4.76/2.19 WORST_CASE(Omega(n^1), O(n^1)) 4.76/2.20 proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat 4.76/2.20 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.76/2.20 4.76/2.20 4.76/2.20 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^1, n^1). 4.76/2.20 4.76/2.20 (0) CpxIntTrs 4.76/2.20 (1) Koat Proof [FINISHED, 315 ms] 4.76/2.20 (2) BOUNDS(1, n^1) 4.76/2.20 (3) Loat Proof [FINISHED, 510 ms] 4.76/2.20 (4) BOUNDS(n^1, INF) 4.76/2.20 4.76/2.20 4.76/2.20 ---------------------------------------- 4.76/2.20 4.76/2.20 (0) 4.76/2.20 Obligation: 4.76/2.20 Complexity Int TRS consisting of the following rules: 4.76/2.20 eval_start_start(v_i_0, v_m, v_n) -> Com_1(eval_start_bb0_in(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_bb0_in(v_i_0, v_m, v_n) -> Com_1(eval_start_0(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_0(v_i_0, v_m, v_n) -> Com_1(eval_start_1(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_1(v_i_0, v_m, v_n) -> Com_1(eval_start_2(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_2(v_i_0, v_m, v_n) -> Com_1(eval_start_3(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_3(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_n, v_m, v_n)) :|: 0 < v_m 4.76/2.20 eval_start_3(v_i_0, v_m, v_n) -> Com_1(eval_start_bb4_in(v_i_0, v_m, v_n)) :|: 0 >= v_m 4.76/2.20 eval_start_bb1_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb2_in(v_i_0, v_m, v_n)) :|: v_i_0 > 0 4.76/2.20 eval_start_bb1_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb3_in(v_i_0, v_m, v_n)) :|: v_i_0 <= 0 4.76/2.20 eval_start_bb2_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_i_0 - 1, v_m, v_n)) :|: v_i_0 < v_m 4.76/2.20 eval_start_bb2_in(v_i_0, v_m, v_n) -> Com_1(eval_start_bb1_in(v_i_0 - v_m, v_m, v_n)) :|: v_i_0 >= v_m 4.76/2.20 eval_start_bb3_in(v_i_0, v_m, v_n) -> Com_1(eval_start_stop(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_bb4_in(v_i_0, v_m, v_n) -> Com_1(eval_start_8(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_8(v_i_0, v_m, v_n) -> Com_1(eval_start_9(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 eval_start_9(v_i_0, v_m, v_n) -> Com_1(eval_start_stop(v_i_0, v_m, v_n)) :|: TRUE 4.76/2.20 4.76/2.20 The start-symbols are:[eval_start_start_3] 4.76/2.20 4.76/2.20 4.76/2.20 ---------------------------------------- 4.76/2.20 4.76/2.20 (1) Koat Proof (FINISHED) 4.76/2.20 YES(?, 6*ar_2 + 14) 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Initial complexity problem: 4.76/2.20 4.76/2.20 1: T: 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.76/2.20 4.76/2.20 start location: koat_start 4.76/2.20 4.76/2.20 leaf cost: 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Repeatedly propagating knowledge in problem 1 produces the following problem: 4.76/2.20 4.76/2.20 2: T: 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.76/2.20 4.76/2.20 start location: koat_start 4.76/2.20 4.76/2.20 leaf cost: 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 A polynomial rank function with 4.76/2.20 4.76/2.20 Pol(evalstartstart) = 2 4.76/2.20 4.76/2.20 Pol(evalstartbb0in) = 2 4.76/2.20 4.76/2.20 Pol(evalstart0) = 2 4.76/2.20 4.76/2.20 Pol(evalstart1) = 2 4.76/2.20 4.76/2.20 Pol(evalstart2) = 2 4.76/2.20 4.76/2.20 Pol(evalstart3) = 2 4.76/2.20 4.76/2.20 Pol(evalstartbb1in) = 2 4.76/2.20 4.76/2.20 Pol(evalstartbb4in) = 0 4.76/2.20 4.76/2.20 Pol(evalstartbb2in) = 2 4.76/2.20 4.76/2.20 Pol(evalstartbb3in) = 1 4.76/2.20 4.76/2.20 Pol(evalstartstop) = 0 4.76/2.20 4.76/2.20 Pol(evalstart8) = 0 4.76/2.20 4.76/2.20 Pol(evalstart9) = 0 4.76/2.20 4.76/2.20 Pol(koat_start) = 2 4.76/2.20 4.76/2.20 orients all transitions weakly and the transitions 4.76/2.20 4.76/2.20 evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 strictly and produces the following problem: 4.76/2.20 4.76/2.20 3: T: 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.76/2.20 4.76/2.20 start location: koat_start 4.76/2.20 4.76/2.20 leaf cost: 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Applied AI with 'oct' on problem 3 to obtain the following invariants: 4.76/2.20 4.76/2.20 For symbol evalstart8: -X_1 >= 0 4.76/2.20 4.76/2.20 For symbol evalstart9: -X_1 >= 0 4.76/2.20 4.76/2.20 For symbol evalstartbb1in: -X_2 + X_3 >= 0 /\ X_1 - 1 >= 0 4.76/2.20 4.76/2.20 For symbol evalstartbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ -X_2 + X_3 >= 0 /\ X_1 + X_3 - 2 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 2 >= 0 /\ X_1 - 1 >= 0 4.76/2.20 4.76/2.20 For symbol evalstartbb3in: -X_2 + X_3 >= 0 /\ -X_2 >= 0 /\ X_1 - X_2 - 1 >= 0 /\ X_1 - 1 >= 0 4.76/2.20 4.76/2.20 For symbol evalstartbb4in: -X_1 >= 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 This yielded the following problem: 4.76/2.20 4.76/2.20 4: T: 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ -ar_1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 (Comp: ?, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 start location: koat_start 4.76/2.20 4.76/2.20 leaf cost: 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 A polynomial rank function with 4.76/2.20 4.76/2.20 Pol(evalstartbb2in) = 2*V_2 - 1 4.76/2.20 4.76/2.20 Pol(evalstartbb1in) = 2*V_2 4.76/2.20 4.76/2.20 and size complexities 4.76/2.20 4.76/2.20 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2))", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2))", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2))", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2))", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 1 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 1 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= 1 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ ar_0 - 1 >= 0 /\\ 0 >= ar_1 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_0 >= ar_1 + 1 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\\ ar_1 + ar_2 - 2 >= 0 /\\ -ar_1 + ar_2 >= 0 /\\ ar_0 + ar_2 - 2 >= 0 /\\ ar_1 - 1 >= 0 /\\ ar_0 + ar_1 - 2 >= 0 /\\ ar_0 - 1 >= 0 /\\ ar_1 >= ar_0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-1) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\\ -ar_1 >= 0 /\\ ar_0 - ar_1 - 1 >= 0 /\\ ar_0 - 1 >= 0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 4.76/2.20 4.76/2.20 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 4.76/2.20 4.76/2.20 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 4.76/2.20 4.76/2.20 orients the transitions 4.76/2.20 4.76/2.20 evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 weakly and the transitions 4.76/2.20 4.76/2.20 evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 strictly and produces the following problem: 4.76/2.20 4.76/2.20 5: T: 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalstartstart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart9(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart8(ar_0, ar_1, ar_2) -> Com_1(evalstart9(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb4in(ar_0, ar_1, ar_2) -> Com_1(evalstart8(ar_0, ar_1, ar_2)) [ -ar_0 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb3in(ar_0, ar_1, ar_2) -> Com_1(evalstartstop(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ -ar_1 >= 0 /\ ar_0 - ar_1 - 1 >= 0 /\ ar_0 - 1 >= 0 ] 4.76/2.20 4.76/2.20 (Comp: 2*ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - ar_0, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: 2*ar_2, Cost: 1) evalstartbb2in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_1 - 1, ar_2)) [ ar_2 - 1 >= 0 /\ ar_1 + ar_2 - 2 >= 0 /\ -ar_1 + ar_2 >= 0 /\ ar_0 + ar_2 - 2 >= 0 /\ ar_1 - 1 >= 0 /\ ar_0 + ar_1 - 2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_0 >= ar_1 + 1 ] 4.76/2.20 4.76/2.20 (Comp: 2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb3in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ 0 >= ar_1 ] 4.76/2.20 4.76/2.20 (Comp: 2*ar_2, Cost: 1) evalstartbb1in(ar_0, ar_1, ar_2) -> Com_1(evalstartbb2in(ar_0, ar_1, ar_2)) [ -ar_1 + ar_2 >= 0 /\ ar_0 - 1 >= 0 /\ ar_1 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb4in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart3(ar_0, ar_1, ar_2) -> Com_1(evalstartbb1in(ar_0, ar_2, ar_2)) [ ar_0 >= 1 ] 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart2(ar_0, ar_1, ar_2) -> Com_1(evalstart3(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart1(ar_0, ar_1, ar_2) -> Com_1(evalstart2(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstart0(ar_0, ar_1, ar_2) -> Com_1(evalstart1(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartbb0in(ar_0, ar_1, ar_2) -> Com_1(evalstart0(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 (Comp: 1, Cost: 1) evalstartstart(ar_0, ar_1, ar_2) -> Com_1(evalstartbb0in(ar_0, ar_1, ar_2)) 4.76/2.20 4.76/2.20 start location: koat_start 4.76/2.20 4.76/2.20 leaf cost: 0 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Complexity upper bound 6*ar_2 + 14 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Time: 0.286 sec (SMT: 0.246 sec) 4.76/2.20 4.76/2.20 4.76/2.20 ---------------------------------------- 4.76/2.20 4.76/2.20 (2) 4.76/2.20 BOUNDS(1, n^1) 4.76/2.20 4.76/2.20 ---------------------------------------- 4.76/2.20 4.76/2.20 (3) Loat Proof (FINISHED) 4.76/2.20 4.76/2.20 4.76/2.20 ### Pre-processing the ITS problem ### 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Initial linear ITS problem 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.76/2.20 4.76/2.20 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.76/2.20 4.76/2.20 2: evalstart0 -> evalstart1 : [], cost: 1 4.76/2.20 4.76/2.20 3: evalstart1 -> evalstart2 : [], cost: 1 4.76/2.20 4.76/2.20 4: evalstart2 -> evalstart3 : [], cost: 1 4.76/2.20 4.76/2.20 5: evalstart3 -> evalstartbb1in : B'=C, [ A>=1 ], cost: 1 4.76/2.20 4.76/2.20 6: evalstart3 -> evalstartbb4in : [ 0>=A ], cost: 1 4.76/2.20 4.76/2.20 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 4.76/2.20 4.76/2.20 8: evalstartbb1in -> evalstartbb3in : [ 0>=B ], cost: 1 4.76/2.20 4.76/2.20 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 4.76/2.20 4.76/2.20 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 4.76/2.20 4.76/2.20 11: evalstartbb3in -> evalstartstop : [], cost: 1 4.76/2.20 4.76/2.20 12: evalstartbb4in -> evalstart8 : [], cost: 1 4.76/2.20 4.76/2.20 13: evalstart8 -> evalstart9 : [], cost: 1 4.76/2.20 4.76/2.20 14: evalstart9 -> evalstartstop : [], cost: 1 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Removed unreachable and leaf rules: 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 0: evalstartstart -> evalstartbb0in : [], cost: 1 4.76/2.20 4.76/2.20 1: evalstartbb0in -> evalstart0 : [], cost: 1 4.76/2.20 4.76/2.20 2: evalstart0 -> evalstart1 : [], cost: 1 4.76/2.20 4.76/2.20 3: evalstart1 -> evalstart2 : [], cost: 1 4.76/2.20 4.76/2.20 4: evalstart2 -> evalstart3 : [], cost: 1 4.76/2.20 4.76/2.20 5: evalstart3 -> evalstartbb1in : B'=C, [ A>=1 ], cost: 1 4.76/2.20 4.76/2.20 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 4.76/2.20 4.76/2.20 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 4.76/2.20 4.76/2.20 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 ### Simplification by acceleration and chaining ### 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Eliminated locations (on linear paths): 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 4.76/2.20 4.76/2.20 7: evalstartbb1in -> evalstartbb2in : [ B>=1 ], cost: 1 4.76/2.20 4.76/2.20 9: evalstartbb2in -> evalstartbb1in : B'=-1+B, [ A>=1+B ], cost: 1 4.76/2.20 4.76/2.20 10: evalstartbb2in -> evalstartbb1in : B'=-A+B, [ B>=A ], cost: 1 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Eliminated locations (on tree-shaped paths): 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 4.76/2.20 4.76/2.20 20: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ B>=1 && A>=1+B ], cost: 2 4.76/2.20 4.76/2.20 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Accelerating simple loops of location 6. 4.76/2.20 4.76/2.20 Accelerating the following rules: 4.76/2.20 4.76/2.20 20: evalstartbb1in -> evalstartbb1in : B'=-1+B, [ B>=1 && A>=1+B ], cost: 2 4.76/2.20 4.76/2.20 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Accelerated rule 20 with metering function B, yielding the new rule 22. 4.76/2.20 4.76/2.20 Found no metering function for rule 21. 4.76/2.20 4.76/2.20 Removing the simple loops: 20. 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Accelerated all simple loops using metering functions (where possible): 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 4.76/2.20 4.76/2.20 21: evalstartbb1in -> evalstartbb1in : B'=-A+B, [ B>=1 && B>=A ], cost: 2 4.76/2.20 4.76/2.20 22: evalstartbb1in -> evalstartbb1in : B'=0, [ B>=1 && A>=1+B ], cost: 2*B 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Chained accelerated rules (with incoming rules): 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 19: evalstartstart -> evalstartbb1in : B'=C, [ A>=1 ], cost: 6 4.76/2.20 4.76/2.20 23: evalstartstart -> evalstartbb1in : B'=C-A, [ A>=1 && C>=1 && C>=A ], cost: 8 4.76/2.20 4.76/2.20 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Removed unreachable locations (and leaf rules with constant cost): 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 ### Computing asymptotic complexity ### 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Fully simplified ITS problem 4.76/2.20 4.76/2.20 Start location: evalstartstart 4.76/2.20 4.76/2.20 24: evalstartstart -> evalstartbb1in : B'=0, [ A>=1 && C>=1 && A>=1+C ], cost: 6+2*C 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Computing asymptotic complexity for rule 24 4.76/2.20 4.76/2.20 Simplified the guard: 4.76/2.20 4.76/2.20 24: evalstartstart -> evalstartbb1in : B'=0, [ C>=1 && A>=1+C ], cost: 6+2*C 4.76/2.20 4.76/2.20 Solved the limit problem by the following transformations: 4.76/2.20 4.76/2.20 Created initial limit problem: 4.76/2.20 4.76/2.20 C (+/+!), -C+A (+/+!), 6+2*C (+) [not solved] 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 removing all constraints (solved by SMT) 4.76/2.20 4.76/2.20 resulting limit problem: [solved] 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 applying transformation rule (C) using substitution {C==n,A==1+n} 4.76/2.20 4.76/2.20 resulting limit problem: 4.76/2.20 4.76/2.20 [solved] 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Solution: 4.76/2.20 4.76/2.20 C / n 4.76/2.20 4.76/2.20 A / 1+n 4.76/2.20 4.76/2.20 Resulting cost 6+2*n has complexity: Poly(n^1) 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Found new complexity Poly(n^1). 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 Obtained the following overall complexity (w.r.t. the length of the input n): 4.76/2.20 4.76/2.20 Complexity: Poly(n^1) 4.76/2.20 4.76/2.20 Cpx degree: 1 4.76/2.20 4.76/2.20 Solved cost: 6+2*n 4.76/2.20 4.76/2.20 Rule cost: 6+2*C 4.76/2.20 4.76/2.20 Rule guard: [ C>=1 && A>=1+C ] 4.76/2.20 4.76/2.20 4.76/2.20 4.76/2.20 WORST_CASE(Omega(n^1),?) 4.76/2.20 4.76/2.20 4.76/2.20 ---------------------------------------- 4.76/2.20 4.76/2.20 (4) 4.76/2.20 BOUNDS(n^1, INF) 4.76/2.21 EOF