NO Initial ITS Start location: l4 0: l0 -> l2 : x^0'=x^post0, 1-x^0+x^post0 == 0, cost: 1 1: l2 -> l1 : x^0'=x^post1, (-299+x^0 <= 0 /\ -x^post1+x^0 == 0), cost: 1 2: l2 -> l1 : x^0'=x^post2, (-x^post2+x^0 == 0 /\ 301-x^0 <= 0), cost: 1 3: l1 -> l0 : x^0'=x^post3, x^0-x^post3 == 0, cost: 1 4: l3 -> l0 : x^0'=x^post4, -100+x^post4 == 0, cost: 1 5: l4 -> l3 : x^0'=x^post5, -x^post5+x^0 == 0, cost: 1 Applied preprocessing Original rule: l0 -> l2 : x^0'=x^post0, 1-x^0+x^post0 == 0, cost: 1 New rule: l0 -> l2 : x^0'=-1+x^0, TRUE, cost: 1 Applied preprocessing Original rule: l2 -> l1 : x^0'=x^post1, (-299+x^0 <= 0 /\ -x^post1+x^0 == 0), cost: 1 New rule: l2 -> l1 : -299+x^0 <= 0, cost: 1 Applied preprocessing Original rule: l2 -> l1 : x^0'=x^post2, (-x^post2+x^0 == 0 /\ 301-x^0 <= 0), cost: 1 New rule: l2 -> l1 : -301+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l1 -> l0 : x^0'=x^post3, x^0-x^post3 == 0, cost: 1 New rule: l1 -> l0 : TRUE, cost: 1 Applied preprocessing Original rule: l3 -> l0 : x^0'=x^post4, -100+x^post4 == 0, cost: 1 New rule: l3 -> l0 : x^0'=100, TRUE, cost: 1 Applied preprocessing Original rule: l4 -> l3 : x^0'=x^post5, -x^post5+x^0 == 0, cost: 1 New rule: l4 -> l3 : TRUE, cost: 1 Simplified rules Start location: l4 6: l0 -> l2 : x^0'=-1+x^0, TRUE, cost: 1 7: l2 -> l1 : -299+x^0 <= 0, cost: 1 8: l2 -> l1 : -301+x^0 >= 0, cost: 1 9: l1 -> l0 : TRUE, cost: 1 10: l3 -> l0 : x^0'=100, 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 -> l0 : x^0'=100, TRUE, cost: 1 New rule: l4 -> l0 : x^0'=100, TRUE, cost: 2 Applied deletion Removed the following rules: 10 11 Eliminated locations on linear paths Start location: l4 6: l0 -> l2 : x^0'=-1+x^0, TRUE, cost: 1 7: l2 -> l1 : -299+x^0 <= 0, cost: 1 8: l2 -> l1 : -301+x^0 >= 0, cost: 1 9: l1 -> l0 : TRUE, cost: 1 12: l4 -> l0 : x^0'=100, TRUE, cost: 2 Eliminating location l2 by chaining: Applied chaining First rule: l0 -> l2 : x^0'=-1+x^0, TRUE, cost: 1 Second rule: l2 -> l1 : -299+x^0 <= 0, cost: 1 New rule: l0 -> l1 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 2 Applied chaining First rule: l0 -> l2 : x^0'=-1+x^0, TRUE, cost: 1 Second rule: l2 -> l1 : -301+x^0 >= 0, cost: 1 New rule: l0 -> l1 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 2 Applied deletion Removed the following rules: 6 7 8 Eliminated locations on tree-shaped paths Start location: l4 13: l0 -> l1 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 2 14: l0 -> l1 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 2 9: l1 -> l0 : TRUE, cost: 1 12: l4 -> l0 : x^0'=100, TRUE, cost: 2 Eliminating location l1 by chaining: Applied chaining First rule: l0 -> l1 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 2 Second rule: l1 -> l0 : TRUE, cost: 1 New rule: l0 -> l0 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 3 Applied chaining First rule: l0 -> l1 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 2 Second rule: l1 -> l0 : TRUE, cost: 1 New rule: l0 -> l0 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 3 Applied deletion Removed the following rules: 9 13 14 Eliminated locations on tree-shaped paths Start location: l4 15: l0 -> l0 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 3 16: l0 -> l0 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 3 12: l4 -> l0 : x^0'=100, TRUE, cost: 2 Applied nonterm Original rule: l0 -> l0 : x^0'=-1+x^0, -300+x^0 <= 0, cost: 3 New rule: l0 -> [5] : (300-x^0 >= 0 /\ n >= 0), cost: NONTERM Sub-proof via acceration calculus written to file:///tmp/tmpnam_ACHgJc.txt Applied acceleration Original rule: l0 -> l0 : x^0'=-1+x^0, -302+x^0 >= 0, cost: 3 New rule: l0 -> l0 : x^0'=x^0-n0, (-301+x^0-n0 >= 0 /\ n0 >= 0), cost: 3*n0 Sub-proof via acceration calculus written to file:///tmp/tmpnam_ecjeJH.txt Applied instantiation Original rule: l0 -> l0 : x^0'=x^0-n0, (-301+x^0-n0 >= 0 /\ n0 >= 0), cost: 3*n0 New rule: l0 -> l0 : x^0'=301, (0 >= 0 /\ -301+x^0 >= 0), cost: -903+3*x^0 Applied simplification Original rule: l0 -> [5] : (300-x^0 >= 0 /\ n >= 0), cost: NONTERM New rule: l0 -> [5] : (-300+x^0 <= 0 /\ n >= 0), cost: NONTERM Applied simplification Original rule: l0 -> l0 : x^0'=301, (0 >= 0 /\ -301+x^0 >= 0), cost: -903+3*x^0 New rule: l0 -> l0 : x^0'=301, -301+x^0 >= 0, cost: -903+3*x^0 Applied deletion Removed the following rules: 15 16 Accelerated simple loops Start location: l4 19: l0 -> [5] : (-300+x^0 <= 0 /\ n >= 0), cost: NONTERM 20: l0 -> l0 : x^0'=301, -301+x^0 >= 0, cost: -903+3*x^0 12: l4 -> l0 : x^0'=100, TRUE, cost: 2 Applied chaining First rule: l4 -> l0 : x^0'=100, TRUE, cost: 2 Second rule: l0 -> [5] : (-300+x^0 <= 0 /\ n >= 0), cost: NONTERM New rule: l4 -> [5] : -200 <= 0, cost: NONTERM Applied deletion Removed the following rules: 19 20 Chained accelerated rules with incoming rules Start location: l4 12: l4 -> l0 : x^0'=100, TRUE, cost: 2 21: l4 -> [5] : -200 <= 0, cost: NONTERM Removed unreachable locations and irrelevant leafs Start location: l4 21: l4 -> [5] : -200 <= 0, cost: NONTERM Computing asymptotic complexity Proved nontermination of rule 21 via SMT. Proved the following lower bound Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: -200 <= 0