NO Initial ITS Start location: l9 0: l0 -> l1 : R^0'=R^post0, n^0'=n^post0, A^0'=A^post0, dobreak^0'=dobreak^post0, (0 == 0 /\ 1-n^post0 <= 0 /\ A^post0 == 0 /\ -R^post0+R^0 == 0 /\ -1+A^10 == 0 /\ -dobreak^post0+dobreak^0 == 0 /\ dobreak^0 <= 0), cost: 1 1: l0 -> l2 : R^0'=R^post1, n^0'=n^post1, A^0'=A^post1, dobreak^0'=dobreak^post1, (-dobreak^post1+dobreak^0 == 0 /\ A^0-A^post1 == 0 /\ n^0-n^post1 == 0 /\ 1-dobreak^0 <= 0 /\ R^0-R^post1 == 0), cost: 1 3: l1 -> l4 : R^0'=R^post3, n^0'=n^post3, A^0'=A^post3, dobreak^0'=dobreak^post3, (-n^post3+n^0 == 0 /\ -A^post3+A^0 == 0 /\ -dobreak^post3+dobreak^0 == 0 /\ R^0-R^post3 == 0), cost: 1 5: l2 -> l7 : R^0'=R^post5, n^0'=n^post5, A^0'=A^post5, dobreak^0'=dobreak^post5, (n^0-n^post5 == 0 /\ R^0-R^post5 == 0 /\ dobreak^0-dobreak^post5 == 0 /\ A^0-A^post5 == 0), cost: 1 2: l3 -> l0 : R^0'=R^post2, n^0'=n^post2, A^0'=A^post2, dobreak^0'=dobreak^post2, (R^0-R^post2 == 0 /\ -dobreak^post2+dobreak^0 == 0 /\ n^0-n^post2 == 0 /\ -A^post2+A^0 == 0), cost: 1 7: l4 -> l1 : R^0'=R^post7, n^0'=n^post7, A^0'=A^post7, dobreak^0'=dobreak^post7, (-A^post7+A^0 == 0 /\ 1-n^0 <= 0 /\ -dobreak^post7+dobreak^0 == 0 /\ -n^post7+n^0 == 0 /\ R^0-R^post7 == 0), cost: 1 8: l4 -> l3 : R^0'=R^post8, n^0'=n^post8, A^0'=A^post8, dobreak^0'=dobreak^post8, (0 == 0 /\ -n^post8+n^0 == 0 /\ n^0 <= 0 /\ -1+R^10 == 0 /\ -A^post8+A^0 == 0 /\ R^post8 == 0), cost: 1 4: l5 -> l6 : R^0'=R^post4, n^0'=n^post4, A^0'=A^post4, dobreak^0'=dobreak^post4, (-n^post4+n^0 == 0 /\ -A^post4+A^0 == 0 /\ R^0-R^post4 == 0 /\ -dobreak^post4+dobreak^0 == 0), cost: 1 6: l7 -> l2 : R^0'=R^post6, n^0'=n^post6, A^0'=A^post6, dobreak^0'=dobreak^post6, (A^0-A^post6 == 0 /\ R^0-R^post6 == 0 /\ -dobreak^post6+dobreak^0 == 0 /\ n^0-n^post6 == 0), cost: 1 9: l8 -> l3 : R^0'=R^post9, n^0'=n^post9, A^0'=A^post9, dobreak^0'=dobreak^post9, (0 == 0 /\ R^post9 == 0 /\ A^post9 == 0 /\ -n^post9+n^0 == 0), cost: 1 10: l9 -> l8 : R^0'=R^post10, n^0'=n^post10, A^0'=A^post10, dobreak^0'=dobreak^post10, (-dobreak^post10+dobreak^0 == 0 /\ n^0-n^post10 == 0 /\ R^0-R^post10 == 0 /\ A^0-A^post10 == 0), cost: 1 Removed unreachable rules and leafs Start location: l9 0: l0 -> l1 : R^0'=R^post0, n^0'=n^post0, A^0'=A^post0, dobreak^0'=dobreak^post0, (0 == 0 /\ 1-n^post0 <= 0 /\ A^post0 == 0 /\ -R^post0+R^0 == 0 /\ -1+A^10 == 0 /\ -dobreak^post0+dobreak^0 == 0 /\ dobreak^0 <= 0), cost: 1 1: l0 -> l2 : R^0'=R^post1, n^0'=n^post1, A^0'=A^post1, dobreak^0'=dobreak^post1, (-dobreak^post1+dobreak^0 == 0 /\ A^0-A^post1 == 0 /\ n^0-n^post1 == 0 /\ 1-dobreak^0 <= 0 /\ R^0-R^post1 == 0), cost: 1 3: l1 -> l4 : R^0'=R^post3, n^0'=n^post3, A^0'=A^post3, dobreak^0'=dobreak^post3, (-n^post3+n^0 == 0 /\ -A^post3+A^0 == 0 /\ -dobreak^post3+dobreak^0 == 0 /\ R^0-R^post3 == 0), cost: 1 5: l2 -> l7 : R^0'=R^post5, n^0'=n^post5, A^0'=A^post5, dobreak^0'=dobreak^post5, (n^0-n^post5 == 0 /\ R^0-R^post5 == 0 /\ dobreak^0-dobreak^post5 == 0 /\ A^0-A^post5 == 0), cost: 1 2: l3 -> l0 : R^0'=R^post2, n^0'=n^post2, A^0'=A^post2, dobreak^0'=dobreak^post2, (R^0-R^post2 == 0 /\ -dobreak^post2+dobreak^0 == 0 /\ n^0-n^post2 == 0 /\ -A^post2+A^0 == 0), cost: 1 7: l4 -> l1 : R^0'=R^post7, n^0'=n^post7, A^0'=A^post7, dobreak^0'=dobreak^post7, (-A^post7+A^0 == 0 /\ 1-n^0 <= 0 /\ -dobreak^post7+dobreak^0 == 0 /\ -n^post7+n^0 == 0 /\ R^0-R^post7 == 0), cost: 1 8: l4 -> l3 : R^0'=R^post8, n^0'=n^post8, A^0'=A^post8, dobreak^0'=dobreak^post8, (0 == 0 /\ -n^post8+n^0 == 0 /\ n^0 <= 0 /\ -1+R^10 == 0 /\ -A^post8+A^0 == 0 /\ R^post8 == 0), cost: 1 6: l7 -> l2 : R^0'=R^post6, n^0'=n^post6, A^0'=A^post6, dobreak^0'=dobreak^post6, (A^0-A^post6 == 0 /\ R^0-R^post6 == 0 /\ -dobreak^post6+dobreak^0 == 0 /\ n^0-n^post6 == 0), cost: 1 9: l8 -> l3 : R^0'=R^post9, n^0'=n^post9, A^0'=A^post9, dobreak^0'=dobreak^post9, (0 == 0 /\ R^post9 == 0 /\ A^post9 == 0 /\ -n^post9+n^0 == 0), cost: 1 10: l9 -> l8 : R^0'=R^post10, n^0'=n^post10, A^0'=A^post10, dobreak^0'=dobreak^post10, (-dobreak^post10+dobreak^0 == 0 /\ n^0-n^post10 == 0 /\ R^0-R^post10 == 0 /\ A^0-A^post10 == 0), cost: 1 Applied preprocessing Original rule: l0 -> l1 : R^0'=R^post0, n^0'=n^post0, A^0'=A^post0, dobreak^0'=dobreak^post0, (0 == 0 /\ 1-n^post0 <= 0 /\ A^post0 == 0 /\ -R^post0+R^0 == 0 /\ -1+A^10 == 0 /\ -dobreak^post0+dobreak^0 == 0 /\ dobreak^0 <= 0), cost: 1 New rule: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 Applied preprocessing Original rule: l0 -> l2 : R^0'=R^post1, n^0'=n^post1, A^0'=A^post1, dobreak^0'=dobreak^post1, (-dobreak^post1+dobreak^0 == 0 /\ A^0-A^post1 == 0 /\ n^0-n^post1 == 0 /\ 1-dobreak^0 <= 0 /\ R^0-R^post1 == 0), cost: 1 New rule: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 Applied preprocessing Original rule: l3 -> l0 : R^0'=R^post2, n^0'=n^post2, A^0'=A^post2, dobreak^0'=dobreak^post2, (R^0-R^post2 == 0 /\ -dobreak^post2+dobreak^0 == 0 /\ n^0-n^post2 == 0 /\ -A^post2+A^0 == 0), cost: 1 New rule: l3 -> l0 : TRUE, cost: 1 Applied preprocessing Original rule: l1 -> l4 : R^0'=R^post3, n^0'=n^post3, A^0'=A^post3, dobreak^0'=dobreak^post3, (-n^post3+n^0 == 0 /\ -A^post3+A^0 == 0 /\ -dobreak^post3+dobreak^0 == 0 /\ R^0-R^post3 == 0), cost: 1 New rule: l1 -> l4 : TRUE, cost: 1 Applied preprocessing Original rule: l2 -> l7 : R^0'=R^post5, n^0'=n^post5, A^0'=A^post5, dobreak^0'=dobreak^post5, (n^0-n^post5 == 0 /\ R^0-R^post5 == 0 /\ dobreak^0-dobreak^post5 == 0 /\ A^0-A^post5 == 0), cost: 1 New rule: l2 -> l7 : TRUE, cost: 1 Applied preprocessing Original rule: l7 -> l2 : R^0'=R^post6, n^0'=n^post6, A^0'=A^post6, dobreak^0'=dobreak^post6, (A^0-A^post6 == 0 /\ R^0-R^post6 == 0 /\ -dobreak^post6+dobreak^0 == 0 /\ n^0-n^post6 == 0), cost: 1 New rule: l7 -> l2 : TRUE, cost: 1 Applied preprocessing Original rule: l4 -> l1 : R^0'=R^post7, n^0'=n^post7, A^0'=A^post7, dobreak^0'=dobreak^post7, (-A^post7+A^0 == 0 /\ 1-n^0 <= 0 /\ -dobreak^post7+dobreak^0 == 0 /\ -n^post7+n^0 == 0 /\ R^0-R^post7 == 0), cost: 1 New rule: l4 -> l1 : -1+n^0 >= 0, cost: 1 Applied preprocessing Original rule: l4 -> l3 : R^0'=R^post8, n^0'=n^post8, A^0'=A^post8, dobreak^0'=dobreak^post8, (0 == 0 /\ -n^post8+n^0 == 0 /\ n^0 <= 0 /\ -1+R^10 == 0 /\ -A^post8+A^0 == 0 /\ R^post8 == 0), cost: 1 New rule: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 Applied preprocessing Original rule: l8 -> l3 : R^0'=R^post9, n^0'=n^post9, A^0'=A^post9, dobreak^0'=dobreak^post9, (0 == 0 /\ R^post9 == 0 /\ A^post9 == 0 /\ -n^post9+n^0 == 0), cost: 1 New rule: l8 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 1 Applied preprocessing Original rule: l9 -> l8 : R^0'=R^post10, n^0'=n^post10, A^0'=A^post10, dobreak^0'=dobreak^post10, (-dobreak^post10+dobreak^0 == 0 /\ n^0-n^post10 == 0 /\ R^0-R^post10 == 0 /\ A^0-A^post10 == 0), cost: 1 New rule: l9 -> l8 : TRUE, cost: 1 Simplified rules Start location: l9 11: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 12: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 14: l1 -> l4 : TRUE, cost: 1 15: l2 -> l7 : TRUE, cost: 1 13: l3 -> l0 : TRUE, cost: 1 17: l4 -> l1 : -1+n^0 >= 0, cost: 1 18: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 16: l7 -> l2 : TRUE, cost: 1 19: l8 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 1 20: l9 -> l8 : TRUE, cost: 1 Eliminating location l8 by chaining: Applied chaining First rule: l9 -> l8 : TRUE, cost: 1 Second rule: l8 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 1 New rule: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Applied deletion Removed the following rules: 19 20 Eliminating location l7 by chaining: Applied chaining First rule: l2 -> l7 : TRUE, cost: 1 Second rule: l7 -> l2 : TRUE, cost: 1 New rule: l2 -> l2 : TRUE, cost: 2 Applied deletion Removed the following rules: 15 16 Eliminated locations on linear paths Start location: l9 11: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 12: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 14: l1 -> l4 : TRUE, cost: 1 22: l2 -> l2 : TRUE, cost: 2 13: l3 -> l0 : TRUE, cost: 1 17: l4 -> l1 : -1+n^0 >= 0, cost: 1 18: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Applied nonterm Original rule: l2 -> l2 : TRUE, cost: 2 New rule: l2 -> [10] : n >= 0, cost: NONTERM Sub-proof via acceration calculus written to file:///tmp/tmpnam_DfMOID.txt Applied deletion Removed the following rules: 22 Accelerated simple loops Start location: l9 11: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 12: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 14: l1 -> l4 : TRUE, cost: 1 23: l2 -> [10] : n >= 0, cost: NONTERM 13: l3 -> l0 : TRUE, cost: 1 17: l4 -> l1 : -1+n^0 >= 0, cost: 1 18: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Applied chaining First rule: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 Second rule: l2 -> [10] : n >= 0, cost: NONTERM New rule: l0 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM Applied deletion Removed the following rules: 23 Chained accelerated rules with incoming rules Start location: l9 11: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 12: l0 -> l2 : -1+dobreak^0 >= 0, cost: 1 24: l0 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 14: l1 -> l4 : TRUE, cost: 1 13: l3 -> l0 : TRUE, cost: 1 17: l4 -> l1 : -1+n^0 >= 0, cost: 1 18: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Removed unreachable locations and irrelevant leafs Start location: l9 11: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 24: l0 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 14: l1 -> l4 : TRUE, cost: 1 13: l3 -> l0 : TRUE, cost: 1 17: l4 -> l1 : -1+n^0 >= 0, cost: 1 18: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Eliminating location l0 by chaining: Applied chaining First rule: l3 -> l0 : TRUE, cost: 1 Second rule: l0 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 1 New rule: l3 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 2 Applied chaining First rule: l3 -> l0 : TRUE, cost: 1 Second rule: l0 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM New rule: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM Applied deletion Removed the following rules: 11 13 24 Eliminating location l4 by chaining: Applied chaining First rule: l1 -> l4 : TRUE, cost: 1 Second rule: l4 -> l1 : -1+n^0 >= 0, cost: 1 New rule: l1 -> l1 : -1+n^0 >= 0, cost: 2 Applied chaining First rule: l1 -> l4 : TRUE, cost: 1 Second rule: l4 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 1 New rule: l1 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 2 Applied deletion Removed the following rules: 14 17 18 Eliminated locations on tree-shaped paths Start location: l9 27: l1 -> l1 : -1+n^0 >= 0, cost: 2 28: l1 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 2 25: l3 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 2 26: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Applied nonterm Original rule: l1 -> l1 : -1+n^0 >= 0, cost: 2 New rule: l1 -> [11] : (-1+n^0 >= 0 /\ n0 >= 0), cost: NONTERM Sub-proof via acceration calculus written to file:///tmp/tmpnam_hmiiOI.txt Applied deletion Removed the following rules: 27 Accelerated simple loops Start location: l9 28: l1 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 2 29: l1 -> [11] : (-1+n^0 >= 0 /\ n0 >= 0), cost: NONTERM 25: l3 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 2 26: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Applied chaining First rule: l3 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 2 Second rule: l1 -> [11] : (-1+n^0 >= 0 /\ n0 >= 0), cost: NONTERM New rule: l3 -> [11] : dobreak^0 <= 0, cost: NONTERM Applied deletion Removed the following rules: 29 Chained accelerated rules with incoming rules Start location: l9 28: l1 -> l3 : R^0'=0, dobreak^0'=dobreak^post8, n^0 <= 0, cost: 2 25: l3 -> l1 : n^0'=n^post0, A^0'=0, (-1+n^post0 >= 0 /\ dobreak^0 <= 0), cost: 2 26: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 30: l3 -> [11] : dobreak^0 <= 0, cost: NONTERM 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Eliminating location l1 by chaining: Applied deletion Removed the following rules: 25 28 Eliminated locations on linear paths Start location: l9 26: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM 30: l3 -> [11] : dobreak^0 <= 0, cost: NONTERM 21: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Eliminating location l3 by chaining: Applied chaining First rule: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Second rule: l3 -> [10] : -1+dobreak^0 >= 0, cost: NONTERM New rule: l9 -> [10] : (0 == 0 /\ -1+dobreak^post9 >= 0), cost: NONTERM Applied simplification Original rule: l9 -> [10] : (0 == 0 /\ -1+dobreak^post9 >= 0), cost: NONTERM New rule: l9 -> [10] : -1+dobreak^post9 >= 0, cost: NONTERM Applied chaining First rule: l9 -> l3 : R^0'=0, A^0'=0, dobreak^0'=dobreak^post9, 0 == 0, cost: 2 Second rule: l3 -> [11] : dobreak^0 <= 0, cost: NONTERM New rule: l9 -> [11] : (0 == 0 /\ dobreak^post9 <= 0), cost: NONTERM Applied simplification Original rule: l9 -> [11] : (0 == 0 /\ dobreak^post9 <= 0), cost: NONTERM New rule: l9 -> [11] : dobreak^post9 <= 0, cost: NONTERM Applied deletion Removed the following rules: 21 26 30 Eliminated locations on tree-shaped paths Start location: l9 31: l9 -> [10] : -1+dobreak^post9 >= 0, cost: NONTERM 32: l9 -> [11] : dobreak^post9 <= 0, cost: NONTERM Computing asymptotic complexity Proved nontermination of rule 31 via SMT. Proved the following lower bound Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: -1+dobreak^post9 >= 0