WORST_CASE(Omega(1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l7 0: l0 -> l1 : Result_4^0'=Result_4^post_1, __disjvr_0^0'=__disjvr_0^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ Result_4^0==Result_4^post_1 && __disjvr_0^0==__disjvr_0^post_1 && x_5^0==x_5^post_1 && y_6^0==y_6^post_1 ], cost: 1 1: l1 -> l3 : Result_4^0'=Result_4^post_2, __disjvr_0^0'=__disjvr_0^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ -x_5^0+y_6^0<=0 && -x_5^0+y_6^0<=0 && Result_4^0==Result_4^post_2 && __disjvr_0^0==__disjvr_0^post_2 && x_5^0==x_5^post_2 && y_6^0==y_6^post_2 ], cost: 1 4: l1 -> l5 : Result_4^0'=Result_4^post_5, __disjvr_0^0'=__disjvr_0^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ -x_5^0+y_6^0<=0 && -x_5^0+y_6^0<=0 && x_5^0<=y_6^0 && y_6^0<=x_5^0 && x_5^post_5==1+x_5^0 && Result_4^0==Result_4^post_5 && __disjvr_0^0==__disjvr_0^post_5 && y_6^0==y_6^post_5 ], cost: 1 6: l1 -> l6 : Result_4^0'=Result_4^post_7, __disjvr_0^0'=__disjvr_0^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ 0<=-1-x_5^0+y_6^0 && x_5^post_7==1+x_5^0 && Result_4^0==Result_4^post_7 && __disjvr_0^0==__disjvr_0^post_7 && y_6^0==y_6^post_7 ], cost: 1 2: l3 -> l4 : Result_4^0'=Result_4^post_3, __disjvr_0^0'=__disjvr_0^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ __disjvr_0^post_3==__disjvr_0^0 && Result_4^0==Result_4^post_3 && __disjvr_0^0==__disjvr_0^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 3: l4 -> l2 : Result_4^0'=Result_4^post_4, __disjvr_0^0'=__disjvr_0^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ Result_4^post_4==Result_4^post_4 && __disjvr_0^0==__disjvr_0^post_4 && x_5^0==x_5^post_4 && y_6^0==y_6^post_4 ], cost: 1 5: l5 -> l1 : Result_4^0'=Result_4^post_6, __disjvr_0^0'=__disjvr_0^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ Result_4^0==Result_4^post_6 && __disjvr_0^0==__disjvr_0^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 7: l6 -> l1 : Result_4^0'=Result_4^post_8, __disjvr_0^0'=__disjvr_0^post_8, x_5^0'=x_5^post_8, y_6^0'=y_6^post_8, [ Result_4^0==Result_4^post_8 && __disjvr_0^0==__disjvr_0^post_8 && x_5^0==x_5^post_8 && y_6^0==y_6^post_8 ], cost: 1 8: l7 -> l0 : Result_4^0'=Result_4^post_9, __disjvr_0^0'=__disjvr_0^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && __disjvr_0^0==__disjvr_0^post_9 && x_5^0==x_5^post_9 && y_6^0==y_6^post_9 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 8: l7 -> l0 : Result_4^0'=Result_4^post_9, __disjvr_0^0'=__disjvr_0^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && __disjvr_0^0==__disjvr_0^post_9 && x_5^0==x_5^post_9 && y_6^0==y_6^post_9 ], cost: 1 Removed unreachable and leaf rules: Start location: l7 0: l0 -> l1 : Result_4^0'=Result_4^post_1, __disjvr_0^0'=__disjvr_0^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ Result_4^0==Result_4^post_1 && __disjvr_0^0==__disjvr_0^post_1 && x_5^0==x_5^post_1 && y_6^0==y_6^post_1 ], cost: 1 4: l1 -> l5 : Result_4^0'=Result_4^post_5, __disjvr_0^0'=__disjvr_0^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ -x_5^0+y_6^0<=0 && -x_5^0+y_6^0<=0 && x_5^0<=y_6^0 && y_6^0<=x_5^0 && x_5^post_5==1+x_5^0 && Result_4^0==Result_4^post_5 && __disjvr_0^0==__disjvr_0^post_5 && y_6^0==y_6^post_5 ], cost: 1 6: l1 -> l6 : Result_4^0'=Result_4^post_7, __disjvr_0^0'=__disjvr_0^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ 0<=-1-x_5^0+y_6^0 && x_5^post_7==1+x_5^0 && Result_4^0==Result_4^post_7 && __disjvr_0^0==__disjvr_0^post_7 && y_6^0==y_6^post_7 ], cost: 1 5: l5 -> l1 : Result_4^0'=Result_4^post_6, __disjvr_0^0'=__disjvr_0^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ Result_4^0==Result_4^post_6 && __disjvr_0^0==__disjvr_0^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 7: l6 -> l1 : Result_4^0'=Result_4^post_8, __disjvr_0^0'=__disjvr_0^post_8, x_5^0'=x_5^post_8, y_6^0'=y_6^post_8, [ Result_4^0==Result_4^post_8 && __disjvr_0^0==__disjvr_0^post_8 && x_5^0==x_5^post_8 && y_6^0==y_6^post_8 ], cost: 1 8: l7 -> l0 : Result_4^0'=Result_4^post_9, __disjvr_0^0'=__disjvr_0^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && __disjvr_0^0==__disjvr_0^post_9 && x_5^0==x_5^post_9 && y_6^0==y_6^post_9 ], cost: 1 Simplified all rules, resulting in: Start location: l7 0: l0 -> l1 : [], cost: 1 4: l1 -> l5 : x_5^0'=1+x_5^0, [ -x_5^0+y_6^0==0 ], cost: 1 6: l1 -> l6 : x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 5: l5 -> l1 : [], cost: 1 7: l6 -> l1 : [], cost: 1 8: l7 -> l0 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l7 10: l1 -> l1 : x_5^0'=1+x_5^0, [ -x_5^0+y_6^0==0 ], cost: 2 11: l1 -> l1 : x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 9: l7 -> l1 : [], cost: 2 Accelerating simple loops of location 1. Accelerating the following rules: 10: l1 -> l1 : x_5^0'=1+x_5^0, [ -x_5^0+y_6^0==0 ], cost: 2 11: l1 -> l1 : x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 Failed to prove monotonicity of the guard of rule 10. Accelerated rule 11 with backward acceleration, yielding the new rule 12. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: l7 10: l1 -> l1 : x_5^0'=1+x_5^0, [ -x_5^0+y_6^0==0 ], cost: 2 12: l1 -> l1 : x_5^0'=y_6^0, [ -x_5^0+y_6^0>=0 ], cost: -2*x_5^0+2*y_6^0 9: l7 -> l1 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l7 9: l7 -> l1 : [], cost: 2 13: l7 -> l1 : x_5^0'=1+x_5^0, [ -x_5^0+y_6^0==0 ], cost: 4 14: l7 -> l1 : x_5^0'=y_6^0, [ -x_5^0+y_6^0>=0 ], cost: 2-2*x_5^0+2*y_6^0 Removed unreachable locations (and leaf rules with constant cost): Start location: l7 14: l7 -> l1 : x_5^0'=y_6^0, [ -x_5^0+y_6^0>=0 ], cost: 2-2*x_5^0+2*y_6^0 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l7 14: l7 -> l1 : x_5^0'=y_6^0, [ -x_5^0+y_6^0>=0 ], cost: 2-2*x_5^0+2*y_6^0 Computing asymptotic complexity for rule 14 Resulting cost 0 has complexity: Unknown Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Constant Cpx degree: 0 Solved cost: 1 Rule cost: 1 Rule guard: [ Result_4^0==Result_4^post_9 && __disjvr_0^0==__disjvr_0^post_9 && x_5^0==x_5^post_9 && y_6^0==y_6^post_9 ] WORST_CASE(Omega(1),?)