NO ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l6 0: l0 -> l1 : L^0'=L^post_1, n^0'=n^post_1, o^0'=o^post_1, [ 1+o^0<=n^0 && L^0==L^post_1 && n^0==n^post_1 && o^0==o^post_1 ], cost: 1 1: l0 -> l2 : L^0'=L^post_2, n^0'=n^post_2, o^0'=o^post_2, [ n^0<=o^0 && L^0==L^post_2 && n^0==n^post_2 && o^0==o^post_2 ], cost: 1 4: l1 -> l4 : L^0'=L^post_5, n^0'=n^post_5, o^0'=o^post_5, [ L^0<=1 && 1<=L^0 && L^0==L^post_5 && n^0==n^post_5 && o^0==o^post_5 ], cost: 1 5: l1 -> l3 : L^0'=L^post_6, n^0'=n^post_6, o^0'=o^post_6, [ L^post_6==1 && o^post_6==n^0 && n^0==n^post_6 ], cost: 1 2: l3 -> l0 : L^0'=L^post_3, n^0'=n^post_3, o^0'=o^post_3, [ n^0<=n^0 && L^0==L^post_3 && n^0==n^post_3 && o^0==o^post_3 ], cost: 1 3: l3 -> l0 : L^0'=L^post_4, n^0'=n^post_4, o^0'=o^post_4, [ L^post_4==0 && n^post_4==1+n^0 && o^0==o^post_4 ], cost: 1 6: l5 -> l1 : L^0'=L^post_7, n^0'=n^post_7, o^0'=o^post_7, [ L^post_7==0 && n^0==n^post_7 && o^0==o^post_7 ], cost: 1 7: l6 -> l5 : L^0'=L^post_8, n^0'=n^post_8, o^0'=o^post_8, [ L^0==L^post_8 && n^0==n^post_8 && o^0==o^post_8 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 7: l6 -> l5 : L^0'=L^post_8, n^0'=n^post_8, o^0'=o^post_8, [ L^0==L^post_8 && n^0==n^post_8 && o^0==o^post_8 ], cost: 1 Removed unreachable and leaf rules: Start location: l6 0: l0 -> l1 : L^0'=L^post_1, n^0'=n^post_1, o^0'=o^post_1, [ 1+o^0<=n^0 && L^0==L^post_1 && n^0==n^post_1 && o^0==o^post_1 ], cost: 1 5: l1 -> l3 : L^0'=L^post_6, n^0'=n^post_6, o^0'=o^post_6, [ L^post_6==1 && o^post_6==n^0 && n^0==n^post_6 ], cost: 1 2: l3 -> l0 : L^0'=L^post_3, n^0'=n^post_3, o^0'=o^post_3, [ n^0<=n^0 && L^0==L^post_3 && n^0==n^post_3 && o^0==o^post_3 ], cost: 1 3: l3 -> l0 : L^0'=L^post_4, n^0'=n^post_4, o^0'=o^post_4, [ L^post_4==0 && n^post_4==1+n^0 && o^0==o^post_4 ], cost: 1 6: l5 -> l1 : L^0'=L^post_7, n^0'=n^post_7, o^0'=o^post_7, [ L^post_7==0 && n^0==n^post_7 && o^0==o^post_7 ], cost: 1 7: l6 -> l5 : L^0'=L^post_8, n^0'=n^post_8, o^0'=o^post_8, [ L^0==L^post_8 && n^0==n^post_8 && o^0==o^post_8 ], cost: 1 Simplified all rules, resulting in: Start location: l6 0: l0 -> l1 : [ 1+o^0<=n^0 ], cost: 1 5: l1 -> l3 : L^0'=1, o^0'=n^0, [], cost: 1 2: l3 -> l0 : [], cost: 1 3: l3 -> l0 : L^0'=0, n^0'=1+n^0, [], cost: 1 6: l5 -> l1 : L^0'=0, [], cost: 1 7: l6 -> l5 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l6 0: l0 -> l1 : [ 1+o^0<=n^0 ], cost: 1 5: l1 -> l3 : L^0'=1, o^0'=n^0, [], cost: 1 2: l3 -> l0 : [], cost: 1 3: l3 -> l0 : L^0'=0, n^0'=1+n^0, [], cost: 1 8: l6 -> l1 : L^0'=0, [], cost: 2 Eliminated locations (on tree-shaped paths): Start location: l6 0: l0 -> l1 : [ 1+o^0<=n^0 ], cost: 1 9: l1 -> l0 : L^0'=1, o^0'=n^0, [], cost: 2 10: l1 -> l0 : L^0'=0, n^0'=1+n^0, o^0'=n^0, [], cost: 2 8: l6 -> l1 : L^0'=0, [], cost: 2 Eliminated locations (on tree-shaped paths): Start location: l6 11: l1 -> l1 : L^0'=0, n^0'=1+n^0, o^0'=n^0, [], cost: 3 8: l6 -> l1 : L^0'=0, [], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 11: l1 -> l1 : L^0'=0, n^0'=1+n^0, o^0'=n^0, [], cost: 3 Accelerated rule 11 with non-termination, yielding the new rule 12. [accelerate] Nesting with 0 inner and 0 outer candidates Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: l6 12: l1 -> [7] : [], cost: NONTERM 8: l6 -> l1 : L^0'=0, [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l6 8: l6 -> l1 : L^0'=0, [], cost: 2 13: l6 -> [7] : [], cost: NONTERM Removed unreachable locations (and leaf rules with constant cost): Start location: l6 13: l6 -> [7] : [], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l6 13: l6 -> [7] : [], cost: NONTERM Computing asymptotic complexity for rule 13 Guard is satisfiable, yielding nontermination Resulting cost NONTERM has complexity: Nonterm Found new complexity Nonterm. Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: [] NO