/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 180 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_speedSimpleMultipleDep_start(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb0_in(v_m, v_n, v_x.0, v_y.0)) :|: TRUE eval_speedSimpleMultipleDep_bb0_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, 0, 0)) :|: TRUE eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x.0, v_y.0)) :|: v_x.0 < v_n eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb3_in(v_m, v_n, v_x.0, v_y.0)) :|: v_x.0 >= v_n eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0, v_y.0 + 1)) :|: v_y.0 < v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0 + 1, v_y.0 + 1)) :|: v_y.0 < v_m && v_y.0 >= v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0, 0)) :|: v_y.0 >= v_m && v_y.0 < v_m eval_speedSimpleMultipleDep_bb2_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_bb1_in(v_m, v_n, v_x.0 + 1, 0)) :|: v_y.0 >= v_m eval_speedSimpleMultipleDep_bb3_in(v_m, v_n, v_x.0, v_y.0) -> Com_1(eval_speedSimpleMultipleDep_stop(v_m, v_n, v_x.0, v_y.0)) :|: TRUE The start-symbols are:[eval_speedSimpleMultipleDep_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_2 + 2*Ar_2*Ar_3 + 2*Ar_3 + 7) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 /\ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 /\ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(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 transitions from problem 1: evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 /\ Ar_1 >= Ar_3 ] evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 /\ Ar_3 >= Ar_1 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedSimpleMultipleDepbb3in) = 1 Pol(evalspeedSimpleMultipleDepstop) = 0 Pol(evalspeedSimpleMultipleDepbb2in) = 2 Pol(evalspeedSimpleMultipleDepbb1in) = 2 Pol(evalspeedSimpleMultipleDepbb0in) = 2 Pol(evalspeedSimpleMultipleDepstart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_1 >= Ar_3 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_3 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol evalspeedSimpleMultipleDepbb1in: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedSimpleMultipleDepbb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalspeedSimpleMultipleDepbb3in: X_1 - X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = V_3 Pol(evalspeedSimpleMultipleDepstart) = V_3 Pol(evalspeedSimpleMultipleDepbb0in) = V_3 Pol(evalspeedSimpleMultipleDepbb1in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepbb2in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepbb3in) = -V_1 + V_3 Pol(evalspeedSimpleMultipleDepstop) = -V_1 + V_3 orients all transitions weakly and the transition evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ] (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] (Comp: Ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedSimpleMultipleDepbb2in) = -V_2 + V_4 Pol(evalspeedSimpleMultipleDepbb1in) = -V_2 + V_4 and size complexities S("evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_2 S("evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_3 ]", 0-0) = Ar_2 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_3 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_3 ]", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= Ar_3 ]", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_1 + 1 ]", 0-0) = Ar_2 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_1 + 1 ]", 0-1) = Ar_3 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_1 + 1 ]", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_3 >= Ar_1 + 1 ]", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 ]", 0-0) = Ar_2 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 ]", 0-1) = 0 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 ]", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 ]", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 + 1 ]", 0-0) = Ar_2 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_3 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 + 1 ]", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3))", 0-0) = 0 S("evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3))", 0-1) = 0 S("evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, 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(evalspeedSimpleMultipleDepstart(Ar_0, 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(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 orients the transitions evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] weakly and the transition evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ] (Comp: Ar_2*Ar_3 + Ar_3, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] (Comp: Ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepstart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedSimpleMultipleDepbb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(0, 0, Ar_2, Ar_3)) (Comp: Ar_2 + Ar_2*Ar_3 + Ar_3 + 1, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 ] (Comp: Ar_2*Ar_3 + Ar_3, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0, Ar_1 + 1, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_3 >= Ar_1 + 1 ] (Comp: Ar_2, Cost: 1) evalspeedSimpleMultipleDepbb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepbb1in(Ar_0 + 1, 0, Ar_2, Ar_3)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= Ar_3 ] (Comp: 2, Cost: 1) evalspeedSimpleMultipleDepbb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedSimpleMultipleDepstop(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 - Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_2 + 2*Ar_2*Ar_3 + 2*Ar_3 + 7 Time: 0.163 sec (SMT: 0.131 sec) ---------------------------------------- (2) BOUNDS(1, n^2)