WORST_CASE(Omega(0),?) Initial ITS Start location: l4 0: l0 -> l1 : y^0'=y^post0, x^0'=x^post0, (-1+y^post0-y^0 == 0 /\ y^0 <= 0 /\ 1+x^post0-x^0 == 0), cost: 1 1: l0 -> l2 : y^0'=y^post1, x^0'=x^post1, (1-y^0+y^post1 == 0 /\ x^0-x^post1 == 0 /\ 1-y^0 <= 0), cost: 1 4: l1 -> l0 : y^0'=y^post4, x^0'=x^post4, (x^0-x^post4 == 0 /\ -x^0+y^post4 == 0 /\ 1-x^0 <= 0), cost: 1 2: l2 -> l0 : y^0'=y^post2, x^0'=x^post2, (x^0-x^post2 == 0 /\ y^0-y^post2 == 0), cost: 1 3: l3 -> l1 : y^0'=y^post3, x^0'=x^post3, (y^0-y^post3 == 0 /\ -x^post3+x^0 == 0), cost: 1 5: l4 -> l3 : y^0'=y^post5, x^0'=x^post5, (y^0-y^post5 == 0 /\ -x^post5+x^0 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l1 : y^0'=y^post0, x^0'=x^post0, (-1+y^post0-y^0 == 0 /\ y^0 <= 0 /\ 1+x^post0-x^0 == 0), cost: 1 New rule: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 Applied preprocessing Original rule: l0 -> l2 : y^0'=y^post1, x^0'=x^post1, (1-y^0+y^post1 == 0 /\ x^0-x^post1 == 0 /\ 1-y^0 <= 0), cost: 1 New rule: l0 -> l2 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 1 Applied preprocessing Original rule: l2 -> l0 : y^0'=y^post2, x^0'=x^post2, (x^0-x^post2 == 0 /\ y^0-y^post2 == 0), cost: 1 New rule: l2 -> l0 : TRUE, cost: 1 Applied preprocessing Original rule: l3 -> l1 : y^0'=y^post3, x^0'=x^post3, (y^0-y^post3 == 0 /\ -x^post3+x^0 == 0), cost: 1 New rule: l3 -> l1 : TRUE, cost: 1 Applied preprocessing Original rule: l1 -> l0 : y^0'=y^post4, x^0'=x^post4, (x^0-x^post4 == 0 /\ -x^0+y^post4 == 0 /\ 1-x^0 <= 0), cost: 1 New rule: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l4 -> l3 : y^0'=y^post5, x^0'=x^post5, (y^0-y^post5 == 0 /\ -x^post5+x^0 == 0), cost: 1 New rule: l4 -> l3 : TRUE, cost: 1 Simplified rules Start location: l4 6: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 7: l0 -> l2 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 1 10: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 8: l2 -> l0 : TRUE, cost: 1 9: l3 -> l1 : TRUE, cost: 1 11: l4 -> l3 : TRUE, cost: 1 Eliminating location l3 by chaining: Applied chaining First rule: l4 -> l3 : TRUE, cost: 1 Second rule: l3 -> l1 : TRUE, cost: 1 New rule: l4 -> l1 : TRUE, cost: 2 Applied deletion Removed the following rules: 9 11 Eliminating location l2 by chaining: Applied chaining First rule: l0 -> l2 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 1 Second rule: l2 -> l0 : TRUE, cost: 1 New rule: l0 -> l0 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 2 Applied deletion Removed the following rules: 7 8 Eliminated locations on linear paths Start location: l4 6: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 13: l0 -> l0 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 2 10: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 12: l4 -> l1 : TRUE, cost: 2 Applied acceleration Original rule: l0 -> l0 : y^0'=-1+y^0, -1+y^0 >= 0, cost: 2 New rule: l0 -> l0 : y^0'=y^0-n, (y^0-n >= 0 /\ n >= 0), cost: 2*n Sub-proof via acceration calculus written to file:///tmp/tmpnam_MhieoO.txt Applied instantiation Original rule: l0 -> l0 : y^0'=y^0-n, (y^0-n >= 0 /\ n >= 0), cost: 2*n New rule: l0 -> l0 : y^0'=0, (0 >= 0 /\ y^0 >= 0), cost: 2*y^0 Applied simplification Original rule: l0 -> l0 : y^0'=0, (0 >= 0 /\ y^0 >= 0), cost: 2*y^0 New rule: l0 -> l0 : y^0'=0, y^0 >= 0, cost: 2*y^0 Applied deletion Removed the following rules: 13 Accelerated simple loops Start location: l4 6: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 15: l0 -> l0 : y^0'=0, y^0 >= 0, cost: 2*y^0 10: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 12: l4 -> l1 : TRUE, cost: 2 Applied chaining First rule: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 Second rule: l0 -> l0 : y^0'=0, y^0 >= 0, cost: 2*y^0 New rule: l1 -> l0 : y^0'=0, -1+x^0 >= 0, cost: 1+2*x^0 Applied deletion Removed the following rules: 15 Chained accelerated rules with incoming rules Start location: l4 6: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 10: l1 -> l0 : y^0'=x^0, -1+x^0 >= 0, cost: 1 16: l1 -> l0 : y^0'=0, -1+x^0 >= 0, cost: 1+2*x^0 12: l4 -> l1 : TRUE, cost: 2 Eliminating location l0 by chaining: Applied chaining First rule: l1 -> l0 : y^0'=0, -1+x^0 >= 0, cost: 1+2*x^0 Second rule: l0 -> l1 : y^0'=1+y^0, x^0'=-1+x^0, y^0 <= 0, cost: 1 New rule: l1 -> l1 : y^0'=1, x^0'=-1+x^0, (0 <= 0 /\ -1+x^0 >= 0), cost: 2+2*x^0 Applied simplification Original rule: l1 -> l1 : y^0'=1, x^0'=-1+x^0, (0 <= 0 /\ -1+x^0 >= 0), cost: 2+2*x^0 New rule: l1 -> l1 : y^0'=1, x^0'=-1+x^0, -1+x^0 >= 0, cost: 2+2*x^0 Applied deletion Removed the following rules: 6 10 16 Eliminated locations on tree-shaped paths Start location: l4 17: l1 -> l1 : y^0'=1, x^0'=-1+x^0, -1+x^0 >= 0, cost: 2+2*x^0 12: l4 -> l1 : TRUE, cost: 2 Applied acceleration Original rule: l1 -> l1 : y^0'=1, x^0'=-1+x^0, -1+x^0 >= 0, cost: 2+2*x^0 New rule: l1 -> l1 : y^0'=1, x^0'=x^0-n0, (x^0-n0 >= 0 /\ -1+n0 >= 0), cost: -n0^2+2*x^0*n0+3*n0 Sub-proof via acceration calculus written to file:///tmp/tmpnam_gakBMD.txt Applied instantiation Original rule: l1 -> l1 : y^0'=1, x^0'=x^0-n0, (x^0-n0 >= 0 /\ -1+n0 >= 0), cost: -n0^2+2*x^0*n0+3*n0 New rule: l1 -> l1 : y^0'=1, x^0'=0, (0 >= 0 /\ -1+x^0 >= 0), cost: 3*x^0+x^0^2 Applied simplification Original rule: l1 -> l1 : y^0'=1, x^0'=0, (0 >= 0 /\ -1+x^0 >= 0), cost: 3*x^0+x^0^2 New rule: l1 -> l1 : y^0'=1, x^0'=0, -1+x^0 >= 0, cost: 3*x^0+x^0^2 Applied deletion Removed the following rules: 17 Accelerated simple loops Start location: l4 19: l1 -> l1 : y^0'=1, x^0'=0, -1+x^0 >= 0, cost: 3*x^0+x^0^2 12: l4 -> l1 : TRUE, cost: 2 Applied chaining First rule: l4 -> l1 : TRUE, cost: 2 Second rule: l1 -> l1 : y^0'=1, x^0'=0, -1+x^0 >= 0, cost: 3*x^0+x^0^2 New rule: l4 -> l1 : y^0'=1, x^0'=0, -1+x^0 >= 0, cost: 2+3*x^0+x^0^2 Applied deletion Removed the following rules: 19 Chained accelerated rules with incoming rules Start location: l4 12: l4 -> l1 : TRUE, cost: 2 20: l4 -> l1 : y^0'=1, x^0'=0, -1+x^0 >= 0, cost: 2+3*x^0+x^0^2 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