5.41/2.61 WORST_CASE(Omega(n^1), O(n^1)) 5.41/2.62 proof of /export/starexec/sandbox/benchmark/theBenchmark.koat 5.41/2.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.41/2.62 5.41/2.62 5.41/2.62 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 5.41/2.62 5.41/2.62 (0) CpxIntTrs 5.41/2.62 (1) Koat Proof [FINISHED, 321 ms] 5.41/2.62 (2) BOUNDS(1, n^1) 5.41/2.62 (3) Loat Proof [FINISHED, 833 ms] 5.41/2.62 (4) BOUNDS(n^1, INF) 5.41/2.62 5.41/2.62 5.41/2.62 ---------------------------------------- 5.41/2.62 5.41/2.62 (0) 5.41/2.62 Obligation: 5.41/2.62 Complexity Int TRS consisting of the following rules: 5.41/2.62 eval_start_start(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb0_in(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_bb0_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_0(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_0(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_1(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_1(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_2(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_2(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_3(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_3(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_4(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_4(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_5(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_5(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb1_in(v_x, v_z, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 eval_start_bb1_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb2_in(v__0, v__01, v_n, v_x, v_z)) :|: v__0 < v_n 5.41/2.62 eval_start_bb1_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb3_in(v__0, v__01, v_n, v_x, v_z)) :|: v__0 >= v_n 5.41/2.62 eval_start_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01, v_n, v_x, v_z)) :|: v__01 > v__0 5.41/2.62 eval_start_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb1_in(v__0, v__01, v_n, v_x, v_z)) :|: v__01 > v__0 && v__01 <= v__0 5.41/2.62 eval_start_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb1_in(v__0 + 1, v__01 + 1, v_n, v_x, v_z)) :|: v__01 <= v__0 && v__01 > v__0 5.41/2.62 eval_start_bb2_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_bb1_in(v__0, v__01 + 1, v_n, v_x, v_z)) :|: v__01 <= v__0 5.41/2.62 eval_start_bb3_in(v__0, v__01, v_n, v_x, v_z) -> Com_1(eval_start_stop(v__0, v__01, v_n, v_x, v_z)) :|: TRUE 5.41/2.62 5.41/2.62 The start-symbols are:[eval_start_start_5] 5.41/2.62 5.41/2.62 5.41/2.62 ---------------------------------------- 5.41/2.62 5.41/2.62 (1) Koat Proof (FINISHED) 5.41/2.62 YES(?, 2*ar_3 + 4*ar_4 + 2*ar_1 + 13) 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Initial complexity problem: 5.41/2.62 5.41/2.62 1: T: 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Testing for reachability in the complexity graph removes the following transitions from problem 1: 5.41/2.62 5.41/2.62 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 We thus obtain the following problem: 5.41/2.62 5.41/2.62 2: T: 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Repeatedly propagating knowledge in problem 2 produces the following problem: 5.41/2.62 5.41/2.62 3: T: 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 A polynomial rank function with 5.41/2.62 5.41/2.62 Pol(evalstartbb3in) = 1 5.41/2.62 5.41/2.62 Pol(evalstartstop) = 0 5.41/2.62 5.41/2.62 Pol(evalstartbb2in) = 2 5.41/2.62 5.41/2.62 Pol(evalstartbb1in) = 2 5.41/2.62 5.41/2.62 Pol(evalstart5) = 2 5.41/2.62 5.41/2.62 Pol(evalstart4) = 2 5.41/2.62 5.41/2.62 Pol(evalstart3) = 2 5.41/2.62 5.41/2.62 Pol(evalstart2) = 2 5.41/2.62 5.41/2.62 Pol(evalstart1) = 2 5.41/2.62 5.41/2.62 Pol(evalstart0) = 2 5.41/2.62 5.41/2.62 Pol(evalstartbb0in) = 2 5.41/2.62 5.41/2.62 Pol(evalstartstart) = 2 5.41/2.62 5.41/2.62 Pol(koat_start) = 2 5.41/2.62 5.41/2.62 orients all transitions weakly and the transitions 5.41/2.62 5.41/2.62 evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 strictly and produces the following problem: 5.41/2.62 5.41/2.62 4: T: 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Applied AI with 'oct' on problem 4 to obtain the following invariants: 5.41/2.62 5.41/2.62 For symbol evalstartbb1in: X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.41/2.62 5.41/2.62 For symbol evalstartbb2in: -X_2 + X_5 - 1 >= 0 /\ -X_1 + X_5 - 1 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.41/2.62 5.41/2.62 For symbol evalstartbb3in: X_1 - X_5 >= 0 /\ X_3 - X_4 >= 0 /\ X_1 - X_2 >= 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 This yielded the following problem: 5.41/2.62 5.41/2.62 5: T: 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 A polynomial rank function with 5.41/2.62 5.41/2.62 Pol(koat_start) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartstart) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb0in) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart0) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart1) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart2) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart3) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart4) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart5) = -V_2 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb1in) = -V_1 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb2in) = -V_1 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb3in) = -V_1 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartstop) = -V_1 + V_5 5.41/2.62 5.41/2.62 orients all transitions weakly and the transition 5.41/2.62 5.41/2.62 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 strictly and produces the following problem: 5.41/2.62 5.41/2.62 6: T: 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ar_1 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 A polynomial rank function with 5.41/2.62 5.41/2.62 Pol(koat_start) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartstart) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb0in) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart0) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart1) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart2) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart3) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart4) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstart5) = -V_4 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb1in) = -V_3 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb2in) = -V_3 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartbb3in) = -V_3 + V_5 5.41/2.62 5.41/2.62 Pol(evalstartstop) = -V_3 + V_5 5.41/2.62 5.41/2.62 orients all transitions weakly and the transition 5.41/2.62 5.41/2.62 evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 strictly and produces the following problem: 5.41/2.62 5.41/2.62 7: T: 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ar_1 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ar_3 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Repeatedly propagating knowledge in problem 7 produces the following problem: 5.41/2.62 5.41/2.62 8: T: 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4)) [ 0 <= 0 ] 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart4(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: 1, Cost: 1) evalstart5(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_1, ar_1, ar_3, ar_3, ar_4)) 5.41/2.62 5.41/2.62 (Comp: ar_3 + 2*ar_4 + ar_1 + 1, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_4 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_4 ] 5.41/2.62 5.41/2.62 (Comp: ar_1 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0 + 1, ar_1, ar_2, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_2 >= ar_0 + 1 ] 5.41/2.62 5.41/2.62 (Comp: ar_3 + ar_4, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartbb1in(ar_0, ar_1, ar_2 + 1, ar_3, ar_4)) [ -ar_1 + ar_4 - 1 >= 0 /\ -ar_0 + ar_4 - 1 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 /\ ar_0 >= ar_2 ] 5.41/2.62 5.41/2.62 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2, ar_3, ar_4) -> Com_1(evalstartstop(ar_0, ar_1, ar_2, ar_3, ar_4)) [ ar_0 - ar_4 >= 0 /\ ar_2 - ar_3 >= 0 /\ ar_0 - ar_1 >= 0 ] 5.41/2.62 5.41/2.62 start location: koat_start 5.41/2.62 5.41/2.62 leaf cost: 0 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Complexity upper bound 2*ar_3 + 4*ar_4 + 2*ar_1 + 13 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Time: 0.328 sec (SMT: 0.249 sec) 5.41/2.62 5.41/2.62 5.41/2.62 ---------------------------------------- 5.41/2.62 5.41/2.62 (2) 5.41/2.62 BOUNDS(1, n^1) 5.41/2.62 5.41/2.62 ---------------------------------------- 5.41/2.62 5.41/2.62 (3) Loat Proof (FINISHED) 5.41/2.62 5.41/2.62 5.41/2.62 ### Pre-processing the ITS problem ### 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Initial linear ITS problem 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.41/2.62 5.41/2.62 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.41/2.62 5.41/2.62 2: evalstart0 -> evalstart1 : [], cost: 1 5.41/2.62 5.41/2.62 3: evalstart1 -> evalstart2 : [], cost: 1 5.41/2.62 5.41/2.62 4: evalstart2 -> evalstart3 : [], cost: 1 5.41/2.62 5.41/2.62 5: evalstart3 -> evalstart4 : [], cost: 1 5.41/2.62 5.41/2.62 6: evalstart4 -> evalstart5 : [], cost: 1 5.41/2.62 5.41/2.62 7: evalstart5 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.41/2.62 5.41/2.62 8: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.41/2.62 5.41/2.62 9: evalstartbb1in -> evalstartbb3in : [ A>=E ], cost: 1 5.41/2.62 5.41/2.62 10: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 11: evalstartbb2in -> evalstartbb1in : [ C>=1+A && A>=C ], cost: 1 5.41/2.62 5.41/2.62 12: evalstartbb2in -> evalstartbb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 13: evalstartbb2in -> evalstartbb1in : C'=1+C, [ A>=C ], cost: 1 5.41/2.62 5.41/2.62 14: evalstartbb3in -> evalstartstop : [], cost: 1 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Removed unreachable and leaf rules: 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.41/2.62 5.41/2.62 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.41/2.62 5.41/2.62 2: evalstart0 -> evalstart1 : [], cost: 1 5.41/2.62 5.41/2.62 3: evalstart1 -> evalstart2 : [], cost: 1 5.41/2.62 5.41/2.62 4: evalstart2 -> evalstart3 : [], cost: 1 5.41/2.62 5.41/2.62 5: evalstart3 -> evalstart4 : [], cost: 1 5.41/2.62 5.41/2.62 6: evalstart4 -> evalstart5 : [], cost: 1 5.41/2.62 5.41/2.62 7: evalstart5 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.41/2.62 5.41/2.62 8: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.41/2.62 5.41/2.62 10: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 11: evalstartbb2in -> evalstartbb1in : [ C>=1+A && A>=C ], cost: 1 5.41/2.62 5.41/2.62 12: evalstartbb2in -> evalstartbb1in : A'=1+A, C'=1+C, [ A>=C && C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 13: evalstartbb2in -> evalstartbb1in : C'=1+C, [ A>=C ], cost: 1 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Removed rules with unsatisfiable guard: 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 0: evalstartstart -> evalstartbb0in : [], cost: 1 5.41/2.62 5.41/2.62 1: evalstartbb0in -> evalstart0 : [], cost: 1 5.41/2.62 5.41/2.62 2: evalstart0 -> evalstart1 : [], cost: 1 5.41/2.62 5.41/2.62 3: evalstart1 -> evalstart2 : [], cost: 1 5.41/2.62 5.41/2.62 4: evalstart2 -> evalstart3 : [], cost: 1 5.41/2.62 5.41/2.62 5: evalstart3 -> evalstart4 : [], cost: 1 5.41/2.62 5.41/2.62 6: evalstart4 -> evalstart5 : [], cost: 1 5.41/2.62 5.41/2.62 7: evalstart5 -> evalstartbb1in : A'=B, C'=D, [], cost: 1 5.41/2.62 5.41/2.62 8: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.41/2.62 5.41/2.62 10: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 13: evalstartbb2in -> evalstartbb1in : C'=1+C, [ A>=C ], cost: 1 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 ### Simplification by acceleration and chaining ### 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Eliminated locations (on linear paths): 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 8 5.41/2.62 5.41/2.62 8: evalstartbb1in -> evalstartbb2in : [ E>=1+A ], cost: 1 5.41/2.62 5.41/2.62 10: evalstartbb2in -> evalstartbb1in : A'=1+A, [ C>=1+A ], cost: 1 5.41/2.62 5.41/2.62 13: evalstartbb2in -> evalstartbb1in : C'=1+C, [ A>=C ], cost: 1 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Eliminated locations (on tree-shaped paths): 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 8 5.41/2.62 5.41/2.62 22: evalstartbb1in -> evalstartbb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 5.41/2.62 5.41/2.62 23: evalstartbb1in -> evalstartbb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Accelerating simple loops of location 8. 5.41/2.62 5.41/2.62 Accelerating the following rules: 5.41/2.62 5.41/2.62 22: evalstartbb1in -> evalstartbb1in : A'=1+A, [ E>=1+A && C>=1+A ], cost: 2 5.41/2.62 5.41/2.62 23: evalstartbb1in -> evalstartbb1in : C'=1+C, [ E>=1+A && A>=C ], cost: 2 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Accelerated rule 22 with backward acceleration, yielding the new rule 24. 5.41/2.62 5.41/2.62 Accelerated rule 22 with backward acceleration, yielding the new rule 25. 5.41/2.62 5.41/2.62 Accelerated rule 23 with metering function 1-C+A, yielding the new rule 26. 5.41/2.62 5.41/2.62 Removing the simple loops: 22 23. 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Accelerated all simple loops using metering functions (where possible): 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 8 5.41/2.62 5.41/2.62 24: evalstartbb1in -> evalstartbb1in : A'=E, [ E>=1+A && C>=1+A && C>=E ], cost: -2*A+2*E 5.41/2.62 5.41/2.62 25: evalstartbb1in -> evalstartbb1in : A'=C, [ E>=1+A && C>=1+A && E>=C ], cost: 2*C-2*A 5.41/2.62 5.41/2.62 26: evalstartbb1in -> evalstartbb1in : C'=1+A, [ E>=1+A && A>=C ], cost: 2-2*C+2*A 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Chained accelerated rules (with incoming rules): 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 21: evalstartstart -> evalstartbb1in : A'=B, C'=D, [], cost: 8 5.41/2.62 5.41/2.62 27: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.41/2.62 5.41/2.62 28: evalstartstart -> evalstartbb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.41/2.62 5.41/2.62 29: evalstartstart -> evalstartbb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Removed unreachable locations (and leaf rules with constant cost): 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 27: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.41/2.62 5.41/2.62 28: evalstartstart -> evalstartbb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.41/2.62 5.41/2.62 29: evalstartstart -> evalstartbb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 ### Computing asymptotic complexity ### 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Fully simplified ITS problem 5.41/2.62 5.41/2.62 Start location: evalstartstart 5.41/2.62 5.41/2.62 27: evalstartstart -> evalstartbb1in : A'=E, C'=D, [ E>=1+B && D>=1+B && D>=E ], cost: 8+2*E-2*B 5.41/2.62 5.41/2.62 28: evalstartstart -> evalstartbb1in : A'=D, C'=D, [ E>=1+B && D>=1+B && E>=D ], cost: 8+2*D-2*B 5.41/2.62 5.41/2.62 29: evalstartstart -> evalstartbb1in : A'=B, C'=1+B, [ E>=1+B && B>=D ], cost: 10-2*D+2*B 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Computing asymptotic complexity for rule 27 5.41/2.62 5.41/2.62 Solved the limit problem by the following transformations: 5.41/2.62 5.41/2.62 Created initial limit problem: 5.41/2.62 5.41/2.62 1+D-E (+/+!), D-B (+/+!), 8+2*E-2*B (+), E-B (+/+!) [not solved] 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 removing all constraints (solved by SMT) 5.41/2.62 5.41/2.62 resulting limit problem: [solved] 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 applying transformation rule (C) using substitution {D==n,E==0,B==-n} 5.41/2.62 5.41/2.62 resulting limit problem: 5.41/2.62 5.41/2.62 [solved] 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Solution: 5.41/2.62 5.41/2.62 D / n 5.41/2.62 5.41/2.62 E / 0 5.41/2.62 5.41/2.62 B / -n 5.41/2.62 5.41/2.62 Resulting cost 8+2*n has complexity: Poly(n^1) 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Found new complexity Poly(n^1). 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 Obtained the following overall complexity (w.r.t. the length of the input n): 5.41/2.62 5.41/2.62 Complexity: Poly(n^1) 5.41/2.62 5.41/2.62 Cpx degree: 1 5.41/2.62 5.41/2.62 Solved cost: 8+2*n 5.41/2.62 5.41/2.62 Rule cost: 8+2*E-2*B 5.41/2.62 5.41/2.62 Rule guard: [ E>=1+B && D>=1+B && D>=E ] 5.41/2.62 5.41/2.62 5.41/2.62 5.41/2.62 WORST_CASE(Omega(n^1),?) 5.41/2.62 5.41/2.62 5.41/2.62 ---------------------------------------- 5.41/2.62 5.41/2.62 (4) 5.41/2.62 BOUNDS(n^1, INF) 5.41/2.64 EOF