NO ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l3 0: l0 -> l1 : t^0'=t^post_1, x^0'=x^post_1, y^0'=y^post_1, [ 0<=x^0 && t^0<=y^0 && x^post_1==x^0+y^0 && t^0==t^post_1 && y^0==y^post_1 ], cost: 1 1: l1 -> l0 : t^0'=t^post_2, x^0'=x^post_2, y^0'=y^post_2, [ t^0==t^post_2 && x^0==x^post_2 && y^0==y^post_2 ], cost: 1 2: l2 -> l0 : t^0'=t^post_3, x^0'=x^post_3, y^0'=y^post_3, [ t^1_1==-1 && t^2_1==-1+t^1_1 && t^3_1==-1+t^2_1 && t^4_1==-1+t^3_1 && t^5_1==-1+t^4_1 && t^6_1==-1+t^5_1 && t^7_1==-1+t^6_1 && t^8_1==-1+t^7_1 && t^9_1==-1+t^8_1 && t^post_3==-1+t^9_1 && -1<=x^0 && t^post_3<=y^0 && x^0==x^post_3 && y^0==y^post_3 ], cost: 1 3: l3 -> l2 : t^0'=t^post_4, x^0'=x^post_4, y^0'=y^post_4, [ t^0==t^post_4 && x^0==x^post_4 && y^0==y^post_4 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 3: l3 -> l2 : t^0'=t^post_4, x^0'=x^post_4, y^0'=y^post_4, [ t^0==t^post_4 && x^0==x^post_4 && y^0==y^post_4 ], cost: 1 Simplified all rules, resulting in: Start location: l3 0: l0 -> l1 : x^0'=x^0+y^0, [ 0<=x^0 && t^0<=y^0 ], cost: 1 1: l1 -> l0 : [], cost: 1 2: l2 -> l0 : t^0'=-10, [ -1<=x^0 && -10<=y^0 ], cost: 1 3: l3 -> l2 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l3 5: l0 -> l0 : x^0'=x^0+y^0, [ 0<=x^0 && t^0<=y^0 ], cost: 2 4: l3 -> l0 : t^0'=-10, [ -1<=x^0 && -10<=y^0 ], cost: 2 Accelerating simple loops of location 0. Accelerating the following rules: 5: l0 -> l0 : x^0'=x^0+y^0, [ 0<=x^0 && t^0<=y^0 ], cost: 2 [test] deduced pseudo-invariant 14-y^0<=0, also trying -14+y^0<=-1 [test] deduced pseudo-invariant 13-y^0<=0, also trying -13+y^0<=-1 [test] deduced pseudo-invariant 12-y^0<=0, also trying -12+y^0<=-1 Accelerated rule 5 with non-termination, yielding the new rule 6. Accelerated rule 5 with non-termination, yielding the new rule 7. Accelerated rule 5 with backward acceleration, yielding the new rule 8. Accelerated rule 5 with non-termination, yielding the new rule 9. Accelerated rule 5 with backward acceleration, yielding the new rule 10. Accelerated rule 5 with non-termination, yielding the new rule 11. Accelerated rule 5 with backward acceleration, yielding the new rule 12. Accelerated rule 5 with non-termination, yielding the new rule 13. [accelerate] Nesting with 0 inner and 1 outer candidates Also removing duplicate rules: 6 7 9 11. Accelerated all simple loops using metering functions (where possible): Start location: l3 5: l0 -> l0 : x^0'=x^0+y^0, [ 0<=x^0 && t^0<=y^0 ], cost: 2 8: l0 -> [4] : [ 0<=x^0 && t^0<=y^0 && 14-y^0<=0 ], cost: NONTERM 10: l0 -> [4] : [ 0<=x^0 && t^0<=y^0 && -14+y^0<=-1 && 13-y^0<=0 ], cost: NONTERM 12: l0 -> [4] : [ 0<=x^0 && t^0<=y^0 && -13+y^0<=-1 && 12-y^0<=0 ], cost: NONTERM 13: l0 -> [4] : [ t^0<=y^0 && x^0==0 && y^0==0 && t^0==0 ], cost: NONTERM 4: l3 -> l0 : t^0'=-10, [ -1<=x^0 && -10<=y^0 ], cost: 2 Chained accelerated rules (with incoming rules): Start location: l3 4: l3 -> l0 : t^0'=-10, [ -1<=x^0 && -10<=y^0 ], cost: 2 14: l3 -> l0 : t^0'=-10, x^0'=x^0+y^0, [ -10<=y^0 && 0<=x^0 ], cost: 4 15: l3 -> [4] : [ 0<=x^0 && 14-y^0<=0 ], cost: NONTERM 16: l3 -> [4] : [ 0<=x^0 && -13+y^0==0 ], cost: NONTERM 17: l3 -> [4] : [ 0<=x^0 && -12+y^0==0 ], cost: NONTERM Removed unreachable locations (and leaf rules with constant cost): Start location: l3 15: l3 -> [4] : [ 0<=x^0 && 14-y^0<=0 ], cost: NONTERM 16: l3 -> [4] : [ 0<=x^0 && -13+y^0==0 ], cost: NONTERM 17: l3 -> [4] : [ 0<=x^0 && -12+y^0==0 ], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l3 15: l3 -> [4] : [ 0<=x^0 && 14-y^0<=0 ], cost: NONTERM 16: l3 -> [4] : [ 0<=x^0 && -13+y^0==0 ], cost: NONTERM 17: l3 -> [4] : [ 0<=x^0 && -12+y^0==0 ], cost: NONTERM Computing asymptotic complexity for rule 15 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: [ 0<=x^0 && 14-y^0<=0 ] NO