/export/starexec/sandbox2/solver/bin/starexec_run_c_complexity /export/starexec/sandbox2/benchmark/theBenchmark.c /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 76 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_y, 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, v_.02, v_c, v_x, v_y)) :|: v_.01 < v_.02 eval_foo_bb1_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.01, v_.02, v_c, v_x, v_y)) :|: v_.01 > v_.02 eval_foo_bb1_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 && v_.01 <= v_.02 eval_foo_bb2_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.01, v_.02 + 1, 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_bb1_in(v_.01 + 1, v_.02 + 1, v_c, v_x, v_y)) :|: v_.01 > v_.02 && v_.01 <= v_.02 eval_foo_bb2_in(v_.01, v_.02, v_c, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_.01, v_.02, v_c, v_x, v_y)) :|: v_.01 <= v_.02 && v_.01 > v_.02 eval_foo_bb2_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)) :|: v_.01 <= v_.02 eval_foo_bb3_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(?, 24*Ar_1^2 + 56*Ar_1*Ar_3 + 32*Ar_3^2 + 176*Ar_1 + 184*Ar_3 + 154) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 /\ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 /\ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(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: evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 /\ Ar_2 >= Ar_0 ] evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 /\ Ar_0 >= Ar_2 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: ?, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(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) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb3in) = 1 Pol(evalfoostop) = 0 Pol(evalfoobb2in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoostart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb1in) = -2*V_1 + 2*V_3 + 2 Pol(evalfoobb2in) = -2*V_1 + 2*V_3 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(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(evalfoostart(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(evalfoostart(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(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-2) = Ar_3 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-2) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-3) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-0) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-2) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-3) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-0) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-2) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-3) = Ar_3 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] weakly and the transitions evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] strictly and produces the following problem: 5: T: (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2*Ar_3 + 2, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2*Ar_3 + 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb2in) = 2*V_1 - 2*V_3 - 1 Pol(evalfoobb1in) = 2*V_1 - 2*V_3 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(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(evalfoostart(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(evalfoostart(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(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ]", 0-3) = Ar_3 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_0 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-2) = Ar_3 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3))", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-0) = 3*Ar_1 + 3*Ar_3 + 18 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-2) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ]", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-0) = 3*Ar_1 + 3*Ar_3 + 54 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-3) = Ar_3 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-0) = 3*Ar_1 + 3*Ar_3 + 162 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ]", 0-3) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-0) = 3*Ar_1 + 3*Ar_3 + 18 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-2) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ]", 0-3) = Ar_3 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-0) = 3*Ar_1 + 3*Ar_3 + 54 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-2) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ]", 0-3) = Ar_3 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 3*Ar_1 + 3*Ar_3 + 486 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = Ar_3 orients the transitions evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] weakly and the transitions evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] strictly and produces the following problem: 6: T: (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 12*Ar_1^2 + 28*Ar_1*Ar_3 + 16*Ar_3^2 + 86*Ar_1 + 90*Ar_3 + 72, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2*Ar_3 + 2, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_0 + 1, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 = Ar_2 ] (Comp: 12*Ar_1^2 + 28*Ar_1*Ar_3 + 16*Ar_3^2 + 86*Ar_1 + 90*Ar_3 + 72, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= Ar_2 + 1 ] (Comp: 2*Ar_1 + 2*Ar_3 + 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_3, Ar_3)) (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostart(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 24*Ar_1^2 + 56*Ar_1*Ar_3 + 32*Ar_3^2 + 176*Ar_1 + 184*Ar_3 + 154 Time: 0.093 sec (SMT: 0.073 sec) ---------------------------------------- (2) BOUNDS(1, n^2)