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) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Proof: Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = 2x0 + x1 + 2x2, [ok](x0) = x0, [active](x0) = x0, [true] = 0, [mark](x0) = x0, [f](x0) = 4x0, [proper](x0) = x0, [top](x0) = x0, [false] = 4, [c] = 0 orientation: active(f(X)) = 4X >= 2X = mark(if(X,c(),f(true()))) active(if(true(),X,Y)) = X + 2Y >= X = mark(X) active(if(false(),X,Y)) = X + 2Y + 8 >= Y = mark(Y) active(f(X)) = 4X >= 4X = f(active(X)) active(if(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = if(active(X1),X2,X3) active(if(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = if(X1,active(X2),X3) f(mark(X)) = 4X >= 4X = mark(f(X)) if(mark(X1),X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = mark(if(X1,X2,X3)) if(X1,mark(X2),X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = mark(if(X1,X2,X3)) proper(f(X)) = 4X >= 4X = f(proper(X)) proper(if(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = if(proper(X1),proper(X2),proper(X3)) proper(c()) = 0 >= 0 = ok(c()) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 4 >= 4 = ok(false()) f(ok(X)) = 4X >= 4X = ok(f(X)) if(ok(X1),ok(X2),ok(X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = ok(if(X1,X2,X3)) top(mark(X)) = X >= X = top(proper(X)) top(ok(X)) = X >= X = top(active(X)) problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(if(true(),X,Y)) -> mark(X) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 1] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [ok](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [active](x0) = [1 0 0]x0 [0 0 0] , [0] [true] = [0] [1], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [f](x0) = [0 0 1]x0 [0 0 1] , [proper](x0) = x0 , [1 0 1] [top](x0) = [0 0 0]x0 [0 0 0] , [1] [false] = [1] [1], [0] [c] = [0] [0] orientation: [1 0 1] [1 0 1] active(f(X)) = [1 0 0]X >= [0 0 0]X = mark(if(X,c(),f(true()))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 1] active(if(true(),X,Y)) = [1 0 1]X + [1 0 0]Y + [1] >= [0 0 0]X = mark(X) [0 0 0] [0 0 0] [0] [0 0 0] [1 0 1] [1 0 1] active(f(X)) = [1 0 0]X >= [0 0 0]X = f(active(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] active(if(X1,X2,X3)) = [1 0 1]X1 + [1 0 1]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] active(if(X1,X2,X3)) = [1 0 1]X1 + [1 0 1]X2 + [1 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,active(X2),X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] f(mark(X)) = [0 0 0]X >= [0 0 0]X = mark(f(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] 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 = mark(if(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] 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 = mark(if(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] proper(f(X)) = [0 0 1]X >= [0 0 1]X = f(proper(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] proper(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 = if(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [1] [1] [1] [1] proper(false()) = [1] >= [0] = ok(false()) [1] [1] [1 0 0] [1 0 0] f(ok(X)) = [0 0 1]X >= [0 0 0]X = ok(f(X)) [0 0 1] [0 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] if(ok(X1),ok(X2),ok(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 = ok(if(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] top(mark(X)) = [0 0 0]X >= [0 0 0]X = top(proper(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) if(X1,mark(X2),X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 1] [1 0 0] [if](x0, x1, x2) = [0 1 1]x0 + [0 1 1]x1 + [0 0 0]x2 [0 1 1] [0 1 1] [0 0 0] , [ok](x0) = x0 , [active](x0) = x0 , [0] [true] = [0] [0], [1 0 0] [0] [mark](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 0 0] [0] [f](x0) = [0 1 1]x0 + [0] [0 1 1] [1], [proper](x0) = x0 , [1 0 0] [0] [top](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [0] [false] = [0] [0], [0] [c] = [0] [0] orientation: [1 0 0] [0] [1 0 0] [0] active(f(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = mark(if(X,c(),f(true()))) [0 1 1] [1] [0 1 1] [1] [1 0 0] [0] [1 0 0] [0] active(f(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = f(active(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 1] [1 0 0] active(if(X1,X2,X3)) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 = if(active(X1),X2,X3) [0 1 1] [0 1 1] [0 0 0] [0 1 1] [0 1 1] [0 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 1] [1 0 0] active(if(X1,X2,X3)) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 = if(X1,active(X2),X3) [0 1 1] [0 1 1] [0 0 0] [0 1 1] [0 1 1] [0 0 0] [1 0 0] [0] [1 0 0] [0] f(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [1] = mark(f(X)) [0 1 1] [2] [0 1 1] [1] [1 0 0] [1 1 1] [1 0 0] [0] [1 0 0] [1 1 1] [1 0 0] [0] if(mark(X1),X2,X3) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 + [1] >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 + [0] = mark(if(X1,X2,X3)) [0 1 1] [0 1 1] [0 0 0] [1] [0 1 1] [0 1 1] [0 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [1] [1 0 0] [1 1 1] [1 0 0] [0] if(X1,mark(X2),X3) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 + [1] >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 + [0] = mark(if(X1,X2,X3)) [0 1 1] [0 1 1] [0 0 0] [1] [0 1 1] [0 1 1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] proper(f(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = f(proper(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 1] [1 0 0] proper(if(X1,X2,X3)) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 = if(proper(X1),proper(X2),proper(X3)) [0 1 1] [0 1 1] [0 0 0] [0 1 1] [0 1 1] [0 0 0] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [0] [0] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [0] [0] [0] [0] proper(false()) = [0] >= [0] = ok(false()) [0] [0] [1 0 0] [0] [1 0 0] [0] f(ok(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = ok(f(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 1] [1 0 0] if(ok(X1),ok(X2),ok(X3)) = [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 >= [0 1 1]X1 + [0 1 1]X2 + [0 0 0]X3 = ok(if(X1,X2,X3)) [0 1 1] [0 1 1] [0 0 0] [0 1 1] [0 1 1] [0 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) f(mark(X)) -> mark(f(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [if](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] , [ok](x0) = x0 , [active](x0) = x0 , [0] [true] = [0] [0], [1 0 0] [0] [mark](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [1 1 1] [0] [f](x0) = [0 0 1]x0 + [0] [0 1 0] [1], [proper](x0) = x0 , [1 0 0] [0] [top](x0) = [0 1 1]x0 + [0] [0 0 0] [1], [1] [false] = [0] [0], [0] [c] = [0] [1] orientation: [1 1 1] [0] [1 0 0] [0] active(f(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = mark(if(X,c(),f(true()))) [0 1 0] [1] [0 1 0] [1] [1 1 1] [0] [1 1 1] [0] active(f(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = f(active(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(if(X1,X2,X3)) = X1 + [0 0 0]X2 + [0 0 0]X3 >= X1 + [0 0 0]X2 + [0 0 0]X3 = if(active(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(if(X1,X2,X3)) = X1 + [0 0 0]X2 + [0 0 0]X3 >= X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,active(X2),X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] [0] f(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(f(X)) [0 0 1] [1] [0 0 1] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] if(mark(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 + [0] = mark(if(X1,X2,X3)) [0 1 0] [0 0 0] [0 0 0] [1] [0 1 0] [0 0 0] [0 0 0] [1] [1 1 1] [0] [1 1 1] [0] proper(f(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = f(proper(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] proper(if(X1,X2,X3)) = X1 + [0 0 0]X2 + [0 0 0]X3 >= X1 + [0 0 0]X2 + [0 0 0]X3 = if(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [1] [1] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [0] [0] [1] [1] proper(false()) = [0] >= [0] = ok(false()) [0] [0] [1 1 1] [0] [1 1 1] [0] f(ok(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = ok(f(X)) [0 1 0] [1] [0 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] if(ok(X1),ok(X2),ok(X3)) = X1 + [0 0 0]X2 + [0 0 0]X3 >= X1 + [0 0 0]X2 + [0 0 0]X3 = ok(if(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] top(mark(X)) = [0 1 1]X + [1] >= [0 1 1]X + [0] = top(proper(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] top(ok(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = top(active(X)) [0 0 0] [1] [0 0 0] [1] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [0 0 0] , [1 0 0] [0] [ok](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [active](x0) = x0 , [0] [true] = [0] [0], [1 1 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 1 0] [0] [f](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [0] [proper](x0) = [0 1 0]x0 + [0] [0 0 0] [1], [1 1 0] [top](x0) = [0 0 0]x0 [0 0 0] , [0] [false] = [0] [0], [0] [c] = [0] [0] orientation: [1 1 0] [0] [1 1 0] [0] active(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(if(X,c(),f(true()))) [0 0 0] [1] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] active(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(active(X)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(if(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(active(X1),X2,X3) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(if(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,active(X2),X3) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [0] [1 1 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 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = mark(if(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] proper(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(proper(X)) [0 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] proper(if(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(proper(X1),proper(X2),proper(X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [0] [0] proper(c()) = [0] >= [0] = ok(c()) [1] [1] [0] [0] proper(true()) = [0] >= [0] = ok(true()) [1] [1] [0] [0] proper(false()) = [0] >= [0] = ok(false()) [1] [1] [1 1 0] [0] [1 1 0] [0] f(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(f(X)) [0 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(ok(X1),ok(X2),ok(X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = ok(if(X1,X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 0] [1] [1 1 0] top(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = top(proper(X)) [0 0 0] [0] [0 0 0] [1 1 0] [1 1 0] top(ok(X)) = [0 0 0]X >= [0 0 0]X = top(active(X)) [0 0 0] [0 0 0] problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) proper(false()) -> ok(false()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = 2x0 + x1 + x2, [ok](x0) = x0, [active](x0) = x0, [true] = 0, [mark](x0) = x0, [f](x0) = 2x0, [proper](x0) = 4x0, [top](x0) = x0 + 1, [false] = 7, [c] = 0 orientation: active(f(X)) = 2X >= 2X = mark(if(X,c(),f(true()))) active(f(X)) = 2X >= 2X = f(active(X)) active(if(X1,X2,X3)) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = if(active(X1),X2,X3) active(if(X1,X2,X3)) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = if(X1,active(X2),X3) if(mark(X1),X2,X3) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = mark(if(X1,X2,X3)) proper(f(X)) = 8X >= 8X = f(proper(X)) proper(if(X1,X2,X3)) = 8X1 + 4X2 + 4X3 >= 8X1 + 4X2 + 4X3 = if(proper(X1),proper(X2),proper(X3)) proper(c()) = 0 >= 0 = ok(c()) proper(true()) = 0 >= 0 = ok(true()) proper(false()) = 28 >= 7 = ok(false()) f(ok(X)) = 2X >= 2X = ok(f(X)) if(ok(X1),ok(X2),ok(X3)) = 2X1 + X2 + X3 >= 2X1 + X2 + X3 = ok(if(X1,X2,X3)) top(ok(X)) = X + 1 >= X + 1 = top(active(X)) problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) active(if(X1,X2,X3)) -> if(X1,active(X2),X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) f(ok(X)) -> ok(f(X)) if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = 4x0 + x1 + x2 + 1, [ok](x0) = 4x0 + 1, [active](x0) = 4x0 + 1, [true] = 0, [mark](x0) = x0, [f](x0) = x0, [proper](x0) = 6x0 + 1, [top](x0) = x0 + 2, [c] = 0 orientation: active(f(X)) = 4X + 1 >= 4X + 1 = mark(if(X,c(),f(true()))) active(f(X)) = 4X + 1 >= 4X + 1 = f(active(X)) active(if(X1,X2,X3)) = 16X1 + 4X2 + 4X3 + 5 >= 16X1 + X2 + X3 + 5 = if(active(X1),X2,X3) active(if(X1,X2,X3)) = 16X1 + 4X2 + 4X3 + 5 >= 4X1 + 4X2 + X3 + 2 = if(X1,active(X2),X3) if(mark(X1),X2,X3) = 4X1 + X2 + X3 + 1 >= 4X1 + X2 + X3 + 1 = mark(if(X1,X2,X3)) proper(f(X)) = 6X + 1 >= 6X + 1 = f(proper(X)) proper(if(X1,X2,X3)) = 24X1 + 6X2 + 6X3 + 7 >= 24X1 + 6X2 + 6X3 + 7 = if(proper(X1),proper(X2),proper(X3)) proper(c()) = 1 >= 1 = ok(c()) proper(true()) = 1 >= 1 = ok(true()) f(ok(X)) = 4X + 1 >= 4X + 1 = ok(f(X)) if(ok(X1),ok(X2),ok(X3)) = 16X1 + 4X2 + 4X3 + 7 >= 16X1 + 4X2 + 4X3 + 5 = ok(if(X1,X2,X3)) top(ok(X)) = 4X + 3 >= 4X + 3 = top(active(X)) problem: active(f(X)) -> mark(if(X,c(),f(true()))) active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) f(ok(X)) -> ok(f(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [if](x0, x1, x2) = x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 0] , [1 0 0] [0] [ok](x0) = [0 0 1]x0 + [1] [0 0 0] [0], [1 0 0] [1] [active](x0) = [0 0 0]x0 + [0] [0 0 1] [0], [0] [true] = [0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [f](x0) = x0 , [1 0 0] [0] [proper](x0) = [0 0 1]x0 + [1] [1 0 0] [0], [1 1 0] [top](x0) = [0 1 1]x0 [0 0 0] , [0] [c] = [0] [0] orientation: [1 0 0] [1] [1 0 0] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X = mark(if(X,c(),f(true()))) [0 0 1] [0] [0 0 0] [1 0 0] [1] [1 0 0] [1] active(f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(active(X)) [0 0 1] [0] [0 0 1] [0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] active(if(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(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 0] [0] [0 0 1] [0 0 1] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] 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 = mark(if(X1,X2,X3)) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] proper(f(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = f(proper(X)) [1 0 0] [0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] proper(if(X1,X2,X3)) = [0 0 1]X1 + [0 0 1]X2 + [0 0 0]X3 + [1] >= [0 0 1]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = if(proper(X1),proper(X2),proper(X3)) [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0 0 0] [0] [0] [0] proper(c()) = [1] >= [1] = ok(c()) [0] [0] [0] [0] proper(true()) = [1] >= [1] = ok(true()) [0] [0] [1 0 0] [0] [1 0 0] [0] f(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = ok(f(X)) [0 0 0] [0] [0 0 0] [0] [1 0 1] [1] [1 0 0] [1] top(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [0] = top(active(X)) [0 0 0] [0] [0 0 0] [0] problem: active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) proper(c()) -> ok(c()) proper(true()) -> ok(true()) f(ok(X)) -> ok(f(X)) top(ok(X)) -> top(active(X)) Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = x0 + 3x1 + x2 + 4, [ok](x0) = x0 + 1, [active](x0) = x0, [true] = 1, [mark](x0) = x0 + 7, [f](x0) = 4x0 + 3, [proper](x0) = 6x0 + 5, [top](x0) = x0 + 3, [c] = 2 orientation: active(f(X)) = 4X + 3 >= 4X + 3 = f(active(X)) active(if(X1,X2,X3)) = X1 + 3X2 + X3 + 4 >= X1 + 3X2 + X3 + 4 = if(active(X1),X2,X3) if(mark(X1),X2,X3) = X1 + 3X2 + X3 + 11 >= X1 + 3X2 + X3 + 11 = mark(if(X1,X2,X3)) proper(f(X)) = 24X + 23 >= 24X + 23 = f(proper(X)) proper(if(X1,X2,X3)) = 6X1 + 18X2 + 6X3 + 29 >= 6X1 + 18X2 + 6X3 + 29 = if(proper(X1),proper(X2),proper(X3)) proper(c()) = 17 >= 3 = ok(c()) proper(true()) = 11 >= 2 = ok(true()) f(ok(X)) = 4X + 7 >= 4X + 4 = ok(f(X)) top(ok(X)) = X + 4 >= X + 3 = top(active(X)) problem: active(f(X)) -> f(active(X)) active(if(X1,X2,X3)) -> if(active(X1),X2,X3) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [0] [if](x0, x1, x2) = x0 + [0 0 1]x1 + x2 + [0] [0 0 1] [1], [1 0 1] [active](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [f](x0) = x0 , [1 1 0] [proper](x0) = [0 0 1]x0 [0 0 1] orientation: [1 0 1] [1 0 1] active(f(X)) = [0 1 0]X >= [0 1 0]X = f(active(X)) [0 0 1] [0 0 1] [1 0 1] [1 1 1] [1 0 1] [1] [1 0 1] [1 1 0] [0] active(if(X1,X2,X3)) = [0 1 0]X1 + [0 0 1]X2 + [0 1 0]X3 + [0] >= [0 1 0]X1 + [0 0 1]X2 + X3 + [0] = if(active(X1),X2,X3) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [1 0 0] if(mark(X1),X2,X3) = [0 1 0]X1 + [0 0 1]X2 + X3 + [0] >= [0 1 0]X1 + [0 0 1]X2 + [0 1 0]X3 = mark(if(X1,X2,X3)) [0 0 0] [0 0 1] [1] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] proper(f(X)) = [0 0 1]X >= [0 0 1]X = f(proper(X)) [0 0 1] [0 0 1] [1 1 0] [1 1 1] [1 1 0] [0] [1 1 0] [1 1 1] [1 1 0] [0] proper(if(X1,X2,X3)) = [0 0 1]X1 + [0 0 1]X2 + [0 0 1]X3 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [0 0 1]X3 + [0] = if(proper(X1),proper(X2),proper(X3)) [0 0 1] [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [0 0 1] [1] problem: active(f(X)) -> f(active(X)) if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) proper(f(X)) -> f(proper(X)) proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3)) Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = 2x0 + x1 + 3x2 + 3, [active](x0) = 2x0, [mark](x0) = x0, [f](x0) = 4x0 + 2, [proper](x0) = 3x0 orientation: active(f(X)) = 8X + 4 >= 8X + 2 = f(active(X)) if(mark(X1),X2,X3) = 2X1 + X2 + 3X3 + 3 >= 2X1 + X2 + 3X3 + 3 = mark(if(X1,X2,X3)) proper(f(X)) = 12X + 6 >= 12X + 2 = f(proper(X)) proper(if(X1,X2,X3)) = 6X1 + 3X2 + 9X3 + 9 >= 6X1 + 3X2 + 9X3 + 3 = if(proper(X1),proper(X2),proper(X3)) problem: if(mark(X1),X2,X3) -> mark(if(X1,X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 0 0] [0 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [0] [0 0 1] [1] orientation: [1 0 1] [1 0 0] [1 0 0] [1] [1 0 1] [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] = mark(if(X1,X2,X3)) [0 0 1] [0 0 0] [0 0 0] [1] [0 0 1] [0 0 0] [0 0 0] [1] problem: Qed