/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(n^2, n^2). (0) CpxIntTrs (1) Koat Proof [FINISHED, 227 ms] (2) BOUNDS(1, n^2) (3) Loat Proof [FINISHED, 710 ms] (4) BOUNDS(n^2, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_speedpldi3_start(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb0_in(v_i_0, v_j_0, v_m, v_n)) :|: TRUE eval_speedpldi3_bb0_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_0(v_i_0, v_j_0, v_m, v_n)) :|: TRUE eval_speedpldi3_0(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_1(v_i_0, v_j_0, v_m, v_n)) :|: TRUE eval_speedpldi3_1(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_2(v_i_0, v_j_0, v_m, v_n)) :|: TRUE eval_speedpldi3_2(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb3_in(v_i_0, v_j_0, v_m, v_n)) :|: v_m <= 0 eval_speedpldi3_2(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb3_in(v_i_0, v_j_0, v_m, v_n)) :|: v_n <= v_m eval_speedpldi3_2(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb1_in(0, 0, v_m, v_n)) :|: v_m > 0 && v_n > v_m eval_speedpldi3_bb1_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb2_in(v_i_0, v_j_0, v_m, v_n)) :|: v_i_0 < v_n eval_speedpldi3_bb1_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb3_in(v_i_0, v_j_0, v_m, v_n)) :|: v_i_0 >= v_n eval_speedpldi3_bb2_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb1_in(v_i_0, v_j_0 + 1, v_m, v_n)) :|: v_j_0 < v_m eval_speedpldi3_bb2_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb1_in(v_i_0 + 1, v_j_0 + 1, v_m, v_n)) :|: v_j_0 < v_m && v_j_0 >= v_m eval_speedpldi3_bb2_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb1_in(v_i_0, 0, v_m, v_n)) :|: v_j_0 >= v_m && v_j_0 < v_m eval_speedpldi3_bb2_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_bb1_in(v_i_0 + 1, 0, v_m, v_n)) :|: v_j_0 >= v_m eval_speedpldi3_bb3_in(v_i_0, v_j_0, v_m, v_n) -> Com_1(eval_speedpldi3_stop(v_i_0, v_j_0, v_m, v_n)) :|: TRUE The start-symbols are:[eval_speedpldi3_start_4] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_1 + 2*Ar_0*Ar_1 + 2*Ar_0 + 12) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 /\ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, 0)) [ Ar_3 >= Ar_0 /\ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3start(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: evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 /\ Ar_3 >= Ar_0 ] evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, 0)) [ Ar_3 >= Ar_0 /\ Ar_0 >= Ar_3 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3start(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) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi3bb1in) = 2 Pol(evalspeedpldi3bb3in) = 1 Pol(evalspeedpldi3bb2in) = 2 Pol(evalspeedpldi3stop) = 0 Pol(evalspeedpldi32) = 2 Pol(evalspeedpldi31) = 2 Pol(evalspeedpldi30) = 2 Pol(evalspeedpldi3bb0in) = 2 Pol(evalspeedpldi3start) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= Ar_0 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_1 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3start(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 evalspeedpldi3bb1in: X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol evalspeedpldi3bb2in: X_4 >= 0 /\ X_3 + X_4 >= 0 /\ X_2 + X_4 - 2 >= 0 /\ X_1 + X_4 - 1 >= 0 /\ X_2 - X_3 - 1 >= 0 /\ X_3 >= 0 /\ X_2 + X_3 - 2 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ -X_1 + X_2 - 1 >= 0 /\ X_1 - 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(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] (Comp: 2, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi3bb2in) = V_2 - V_3 Pol(evalspeedpldi3bb1in) = V_2 - V_3 and size complexities S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-3) = 0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-0) = Ar_0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-1) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-2) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-3) = 0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-3) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-3) = Ar_0 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_1 + Ar_2 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-2) = 0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-3) = 0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-2) = Ar_2 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-3) = Ar_3 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(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(evalspeedpldi3start(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(evalspeedpldi3start(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(evalspeedpldi3start(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(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 orients the transitions evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] weakly and the transition evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] (Comp: 2, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalspeedpldi3bb2in) = V_1 - V_4 Pol(evalspeedpldi3bb1in) = V_1 - V_4 and size complexities S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_2 >= Ar_1 ]", 0-3) = 0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-0) = Ar_0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-1) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-2) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_3 >= Ar_0 ]", 0-3) = 0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= Ar_3 + 1 ]", 0-3) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-2) = Ar_1 S("evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\\ Ar_2 + Ar_3 >= 0 /\\ Ar_1 + Ar_3 - 2 >= 0 /\\ Ar_0 + Ar_3 - 1 >= 0 /\\ Ar_2 >= 0 /\\ Ar_1 + Ar_2 - 2 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ -Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= Ar_2 + 1 ]", 0-3) = Ar_0 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_1 + Ar_2 S("evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-2) = 0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\\ Ar_1 >= Ar_0 + 1 ]", 0-3) = 0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-2) = Ar_2 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ]", 0-3) = Ar_3 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-2) = Ar_2 S("evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ]", 0-3) = Ar_3 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(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(evalspeedpldi3start(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(evalspeedpldi3start(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(evalspeedpldi3start(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(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 orients the transitions evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] weakly and the transition evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 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(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: Ar_0*Ar_1 + Ar_0, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] (Comp: 2, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 ] 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(evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] (Comp: 1, Cost: 1) evalspeedpldi3start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi3bb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi30(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi31(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_1 ] (Comp: 1, Cost: 1) evalspeedpldi32(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, 0, 0)) [ Ar_0 >= 1 /\ Ar_1 >= Ar_0 + 1 ] (Comp: 2, Cost: 1) evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3stop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: Ar_1 + Ar_0*Ar_1 + Ar_0 + 1, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= Ar_2 + 1 ] (Comp: Ar_0*Ar_1 + Ar_0, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3 + 1)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= Ar_3 + 1 ] (Comp: Ar_1, Cost: 1) evalspeedpldi3bb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2 + 1, 0)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_3 >= Ar_0 ] (Comp: 2, Cost: 1) evalspeedpldi3bb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalspeedpldi3bb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= 0 /\ Ar_2 + Ar_3 >= 0 /\ Ar_1 + Ar_3 - 2 >= 0 /\ Ar_0 + Ar_3 - 1 >= 0 /\ Ar_2 >= 0 /\ Ar_1 + Ar_2 - 2 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ -Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_2 >= Ar_1 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_1 + 2*Ar_0*Ar_1 + 2*Ar_0 + 12 Time: 0.260 sec (SMT: 0.196 sec) ---------------------------------------- (2) BOUNDS(1, n^2) ---------------------------------------- (3) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalspeedpldi3start 0: evalspeedpldi3start -> evalspeedpldi3bb0in : [], cost: 1 1: evalspeedpldi3bb0in -> evalspeedpldi30 : [], cost: 1 2: evalspeedpldi30 -> evalspeedpldi31 : [], cost: 1 3: evalspeedpldi31 -> evalspeedpldi32 : [], cost: 1 4: evalspeedpldi32 -> evalspeedpldi3bb3in : [ 0>=A ], cost: 1 5: evalspeedpldi32 -> evalspeedpldi3bb3in : [ A>=B ], cost: 1 6: evalspeedpldi32 -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 7: evalspeedpldi3bb1in -> evalspeedpldi3bb2in : [ B>=1+C ], cost: 1 8: evalspeedpldi3bb1in -> evalspeedpldi3bb3in : [ C>=B ], cost: 1 9: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=1+D, [ A>=1+D ], cost: 1 10: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=1+D, [ A>=1+D && D>=A ], cost: 1 11: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=0, [ D>=A && A>=1+D ], cost: 1 12: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ D>=A ], cost: 1 13: evalspeedpldi3bb3in -> evalspeedpldi3stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalspeedpldi3start -> evalspeedpldi3bb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalspeedpldi3start 0: evalspeedpldi3start -> evalspeedpldi3bb0in : [], cost: 1 1: evalspeedpldi3bb0in -> evalspeedpldi30 : [], cost: 1 2: evalspeedpldi30 -> evalspeedpldi31 : [], cost: 1 3: evalspeedpldi31 -> evalspeedpldi32 : [], cost: 1 6: evalspeedpldi32 -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 7: evalspeedpldi3bb1in -> evalspeedpldi3bb2in : [ B>=1+C ], cost: 1 9: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=1+D, [ A>=1+D ], cost: 1 10: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=1+D, [ A>=1+D && D>=A ], cost: 1 11: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=0, [ D>=A && A>=1+D ], cost: 1 12: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ D>=A ], cost: 1 Removed rules with unsatisfiable guard: Start location: evalspeedpldi3start 0: evalspeedpldi3start -> evalspeedpldi3bb0in : [], cost: 1 1: evalspeedpldi3bb0in -> evalspeedpldi30 : [], cost: 1 2: evalspeedpldi30 -> evalspeedpldi31 : [], cost: 1 3: evalspeedpldi31 -> evalspeedpldi32 : [], cost: 1 6: evalspeedpldi32 -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 1 7: evalspeedpldi3bb1in -> evalspeedpldi3bb2in : [ B>=1+C ], cost: 1 9: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=1+D, [ A>=1+D ], cost: 1 12: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ D>=A ], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalspeedpldi3start 17: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 5 7: evalspeedpldi3bb1in -> evalspeedpldi3bb2in : [ B>=1+C ], cost: 1 9: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : D'=1+D, [ A>=1+D ], cost: 1 12: evalspeedpldi3bb2in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ D>=A ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalspeedpldi3start 17: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 5 18: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : D'=1+D, [ B>=1+C && A>=1+D ], cost: 2 19: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ B>=1+C && D>=A ], cost: 2 Accelerating simple loops of location 5. Accelerating the following rules: 18: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : D'=1+D, [ B>=1+C && A>=1+D ], cost: 2 19: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : C'=1+C, D'=0, [ B>=1+C && D>=A ], cost: 2 Accelerated rule 18 with metering function -D+A, yielding the new rule 20. Accelerated rule 19 with metering function -C+B (after strengthening guard), yielding the new rule 21. Nested simple loops 19 (outer loop) and 20 (inner loop) with metering function -1-C+B, resulting in the new rules: 22, 23. Removing the simple loops: 18 19. Accelerated all simple loops using metering functions (where possible): Start location: evalspeedpldi3start 17: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 5 20: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : D'=A, [ B>=1+C && A>=1+D ], cost: -2*D+2*A 21: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : C'=B, D'=0, [ B>=1+C && D>=A && 0>=A ], cost: -2*C+2*B 22: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : C'=-1+B, D'=A, [ D>=A && B>=2+C && A>=1 ], cost: -2-2*C+2*B-2*(1+C-B)*A 23: evalspeedpldi3bb1in -> evalspeedpldi3bb1in : C'=-1+B, D'=A, [ A>=1+D && B>=2+C && A>=1 ], cost: -2-2*C-2*D+2*A+2*B-2*(1+C-B)*A Chained accelerated rules (with incoming rules): Start location: evalspeedpldi3start 17: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=0, [ A>=1 && B>=1+A ], cost: 5 24: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=A, [ A>=1 && B>=1+A && B>=1 ], cost: 5+2*A 25: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=-1+B, D'=A, [ A>=1 && B>=1+A && B>=2 ], cost: 3+2*A+2*B+2*(-1+B)*A Removed unreachable locations (and leaf rules with constant cost): Start location: evalspeedpldi3start 24: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=A, [ A>=1 && B>=1+A && B>=1 ], cost: 5+2*A 25: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=-1+B, D'=A, [ A>=1 && B>=1+A && B>=2 ], cost: 3+2*A+2*B+2*(-1+B)*A ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalspeedpldi3start 24: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=0, D'=A, [ A>=1 && B>=1+A && B>=1 ], cost: 5+2*A 25: evalspeedpldi3start -> evalspeedpldi3bb1in : C'=-1+B, D'=A, [ A>=1 && B>=1+A && B>=2 ], cost: 3+2*A+2*B+2*(-1+B)*A Computing asymptotic complexity for rule 24 Solved the limit problem by the following transformations: Created initial limit problem: 5+2*A (+), A (+/+!), -A+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==n,B==2*n} resulting limit problem: [solved] Solution: A / n B / 2*n Resulting cost 5+2*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 25 Solved the limit problem by the following transformations: Created initial limit problem: A (+/+!), 3+2*A*B+2*B (+), -A+B (+/+!) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {A==1+n,B==2+n} resulting limit problem: [solved] Solution: A / 1+n B / 2+n Resulting cost 11+8*n+2*n^2 has complexity: Poly(n^2) Found new complexity Poly(n^2). Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Poly(n^2) Cpx degree: 2 Solved cost: 11+8*n+2*n^2 Rule cost: 3+2*A+2*B+2*(-1+B)*A Rule guard: [ A>=1 && B>=1+A ] WORST_CASE(Omega(n^2),?) ---------------------------------------- (4) BOUNDS(n^2, INF)