [0] [1 0 1] [0] f(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(X) [0 1 0] [1] [0 1 0] [1] [1 0 1] [0] [1 0 1] [0] f(active(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(X) [0 1 0] [1] [0 1 0] [1] [1 1 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(mark(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [1 0 1] [1 1 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(X1,mark(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 1 1] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(X1,X2,mark(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(active(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(X1,active(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] if(X1,X2,active(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 1 0] [0 1 0] [0 0 0] [0] [0 1 0] [0 1 0] [0 0 0] [0] problem: mark(if(X1,X2,X3)) -> active(if(mark(X1),mark(X2),X3)) mark(c()) -> active(c()) mark(true()) -> active(true()) mark(false()) -> active(false()) f(mark(X)) -> f(X) f(active(X)) -> f(X) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [0], [1 0 1] [0] [mark](x0) = [0 1 0]x0 + [1] [1 0 1] [0], [1 0 0] [1 0 0] [1 0 0] [0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 0 1] [0 0 0] [1], [0] [true] = [0] [0], [1] [c] = [0] [0], [active](x0) = x0 , [1 1 0] [f](x0) = [0 0 0]x0 [0 0 1] orientation: [1 0 1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 1] [1 0 0] [0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = active(if(mark(X1),mark(X2),X3)) [1 0 1] [1 0 1] [1 0 0] [1] [1 0 1] [1 0 1] [0 0 0] [1] [1] [1] mark(c()) = [1] >= [0] = active(c()) [1] [0] [0] [0] mark(true()) = [1] >= [0] = active(true()) [0] [0] [0] [0] mark(false()) = [1] >= [0] = active(false()) [0] [0] [1 1 1] [1] [1 1 0] f(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = f(X) [1 0 1] [0] [0 0 1] [1 1 0] [1 1 0] f(active(X)) = [0 0 0]X >= [0 0 0]X = f(X) [0 0 1] [0 0 1] [1 0 1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [1 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [1 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] problem: mark(c()) -> active(c()) mark(true()) -> active(true()) mark(false()) -> active(false()) f(active(X)) -> f(X) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [0], [1 0 0] [1] [mark](x0) = [0 0 1]x0 + [0] [0 1 0] [0], [1 0 0] [1 1 1] [1 1 1] [if](x0, x1, x2) = [0 1 1]x0 + [0 0 0]x1 + [1 1 1]x2 [1 0 0] [0 0 0] [0 0 0] , [0] [true] = [0] [1], [0] [c] = [0] [0], [1 0 0] [active](x0) = [0 0 1]x0 [0 1 0] , [1 0 0] [f](x0) = [0 1 1]x0 [0 0 0] orientation: [1] [0] mark(c()) = [0] >= [0] = active(c()) [0] [0] [1] [0] mark(true()) = [1] >= [1] = active(true()) [0] [0] [1] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [1 0 0] [1 0 0] f(active(X)) = [0 1 1]X >= [0 1 1]X = f(X) [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1] [1 0 0] [1 1 1] [1 1 1] if(mark(X1),X2,X3) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1] [1 0 0] [1 1 1] [1 1 1] if(X1,mark(X2),X3) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 + [0] >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1] [1 0 0] [1 1 1] [1 1 1] if(X1,X2,mark(X3)) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 + [1] >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] if(active(X1),X2,X3) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] if(X1,active(X2),X3) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [1 1 1] [1 0 0] [1 1 1] [1 1 1] if(X1,X2,active(X3)) = [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 >= [0 1 1]X1 + [0 0 0]X2 + [1 1 1]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] problem: f(active(X)) -> f(X) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 1]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 1 0] [0 0 0] , [1] [active](x0) = x0 + [0] [0], [1 0 0] [f](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1] [1 0 0] f(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = f(X) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] if(active(X1),X2,X3) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] if(X1,active(X2),X3) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0] [0 0 0] [0 1 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] if(X1,X2,active(X3)) = [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 1 0] [0 0 0] [0] [0 0 0] [0 1 0] [0 0 0] problem: Qed