WORST_CASE(Omega(1),?) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: l10 0: l0 -> l1 : i^0'=i^post_1, j^0'=j^post_1, k^0'=k^post_1, tmp^0'=tmp^post_1, tmp___0^0'=tmp___0^post_1, [ k^0<=100 && tmp___0^post_1==tmp___0^post_1 && i^post_1==0 && j^0==j^post_1 && k^0==k^post_1 && tmp^0==tmp^post_1 ], cost: 1 1: l0 -> l2 : i^0'=i^post_2, j^0'=j^post_2, k^0'=k^post_2, tmp^0'=tmp^post_2, tmp___0^0'=tmp___0^post_2, [ 101<=k^0 && i^0==i^post_2 && j^0==j^post_2 && k^0==k^post_2 && tmp^0==tmp^post_2 && tmp___0^0==tmp___0^post_2 ], cost: 1 4: l1 -> l4 : i^0'=i^post_5, j^0'=j^post_5, k^0'=k^post_5, tmp^0'=tmp^post_5, tmp___0^0'=tmp___0^post_5, [ i^0==i^post_5 && j^0==j^post_5 && k^0==k^post_5 && tmp^0==tmp^post_5 && tmp___0^0==tmp___0^post_5 ], cost: 1 2: l3 -> l0 : i^0'=i^post_3, j^0'=j^post_3, k^0'=k^post_3, tmp^0'=tmp^post_3, tmp___0^0'=tmp___0^post_3, [ 1<=k^0 && i^0==i^post_3 && j^0==j^post_3 && k^0==k^post_3 && tmp^0==tmp^post_3 && tmp___0^0==tmp___0^post_3 ], cost: 1 3: l3 -> l2 : i^0'=i^post_4, j^0'=j^post_4, k^0'=k^post_4, tmp^0'=tmp^post_4, tmp___0^0'=tmp___0^post_4, [ k^0<=0 && i^0==i^post_4 && j^0==j^post_4 && k^0==k^post_4 && tmp^0==tmp^post_4 && tmp___0^0==tmp___0^post_4 ], cost: 1 11: l4 -> l7 : i^0'=i^post_12, j^0'=j^post_12, k^0'=k^post_12, tmp^0'=tmp^post_12, tmp___0^0'=tmp___0^post_12, [ i^0<=k^0 && k^0<=i^0 && i^0==i^post_12 && j^0==j^post_12 && k^0==k^post_12 && tmp^0==tmp^post_12 && tmp___0^0==tmp___0^post_12 ], cost: 1 12: l4 -> l8 : i^0'=i^post_13, j^0'=j^post_13, k^0'=k^post_13, tmp^0'=tmp^post_13, tmp___0^0'=tmp___0^post_13, [ 1+k^0<=i^0 && i^0==i^post_13 && j^0==j^post_13 && k^0==k^post_13 && tmp^0==tmp^post_13 && tmp___0^0==tmp___0^post_13 ], cost: 1 13: l4 -> l8 : i^0'=i^post_14, j^0'=j^post_14, k^0'=k^post_14, tmp^0'=tmp^post_14, tmp___0^0'=tmp___0^post_14, [ 1+i^0<=k^0 && i^0==i^post_14 && j^0==j^post_14 && k^0==k^post_14 && tmp^0==tmp^post_14 && tmp___0^0==tmp___0^post_14 ], cost: 1 5: l5 -> l2 : i^0'=i^post_6, j^0'=j^post_6, k^0'=k^post_6, tmp^0'=tmp^post_6, tmp___0^0'=tmp___0^post_6, [ i^0<=j^0 && i^0==i^post_6 && j^0==j^post_6 && k^0==k^post_6 && tmp^0==tmp^post_6 && tmp___0^0==tmp___0^post_6 ], cost: 1 6: l5 -> l6 : i^0'=i^post_7, j^0'=j^post_7, k^0'=k^post_7, tmp^0'=tmp^post_7, tmp___0^0'=tmp___0^post_7, [ 1+j^0<=i^0 && j^post_7==1+j^0 && i^0==i^post_7 && k^0==k^post_7 && tmp^0==tmp^post_7 && tmp___0^0==tmp___0^post_7 ], cost: 1 8: l6 -> l5 : i^0'=i^post_9, j^0'=j^post_9, k^0'=k^post_9, tmp^0'=tmp^post_9, tmp___0^0'=tmp___0^post_9, [ i^0==i^post_9 && j^0==j^post_9 && k^0==k^post_9 && tmp^0==tmp^post_9 && tmp___0^0==tmp___0^post_9 ], cost: 1 7: l7 -> l6 : i^0'=i^post_8, j^0'=j^post_8, k^0'=k^post_8, tmp^0'=tmp^post_8, tmp___0^0'=tmp___0^post_8, [ i^post_8==-1+i^0 && j^post_8==0 && k^0==k^post_8 && tmp^0==tmp^post_8 && tmp___0^0==tmp___0^post_8 ], cost: 1 9: l8 -> l1 : i^0'=i^post_10, j^0'=j^post_10, k^0'=k^post_10, tmp^0'=tmp^post_10, tmp___0^0'=tmp___0^post_10, [ i^post_10==1+i^0 && j^0==j^post_10 && k^0==k^post_10 && tmp^0==tmp^post_10 && tmp___0^0==tmp___0^post_10 ], cost: 1 10: l8 -> l7 : i^0'=i^post_11, j^0'=j^post_11, k^0'=k^post_11, tmp^0'=tmp^post_11, tmp___0^0'=tmp___0^post_11, [ i^0==i^post_11 && j^0==j^post_11 && k^0==k^post_11 && tmp^0==tmp^post_11 && tmp___0^0==tmp___0^post_11 ], cost: 1 14: l9 -> l3 : i^0'=i^post_15, j^0'=j^post_15, k^0'=k^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_15, [ tmp^post_15==tmp^post_15 && k^post_15==tmp^post_15 && i^0==i^post_15 && j^0==j^post_15 && tmp___0^0==tmp___0^post_15 ], cost: 1 15: l10 -> l9 : i^0'=i^post_16, j^0'=j^post_16, k^0'=k^post_16, tmp^0'=tmp^post_16, tmp___0^0'=tmp___0^post_16, [ i^0==i^post_16 && j^0==j^post_16 && k^0==k^post_16 && tmp^0==tmp^post_16 && tmp___0^0==tmp___0^post_16 ], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 15: l10 -> l9 : i^0'=i^post_16, j^0'=j^post_16, k^0'=k^post_16, tmp^0'=tmp^post_16, tmp___0^0'=tmp___0^post_16, [ i^0==i^post_16 && j^0==j^post_16 && k^0==k^post_16 && tmp^0==tmp^post_16 && tmp___0^0==tmp___0^post_16 ], cost: 1 Removed unreachable and leaf rules: Start location: l10 0: l0 -> l1 : i^0'=i^post_1, j^0'=j^post_1, k^0'=k^post_1, tmp^0'=tmp^post_1, tmp___0^0'=tmp___0^post_1, [ k^0<=100 && tmp___0^post_1==tmp___0^post_1 && i^post_1==0 && j^0==j^post_1 && k^0==k^post_1 && tmp^0==tmp^post_1 ], cost: 1 4: l1 -> l4 : i^0'=i^post_5, j^0'=j^post_5, k^0'=k^post_5, tmp^0'=tmp^post_5, tmp___0^0'=tmp___0^post_5, [ i^0==i^post_5 && j^0==j^post_5 && k^0==k^post_5 && tmp^0==tmp^post_5 && tmp___0^0==tmp___0^post_5 ], cost: 1 2: l3 -> l0 : i^0'=i^post_3, j^0'=j^post_3, k^0'=k^post_3, tmp^0'=tmp^post_3, tmp___0^0'=tmp___0^post_3, [ 1<=k^0 && i^0==i^post_3 && j^0==j^post_3 && k^0==k^post_3 && tmp^0==tmp^post_3 && tmp___0^0==tmp___0^post_3 ], cost: 1 11: l4 -> l7 : i^0'=i^post_12, j^0'=j^post_12, k^0'=k^post_12, tmp^0'=tmp^post_12, tmp___0^0'=tmp___0^post_12, [ i^0<=k^0 && k^0<=i^0 && i^0==i^post_12 && j^0==j^post_12 && k^0==k^post_12 && tmp^0==tmp^post_12 && tmp___0^0==tmp___0^post_12 ], cost: 1 12: l4 -> l8 : i^0'=i^post_13, j^0'=j^post_13, k^0'=k^post_13, tmp^0'=tmp^post_13, tmp___0^0'=tmp___0^post_13, [ 1+k^0<=i^0 && i^0==i^post_13 && j^0==j^post_13 && k^0==k^post_13 && tmp^0==tmp^post_13 && tmp___0^0==tmp___0^post_13 ], cost: 1 13: l4 -> l8 : i^0'=i^post_14, j^0'=j^post_14, k^0'=k^post_14, tmp^0'=tmp^post_14, tmp___0^0'=tmp___0^post_14, [ 1+i^0<=k^0 && i^0==i^post_14 && j^0==j^post_14 && k^0==k^post_14 && tmp^0==tmp^post_14 && tmp___0^0==tmp___0^post_14 ], cost: 1 6: l5 -> l6 : i^0'=i^post_7, j^0'=j^post_7, k^0'=k^post_7, tmp^0'=tmp^post_7, tmp___0^0'=tmp___0^post_7, [ 1+j^0<=i^0 && j^post_7==1+j^0 && i^0==i^post_7 && k^0==k^post_7 && tmp^0==tmp^post_7 && tmp___0^0==tmp___0^post_7 ], cost: 1 8: l6 -> l5 : i^0'=i^post_9, j^0'=j^post_9, k^0'=k^post_9, tmp^0'=tmp^post_9, tmp___0^0'=tmp___0^post_9, [ i^0==i^post_9 && j^0==j^post_9 && k^0==k^post_9 && tmp^0==tmp^post_9 && tmp___0^0==tmp___0^post_9 ], cost: 1 7: l7 -> l6 : i^0'=i^post_8, j^0'=j^post_8, k^0'=k^post_8, tmp^0'=tmp^post_8, tmp___0^0'=tmp___0^post_8, [ i^post_8==-1+i^0 && j^post_8==0 && k^0==k^post_8 && tmp^0==tmp^post_8 && tmp___0^0==tmp___0^post_8 ], cost: 1 9: l8 -> l1 : i^0'=i^post_10, j^0'=j^post_10, k^0'=k^post_10, tmp^0'=tmp^post_10, tmp___0^0'=tmp___0^post_10, [ i^post_10==1+i^0 && j^0==j^post_10 && k^0==k^post_10 && tmp^0==tmp^post_10 && tmp___0^0==tmp___0^post_10 ], cost: 1 10: l8 -> l7 : i^0'=i^post_11, j^0'=j^post_11, k^0'=k^post_11, tmp^0'=tmp^post_11, tmp___0^0'=tmp___0^post_11, [ i^0==i^post_11 && j^0==j^post_11 && k^0==k^post_11 && tmp^0==tmp^post_11 && tmp___0^0==tmp___0^post_11 ], cost: 1 14: l9 -> l3 : i^0'=i^post_15, j^0'=j^post_15, k^0'=k^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_15, [ tmp^post_15==tmp^post_15 && k^post_15==tmp^post_15 && i^0==i^post_15 && j^0==j^post_15 && tmp___0^0==tmp___0^post_15 ], cost: 1 15: l10 -> l9 : i^0'=i^post_16, j^0'=j^post_16, k^0'=k^post_16, tmp^0'=tmp^post_16, tmp___0^0'=tmp___0^post_16, [ i^0==i^post_16 && j^0==j^post_16 && k^0==k^post_16 && tmp^0==tmp^post_16 && tmp___0^0==tmp___0^post_16 ], cost: 1 Simplified all rules, resulting in: Start location: l10 0: l0 -> l1 : i^0'=0, tmp___0^0'=tmp___0^post_1, [ k^0<=100 ], cost: 1 4: l1 -> l4 : [], cost: 1 2: l3 -> l0 : [ 1<=k^0 ], cost: 1 11: l4 -> l7 : [ -k^0+i^0==0 ], cost: 1 12: l4 -> l8 : [ 1+k^0<=i^0 ], cost: 1 13: l4 -> l8 : [ 1+i^0<=k^0 ], cost: 1 6: l5 -> l6 : j^0'=1+j^0, [ 1+j^0<=i^0 ], cost: 1 8: l6 -> l5 : [], cost: 1 7: l7 -> l6 : i^0'=-1+i^0, j^0'=0, [], cost: 1 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 14: l9 -> l3 : k^0'=tmp^post_15, tmp^0'=tmp^post_15, [], cost: 1 15: l10 -> l9 : [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: l10 4: l1 -> l4 : [], cost: 1 11: l4 -> l7 : [ -k^0+i^0==0 ], cost: 1 12: l4 -> l8 : [ 1+k^0<=i^0 ], cost: 1 13: l4 -> l8 : [ 1+i^0<=k^0 ], cost: 1 19: l6 -> l6 : j^0'=1+j^0, [ 1+j^0<=i^0 ], cost: 2 7: l7 -> l6 : i^0'=-1+i^0, j^0'=0, [], cost: 1 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Accelerating simple loops of location 6. Accelerating the following rules: 19: l6 -> l6 : j^0'=1+j^0, [ 1+j^0<=i^0 ], cost: 2 Accelerated rule 19 with backward acceleration, yielding the new rule 20. [accelerate] Nesting with 1 inner and 1 outer candidates Removing the simple loops: 19. Accelerated all simple loops using metering functions (where possible): Start location: l10 4: l1 -> l4 : [], cost: 1 11: l4 -> l7 : [ -k^0+i^0==0 ], cost: 1 12: l4 -> l8 : [ 1+k^0<=i^0 ], cost: 1 13: l4 -> l8 : [ 1+i^0<=k^0 ], cost: 1 20: l6 -> l6 : j^0'=i^0, [ -j^0+i^0>=0 ], cost: -2*j^0+2*i^0 7: l7 -> l6 : i^0'=-1+i^0, j^0'=0, [], cost: 1 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Chained accelerated rules (with incoming rules): Start location: l10 4: l1 -> l4 : [], cost: 1 11: l4 -> l7 : [ -k^0+i^0==0 ], cost: 1 12: l4 -> l8 : [ 1+k^0<=i^0 ], cost: 1 13: l4 -> l8 : [ 1+i^0<=k^0 ], cost: 1 7: l7 -> l6 : i^0'=-1+i^0, j^0'=0, [], cost: 1 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Removed unreachable locations (and leaf rules with constant cost): Start location: l10 4: l1 -> l4 : [], cost: 1 11: l4 -> l7 : [ -k^0+i^0==0 ], cost: 1 12: l4 -> l8 : [ 1+k^0<=i^0 ], cost: 1 13: l4 -> l8 : [ 1+i^0<=k^0 ], cost: 1 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: l10 22: l1 -> l7 : [ -k^0+i^0==0 ], cost: 2 23: l1 -> l8 : [ 1+k^0<=i^0 ], cost: 2 24: l1 -> l8 : [ 1+i^0<=k^0 ], cost: 2 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 9: l8 -> l1 : i^0'=1+i^0, [], cost: 1 10: l8 -> l7 : [], cost: 1 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Eliminated locations (on tree-shaped paths): Start location: l10 22: l1 -> l7 : [ -k^0+i^0==0 ], cost: 2 25: l1 -> l1 : i^0'=1+i^0, [ 1+k^0<=i^0 ], cost: 3 26: l1 -> l7 : [ 1+k^0<=i^0 ], cost: 3 27: l1 -> l1 : i^0'=1+i^0, [ 1+i^0<=k^0 ], cost: 3 28: l1 -> l7 : [ 1+i^0<=k^0 ], cost: 3 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Accelerating simple loops of location 1. Accelerating the following rules: 25: l1 -> l1 : i^0'=1+i^0, [ 1+k^0<=i^0 ], cost: 3 27: l1 -> l1 : i^0'=1+i^0, [ 1+i^0<=k^0 ], cost: 3 Accelerated rule 25 with non-termination, yielding the new rule 29. Accelerated rule 27 with backward acceleration, yielding the new rule 30. [accelerate] Nesting with 1 inner and 1 outer candidates Removing the simple loops: 25 27. Accelerated all simple loops using metering functions (where possible): Start location: l10 22: l1 -> l7 : [ -k^0+i^0==0 ], cost: 2 26: l1 -> l7 : [ 1+k^0<=i^0 ], cost: 3 28: l1 -> l7 : [ 1+i^0<=k^0 ], cost: 3 29: l1 -> [12] : [ 1+k^0<=i^0 ], cost: NONTERM 30: l1 -> l1 : i^0'=k^0, [ k^0-i^0>=0 ], cost: 3*k^0-3*i^0 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 Chained accelerated rules (with incoming rules): Start location: l10 22: l1 -> l7 : [ -k^0+i^0==0 ], cost: 2 26: l1 -> l7 : [ 1+k^0<=i^0 ], cost: 3 28: l1 -> l7 : [ 1+i^0<=k^0 ], cost: 3 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 31: l10 -> l1 : i^0'=tmp^post_15, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4+3*tmp^post_15 Removed unreachable locations (and leaf rules with constant cost): Start location: l10 22: l1 -> l7 : [ -k^0+i^0==0 ], cost: 2 26: l1 -> l7 : [ 1+k^0<=i^0 ], cost: 3 28: l1 -> l7 : [ 1+i^0<=k^0 ], cost: 3 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 18: l10 -> l1 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4 31: l10 -> l1 : i^0'=tmp^post_15, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4+3*tmp^post_15 Eliminated locations (on tree-shaped paths): Start location: l10 21: l7 -> l6 : i^0'=-1+i^0, j^0'=-1+i^0, [ -1+i^0>=0 ], cost: -1+2*i^0 32: l10 -> l7 : i^0'=0, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 7 33: l10 -> l7 : i^0'=tmp^post_15, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 6+3*tmp^post_15 34: l10 -> [13] : [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4+3*tmp^post_15 Eliminated locations (on tree-shaped paths): Start location: l10 34: l10 -> [13] : [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4+3*tmp^post_15 35: l10 -> l6 : i^0'=-1+tmp^post_15, j^0'=-1+tmp^post_15, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 5+5*tmp^post_15 ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: l10 34: l10 -> [13] : [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 4+3*tmp^post_15 35: l10 -> l6 : i^0'=-1+tmp^post_15, j^0'=-1+tmp^post_15, k^0'=tmp^post_15, tmp^0'=tmp^post_15, tmp___0^0'=tmp___0^post_1, [ 1<=tmp^post_15 && tmp^post_15<=100 ], cost: 5+5*tmp^post_15 Computing asymptotic complexity for rule 34 Resulting cost 0 has complexity: Unknown Computing asymptotic complexity for rule 35 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: [ i^0==i^post_16 && j^0==j^post_16 && k^0==k^post_16 && tmp^0==tmp^post_16 && tmp___0^0==tmp___0^post_16 ] WORST_CASE(Omega(1),?)