WORST_CASE(Omega(0),?) Initial ITS Start location: l5 0: l0 -> l1 : x^0'=x^post0, __const_5^0'=__const_5^post0, (x^0 <= 0 /\ -__const_5^post0+__const_5^0 == 0 /\ -1+x^post0 == 0), cost: 1 1: l0 -> l1 : x^0'=x^post1, __const_5^0'=__const_5^post1, (__const_5^0-__const_5^post1 == 0 /\ 1-x^0 <= 0 /\ -1-x^0+x^post1 == 0), cost: 1 4: l1 -> l2 : x^0'=x^post4, __const_5^0'=__const_5^post4, (x^0-x^post4 == 0 /\ __const_5^0-__const_5^post4 == 0), cost: 1 2: l2 -> l3 : x^0'=x^post2, __const_5^0'=__const_5^post2, (4-x^0 <= 0 /\ __const_5^0-__const_5^post2 == 0 /\ x^0-x^post2 == 0), cost: 1 3: l2 -> l0 : x^0'=x^post3, __const_5^0'=__const_5^post3, (-3+x^0 <= 0 /\ x^0-x^post3 == 0 /\ -__const_5^post3+__const_5^0 == 0), cost: 1 5: l4 -> l1 : x^0'=x^post5, __const_5^0'=__const_5^post5, (0 == 0 /\ -__const_5^post5+__const_5^0 == 0 /\ -__const_5^0+x^10 == 0), cost: 1 6: l5 -> l4 : x^0'=x^post6, __const_5^0'=__const_5^post6, (-__const_5^post6+__const_5^0 == 0 /\ x^0-x^post6 == 0), cost: 1 Removed unreachable rules and leafs Start location: l5 0: l0 -> l1 : x^0'=x^post0, __const_5^0'=__const_5^post0, (x^0 <= 0 /\ -__const_5^post0+__const_5^0 == 0 /\ -1+x^post0 == 0), cost: 1 1: l0 -> l1 : x^0'=x^post1, __const_5^0'=__const_5^post1, (__const_5^0-__const_5^post1 == 0 /\ 1-x^0 <= 0 /\ -1-x^0+x^post1 == 0), cost: 1 4: l1 -> l2 : x^0'=x^post4, __const_5^0'=__const_5^post4, (x^0-x^post4 == 0 /\ __const_5^0-__const_5^post4 == 0), cost: 1 3: l2 -> l0 : x^0'=x^post3, __const_5^0'=__const_5^post3, (-3+x^0 <= 0 /\ x^0-x^post3 == 0 /\ -__const_5^post3+__const_5^0 == 0), cost: 1 5: l4 -> l1 : x^0'=x^post5, __const_5^0'=__const_5^post5, (0 == 0 /\ -__const_5^post5+__const_5^0 == 0 /\ -__const_5^0+x^10 == 0), cost: 1 6: l5 -> l4 : x^0'=x^post6, __const_5^0'=__const_5^post6, (-__const_5^post6+__const_5^0 == 0 /\ x^0-x^post6 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l1 : x^0'=x^post0, __const_5^0'=__const_5^post0, (x^0 <= 0 /\ -__const_5^post0+__const_5^0 == 0 /\ -1+x^post0 == 0), cost: 1 New rule: l0 -> l1 : x^0'=1, x^0 <= 0, cost: 1 Applied preprocessing Original rule: l0 -> l1 : x^0'=x^post1, __const_5^0'=__const_5^post1, (__const_5^0-__const_5^post1 == 0 /\ 1-x^0 <= 0 /\ -1-x^0+x^post1 == 0), cost: 1 New rule: l0 -> l1 : x^0'=1+x^0, -1+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l2 -> l0 : x^0'=x^post3, __const_5^0'=__const_5^post3, (-3+x^0 <= 0 /\ x^0-x^post3 == 0 /\ -__const_5^post3+__const_5^0 == 0), cost: 1 New rule: l2 -> l0 : -3+x^0 <= 0, cost: 1 Applied preprocessing Original rule: l1 -> l2 : x^0'=x^post4, __const_5^0'=__const_5^post4, (x^0-x^post4 == 0 /\ __const_5^0-__const_5^post4 == 0), cost: 1 New rule: l1 -> l2 : TRUE, cost: 1 Applied preprocessing Original rule: l4 -> l1 : x^0'=x^post5, __const_5^0'=__const_5^post5, (0 == 0 /\ -__const_5^post5+__const_5^0 == 0 /\ -__const_5^0+x^10 == 0), cost: 1 New rule: l4 -> l1 : x^0'=x^post5, 0 == 0, cost: 1 Applied preprocessing Original rule: l5 -> l4 : x^0'=x^post6, __const_5^0'=__const_5^post6, (-__const_5^post6+__const_5^0 == 0 /\ x^0-x^post6 == 0), cost: 1 New rule: l5 -> l4 : TRUE, cost: 1 Simplified rules Start location: l5 7: l0 -> l1 : x^0'=1, x^0 <= 0, cost: 1 8: l0 -> l1 : x^0'=1+x^0, -1+x^0 >= 0, cost: 1 10: l1 -> l2 : TRUE, cost: 1 9: l2 -> l0 : -3+x^0 <= 0, cost: 1 11: l4 -> l1 : x^0'=x^post5, 0 == 0, cost: 1 12: l5 -> l4 : TRUE, cost: 1 Eliminating location l4 by chaining: Applied chaining First rule: l5 -> l4 : TRUE, cost: 1 Second rule: l4 -> l1 : x^0'=x^post5, 0 == 0, cost: 1 New rule: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Applied deletion Removed the following rules: 11 12 Eliminating location l2 by chaining: Applied chaining First rule: l1 -> l2 : TRUE, cost: 1 Second rule: l2 -> l0 : -3+x^0 <= 0, cost: 1 New rule: l1 -> l0 : -3+x^0 <= 0, cost: 2 Applied deletion Removed the following rules: 9 10 Eliminated locations on linear paths Start location: l5 7: l0 -> l1 : x^0'=1, x^0 <= 0, cost: 1 8: l0 -> l1 : x^0'=1+x^0, -1+x^0 >= 0, cost: 1 14: l1 -> l0 : -3+x^0 <= 0, cost: 2 13: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Eliminating location l0 by chaining: Applied chaining First rule: l1 -> l0 : -3+x^0 <= 0, cost: 2 Second rule: l0 -> l1 : x^0'=1, x^0 <= 0, cost: 1 New rule: l1 -> l1 : x^0'=1, (x^0 <= 0 /\ -3+x^0 <= 0), cost: 3 Applied simplification Original rule: l1 -> l1 : x^0'=1, (x^0 <= 0 /\ -3+x^0 <= 0), cost: 3 New rule: l1 -> l1 : x^0'=1, x^0 <= 0, cost: 3 Applied chaining First rule: l1 -> l0 : -3+x^0 <= 0, cost: 2 Second rule: l0 -> l1 : x^0'=1+x^0, -1+x^0 >= 0, cost: 1 New rule: l1 -> l1 : x^0'=1+x^0, (-3+x^0 <= 0 /\ -1+x^0 >= 0), cost: 3 Applied deletion Removed the following rules: 7 8 14 Eliminated locations on tree-shaped paths Start location: l5 15: l1 -> l1 : x^0'=1, x^0 <= 0, cost: 3 16: l1 -> l1 : x^0'=1+x^0, (-3+x^0 <= 0 /\ -1+x^0 >= 0), cost: 3 13: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Applied acceleration Original rule: l1 -> l1 : x^0'=1+x^0, (-3+x^0 <= 0 /\ -1+x^0 >= 0), cost: 3 New rule: l1 -> l1 : x^0'=n0+x^0, (n0 >= 0 /\ 4-n0-x^0 >= 0 /\ -1+x^0 >= 0), cost: 3*n0 Sub-proof via acceration calculus written to file:///tmp/tmpnam_cBfEPb.txt Applied instantiation Original rule: l1 -> l1 : x^0'=n0+x^0, (n0 >= 0 /\ 4-n0-x^0 >= 0 /\ -1+x^0 >= 0), cost: 3*n0 New rule: l1 -> l1 : x^0'=4, (0 >= 0 /\ 4-x^0 >= 0 /\ -1+x^0 >= 0), cost: 12-3*x^0 Applied simplification Original rule: l1 -> l1 : x^0'=4, (0 >= 0 /\ 4-x^0 >= 0 /\ -1+x^0 >= 0), cost: 12-3*x^0 New rule: l1 -> l1 : x^0'=4, (-4+x^0 <= 0 /\ -1+x^0 >= 0), cost: 12-3*x^0 Applied deletion Removed the following rules: 16 Accelerated simple loops Start location: l5 15: l1 -> l1 : x^0'=1, x^0 <= 0, cost: 3 18: l1 -> l1 : x^0'=4, (-4+x^0 <= 0 /\ -1+x^0 >= 0), cost: 12-3*x^0 13: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Applied chaining First rule: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Second rule: l1 -> l1 : x^0'=1, x^0 <= 0, cost: 3 New rule: l5 -> l1 : x^0'=1, 0 == 0, cost: 5 Applied chaining First rule: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 Second rule: l1 -> l1 : x^0'=4, (-4+x^0 <= 0 /\ -1+x^0 >= 0), cost: 12-3*x^0 New rule: l5 -> l1 : x^0'=4, (-4+x^post5 <= 0 /\ -1+x^post5 >= 0), cost: 14-3*x^post5 Applied deletion Removed the following rules: 15 18 Chained accelerated rules with incoming rules Start location: l5 13: l5 -> l1 : x^0'=x^post5, 0 == 0, cost: 2 19: l5 -> l1 : x^0'=1, 0 == 0, cost: 5 20: l5 -> l1 : x^0'=4, (-4+x^post5 <= 0 /\ -1+x^post5 >= 0), cost: 14-3*x^post5 Removed unreachable locations and irrelevant leafs Start location: l5 Computing asymptotic complexity Proved the following lower bound Complexity: Unknown Cpx degree: ? Solved cost: 0 Rule cost: 0