WORST_CASE(Omega(n^1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l5 0: l0 -> l1 : Result_4^0'=Result_4^post_1, __const_21^0'=__const_21^post_1, __const_31^0'=__const_31^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ 1+x_5^0-__const_21^0<=0 && Result_4^post_1==Result_4^post_1 && __const_21^0==__const_21^post_1 && __const_31^0==__const_31^post_1 && x_5^0==x_5^post_1 && y_6^0==y_6^post_1 ], cost: 1 1: l0 -> l2 : Result_4^0'=Result_4^post_2, __const_21^0'=__const_21^post_2, __const_31^0'=__const_31^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ 0<=x_5^0-__const_21^0 && Result_4^0==Result_4^post_2 && __const_21^0==__const_21^post_2 && __const_31^0==__const_31^post_2 && x_5^0==x_5^post_2 && y_6^0==y_6^post_2 ], cost: 1 3: l2 -> l0 : Result_4^0'=Result_4^post_4, __const_21^0'=__const_21^post_4, __const_31^0'=__const_31^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ 1-__const_31^0+y_6^0<=0 && x_5^post_4==-1+x_5^0 && Result_4^0==Result_4^post_4 && __const_21^0==__const_21^post_4 && __const_31^0==__const_31^post_4 && y_6^0==y_6^post_4 ], cost: 1 4: l2 -> l4 : Result_4^0'=Result_4^post_5, __const_21^0'=__const_21^post_5, __const_31^0'=__const_31^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ 0<=-__const_31^0+y_6^0 && y_6^post_5==-1+y_6^0 && Result_4^0==Result_4^post_5 && __const_21^0==__const_21^post_5 && __const_31^0==__const_31^post_5 && x_5^0==x_5^post_5 ], cost: 1 2: l3 -> l0 : Result_4^0'=Result_4^post_3, __const_21^0'=__const_21^post_3, __const_31^0'=__const_31^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && __const_21^0==__const_21^post_3 && __const_31^0==__const_31^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 5: l4 -> l2 : Result_4^0'=Result_4^post_6, __const_21^0'=__const_21^post_6, __const_31^0'=__const_31^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ Result_4^0==Result_4^post_6 && __const_21^0==__const_21^post_6 && __const_31^0==__const_31^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 6: l5 -> l3 : Result_4^0'=Result_4^post_7, __const_21^0'=__const_21^post_7, __const_31^0'=__const_31^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ Result_4^0==Result_4^post_7 && __const_21^0==__const_21^post_7 && __const_31^0==__const_31^post_7 && x_5^0==x_5^post_7 && y_6^0==y_6^post_7 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 6: l5 -> l3 : Result_4^0'=Result_4^post_7, __const_21^0'=__const_21^post_7, __const_31^0'=__const_31^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ Result_4^0==Result_4^post_7 && __const_21^0==__const_21^post_7 && __const_31^0==__const_31^post_7 && x_5^0==x_5^post_7 && y_6^0==y_6^post_7 ], cost: 1 Removed unreachable and leaf rules: Start location: l5 1: l0 -> l2 : Result_4^0'=Result_4^post_2, __const_21^0'=__const_21^post_2, __const_31^0'=__const_31^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ 0<=x_5^0-__const_21^0 && Result_4^0==Result_4^post_2 && __const_21^0==__const_21^post_2 && __const_31^0==__const_31^post_2 && x_5^0==x_5^post_2 && y_6^0==y_6^post_2 ], cost: 1 3: l2 -> l0 : Result_4^0'=Result_4^post_4, __const_21^0'=__const_21^post_4, __const_31^0'=__const_31^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ 1-__const_31^0+y_6^0<=0 && x_5^post_4==-1+x_5^0 && Result_4^0==Result_4^post_4 && __const_21^0==__const_21^post_4 && __const_31^0==__const_31^post_4 && y_6^0==y_6^post_4 ], cost: 1 4: l2 -> l4 : Result_4^0'=Result_4^post_5, __const_21^0'=__const_21^post_5, __const_31^0'=__const_31^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ 0<=-__const_31^0+y_6^0 && y_6^post_5==-1+y_6^0 && Result_4^0==Result_4^post_5 && __const_21^0==__const_21^post_5 && __const_31^0==__const_31^post_5 && x_5^0==x_5^post_5 ], cost: 1 2: l3 -> l0 : Result_4^0'=Result_4^post_3, __const_21^0'=__const_21^post_3, __const_31^0'=__const_31^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && __const_21^0==__const_21^post_3 && __const_31^0==__const_31^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 5: l4 -> l2 : Result_4^0'=Result_4^post_6, __const_21^0'=__const_21^post_6, __const_31^0'=__const_31^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ Result_4^0==Result_4^post_6 && __const_21^0==__const_21^post_6 && __const_31^0==__const_31^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 6: l5 -> l3 : Result_4^0'=Result_4^post_7, __const_21^0'=__const_21^post_7, __const_31^0'=__const_31^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ Result_4^0==Result_4^post_7 && __const_21^0==__const_21^post_7 && __const_31^0==__const_31^post_7 && x_5^0==x_5^post_7 && y_6^0==y_6^post_7 ], cost: 1 Simplified all rules, resulting in: Start location: l5 1: l0 -> l2 : [ 0<=x_5^0-__const_21^0 ], cost: 1 3: l2 -> l0 : x_5^0'=-1+x_5^0, [ 1-__const_31^0+y_6^0<=0 ], cost: 1 4: l2 -> l4 : y_6^0'=-1+y_6^0, [ 0<=-__const_31^0+y_6^0 ], cost: 1 2: l3 -> l0 : [], cost: 1 5: l4 -> l2 : [], cost: 1 6: l5 -> l3 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l5 1: l0 -> l2 : [ 0<=x_5^0-__const_21^0 ], cost: 1 3: l2 -> l0 : x_5^0'=-1+x_5^0, [ 1-__const_31^0+y_6^0<=0 ], cost: 1 8: l2 -> l2 : y_6^0'=-1+y_6^0, [ 0<=-__const_31^0+y_6^0 ], cost: 2 7: l5 -> l0 : [], cost: 2 Accelerating simple loops of location 2. Accelerating the following rules: 8: l2 -> l2 : y_6^0'=-1+y_6^0, [ 0<=-__const_31^0+y_6^0 ], cost: 2 Accelerated rule 8 with metering function 1-__const_31^0+y_6^0, yielding the new rule 9. Removing the simple loops: 8. Accelerated all simple loops using metering functions (where possible): Start location: l5 1: l0 -> l2 : [ 0<=x_5^0-__const_21^0 ], cost: 1 3: l2 -> l0 : x_5^0'=-1+x_5^0, [ 1-__const_31^0+y_6^0<=0 ], cost: 1 9: l2 -> l2 : y_6^0'=-1+__const_31^0, [ 0<=-__const_31^0+y_6^0 ], cost: 2-2*__const_31^0+2*y_6^0 7: l5 -> l0 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l5 1: l0 -> l2 : [ 0<=x_5^0-__const_21^0 ], cost: 1 10: l0 -> l2 : y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 3-2*__const_31^0+2*y_6^0 3: l2 -> l0 : x_5^0'=-1+x_5^0, [ 1-__const_31^0+y_6^0<=0 ], cost: 1 7: l5 -> l0 : [], cost: 2 Eliminated locations (on tree-shaped paths): Start location: l5 11: l0 -> l0 : x_5^0'=-1+x_5^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 2 12: l0 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 4-2*__const_31^0+2*y_6^0 7: l5 -> l0 : [], cost: 2 Accelerating simple loops of location 0. Accelerating the following rules: 11: l0 -> l0 : x_5^0'=-1+x_5^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 2 12: l0 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 4-2*__const_31^0+2*y_6^0 Accelerated rule 11 with metering function 1+x_5^0-__const_21^0, yielding the new rule 13. Found no metering function for rule 12. Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: l5 12: l0 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 4-2*__const_31^0+2*y_6^0 13: l0 -> l0 : x_5^0'=-1+__const_21^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 2+2*x_5^0-2*__const_21^0 7: l5 -> l0 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l5 7: l5 -> l0 : [], cost: 2 14: l5 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 6-2*__const_31^0+2*y_6^0 15: l5 -> l0 : x_5^0'=-1+__const_21^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 4+2*x_5^0-2*__const_21^0 Removed unreachable locations (and leaf rules with constant cost): Start location: l5 14: l5 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 6-2*__const_31^0+2*y_6^0 15: l5 -> l0 : x_5^0'=-1+__const_21^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 4+2*x_5^0-2*__const_21^0 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l5 14: l5 -> l0 : x_5^0'=-1+x_5^0, y_6^0'=-1+__const_31^0, [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ], cost: 6-2*__const_31^0+2*y_6^0 15: l5 -> l0 : x_5^0'=-1+__const_21^0, [ 0<=x_5^0-__const_21^0 && 1-__const_31^0+y_6^0<=0 ], cost: 4+2*x_5^0-2*__const_21^0 Computing asymptotic complexity for rule 14 Solved the limit problem by the following transformations: Created initial limit problem: 1+x_5^0-__const_21^0 (+/+!), 6-2*__const_31^0+2*y_6^0 (+), 1-__const_31^0+y_6^0 (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {__const_31^0==0,x_5^0==0,y_6^0==n,__const_21^0==0} resulting limit problem: [solved] Solution: __const_31^0 / 0 x_5^0 / 0 y_6^0 / n __const_21^0 / 0 Resulting cost 6+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^1) Cpx degree: 1 Solved cost: 6+2*n Rule cost: 6-2*__const_31^0+2*y_6^0 Rule guard: [ 0<=x_5^0-__const_21^0 && 0<=-__const_31^0+y_6^0 ] WORST_CASE(Omega(n^1),?)