NO ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l7 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 && Result_4^0==Result_4^post_2 && x_5^0==x_5^post_2 && y_6^0==y_6^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 -> l5 : 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 -> l5 : 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: l5 -> l3 : 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, [ x_5^post_7==1+x_5^0 && Result_4^0==Result_4^post_7 && tmp_7^0==tmp_7^post_7 && y_6^0==y_6^post_7 ], cost: 1 7: l3 -> 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 -> l0 : 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 9: l7 -> l6 : Result_4^0'=Result_4^post_10, tmp_7^0'=tmp_7^post_10, x_5^0'=x_5^post_10, y_6^0'=y_6^post_10, [ Result_4^0==Result_4^post_10 && tmp_7^0==tmp_7^post_10 && x_5^0==x_5^post_10 && y_6^0==y_6^post_10 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 9: l7 -> l6 : Result_4^0'=Result_4^post_10, tmp_7^0'=tmp_7^post_10, x_5^0'=x_5^post_10, y_6^0'=y_6^post_10, [ Result_4^0==Result_4^post_10 && tmp_7^0==tmp_7^post_10 && x_5^0==x_5^post_10 && y_6^0==y_6^post_10 ], cost: 1 Removed unreachable and leaf rules: Start location: l7 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 && Result_4^0==Result_4^post_2 && x_5^0==x_5^post_2 && y_6^0==y_6^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 -> l5 : 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 -> l5 : 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: l5 -> l3 : 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, [ x_5^post_7==1+x_5^0 && Result_4^0==Result_4^post_7 && tmp_7^0==tmp_7^post_7 && y_6^0==y_6^post_7 ], cost: 1 7: l3 -> 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 -> l0 : 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 9: l7 -> l6 : Result_4^0'=Result_4^post_10, tmp_7^0'=tmp_7^post_10, x_5^0'=x_5^post_10, y_6^0'=y_6^post_10, [ Result_4^0==Result_4^post_10 && tmp_7^0==tmp_7^post_10 && x_5^0==x_5^post_10 && y_6^0==y_6^post_10 ], cost: 1 Simplified all rules, resulting in: Start location: l7 1: l0 -> l2 : tmp_7^0'=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 -> l5 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l5 : [ 1<=tmp_7^0 ], cost: 1 6: l5 -> l3 : x_5^0'=1+x_5^0, [], cost: 1 7: l3 -> l0 : [], cost: 1 8: l6 -> l0 : [], cost: 1 9: l7 -> l6 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l7 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, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 4: l4 -> l5 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l5 : [ 1<=tmp_7^0 ], cost: 1 12: l5 -> l0 : x_5^0'=1+x_5^0, [], cost: 2 10: l7 -> l0 : [], cost: 2 Accelerating simple loops of location 0. Accelerating the following rules: 11: l0 -> l0 : tmp_7^0'=0, [ 0<=-1-x_5^0+y_6^0 ], cost: 2 Accelerated rule 11 with non-termination, yielding the new rule 13. [accelerate] Nesting with 0 inner and 0 outer candidates Removing the simple loops: 11. Accelerated all simple loops using metering functions (where possible): Start location: l7 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 13: l0 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 4: l4 -> l5 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l5 : [ 1<=tmp_7^0 ], cost: 1 12: l5 -> l0 : x_5^0'=1+x_5^0, [], cost: 2 10: l7 -> l0 : [], cost: 2 Chained accelerated rules (with incoming rules): Start location: l7 3: l0 -> l4 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 ], cost: 1 4: l4 -> l5 : [ 1+tmp_7^0<=0 ], cost: 1 5: l4 -> l5 : [ 1<=tmp_7^0 ], cost: 1 12: l5 -> l0 : x_5^0'=1+x_5^0, [], cost: 2 15: l5 -> [8] : [ 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 10: l7 -> l0 : [], cost: 2 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Eliminated locations (on tree-shaped paths): Start location: l7 16: l0 -> l5 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 2 17: l0 -> l5 : tmp_7^0'=tmp_7^post_4, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 2 12: l5 -> l0 : x_5^0'=1+x_5^0, [], cost: 2 15: l5 -> [8] : [ 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 10: l7 -> l0 : [], cost: 2 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Eliminated locations (on tree-shaped paths): Start location: l7 18: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 4 19: l0 -> [8] : [ 1+tmp_7^post_4<=0 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 20: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 4 21: l0 -> [8] : [ 1<=tmp_7^post_4 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 10: l7 -> l0 : [], cost: 2 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Accelerating simple loops of location 0. Accelerating the following rules: 18: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1+tmp_7^post_4<=0 ], cost: 4 20: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=1+x_5^0, [ 0<=-1-x_5^0+y_6^0 && 1<=tmp_7^post_4 ], cost: 4 Accelerated rule 18 with backward acceleration, yielding the new rule 22. Accelerated rule 20 with backward acceleration, yielding the new rule 23. [accelerate] Nesting with 2 inner and 2 outer candidates Removing the simple loops: 18 20. Accelerated all simple loops using metering functions (where possible): Start location: l7 19: l0 -> [8] : [ 1+tmp_7^post_4<=0 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 21: l0 -> [8] : [ 1<=tmp_7^post_4 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 22: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=y_6^0, [ 1+tmp_7^post_4<=0 && -x_5^0+y_6^0>=1 ], cost: -4*x_5^0+4*y_6^0 23: l0 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=y_6^0, [ 1<=tmp_7^post_4 && -x_5^0+y_6^0>=1 ], cost: -4*x_5^0+4*y_6^0 10: l7 -> l0 : [], cost: 2 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM Chained accelerated rules (with incoming rules): Start location: l7 19: l0 -> [8] : [ 1+tmp_7^post_4<=0 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 21: l0 -> [8] : [ 1<=tmp_7^post_4 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 10: l7 -> l0 : [], cost: 2 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 24: l7 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=y_6^0, [ 1+tmp_7^post_4<=0 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 25: l7 -> l0 : tmp_7^0'=tmp_7^post_4, x_5^0'=y_6^0, [ 1<=tmp_7^post_4 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 Eliminated locations (on tree-shaped paths): Start location: l7 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 26: l7 -> [8] : [ 1+tmp_7^post_4<=0 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 27: l7 -> [8] : [ 1<=tmp_7^post_4 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 28: l7 -> [10] : [ 1+tmp_7^post_4<=0 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 29: l7 -> [10] : [ 1<=tmp_7^post_4 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l7 14: l7 -> [8] : [ 0<=-1-x_5^0+y_6^0 ], cost: NONTERM 26: l7 -> [8] : [ 1+tmp_7^post_4<=0 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 27: l7 -> [8] : [ 1<=tmp_7^post_4 && 0<=-2-x_5^0+y_6^0 ], cost: NONTERM 28: l7 -> [10] : [ 1+tmp_7^post_4<=0 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 29: l7 -> [10] : [ 1<=tmp_7^post_4 && -x_5^0+y_6^0>=1 ], cost: 2-4*x_5^0+4*y_6^0 Computing asymptotic complexity for rule 14 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