WORST_CASE(Omega(0),?) Initial ITS Start location: l4 0: l0 -> l1 : rt_11^0'=rt_11^post0, x_15^0'=x_15^post0, st_14^0'=st_14^post0, y_13^0'=y_13^post0, (-y_13^post0+y_13^0 == 0 /\ st_14^0-st_14^post0 == 0 /\ rt_11^post0-st_14^0 == 0 /\ x_15^0-x_15^post0 == 0 /\ y_13^0 <= 0), cost: 1 1: l0 -> l2 : rt_11^0'=rt_11^post1, x_15^0'=x_15^post1, st_14^0'=st_14^post1, y_13^0'=y_13^post1, (-st_14^post1+st_14^0 == 0 /\ rt_11^0-rt_11^post1 == 0 /\ -x_15^post1+x_15^0 == 0 /\ -y_13^post1+y_13^0 == 0 /\ 1-y_13^0 <= 0), cost: 1 2: l2 -> l1 : rt_11^0'=rt_11^post2, x_15^0'=x_15^post2, st_14^0'=st_14^post2, y_13^0'=y_13^post2, (x_15^0 <= 0 /\ -st_14^post2+st_14^0 == 0 /\ x_15^0-x_15^post2 == 0 /\ -y_13^post2+y_13^0 == 0 /\ rt_11^post2-st_14^0 == 0), cost: 1 3: l2 -> l3 : rt_11^0'=rt_11^post3, x_15^0'=x_15^post3, st_14^0'=st_14^post3, y_13^0'=y_13^post3, (rt_11^0-rt_11^post3 == 0 /\ -x_15^0+x_15^post3+y_13^0 == 0 /\ 1-x_15^0 <= 0 /\ -1+y_13^post3-y_13^0 == 0 /\ st_14^0-st_14^post3 == 0), cost: 1 4: l3 -> l2 : rt_11^0'=rt_11^post4, x_15^0'=x_15^post4, st_14^0'=st_14^post4, y_13^0'=y_13^post4, (rt_11^0-rt_11^post4 == 0 /\ -y_13^post4+y_13^0 == 0 /\ -st_14^post4+st_14^0 == 0 /\ -x_15^post4+x_15^0 == 0), cost: 1 5: l4 -> l0 : rt_11^0'=rt_11^post5, x_15^0'=x_15^post5, st_14^0'=st_14^post5, y_13^0'=y_13^post5, (rt_11^0-rt_11^post5 == 0 /\ x_15^0-x_15^post5 == 0 /\ -y_13^post5+y_13^0 == 0 /\ -st_14^post5+st_14^0 == 0), cost: 1 Removed unreachable rules and leafs Start location: l4 1: l0 -> l2 : rt_11^0'=rt_11^post1, x_15^0'=x_15^post1, st_14^0'=st_14^post1, y_13^0'=y_13^post1, (-st_14^post1+st_14^0 == 0 /\ rt_11^0-rt_11^post1 == 0 /\ -x_15^post1+x_15^0 == 0 /\ -y_13^post1+y_13^0 == 0 /\ 1-y_13^0 <= 0), cost: 1 3: l2 -> l3 : rt_11^0'=rt_11^post3, x_15^0'=x_15^post3, st_14^0'=st_14^post3, y_13^0'=y_13^post3, (rt_11^0-rt_11^post3 == 0 /\ -x_15^0+x_15^post3+y_13^0 == 0 /\ 1-x_15^0 <= 0 /\ -1+y_13^post3-y_13^0 == 0 /\ st_14^0-st_14^post3 == 0), cost: 1 4: l3 -> l2 : rt_11^0'=rt_11^post4, x_15^0'=x_15^post4, st_14^0'=st_14^post4, y_13^0'=y_13^post4, (rt_11^0-rt_11^post4 == 0 /\ -y_13^post4+y_13^0 == 0 /\ -st_14^post4+st_14^0 == 0 /\ -x_15^post4+x_15^0 == 0), cost: 1 5: l4 -> l0 : rt_11^0'=rt_11^post5, x_15^0'=x_15^post5, st_14^0'=st_14^post5, y_13^0'=y_13^post5, (rt_11^0-rt_11^post5 == 0 /\ x_15^0-x_15^post5 == 0 /\ -y_13^post5+y_13^0 == 0 /\ -st_14^post5+st_14^0 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l2 : rt_11^0'=rt_11^post1, x_15^0'=x_15^post1, st_14^0'=st_14^post1, y_13^0'=y_13^post1, (-st_14^post1+st_14^0 == 0 /\ rt_11^0-rt_11^post1 == 0 /\ -x_15^post1+x_15^0 == 0 /\ -y_13^post1+y_13^0 == 0 /\ 1-y_13^0 <= 0), cost: 1 New rule: l0 -> l2 : -1+y_13^0 >= 0, cost: 1 Applied preprocessing Original rule: l2 -> l3 : rt_11^0'=rt_11^post3, x_15^0'=x_15^post3, st_14^0'=st_14^post3, y_13^0'=y_13^post3, (rt_11^0-rt_11^post3 == 0 /\ -x_15^0+x_15^post3+y_13^0 == 0 /\ 1-x_15^0 <= 0 /\ -1+y_13^post3-y_13^0 == 0 /\ st_14^0-st_14^post3 == 0), cost: 1 New rule: l2 -> l3 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 1 Applied preprocessing Original rule: l3 -> l2 : rt_11^0'=rt_11^post4, x_15^0'=x_15^post4, st_14^0'=st_14^post4, y_13^0'=y_13^post4, (rt_11^0-rt_11^post4 == 0 /\ -y_13^post4+y_13^0 == 0 /\ -st_14^post4+st_14^0 == 0 /\ -x_15^post4+x_15^0 == 0), cost: 1 New rule: l3 -> l2 : TRUE, cost: 1 Applied preprocessing Original rule: l4 -> l0 : rt_11^0'=rt_11^post5, x_15^0'=x_15^post5, st_14^0'=st_14^post5, y_13^0'=y_13^post5, (rt_11^0-rt_11^post5 == 0 /\ x_15^0-x_15^post5 == 0 /\ -y_13^post5+y_13^0 == 0 /\ -st_14^post5+st_14^0 == 0), cost: 1 New rule: l4 -> l0 : TRUE, cost: 1 Simplified rules Start location: l4 6: l0 -> l2 : -1+y_13^0 >= 0, cost: 1 7: l2 -> l3 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 1 8: l3 -> l2 : TRUE, cost: 1 9: l4 -> l0 : TRUE, cost: 1 Eliminating location l0 by chaining: Applied chaining First rule: l4 -> l0 : TRUE, cost: 1 Second rule: l0 -> l2 : -1+y_13^0 >= 0, cost: 1 New rule: l4 -> l2 : -1+y_13^0 >= 0, cost: 2 Applied deletion Removed the following rules: 6 9 Eliminating location l3 by chaining: Applied chaining First rule: l2 -> l3 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 1 Second rule: l3 -> l2 : TRUE, cost: 1 New rule: l2 -> l2 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 2 Applied deletion Removed the following rules: 7 8 Eliminated locations on linear paths Start location: l4 11: l2 -> l2 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 2 10: l4 -> l2 : -1+y_13^0 >= 0, cost: 2 Applied acceleration Original rule: l2 -> l2 : x_15^0'=x_15^0-y_13^0, y_13^0'=1+y_13^0, -1+x_15^0 >= 0, cost: 2 New rule: l2 -> l2 : x_15^0'=x_15^0-1/2*n0^2+1/2*n0-n0*y_13^0, y_13^0'=n0+y_13^0, (-1+x_15^0 >= 0 /\ n0 >= 0 /\ -3/2-1/2*(-1+n0)^2+x_15^0+1/2*n0-(-1+n0)*y_13^0 >= 0), cost: 2*n0 Applied deletion Removed the following rules: 11 Accelerated simple loops Start location: l4 12: l2 -> l2 : x_15^0'=x_15^0-1/2*n0^2+1/2*n0-n0*y_13^0, y_13^0'=n0+y_13^0, (-1+x_15^0 >= 0 /\ n0 >= 0 /\ -3/2-1/2*(-1+n0)^2+x_15^0+1/2*n0-(-1+n0)*y_13^0 >= 0), cost: 2*n0 10: l4 -> l2 : -1+y_13^0 >= 0, cost: 2 Applied chaining First rule: l4 -> l2 : -1+y_13^0 >= 0, cost: 2 Second rule: l2 -> l2 : x_15^0'=x_15^0-1/2*n0^2+1/2*n0-n0*y_13^0, y_13^0'=n0+y_13^0, (-1+x_15^0 >= 0 /\ n0 >= 0 /\ -3/2-1/2*(-1+n0)^2+x_15^0+1/2*n0-(-1+n0)*y_13^0 >= 0), cost: 2*n0 New rule: l4 -> l2 : x_15^0'=x_15^0-1/2*n0^2+1/2*n0-n0*y_13^0, y_13^0'=n0+y_13^0, (-1+x_15^0 >= 0 /\ n0 >= 0 /\ -3/2-1/2*(-1+n0)^2+x_15^0+1/2*n0-(-1+n0)*y_13^0 >= 0 /\ -1+y_13^0 >= 0), cost: 2+2*n0 Applied deletion Removed the following rules: 12 Chained accelerated rules with incoming rules Start location: l4 10: l4 -> l2 : -1+y_13^0 >= 0, cost: 2 13: l4 -> l2 : x_15^0'=x_15^0-1/2*n0^2+1/2*n0-n0*y_13^0, y_13^0'=n0+y_13^0, (-1+x_15^0 >= 0 /\ n0 >= 0 /\ -3/2-1/2*(-1+n0)^2+x_15^0+1/2*n0-(-1+n0)*y_13^0 >= 0 /\ -1+y_13^0 >= 0), cost: 2+2*n0 Removed unreachable locations and irrelevant leafs Start location: l4 Computing asymptotic complexity Proved the following lower bound Complexity: Unknown Cpx degree: ? Solved cost: 0 Rule cost: 0