NO Initial ITS Start location: l6 0: l0 -> l1 : n^0'=n^post0, L^0'=L^post0, o^0'=o^post0, (o^0-o^post0 == 0 /\ L^0-L^post0 == 0 /\ n^0-n^post0 == 0 /\ 1-n^0+o^0 <= 0), cost: 1 1: l0 -> l2 : n^0'=n^post1, L^0'=L^post1, o^0'=o^post1, (o^0-o^post1 == 0 /\ L^0-L^post1 == 0 /\ n^0-o^0 <= 0 /\ n^0-n^post1 == 0), cost: 1 4: l1 -> l4 : n^0'=n^post4, L^0'=L^post4, o^0'=o^post4, (-L^post4+L^0 == 0 /\ -o^post4+o^0 == 0 /\ -n^post4+n^0 == 0 /\ 1-L^0 <= 0 /\ -1+L^0 <= 0), cost: 1 5: l1 -> l3 : n^0'=n^post5, L^0'=L^post5, o^0'=o^post5, (-n^0+o^post5 == 0 /\ -n^post5+n^0 == 0 /\ -1+L^post5 == 0), cost: 1 2: l3 -> l0 : n^0'=n^post2, L^0'=L^post2, o^0'=o^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ -L^post2+L^0 == 0), cost: 1 3: l3 -> l0 : n^0'=n^post3, L^0'=L^post3, o^0'=o^post3, (-o^post3+o^0 == 0 /\ L^post3 == 0 /\ -1+n^post3-n^0 == 0), cost: 1 6: l5 -> l1 : n^0'=n^post6, L^0'=L^post6, o^0'=o^post6, (L^post6 == 0 /\ n^0-n^post6 == 0 /\ -o^post6+o^0 == 0), cost: 1 7: l6 -> l5 : n^0'=n^post7, L^0'=L^post7, o^0'=o^post7, (-o^post7+o^0 == 0 /\ -L^post7+L^0 == 0 /\ n^0-n^post7 == 0), cost: 1 Removed unreachable rules and leafs Start location: l6 0: l0 -> l1 : n^0'=n^post0, L^0'=L^post0, o^0'=o^post0, (o^0-o^post0 == 0 /\ L^0-L^post0 == 0 /\ n^0-n^post0 == 0 /\ 1-n^0+o^0 <= 0), cost: 1 5: l1 -> l3 : n^0'=n^post5, L^0'=L^post5, o^0'=o^post5, (-n^0+o^post5 == 0 /\ -n^post5+n^0 == 0 /\ -1+L^post5 == 0), cost: 1 2: l3 -> l0 : n^0'=n^post2, L^0'=L^post2, o^0'=o^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ -L^post2+L^0 == 0), cost: 1 3: l3 -> l0 : n^0'=n^post3, L^0'=L^post3, o^0'=o^post3, (-o^post3+o^0 == 0 /\ L^post3 == 0 /\ -1+n^post3-n^0 == 0), cost: 1 6: l5 -> l1 : n^0'=n^post6, L^0'=L^post6, o^0'=o^post6, (L^post6 == 0 /\ n^0-n^post6 == 0 /\ -o^post6+o^0 == 0), cost: 1 7: l6 -> l5 : n^0'=n^post7, L^0'=L^post7, o^0'=o^post7, (-o^post7+o^0 == 0 /\ -L^post7+L^0 == 0 /\ n^0-n^post7 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l1 : n^0'=n^post0, L^0'=L^post0, o^0'=o^post0, (o^0-o^post0 == 0 /\ L^0-L^post0 == 0 /\ n^0-n^post0 == 0 /\ 1-n^0+o^0 <= 0), cost: 1 New rule: l0 -> l1 : 1-n^0+o^0 <= 0, cost: 1 Applied preprocessing Original rule: l3 -> l0 : n^0'=n^post2, L^0'=L^post2, o^0'=o^post2, (0 <= 0 /\ o^0-o^post2 == 0 /\ n^0-n^post2 == 0 /\ -L^post2+L^0 == 0), cost: 1 New rule: l3 -> l0 : 0 <= 0, cost: 1 Applied preprocessing Original rule: l3 -> l0 : n^0'=n^post3, L^0'=L^post3, o^0'=o^post3, (-o^post3+o^0 == 0 /\ L^post3 == 0 /\ -1+n^post3-n^0 == 0), cost: 1 New rule: l3 -> l0 : n^0'=1+n^0, L^0'=0, TRUE, cost: 1 Applied preprocessing Original rule: l1 -> l3 : n^0'=n^post5, L^0'=L^post5, o^0'=o^post5, (-n^0+o^post5 == 0 /\ -n^post5+n^0 == 0 /\ -1+L^post5 == 0), cost: 1 New rule: l1 -> l3 : L^0'=1, o^0'=n^0, TRUE, cost: 1 Applied preprocessing Original rule: l5 -> l1 : n^0'=n^post6, L^0'=L^post6, o^0'=o^post6, (L^post6 == 0 /\ n^0-n^post6 == 0 /\ -o^post6+o^0 == 0), cost: 1 New rule: l5 -> l1 : L^0'=0, TRUE, cost: 1 Applied preprocessing Original rule: l6 -> l5 : n^0'=n^post7, L^0'=L^post7, o^0'=o^post7, (-o^post7+o^0 == 0 /\ -L^post7+L^0 == 0 /\ n^0-n^post7 == 0), cost: 1 New rule: l6 -> l5 : TRUE, cost: 1 Simplified rules Start location: l6 8: l0 -> l1 : 1-n^0+o^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 : n^0'=1+n^0, L^0'=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-n^0+o^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 : n^0'=1+n^0, L^0'=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 : n^0'=1+n^0, L^0'=0, TRUE, cost: 1 New rule: l1 -> l0 : n^0'=1+n^0, L^0'=0, o^0'=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-n^0+o^0 <= 0, cost: 1 15: l1 -> l0 : L^0'=1, o^0'=n^0, 0 <= 0, cost: 2 16: l1 -> l0 : n^0'=1+n^0, L^0'=0, o^0'=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 : n^0'=1+n^0, L^0'=0, o^0'=n^0, TRUE, cost: 2 Second rule: l0 -> l1 : 1-n^0+o^0 <= 0, cost: 1 New rule: l1 -> l1 : n^0'=1+n^0, L^0'=0, o^0'=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 : n^0'=1+n^0, L^0'=0, o^0'=n^0, 0 <= 0, cost: 3 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 Applied nonterm Original rule: l1 -> l1 : n^0'=1+n^0, L^0'=0, o^0'=n^0, 0 <= 0, cost: 3 New rule: l1 -> [7] : (0 >= 0 /\ -1+n >= 0), cost: NONTERM Sub-proof via acceration calculus written to file:///tmp/tmpnam_NMgEPc.txt Applied simplification Original rule: l1 -> [7] : (0 >= 0 /\ -1+n >= 0), cost: NONTERM New rule: l1 -> [7] : -1+n >= 0, cost: NONTERM Applied deletion Removed the following rules: 17 Accelerated simple loops Start location: l6 19: l1 -> [7] : -1+n >= 0, cost: NONTERM 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] : -1+n >= 0, cost: NONTERM New rule: l6 -> [7] : TRUE, cost: NONTERM Applied deletion Removed the following rules: 19 Chained accelerated rules with incoming rules Start location: l6 14: l6 -> l1 : L^0'=0, TRUE, cost: 2 20: l6 -> [7] : TRUE, cost: NONTERM Removed unreachable locations and irrelevant leafs Start location: l6 20: l6 -> [7] : TRUE, 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: TRUE