NO ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l4 0: l0 -> l1 : Result_4^0'=Result_4^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ Result_4^0==Result_4^post_1 && x_5^0==x_5^post_1 && y_6^0==y_6^post_1 ], cost: 1 1: l1 -> l2 : Result_4^0'=Result_4^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ x_5^post_2==x_5^0+y_6^0 && -x_5^post_2<=0 && Result_4^0==Result_4^post_2 && y_6^0==y_6^post_2 ], cost: 1 3: l1 -> l3 : Result_4^0'=Result_4^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ x_5^post_4==x_5^0+y_6^0 && 0<=-1-x_5^post_4 && Result_4^post_4==Result_4^post_4 && y_6^0==y_6^post_4 ], cost: 1 2: l2 -> l1 : Result_4^0'=Result_4^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 4: l4 -> l0 : Result_4^0'=Result_4^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ Result_4^0==Result_4^post_5 && x_5^0==x_5^post_5 && y_6^0==y_6^post_5 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 4: l4 -> l0 : Result_4^0'=Result_4^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ Result_4^0==Result_4^post_5 && x_5^0==x_5^post_5 && y_6^0==y_6^post_5 ], cost: 1 Removed unreachable and leaf rules: Start location: l4 0: l0 -> l1 : Result_4^0'=Result_4^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ Result_4^0==Result_4^post_1 && x_5^0==x_5^post_1 && y_6^0==y_6^post_1 ], cost: 1 1: l1 -> l2 : Result_4^0'=Result_4^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ x_5^post_2==x_5^0+y_6^0 && -x_5^post_2<=0 && Result_4^0==Result_4^post_2 && y_6^0==y_6^post_2 ], cost: 1 2: l2 -> l1 : Result_4^0'=Result_4^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 4: l4 -> l0 : Result_4^0'=Result_4^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ Result_4^0==Result_4^post_5 && x_5^0==x_5^post_5 && y_6^0==y_6^post_5 ], cost: 1 Simplified all rules, resulting in: Start location: l4 0: l0 -> l1 : [], cost: 1 1: l1 -> l2 : x_5^0'=x_5^0+y_6^0, [ -x_5^0-y_6^0<=0 ], cost: 1 2: l2 -> l1 : [], cost: 1 4: l4 -> l0 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l4 6: l1 -> l1 : x_5^0'=x_5^0+y_6^0, [ -x_5^0-y_6^0<=0 ], cost: 2 5: l4 -> l1 : [], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 6: l1 -> l1 : x_5^0'=x_5^0+y_6^0, [ -x_5^0-y_6^0<=0 ], cost: 2 [test] deduced pseudo-invariant -y_6^0<=0, also trying y_6^0<=-1 Accelerated rule 6 with non-termination, yielding the new rule 7. Accelerated rule 6 with non-termination, yielding the new rule 8. Accelerated rule 6 with backward acceleration, yielding the new rule 9. Accelerated rule 6 with backward acceleration, yielding the new rule 10. [accelerate] Nesting with 1 inner and 1 outer candidates Also removing duplicate rules: 8. Accelerated all simple loops using metering functions (where possible): Start location: l4 6: l1 -> l1 : x_5^0'=x_5^0+y_6^0, [ -x_5^0-y_6^0<=0 ], cost: 2 7: l1 -> [5] : [ -x_5^0-y_6^0<=0 && x_5^0==0 && y_6^0==0 ], cost: NONTERM 9: l1 -> [5] : [ -x_5^0-y_6^0<=0 && -y_6^0<=0 ], cost: NONTERM 10: l1 -> l1 : x_5^0'=x_5^0+y_6^0*k, [ y_6^0<=-1 && k>=0 && -x_5^0-y_6^0-(-1+k)*y_6^0<=0 ], cost: 2*k 5: l4 -> l1 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l4 5: l4 -> l1 : [], cost: 2 11: l4 -> l1 : x_5^0'=x_5^0+y_6^0, [ -x_5^0-y_6^0<=0 ], cost: 4 12: l4 -> [5] : [ -x_5^0-y_6^0<=0 && x_5^0==0 && y_6^0==0 ], cost: NONTERM 13: l4 -> [5] : [ -x_5^0-y_6^0<=0 && -y_6^0<=0 ], cost: NONTERM 14: l4 -> l1 : x_5^0'=x_5^0+y_6^0*k, [ y_6^0<=-1 && k>=0 && -x_5^0-y_6^0-(-1+k)*y_6^0<=0 ], cost: 2+2*k Removed unreachable locations (and leaf rules with constant cost): Start location: l4 12: l4 -> [5] : [ -x_5^0-y_6^0<=0 && x_5^0==0 && y_6^0==0 ], cost: NONTERM 13: l4 -> [5] : [ -x_5^0-y_6^0<=0 && -y_6^0<=0 ], cost: NONTERM 14: l4 -> l1 : x_5^0'=x_5^0+y_6^0*k, [ y_6^0<=-1 && k>=0 && -x_5^0-y_6^0-(-1+k)*y_6^0<=0 ], cost: 2+2*k ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l4 12: l4 -> [5] : [ -x_5^0-y_6^0<=0 && x_5^0==0 && y_6^0==0 ], cost: NONTERM 13: l4 -> [5] : [ -x_5^0-y_6^0<=0 && -y_6^0<=0 ], cost: NONTERM 14: l4 -> l1 : x_5^0'=x_5^0+y_6^0*k, [ y_6^0<=-1 && k>=0 && -x_5^0-y_6^0-(-1+k)*y_6^0<=0 ], cost: 2+2*k 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: [ -x_5^0-y_6^0<=0 && -y_6^0<=0 ] NO