NO Initial ITS Start location: l3 0: l0 -> l1 : x^0'=x^post0, (-300-x^0+x^post0 == 0 /\ 101-x^post0 <= 0), cost: 1 1: l1 -> l0 : x^0'=x^post1, (101-x^post1 <= 0 /\ -400-x^0+x^post1 == 0), cost: 1 2: l2 -> l0 : x^0'=x^post2, (-200-x^0+x^post2 == 0 /\ 101-x^post2 <= 0), cost: 1 3: l2 -> l1 : x^0'=x^post3, (-100-x^0+x^post3 == 0 /\ 101-x^post3 <= 0), cost: 1 4: l3 -> l2 : x^0'=x^post4, x^0-x^post4 == 0, cost: 1 Applied preprocessing Original rule: l0 -> l1 : x^0'=x^post0, (-300-x^0+x^post0 == 0 /\ 101-x^post0 <= 0), cost: 1 New rule: l0 -> l1 : x^0'=300+x^0, 199+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l1 -> l0 : x^0'=x^post1, (101-x^post1 <= 0 /\ -400-x^0+x^post1 == 0), cost: 1 New rule: l1 -> l0 : x^0'=400+x^0, 299+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l2 -> l0 : x^0'=x^post2, (-200-x^0+x^post2 == 0 /\ 101-x^post2 <= 0), cost: 1 New rule: l2 -> l0 : x^0'=200+x^0, 99+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l2 -> l1 : x^0'=x^post3, (-100-x^0+x^post3 == 0 /\ 101-x^post3 <= 0), cost: 1 New rule: l2 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 1 Applied preprocessing Original rule: l3 -> l2 : x^0'=x^post4, x^0-x^post4 == 0, cost: 1 New rule: l3 -> l2 : TRUE, cost: 1 Simplified rules Start location: l3 5: l0 -> l1 : x^0'=300+x^0, 199+x^0 >= 0, cost: 1 6: l1 -> l0 : x^0'=400+x^0, 299+x^0 >= 0, cost: 1 7: l2 -> l0 : x^0'=200+x^0, 99+x^0 >= 0, cost: 1 8: l2 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 1 9: l3 -> l2 : TRUE, cost: 1 Eliminating location l2 by chaining: Applied chaining First rule: l3 -> l2 : TRUE, cost: 1 Second rule: l2 -> l0 : x^0'=200+x^0, 99+x^0 >= 0, cost: 1 New rule: l3 -> l0 : x^0'=200+x^0, 99+x^0 >= 0, cost: 2 Applied chaining First rule: l3 -> l2 : TRUE, cost: 1 Second rule: l2 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 1 New rule: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 Applied deletion Removed the following rules: 7 8 9 Eliminated locations on tree-shaped paths Start location: l3 5: l0 -> l1 : x^0'=300+x^0, 199+x^0 >= 0, cost: 1 6: l1 -> l0 : x^0'=400+x^0, 299+x^0 >= 0, cost: 1 10: l3 -> l0 : x^0'=200+x^0, 99+x^0 >= 0, cost: 2 11: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 Eliminated location l0 Start location: l3 12: l1 -> l1 : x^0'=700+x^0, 299+x^0 >= 0, cost: 2 11: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 13: l3 -> l1 : x^0'=500+x^0, 99+x^0 >= 0, cost: 3 Applied nonterm Original rule: l1 -> l1 : x^0'=700+x^0, 299+x^0 >= 0, cost: 2 New rule: l1 -> [4] : 299+x^0 >= 0, cost: NONTERM Applied acceleration Original rule: l1 -> l1 : x^0'=700+x^0, 299+x^0 >= 0, cost: 2 New rule: l1 -> l1 : x^0'=x^0+700*n0, 299+x^0 >= 0, cost: 2*n0 Applied deletion Removed the following rules: 12 Accelerated simple loops Start location: l3 14: l1 -> [4] : 299+x^0 >= 0, cost: NONTERM 15: l1 -> l1 : x^0'=x^0+700*n0, 299+x^0 >= 0, cost: 2*n0 11: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 13: l3 -> l1 : x^0'=500+x^0, 99+x^0 >= 0, cost: 3 Applied chaining First rule: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 Second rule: l1 -> [4] : 299+x^0 >= 0, cost: NONTERM New rule: l3 -> [4] : -1+x^0 >= 0, cost: NONTERM Applied chaining First rule: l3 -> l1 : x^0'=500+x^0, 99+x^0 >= 0, cost: 3 Second rule: l1 -> [4] : 299+x^0 >= 0, cost: NONTERM New rule: l3 -> [4] : 99+x^0 >= 0, cost: NONTERM Applied chaining First rule: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 Second rule: l1 -> l1 : x^0'=x^0+700*n0, 299+x^0 >= 0, cost: 2*n0 New rule: l3 -> l1 : x^0'=100+x^0+700*n0, -1+x^0 >= 0, cost: 2+2*n0 Applied chaining First rule: l3 -> l1 : x^0'=500+x^0, 99+x^0 >= 0, cost: 3 Second rule: l1 -> l1 : x^0'=x^0+700*n0, 299+x^0 >= 0, cost: 2*n0 New rule: l3 -> l1 : x^0'=500+x^0+700*n0, 99+x^0 >= 0, cost: 3+2*n0 Applied deletion Removed the following rules: 14 15 Chained accelerated rules with incoming rules Start location: l3 11: l3 -> l1 : x^0'=100+x^0, -1+x^0 >= 0, cost: 2 13: l3 -> l1 : x^0'=500+x^0, 99+x^0 >= 0, cost: 3 16: l3 -> [4] : -1+x^0 >= 0, cost: NONTERM 17: l3 -> [4] : 99+x^0 >= 0, cost: NONTERM 18: l3 -> l1 : x^0'=100+x^0+700*n0, -1+x^0 >= 0, cost: 2+2*n0 19: l3 -> l1 : x^0'=500+x^0+700*n0, 99+x^0 >= 0, cost: 3+2*n0 Removed unreachable locations and irrelevant leafs Start location: l3 16: l3 -> [4] : -1+x^0 >= 0, cost: NONTERM 17: l3 -> [4] : 99+x^0 >= 0, cost: NONTERM Computing asymptotic complexity Proved nontermination of rule 16 via SMT. Proved the following lower bound Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: -1+x^0 >= 0