YES(?,O(n^2)) 63.85/20.89 YES(?,O(n^2)) 63.85/20.89 63.85/20.89 Problem: 63.85/20.89 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.89 active(b()) -> mark(a()) 63.85/20.89 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.89 mark(a()) -> active(a()) 63.85/20.89 mark(b()) -> active(b()) 63.85/20.89 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.89 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.89 63.85/20.89 Proof: 63.85/20.89 Complexity Transformation Processor: 63.85/20.89 strict: 63.85/20.89 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.89 active(b()) -> mark(a()) 63.85/20.89 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.89 mark(a()) -> active(a()) 63.85/20.89 mark(b()) -> active(b()) 63.85/20.89 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.89 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.89 weak: 63.85/20.89 63.85/20.89 Matrix Interpretation Processor: dim=1 63.85/20.89 63.85/20.89 max_matrix: 63.85/20.89 1 63.85/20.89 interpretation: 63.85/20.89 [mark](x0) = x0 + 29, 63.85/20.89 63.85/20.89 [b] = 32, 63.85/20.89 63.85/20.89 [active](x0) = x0 + 1, 63.85/20.89 63.85/20.89 [f](x0, x1, x2) = x0 + x1 + x2 + 72, 63.85/20.89 63.85/20.89 [a] = 0 63.85/20.89 orientation: 63.85/20.89 active(f(a(),X,X)) = 2X + 73 >= X + 165 = mark(f(X,b(),b())) 63.85/20.89 63.85/20.89 active(b()) = 33 >= 29 = mark(a()) 63.85/20.89 63.85/20.89 mark(f(X1,X2,X3)) = X1 + X2 + X3 + 101 >= X1 + X2 + X3 + 102 = active(f(X1,mark(X2),X3)) 63.85/20.89 63.85/20.89 mark(a()) = 29 >= 1 = active(a()) 63.85/20.89 63.85/20.89 mark(b()) = 61 >= 33 = active(b()) 63.85/20.89 63.85/20.89 f(mark(X1),X2,X3) = X1 + X2 + X3 + 101 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 63.85/20.89 f(X1,mark(X2),X3) = X1 + X2 + X3 + 101 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 63.85/20.89 f(X1,X2,mark(X3)) = X1 + X2 + X3 + 101 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 63.85/20.89 f(active(X1),X2,X3) = X1 + X2 + X3 + 73 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 63.85/20.89 f(X1,active(X2),X3) = X1 + X2 + X3 + 73 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 63.85/20.89 f(X1,X2,active(X3)) = X1 + X2 + X3 + 73 >= X1 + X2 + X3 + 72 = f(X1,X2,X3) 63.85/20.89 problem: 63.85/20.89 strict: 63.85/20.89 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.89 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.89 weak: 63.85/20.89 active(b()) -> mark(a()) 63.85/20.89 mark(a()) -> active(a()) 63.85/20.89 mark(b()) -> active(b()) 63.85/20.89 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.89 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.89 Splitting Processor: 63.85/20.89 strict: 63.85/20.89 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.89 weak: 63.85/20.89 active(b()) -> mark(a()) 63.85/20.89 mark(a()) -> active(a()) 63.85/20.89 mark(b()) -> active(b()) 63.85/20.89 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.89 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.89 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.89 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.89 Matrix Interpretation Processor: dim=3 63.85/20.89 63.85/20.89 max_matrix: 63.85/20.89 [1 1 0] 63.85/20.89 [0 1 1] 63.85/20.89 [0 0 0] 63.85/20.89 interpretation: 63.85/20.89 [1 0 0] 63.85/20.89 [mark](x0) = [0 1 1]x0 63.85/20.89 [0 0 0] , 63.85/20.89 63.85/20.89 [0] 63.85/20.89 [b] = [0] 63.85/20.89 [1], 63.85/20.89 63.85/20.89 [1 0 0] 63.85/20.89 [active](x0) = [0 1 1]x0 63.85/20.89 [0 0 0] , 63.85/20.89 63.85/20.89 [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.89 [f](x0, x1, x2) = [0 1 0]x0 + [0 0 0]x1 + [0 1 0]x2 + [1] 63.85/20.89 [0 0 0] [0 0 0] [0 0 0] [0], 63.85/20.89 63.85/20.89 [0] 63.85/20.89 [a] = [1] 63.85/20.89 [0] 63.85/20.89 orientation: 63.85/20.89 [2 1 0] [1] [1 1 0] [0] 63.85/20.89 active(f(a(),X,X)) = [0 1 0]X + [2] >= [0 1 0]X + [1] = mark(f(X,b(),b())) 63.85/20.89 [0 0 0] [0] [0 0 0] [0] 63.85/20.89 63.85/20.89 [0] [0] 63.85/20.89 active(b()) = [1] >= [1] = mark(a()) 63.85/20.89 [0] [0] 63.85/20.89 63.85/20.89 [0] [0] 63.85/20.89 mark(a()) = [1] >= [1] = active(a()) 63.85/20.89 [0] [0] 63.85/20.89 63.85/20.89 [0] [0] 63.85/20.89 mark(b()) = [1] >= [1] = active(b()) 63.85/20.89 [0] [0] 63.85/20.90 63.85/20.90 [1 1 1] [1 0 0] [1 1 0] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(mark(X1),X2,X3) = [0 1 1]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 0] [1 0 0] [1 1 0] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(X1,mark(X2),X3) = [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 0] [1 0 0] [1 1 1] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(X1,X2,mark(X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 1 1]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 1] [1 0 0] [1 1 0] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(active(X1),X2,X3) = [0 1 1]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 0] [1 0 0] [1 1 0] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(X1,active(X2),X3) = [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 0] [1 0 0] [1 1 1] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 f(X1,X2,active(X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 1 1]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = f(X1,X2,X3) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 63.85/20.90 [1 1 0] [1 0 0] [1 1 0] [0] [1 1 0] [1 0 0] [1 1 0] [0] 63.85/20.90 mark(f(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 1 0]X3 + [1] = active(f(X1,mark(X2),X3)) 63.85/20.90 [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] 63.85/20.90 problem: 63.85/20.90 strict: 63.85/20.90 63.85/20.90 weak: 63.85/20.90 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.90 active(b()) -> mark(a()) 63.85/20.90 mark(a()) -> active(a()) 63.85/20.90 mark(b()) -> active(b()) 63.85/20.90 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.90 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.90 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.90 Qed 63.85/20.90 63.85/20.90 strict: 63.85/20.90 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.85/20.90 weak: 63.85/20.90 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.85/20.90 active(b()) -> mark(a()) 63.85/20.90 mark(a()) -> active(a()) 63.85/20.90 mark(b()) -> active(b()) 63.85/20.90 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.85/20.90 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.85/20.90 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.85/20.90 Matrix Interpretation Processor: dim=4 63.85/20.90 63.85/20.90 max_matrix: 63.85/20.90 [1 1 1 1] 63.85/20.90 [0 0 1 0] 63.85/20.90 [0 0 1 1] 63.85/20.90 [0 0 0 0] 63.85/20.90 interpretation: 63.85/20.90 [1 0 1 1] 63.85/20.90 [0 0 0 0] 63.85/20.90 [mark](x0) = [0 0 1 1]x0 63.85/20.90 [0 0 0 0] , 63.85/20.90 63.85/20.90 [0] 63.85/20.90 [1] 63.85/20.90 [b] = [0] 63.85/20.90 [1], 63.85/20.90 63.85/20.90 [1 1 0 0] 63.85/20.90 [0 0 0 0] 63.85/20.90 [active](x0) = [0 0 1 1]x0 63.95/20.91 [0 0 0 0] , 63.95/20.91 63.95/20.91 [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 [f](x0, x1, x2) = [0 0 1 0]x0 + [0 0 1 1]x1 + [0 0 1 0]x2 + [0] 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1], 63.95/20.91 63.95/20.91 [0] 63.95/20.91 [0] 63.95/20.91 [a] = [1] 63.95/20.91 [0] 63.95/20.91 orientation: 63.95/20.91 [1 0 2 0] [1 0 1 1] [1 0 2 0] [1] [1 0 2 0] [1 0 1 1] [1 0 2 0] [0] 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 63.95/20.91 mark(f(X1,X2,X3)) = [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [1] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [1] = active(f(X1,mark(X2),X3)) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 63.95/20.91 63.95/20.91 [2 0 2 0] [2] [1 0 2 0] [2] 63.95/20.91 [0 0 0 0] [0] [0 0 0 0] [0] 63.95/20.91 active(f(a(),X,X)) = [0 0 2 1]X + [2] >= [0 0 1 0]X + [2] = mark(f(X,b(),b())) 63.95/20.91 [0 0 0 0] [0] [0 0 0 0] [0] 63.95/20.91 63.95/20.91 [1] [1] 63.95/20.91 [0] [0] 63.95/20.91 active(b()) = [1] >= [1] = mark(a()) 63.95/20.91 [0] [0] 63.95/20.91 63.95/20.91 [1] [0] 63.95/20.91 [0] [0] 63.95/20.91 mark(a()) = [1] >= [1] = active(a()) 63.95/20.91 [0] [0] 63.95/20.91 63.95/20.91 [1] [1] 63.95/20.91 [0] [0] 63.95/20.91 mark(b()) = [1] >= [1] = active(b()) 63.95/20.91 [0] [0] 63.95/20.91 63.95/20.91 [1 0 2 2] [1 0 0 0] [1 0 1 0] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 1] [0 0 0 0] [0 0 1 0] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(mark(X1),X2,X3) = [0 0 1 1]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 63.95/20.91 [1 0 1 0] [1 0 1 1] [1 0 1 0] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(X1,mark(X2),X3) = [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 63.95/20.91 [1 0 1 0] [1 0 0 0] [1 0 2 2] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 0] [0 0 0 0] [0 0 1 1] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(X1,X2,mark(X3)) = [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 1]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 63.95/20.91 [1 1 1 1] [1 0 0 0] [1 0 1 0] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 1] [0 0 0 0] [0 0 1 0] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(active(X1),X2,X3) = [0 0 1 1]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 63.95/20.91 [1 0 1 0] [1 1 0 0] [1 0 1 0] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(X1,active(X2),X3) = [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 63.95/20.91 [1 0 1 0] [1 0 0 0] [1 1 1 1] [0] [1 0 1 0] [1 0 0 0] [1 0 1 0] [0] 63.95/20.91 [0 0 1 0] [0 0 0 0] [0 0 1 1] [0] [0 0 1 0] [0 0 0 0] [0 0 1 0] [0] 63.95/20.91 f(X1,X2,active(X3)) = [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 1]X3 + [0] >= [0 0 1 0]X1 + [0 0 1 1]X2 + [0 0 1 0]X3 + [0] = f(X1,X2,X3) 63.95/20.91 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 63.95/20.91 problem: 63.95/20.91 strict: 63.95/20.91 63.95/20.91 weak: 63.95/20.91 mark(f(X1,X2,X3)) -> active(f(X1,mark(X2),X3)) 63.95/20.91 active(f(a(),X,X)) -> mark(f(X,b(),b())) 63.95/20.91 active(b()) -> mark(a()) 63.95/20.91 mark(a()) -> active(a()) 63.95/20.91 mark(b()) -> active(b()) 63.95/20.91 f(mark(X1),X2,X3) -> f(X1,X2,X3) 63.95/20.91 f(X1,mark(X2),X3) -> f(X1,X2,X3) 63.95/20.91 f(X1,X2,mark(X3)) -> f(X1,X2,X3) 63.95/20.91 f(active(X1),X2,X3) -> f(X1,X2,X3) 63.95/20.91 f(X1,active(X2),X3) -> f(X1,X2,X3) 63.95/20.91 f(X1,X2,active(X3)) -> f(X1,X2,X3) 63.95/20.91 Qed 63.95/20.92 EOF