3.45/2.08 WORST_CASE(?, O(1)) 3.45/2.09 proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat 3.45/2.09 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.45/2.09 3.45/2.09 3.45/2.09 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, 1). 3.45/2.09 3.45/2.09 (0) CpxIntTrs 3.45/2.09 (1) Koat Proof [FINISHED, 24 ms] 3.45/2.09 (2) BOUNDS(1, 1) 3.45/2.09 3.45/2.09 3.45/2.09 ---------------------------------------- 3.45/2.09 3.45/2.09 (0) 3.45/2.09 Obligation: 3.45/2.09 Complexity Int TRS consisting of the following rules: 3.45/2.09 eval_easy1_start(v_0, v_x_0) -> Com_1(eval_easy1_bb0_in(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_bb0_in(v_0, v_x_0) -> Com_1(eval_easy1_1(nondef_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_1(v_0, v_x_0) -> Com_1(eval_easy1_2(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_2(v_0, v_x_0) -> Com_1(eval_easy1_3(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_3(v_0, v_x_0) -> Com_1(eval_easy1_4(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_4(v_0, v_x_0) -> Com_1(eval_easy1_5(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_5(v_0, v_x_0) -> Com_1(eval_easy1_6(v_0, v_x_0)) :|: TRUE 3.45/2.09 eval_easy1_6(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, 0)) :|: TRUE 3.45/2.09 eval_easy1_bb1_in(v_0, v_x_0) -> Com_1(eval_easy1_bb2_in(v_0, v_x_0)) :|: v_x_0 < 40 3.45/2.09 eval_easy1_bb1_in(v_0, v_x_0) -> Com_1(eval_easy1_bb3_in(v_0, v_x_0)) :|: v_x_0 >= 40 3.45/2.09 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 1)) :|: v_0 >= 0 && v_0 <= 0 3.45/2.09 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 2)) :|: v_0 < 0 3.45/2.09 eval_easy1_bb2_in(v_0, v_x_0) -> Com_1(eval_easy1_bb1_in(v_0, v_x_0 + 2)) :|: v_0 > 0 3.45/2.09 eval_easy1_bb3_in(v_0, v_x_0) -> Com_1(eval_easy1_stop(v_0, v_x_0)) :|: TRUE 3.45/2.09 3.45/2.09 The start-symbols are:[eval_easy1_start_2] 3.45/2.09 3.45/2.09 3.45/2.09 ---------------------------------------- 3.45/2.09 3.45/2.09 (1) Koat Proof (FINISHED) 3.45/2.09 YES(?, 172) 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 Initial complexity problem: 3.45/2.09 3.45/2.09 1: T: 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1start(ar_0, ar_1) -> Com_1(evaleasy1bb0in(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb0in(ar_0, ar_1) -> Com_1(evaleasy11(c, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy11(ar_0, ar_1) -> Com_1(evaleasy12(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy12(ar_0, ar_1) -> Com_1(evaleasy13(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy13(ar_0, ar_1) -> Com_1(evaleasy14(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy14(ar_0, ar_1) -> Com_1(evaleasy15(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy15(ar_0, ar_1) -> Com_1(evaleasy16(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy16(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, 0)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 1)) [ ar_0 = 0 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ 0 >= ar_0 + 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ ar_0 >= 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy1start(ar_0, ar_1)) [ 0 <= 0 ] 3.45/2.09 3.45/2.09 start location: koat_start 3.45/2.09 3.45/2.09 leaf cost: 0 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 Repeatedly propagating knowledge in problem 1 produces the following problem: 3.45/2.09 3.45/2.09 2: T: 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1start(ar_0, ar_1) -> Com_1(evaleasy1bb0in(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1bb0in(ar_0, ar_1) -> Com_1(evaleasy11(c, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy11(ar_0, ar_1) -> Com_1(evaleasy12(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy12(ar_0, ar_1) -> Com_1(evaleasy13(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy13(ar_0, ar_1) -> Com_1(evaleasy14(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy14(ar_0, ar_1) -> Com_1(evaleasy15(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy15(ar_0, ar_1) -> Com_1(evaleasy16(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy16(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, 0)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 1)) [ ar_0 = 0 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ 0 >= ar_0 + 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ ar_0 >= 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy1start(ar_0, ar_1)) [ 0 <= 0 ] 3.45/2.09 3.45/2.09 start location: koat_start 3.45/2.09 3.45/2.09 leaf cost: 0 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 A polynomial rank function with 3.45/2.09 3.45/2.09 Pol(evaleasy1start) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy1bb0in) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy11) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy12) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy13) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy14) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy15) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy16) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy1bb1in) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy1bb2in) = 2 3.45/2.09 3.45/2.09 Pol(evaleasy1bb3in) = 1 3.45/2.09 3.45/2.09 Pol(evaleasy1stop) = 0 3.45/2.09 3.45/2.09 Pol(koat_start) = 2 3.45/2.09 3.45/2.09 orients all transitions weakly and the transitions 3.45/2.09 3.45/2.09 evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 strictly and produces the following problem: 3.45/2.09 3.45/2.09 3: T: 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1start(ar_0, ar_1) -> Com_1(evaleasy1bb0in(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1bb0in(ar_0, ar_1) -> Com_1(evaleasy11(c, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy11(ar_0, ar_1) -> Com_1(evaleasy12(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy12(ar_0, ar_1) -> Com_1(evaleasy13(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy13(ar_0, ar_1) -> Com_1(evaleasy14(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy14(ar_0, ar_1) -> Com_1(evaleasy15(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy15(ar_0, ar_1) -> Com_1(evaleasy16(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy16(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, 0)) 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 1)) [ ar_0 = 0 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ 0 >= ar_0 + 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ ar_0 >= 1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy1start(ar_0, ar_1)) [ 0 <= 0 ] 3.45/2.09 3.45/2.09 start location: koat_start 3.45/2.09 3.45/2.09 leaf cost: 0 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 A polynomial rank function with 3.45/2.09 3.45/2.09 Pol(evaleasy1start) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy1bb0in) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy11) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy12) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy13) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy14) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy15) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy16) = 40 3.45/2.09 3.45/2.09 Pol(evaleasy1bb1in) = -V_2 + 40 3.45/2.09 3.45/2.09 Pol(evaleasy1bb2in) = -V_2 + 39 3.45/2.09 3.45/2.09 Pol(evaleasy1bb3in) = -V_2 3.45/2.09 3.45/2.09 Pol(evaleasy1stop) = -V_2 3.45/2.09 3.45/2.09 Pol(koat_start) = 40 3.45/2.09 3.45/2.09 orients all transitions weakly and the transition 3.45/2.09 3.45/2.09 evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 strictly and produces the following problem: 3.45/2.09 3.45/2.09 4: T: 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1start(ar_0, ar_1) -> Com_1(evaleasy1bb0in(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1bb0in(ar_0, ar_1) -> Com_1(evaleasy11(c, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy11(ar_0, ar_1) -> Com_1(evaleasy12(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy12(ar_0, ar_1) -> Com_1(evaleasy13(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy13(ar_0, ar_1) -> Com_1(evaleasy14(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy14(ar_0, ar_1) -> Com_1(evaleasy15(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy15(ar_0, ar_1) -> Com_1(evaleasy16(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy16(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, 0)) 3.45/2.09 3.45/2.09 (Comp: 40, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 1)) [ ar_0 = 0 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ 0 >= ar_0 + 1 ] 3.45/2.09 3.45/2.09 (Comp: ?, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ ar_0 >= 1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy1start(ar_0, ar_1)) [ 0 <= 0 ] 3.45/2.09 3.45/2.09 start location: koat_start 3.45/2.09 3.45/2.09 leaf cost: 0 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 Repeatedly propagating knowledge in problem 4 produces the following problem: 3.45/2.09 3.45/2.09 5: T: 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1start(ar_0, ar_1) -> Com_1(evaleasy1bb0in(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy1bb0in(ar_0, ar_1) -> Com_1(evaleasy11(c, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy11(ar_0, ar_1) -> Com_1(evaleasy12(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy12(ar_0, ar_1) -> Com_1(evaleasy13(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy13(ar_0, ar_1) -> Com_1(evaleasy14(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy14(ar_0, ar_1) -> Com_1(evaleasy15(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy15(ar_0, ar_1) -> Com_1(evaleasy16(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 1) evaleasy16(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, 0)) 3.45/2.09 3.45/2.09 (Comp: 40, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb2in(ar_0, ar_1)) [ 39 >= ar_1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb1in(ar_0, ar_1) -> Com_1(evaleasy1bb3in(ar_0, ar_1)) [ ar_1 >= 40 ] 3.45/2.09 3.45/2.09 (Comp: 40, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 1)) [ ar_0 = 0 ] 3.45/2.09 3.45/2.09 (Comp: 40, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ 0 >= ar_0 + 1 ] 3.45/2.09 3.45/2.09 (Comp: 40, Cost: 1) evaleasy1bb2in(ar_0, ar_1) -> Com_1(evaleasy1bb1in(ar_0, ar_1 + 2)) [ ar_0 >= 1 ] 3.45/2.09 3.45/2.09 (Comp: 2, Cost: 1) evaleasy1bb3in(ar_0, ar_1) -> Com_1(evaleasy1stop(ar_0, ar_1)) 3.45/2.09 3.45/2.09 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1) -> Com_1(evaleasy1start(ar_0, ar_1)) [ 0 <= 0 ] 3.45/2.09 3.45/2.09 start location: koat_start 3.45/2.09 3.45/2.09 leaf cost: 0 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 Complexity upper bound 172 3.45/2.09 3.45/2.09 3.45/2.09 3.45/2.09 Time: 0.096 sec (SMT: 0.087 sec) 3.45/2.09 3.45/2.09 3.45/2.09 ---------------------------------------- 3.45/2.09 3.45/2.09 (2) 3.45/2.09 BOUNDS(1, 1) 3.71/2.11 EOF