NO ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l6 0: l0 -> l1 : Result_4^0'=Result_4^post_1, tmp_7^0'=tmp_7^post_1, x_5^0'=x_5^post_1, y_6^0'=y_6^post_1, [ -x_5^0+y_6^0<=0 && Result_4^post_1==Result_4^post_1 && tmp_7^0==tmp_7^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, tmp_7^0'=tmp_7^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ 0<=-1-x_5^0+y_6^0 && tmp_7^post_2==tmp_7^post_2 && tmp_7^post_2<=0 && 0<=tmp_7^post_2 && y_6^post_2==-1+y_6^0 && Result_4^0==Result_4^post_2 && x_5^0==x_5^post_2 ], cost: 1 3: l0 -> l4 : Result_4^0'=Result_4^post_4, tmp_7^0'=tmp_7^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ 0<=-1-x_5^0+y_6^0 && tmp_7^post_4==tmp_7^post_4 && Result_4^0==Result_4^post_4 && x_5^0==x_5^post_4 && y_6^0==y_6^post_4 ], cost: 1 2: l2 -> l0 : Result_4^0'=Result_4^post_3, tmp_7^0'=tmp_7^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && tmp_7^0==tmp_7^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 4: l4 -> l3 : Result_4^0'=Result_4^post_5, tmp_7^0'=tmp_7^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ 1+tmp_7^0<=0 && Result_4^0==Result_4^post_5 && tmp_7^0==tmp_7^post_5 && x_5^0==x_5^post_5 && y_6^0==y_6^post_5 ], cost: 1 5: l4 -> l3 : Result_4^0'=Result_4^post_6, tmp_7^0'=tmp_7^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ 1<=tmp_7^0 && Result_4^0==Result_4^post_6 && tmp_7^0==tmp_7^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 6: l3 -> l0 : Result_4^0'=Result_4^post_7, tmp_7^0'=tmp_7^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ Result_4^0==Result_4^post_7 && tmp_7^0==tmp_7^post_7 && x_5^0==x_5^post_7 && y_6^0==y_6^post_7 ], cost: 1 7: l5 -> l0 : Result_4^0'=Result_4^post_8, tmp_7^0'=tmp_7^post_8, x_5^0'=x_5^post_8, y_6^0'=y_6^post_8, [ Result_4^0==Result_4^post_8 && tmp_7^0==tmp_7^post_8 && x_5^0==x_5^post_8 && y_6^0==y_6^post_8 ], cost: 1 8: l6 -> l5 : Result_4^0'=Result_4^post_9, tmp_7^0'=tmp_7^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && tmp_7^0==tmp_7^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: l6 -> l5 : Result_4^0'=Result_4^post_9, tmp_7^0'=tmp_7^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && tmp_7^0==tmp_7^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: l6 1: l0 -> l2 : Result_4^0'=Result_4^post_2, tmp_7^0'=tmp_7^post_2, x_5^0'=x_5^post_2, y_6^0'=y_6^post_2, [ 0<=-1-x_5^0+y_6^0 && tmp_7^post_2==tmp_7^post_2 && tmp_7^post_2<=0 && 0<=tmp_7^post_2 && y_6^post_2==-1+y_6^0 && Result_4^0==Result_4^post_2 && x_5^0==x_5^post_2 ], cost: 1 3: l0 -> l4 : Result_4^0'=Result_4^post_4, tmp_7^0'=tmp_7^post_4, x_5^0'=x_5^post_4, y_6^0'=y_6^post_4, [ 0<=-1-x_5^0+y_6^0 && tmp_7^post_4==tmp_7^post_4 && Result_4^0==Result_4^post_4 && x_5^0==x_5^post_4 && y_6^0==y_6^post_4 ], cost: 1 2: l2 -> l0 : Result_4^0'=Result_4^post_3, tmp_7^0'=tmp_7^post_3, x_5^0'=x_5^post_3, y_6^0'=y_6^post_3, [ Result_4^0==Result_4^post_3 && tmp_7^0==tmp_7^post_3 && x_5^0==x_5^post_3 && y_6^0==y_6^post_3 ], cost: 1 4: l4 -> l3 : Result_4^0'=Result_4^post_5, tmp_7^0'=tmp_7^post_5, x_5^0'=x_5^post_5, y_6^0'=y_6^post_5, [ 1+tmp_7^0<=0 && Result_4^0==Result_4^post_5 && tmp_7^0==tmp_7^post_5 && x_5^0==x_5^post_5 && y_6^0==y_6^post_5 ], cost: 1 5: l4 -> l3 : Result_4^0'=Result_4^post_6, tmp_7^0'=tmp_7^post_6, x_5^0'=x_5^post_6, y_6^0'=y_6^post_6, [ 1<=tmp_7^0 && Result_4^0==Result_4^post_6 && tmp_7^0==tmp_7^post_6 && x_5^0==x_5^post_6 && y_6^0==y_6^post_6 ], cost: 1 6: l3 -> l0 : Result_4^0'=Result_4^post_7, tmp_7^0'=tmp_7^post_7, x_5^0'=x_5^post_7, y_6^0'=y_6^post_7, [ Result_4^0==Result_4^post_7 && tmp_7^0==tmp_7^post_7 && x_5^0==x_5^post_7 && y_6^0==y_6^post_7 ], cost: 1 7: l5 -> l0 : Result_4^0'=Result_4^post_8, tmp_7^0'=tmp_7^post_8, x_5^0'=x_5^post_8, y_6^0'=y_6^post_8, [ Result_4^0==Result_4^post_8 && tmp_7^0==tmp_7^post_8 && x_5^0==x_5^post_8 && y_6^0==y_6^post_8 ], cost: 1 8: l6 -> l5 : Result_4^0'=Result_4^post_9, tmp_7^0'=tmp_7^post_9, x_5^0'=x_5^post_9, y_6^0'=y_6^post_9, [ Result_4^0==Result_4^post_9 && tmp_7^0==tmp_7^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: l6 1: l0 -> l2 : tmp_7^0'=0, y_6^0'=-1+y_6^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 2: l2 -> l0 : [], cost: 1 4: l4 -> l3 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l3 : [ 1<=tmp_7^0 ], cost: 1 6: l3 -> l0 : [], cost: 1 7: l5 -> l0 : [], cost: 1 8: l6 -> l5 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l6 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 10: l0 -> l0 : tmp_7^0'=0, y_6^0'=-1+y_6^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 4: l4 -> l3 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l3 : [ 1<=tmp_7^0 ], cost: 1 6: l3 -> l0 : [], cost: 1 9: l6 -> l0 : [], cost: 2 Accelerating simple loops of location 0. Accelerating the following rules: 10: l0 -> l0 : tmp_7^0'=0, y_6^0'=-1+y_6^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 Accelerated rule 10 with metering function -x_5^0+y_6^0, yielding the new rule 11. Removing the simple loops: 10. Accelerated all simple loops using metering functions (where possible): Start location: l6 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 11: l0 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: -2*x_5^0+2*y_6^0 4: l4 -> l3 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l3 : [ 1<=tmp_7^0 ], cost: 1 6: l3 -> l0 : [], cost: 1 9: l6 -> l0 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l6 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 4: l4 -> l3 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l3 : [ 1<=tmp_7^0 ], cost: 1 6: l3 -> l0 : [], cost: 1 12: l3 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 1-2*x_5^0+2*y_6^0 9: l6 -> l0 : [], cost: 2 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 Eliminated locations (on tree-shaped paths): Start location: l6 14: l0 -> l3 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 2 15: l0 -> l3 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 2 6: l3 -> l0 : [], cost: 1 12: l3 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 1-2*x_5^0+2*y_6^0 9: l6 -> l0 : [], cost: 2 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 Eliminated locations (on tree-shaped paths): Start location: l6 16: l0 -> l0 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 3 17: l0 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 3-2*x_5^0+2*y_6^0 18: l0 -> l0 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 3 19: l0 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 3-2*x_5^0+2*y_6^0 9: l6 -> l0 : [], cost: 2 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 Accelerating simple loops of location 0. Simplified some of the simple loops (and removed duplicate rules). Accelerating the following rules: 16: l0 -> l0 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 3 18: l0 -> l0 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 3 19: l0 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 3-2*x_5^0+2*y_6^0 Accelerated rule 16 with NONTERM, yielding the new rule 20. Accelerated rule 18 with NONTERM, yielding the new rule 21. Found no metering function for rule 19. Removing the simple loops: 16 18. Accelerated all simple loops using metering functions (where possible): Start location: l6 19: l0 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 3-2*x_5^0+2*y_6^0 20: l0 -> [8] : [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: NONTERM 21: l0 -> [8] : [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: NONTERM 9: l6 -> l0 : [], cost: 2 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 Chained accelerated rules (with incoming rules): Start location: l6 9: l6 -> l0 : [], cost: 2 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 22: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 5-2*x_5^0+2*y_6^0 23: l6 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 24: l6 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Removed unreachable locations (and leaf rules with constant cost): Start location: l6 13: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2-2*x_5^0+2*y_6^0 22: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 5-2*x_5^0+2*y_6^0 23: l6 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 24: l6 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l6 22: l6 -> l0 : tmp_7^0'=0, y_6^0'=x_5^0, [ 0<=-1-x_5^0+y_6^0 ], cost: 5-2*x_5^0+2*y_6^0 24: l6 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Computing asymptotic complexity for rule 22 Solved the limit problem by the following transformations: Created initial limit problem: -x_5^0+y_6^0 (+/+!), 5-2*x_5^0+2*y_6^0 (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {x_5^0==0,y_6^0==n} resulting limit problem: [solved] Solution: x_5^0 / 0 y_6^0 / n Resulting cost 5+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 24 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<=-1-x_5^0+y_6^0 ] NO