/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 185 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_foo_start(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.01, v_.02, v_c, v_x, v_y)) :|: TRUE eval_foo_bb0_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_.02, v_c, v_x, v_y)) :|: TRUE eval_foo_bb1_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.01, 1, v_c, v_x, v_y)) :|: v_.01 >= 0 eval_foo_bb1_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb5_in(v_.01, v_.02, v_c, v_x, v_y)) :|: v_.01 < 0 eval_foo_bb2_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.01, v_.02, v_c, v_x, v_y)) :|: v_.01 > v_.02 eval_foo_bb2_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb4_in(v_.01, v_.02, v_c, v_x, v_y)) :|: v_.01 <= v_.02 eval_foo_bb3_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.01, 2 * v_.02, v_c, v_x, v_y)) :|: TRUE eval_foo_bb4_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.01 - 1, v_.02, v_c, v_x, v_y)) :|: TRUE eval_foo_bb5_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_stop(v_.01, v_.02, v_c, v_x, v_y)) :|: TRUE The start-symbols are:[eval_foo_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 17*Ar_1 + 4*Ar_1^2 + 19) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 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) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb2in) = 2 Pol(evalfoobb5in) = 1 Pol(evalfoobb3in) = 2 Pol(evalfoobb4in) = 2 Pol(evalfoostop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = V_2 + 1 Pol(evalfoobb0in) = V_2 + 1 Pol(evalfoobb1in) = V_1 + 1 Pol(evalfoobb2in) = V_1 Pol(evalfoobb5in) = V_1 Pol(evalfoobb3in) = V_1 Pol(evalfoobb4in) = V_1 Pol(evalfoostop) = V_1 Pol(koat_start) = V_2 + 1 orients all transitions weakly and the transition evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb4in) = 1 Pol(evalfoobb1in) = 0 Pol(evalfoobb3in) = 2 Pol(evalfoobb2in) = 2 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2))", 0-0) = ? S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2))", 0-2) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2))", 0-0) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2))", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2))", 0-0) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2))", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2))", 0-2) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ]", 0-0) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ]", 0-2) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ]", 0-0) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ]", 0-2) = 1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] weakly and the transitions evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 5 to obtain the following invariants: For symbol evalfoobb1in: -X_1 + X_2 >= 0 For symbol evalfoobb2in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalfoobb3in: X_2 - X_3 - 1 >= 0 /\ X_1 - X_3 - 1 >= 0 /\ X_3 - 1 >= 0 /\ X_2 + X_3 - 3 >= 0 /\ X_1 + X_3 - 3 >= 0 /\ X_2 - 2 >= 0 /\ X_1 + X_2 - 4 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 - 2 >= 0 For symbol evalfoobb4in: X_3 - 1 >= 0 /\ X_2 + X_3 - 1 >= 0 /\ X_1 + X_3 - 1 >= 0 /\ -X_1 + X_3 >= 0 /\ X_2 >= 0 /\ X_1 + X_2 >= 0 /\ -X_1 + X_2 >= 0 /\ X_1 >= 0 For symbol evalfoobb5in: -X_1 + X_2 >= 0 /\ -X_1 - 1 >= 0 This yielded the following problem: 6: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\ -Ar_0 - 1 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\ 0 >= Ar_0 + 1 ] (Comp: Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb3in) = 2*V_1 - 2*V_3 - 1 Pol(evalfoobb2in) = 2*V_1 - 2*V_3 and size complexities S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_1 + 1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = 1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 ]", 0-0) = Ar_1 + 1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 ]", 0-2) = 2*Ar_1 + 2*Ar_2 + 24 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 + 1 ]", 0-0) = Ar_1 + 1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= Ar_2 + 1 ]", 0-2) = 2*Ar_1 + 6 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 ]", 0-0) = Ar_1 + 1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_2 >= Ar_0 ]", 0-2) = 2*Ar_1 + 6 S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-0) = Ar_1 + 1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\\ Ar_0 - Ar_2 - 1 >= 0 /\\ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 3 >= 0 /\\ Ar_0 + Ar_2 - 3 >= 0 /\\ Ar_1 - 2 >= 0 /\\ Ar_0 + Ar_1 - 4 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-2) = 2*Ar_1 + 2 S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-0) = Ar_1 + 1 S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\\ Ar_1 + Ar_2 - 1 >= 0 /\\ Ar_0 + Ar_2 - 1 >= 0 /\\ -Ar_0 + Ar_2 >= 0 /\\ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ -Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 ]", 0-2) = 2*Ar_1 + 12 S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ -Ar_0 - 1 >= 0 ]", 0-0) = Ar_1 + 1 S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ -Ar_0 - 1 >= 0 ]", 0-1) = Ar_1 S("evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\\ -Ar_0 - 1 >= 0 ]", 0-2) = 2*Ar_1 + 2*Ar_2 + 48 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 orients the transitions evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 2 >= 0 ] evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 + 1 ] weakly and the transitions evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 2 >= 0 ] evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\ -Ar_0 - 1 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_0 - 1, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ -Ar_0 + Ar_2 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: 2*Ar_1^2 + 6*Ar_1 + 4, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 2*Ar_2)) [ Ar_1 - Ar_2 - 1 >= 0 /\ Ar_0 - Ar_2 - 1 >= 0 /\ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 3 >= 0 /\ Ar_0 + Ar_2 - 3 >= 0 /\ Ar_1 - 2 >= 0 /\ Ar_0 + Ar_1 - 4 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 2*Ar_1 + 2, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_2 >= Ar_0 ] (Comp: 2*Ar_1^2 + 6*Ar_1 + 4, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2)) [ Ar_2 - 1 >= 0 /\ Ar_1 + Ar_2 - 1 >= 0 /\ Ar_0 + Ar_2 - 1 >= 0 /\ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2)) [ -Ar_0 + Ar_1 >= 0 /\ 0 >= Ar_0 + 1 ] (Comp: Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb2in(Ar_0, Ar_1, 1)) [ -Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2)) start location: koat_start leaf cost: 0 Complexity upper bound 17*Ar_1 + 4*Ar_1^2 + 19 Time: 0.189 sec (SMT: 0.160 sec) ---------------------------------------- (2) BOUNDS(1, n^2)