NO Initial ITS Start location: l6 0: l0 -> l1 : L^0'=L^post0, o^0'=o^post0, n^0'=n^post0, (1+o^0-n^0 <= 0 /\ L^0-L^post0 == 0 /\ o^0-o^post0 == 0 /\ -n^post0+n^0 == 0), cost: 1 1: l0 -> l2 : L^0'=L^post1, o^0'=o^post1, n^0'=n^post1, (o^0-o^post1 == 0 /\ -o^0+n^0 <= 0 /\ L^0-L^post1 == 0 /\ n^0-n^post1 == 0), cost: 1 4: l1 -> l4 : L^0'=L^post4, o^0'=o^post4, n^0'=n^post4, (1-L^0 <= 0 /\ -1+L^0 <= 0 /\ o^0-o^post4 == 0 /\ n^0-n^post4 == 0 /\ L^0-L^post4 == 0), cost: 1 5: l1 -> l3 : L^0'=L^post5, o^0'=o^post5, n^0'=n^post5, (-1+L^post5 == 0 /\ o^post5-n^0 == 0 /\ n^0-n^post5 == 0), cost: 1 2: l3 -> l0 : L^0'=L^post2, o^0'=o^post2, n^0'=n^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ L^0-L^post2 == 0), cost: 1 3: l3 -> l0 : L^0'=L^post3, o^0'=o^post3, n^0'=n^post3, (L^post3 == 0 /\ -1-n^0+n^post3 == 0 /\ o^0-o^post3 == 0), cost: 1 6: l5 -> l1 : L^0'=L^post6, o^0'=o^post6, n^0'=n^post6, (n^0-n^post6 == 0 /\ L^post6 == 0 /\ o^0-o^post6 == 0), cost: 1 7: l6 -> l5 : L^0'=L^post7, o^0'=o^post7, n^0'=n^post7, (o^0-o^post7 == 0 /\ n^0-n^post7 == 0 /\ L^0-L^post7 == 0), cost: 1 Removed unreachable rules and leafs Start location: l6 0: l0 -> l1 : L^0'=L^post0, o^0'=o^post0, n^0'=n^post0, (1+o^0-n^0 <= 0 /\ L^0-L^post0 == 0 /\ o^0-o^post0 == 0 /\ -n^post0+n^0 == 0), cost: 1 5: l1 -> l3 : L^0'=L^post5, o^0'=o^post5, n^0'=n^post5, (-1+L^post5 == 0 /\ o^post5-n^0 == 0 /\ n^0-n^post5 == 0), cost: 1 2: l3 -> l0 : L^0'=L^post2, o^0'=o^post2, n^0'=n^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ L^0-L^post2 == 0), cost: 1 3: l3 -> l0 : L^0'=L^post3, o^0'=o^post3, n^0'=n^post3, (L^post3 == 0 /\ -1-n^0+n^post3 == 0 /\ o^0-o^post3 == 0), cost: 1 6: l5 -> l1 : L^0'=L^post6, o^0'=o^post6, n^0'=n^post6, (n^0-n^post6 == 0 /\ L^post6 == 0 /\ o^0-o^post6 == 0), cost: 1 7: l6 -> l5 : L^0'=L^post7, o^0'=o^post7, n^0'=n^post7, (o^0-o^post7 == 0 /\ n^0-n^post7 == 0 /\ L^0-L^post7 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l1 : L^0'=L^post0, o^0'=o^post0, n^0'=n^post0, (1+o^0-n^0 <= 0 /\ L^0-L^post0 == 0 /\ o^0-o^post0 == 0 /\ -n^post0+n^0 == 0), cost: 1 New rule: l0 -> l1 : 1+o^0-n^0 <= 0, cost: 1 Applied preprocessing Original rule: l3 -> l0 : L^0'=L^post2, o^0'=o^post2, n^0'=n^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ L^0-L^post2 == 0), cost: 1 New rule: l3 -> l0 : 0 <= 0, cost: 1 Applied preprocessing Original rule: l3 -> l0 : L^0'=L^post3, o^0'=o^post3, n^0'=n^post3, (L^post3 == 0 /\ -1-n^0+n^post3 == 0 /\ o^0-o^post3 == 0), cost: 1 New rule: l3 -> l0 : L^0'=0, n^0'=1+n^0, TRUE, cost: 1 Applied preprocessing Original rule: l1 -> l3 : L^0'=L^post5, o^0'=o^post5, n^0'=n^post5, (-1+L^post5 == 0 /\ o^post5-n^0 == 0 /\ n^0-n^post5 == 0), cost: 1 New rule: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 Applied preprocessing Original rule: l5 -> l1 : L^0'=L^post6, o^0'=o^post6, n^0'=n^post6, (n^0-n^post6 == 0 /\ L^post6 == 0 /\ o^0-o^post6 == 0), cost: 1 New rule: l5 -> l1 : L^0'=0, TRUE, cost: 1 Applied preprocessing Original rule: l6 -> l5 : L^0'=L^post7, o^0'=o^post7, n^0'=n^post7, (o^0-o^post7 == 0 /\ n^0-n^post7 == 0 /\ L^0-L^post7 == 0), cost: 1 New rule: l6 -> l5 : TRUE, cost: 1 Simplified rules Start location: l6 8: l0 -> l1 : 1+o^0-n^0 <= 0, cost: 1 11: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 9: l3 -> l0 : 0 <= 0, cost: 1 10: l3 -> l0 : L^0'=0, n^0'=1+n^0, TRUE, cost: 1 12: l5 -> l1 : L^0'=0, TRUE, cost: 1 13: l6 -> l5 : TRUE, cost: 1 Eliminating location l5 by chaining: Applied chaining First rule: l6 -> l5 : TRUE, cost: 1 Second rule: l5 -> l1 : L^0'=0, TRUE, cost: 1 New rule: l6 -> l1 : L^0'=0, TRUE, cost: 2 Applied deletion Removed the following rules: 12 13 Eliminated locations on linear paths Start location: l6 8: l0 -> l1 : 1+o^0-n^0 <= 0, cost: 1 11: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 9: l3 -> l0 : 0 <= 0, cost: 1 10: l3 -> l0 : L^0'=0, n^0'=1+n^0, TRUE, cost: 1 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 Eliminating location l3 by chaining: Applied chaining First rule: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 Second rule: l3 -> l0 : 0 <= 0, cost: 1 New rule: l1 -> l0 : L^0'=1, o^0'=n^0, 0 <= 0, cost: 2 Applied chaining First rule: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 Second rule: l3 -> l0 : L^0'=0, n^0'=1+n^0, TRUE, cost: 1 New rule: l1 -> l0 : L^0'=0, o^0'=n^0, n^0'=1+n^0, TRUE, cost: 2 Applied deletion Removed the following rules: 9 10 11 Eliminated locations on tree-shaped paths Start location: l6 8: l0 -> l1 : 1+o^0-n^0 <= 0, cost: 1 15: l1 -> l0 : L^0'=1, o^0'=n^0, 0 <= 0, cost: 2 16: l1 -> l0 : L^0'=0, o^0'=n^0, n^0'=1+n^0, TRUE, cost: 2 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 Eliminating location l0 by chaining: Applied chaining First rule: l1 -> l0 : L^0'=0, o^0'=n^0, n^0'=1+n^0, TRUE, cost: 2 Second rule: l0 -> l1 : 1+o^0-n^0 <= 0, cost: 1 New rule: l1 -> l1 : L^0'=0, o^0'=n^0, n^0'=1+n^0, 0 <= 0, cost: 3 Applied deletion Removed the following rules: 8 15 16 Eliminated locations on tree-shaped paths Start location: l6 17: l1 -> l1 : L^0'=0, o^0'=n^0, n^0'=1+n^0, 0 <= 0, cost: 3 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 Applied nonterm Original rule: l1 -> l1 : L^0'=0, o^0'=n^0, n^0'=1+n^0, 0 <= 0, cost: 3 New rule: l1 -> [7] : 0 >= 0, cost: NONTERM Applied acceleration Original rule: l1 -> l1 : L^0'=0, o^0'=n^0, n^0'=1+n^0, 0 <= 0, cost: 3 New rule: l1 -> l1 : L^0'=0, o^0'=-1+n^0+n0, n^0'=n^0+n0, 0 >= 0, cost: 3*n0 Applied deletion Removed the following rules: 17 Accelerated simple loops Start location: l6 18: l1 -> [7] : 0 >= 0, cost: NONTERM 19: l1 -> l1 : L^0'=0, o^0'=-1+n^0+n0, n^0'=n^0+n0, 0 >= 0, cost: 3*n0 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 Applied chaining First rule: l6 -> l1 : L^0'=0, TRUE, cost: 2 Second rule: l1 -> [7] : 0 >= 0, cost: NONTERM New rule: l6 -> [7] : 0 >= 0, cost: NONTERM Applied chaining First rule: l6 -> l1 : L^0'=0, TRUE, cost: 2 Second rule: l1 -> l1 : L^0'=0, o^0'=-1+n^0+n0, n^0'=n^0+n0, 0 >= 0, cost: 3*n0 New rule: l6 -> l1 : L^0'=0, o^0'=-1+n^0+n0, n^0'=n^0+n0, 0 >= 0, cost: 2+3*n0 Applied deletion Removed the following rules: 18 19 Chained accelerated rules with incoming rules Start location: l6 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 20: l6 -> [7] : 0 >= 0, cost: NONTERM 21: l6 -> l1 : L^0'=0, o^0'=-1+n^0+n0, n^0'=n^0+n0, 0 >= 0, cost: 2+3*n0 Removed unreachable locations and irrelevant leafs Start location: l6 20: l6 -> [7] : 0 >= 0, cost: NONTERM Computing asymptotic complexity Proved nontermination of rule 20 via SMT. Proved the following lower bound Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: 0 >= 0