/export/starexec/sandbox/solver/bin/starexec_run_c_complexity /export/starexec/sandbox/benchmark/theBenchmark.c /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox/output/output_files/bench.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 276 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_ex_paper1_start(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb0_in(v_.0, v_fwd, v_i, v_n)) :|: TRUE eval_ex_paper1_bb0_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb1_in(v_i, v_fwd, v_i, v_n)) :|: TRUE eval_ex_paper1_bb1_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb2_in(v_.0, v_fwd, v_i, v_n)) :|: 0 < v_.0 && v_.0 < v_n eval_ex_paper1_bb1_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb3_in(v_.0, v_fwd, v_i, v_n)) :|: 0 >= v_.0 eval_ex_paper1_bb1_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb3_in(v_.0, v_fwd, v_i, v_n)) :|: v_.0 >= v_n eval_ex_paper1_bb2_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb1_in(v_.0 + 1, v_fwd, v_i, v_n)) :|: v_fwd > 0 eval_ex_paper1_bb2_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_bb1_in(v_.0 - 1, v_fwd, v_i, v_n)) :|: v_fwd <= 0 eval_ex_paper1_bb3_in(v_.0, v_fwd, v_i, v_n) -> Com_1(eval_ex_paper1_stop(v_.0, v_fwd, v_i, v_n)) :|: TRUE The start-symbols are:[eval_ex_paper1_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 4*Ar_1^2 + 4*Ar_1*Ar_2 + 4*Ar_1 + 13) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (Comp: 1, Cost: 1) evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper1start) = 2 Pol(evalexpaper1bb0in) = 2 Pol(evalexpaper1bb1in) = 2 Pol(evalexpaper1bb2in) = 2 Pol(evalexpaper1bb3in) = 1 Pol(evalexpaper1stop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_3 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 3 to obtain the following invariants: For symbol evalexpaper1bb2in: X_3 - 2 >= 0 /\ X_1 + X_3 - 3 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) start location: koat_start leaf cost: 0 By chaining the transition koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] with all transitions in problem 4, the following new transition is obtained: koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] We thus obtain the following problem: 5: T: (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 5: evalexpaper1start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) We thus obtain the following problem: 6: T: (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] with all transitions in problem 6, the following new transition is obtained: evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] We thus obtain the following problem: 7: T: (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] with all transitions in problem 7, the following new transition is obtained: evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] We thus obtain the following problem: 8: T: (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 1) evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 8: evalexpaper1bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) We thus obtain the following problem: 9: T: (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] with all transitions in problem 9, the following new transition is obtained: koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] We thus obtain the following problem: 10: T: (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 10: evalexpaper1bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) We thus obtain the following problem: 11: T: (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 ] with all transitions in problem 11, the following new transitions are obtained: evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] We thus obtain the following problem: 12: T: (Comp: ?, Cost: 3) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] (Comp: ?, Cost: 2) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper1bb2in) = 1 Pol(evalexpaper1stop) = 0 Pol(evalexpaper1bb1in) = 1 Pol(koat_start) = 1 orients all transitions weakly and the transition evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] strictly and produces the following problem: 13: T: (Comp: 1, Cost: 3) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] (Comp: ?, Cost: 2) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] (Comp: ?, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper1bb2in) = 2*V_1 - 1 Pol(evalexpaper1bb1in) = 2*V_1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-0) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-0) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-0) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-3) = Ar_3 orients the transitions evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] weakly and the transitions evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] strictly and produces the following problem: 14: T: (Comp: 1, Cost: 3) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] (Comp: ?, Cost: 2) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalexpaper1bb2in) = -V_1 + V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-2) = Ar_2 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\\ Ar_2 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-0) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-2) = Ar_2 S("evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-0) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ 0 >= Ar_3 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-0) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= 1 /\\ Ar_2 >= Ar_0 + 2 ]", 0-3) = Ar_3 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-0) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-1) = Ar_1 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= 1 /\\ Ar_0 + 1 >= Ar_2 ]", 0-3) = Ar_3 orients the transitions evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] weakly and the transition evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] strictly and produces the following problem: 15: T: (Comp: 1, Cost: 3) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= Ar_2 ] (Comp: 2*Ar_1^2 + 2*Ar_1*Ar_2, Cost: 2) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= 1 /\ Ar_0 + 1 >= 1 /\ Ar_2 >= Ar_0 + 2 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_0 - 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_3 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: 2, Cost: 2) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1stop(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 2*Ar_1, Cost: 1) evalexpaper1bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 1 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 2) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalexpaper1bb1in(Ar_1, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 4*Ar_1^2 + 4*Ar_1*Ar_2 + 4*Ar_1 + 13 Time: 0.326 sec (SMT: 0.263 sec) ---------------------------------------- (2) BOUNDS(1, n^2)