/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 85 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_foo_start(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb0_in(v_.0, v_.01, v_1, v_x, v_y)) :|: TRUE eval_foo_bb0_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_x, v_.01, v_1, v_x, v_y)) :|: TRUE eval_foo_bb1_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb2_in(v_.0, v_.01, v_1, v_x, v_y)) :|: v_.0 >= 0 eval_foo_bb1_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb6_in(v_.0, v_.01, v_1, v_x, v_y)) :|: v_.0 < 0 eval_foo_bb2_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, 1, v_.0 + 1, v_x, v_y)) :|: TRUE eval_foo_bb3_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb4_in(v_.0, v_.01, v_1, v_x, v_y)) :|: v_1 >= v_.01 eval_foo_bb3_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb5_in(v_.0, v_.01, v_1, v_x, v_y)) :|: v_1 < v_.01 eval_foo_bb4_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb3_in(v_.0, v_.01 + 1, v_1, v_x, v_y)) :|: TRUE eval_foo_bb5_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_bb1_in(v_1 - 2, v_.01, v_1, v_x, v_y)) :|: TRUE eval_foo_bb6_in(v_.0, v_.01, v_1, v_x, v_y) -> Com_1(eval_foo_stop(v_.0, v_.01, v_1, v_x, v_y)) :|: TRUE The start-symbols are:[eval_foo_start_5] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2462*Ar_1 + 44*Ar_1^2 + 1226) 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_2, 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_0 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb6in(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 Repeatedly propagating knowledge in problem 1 produces the following problem: 2: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, 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_0 >= 0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: ?, Cost: 1) evalfoobb6in(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 A polynomial rank function with Pol(evalfoostart) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb2in) = 2 Pol(evalfoobb6in) = 1 Pol(evalfoobb3in) = 2 Pol(evalfoobb4in) = 2 Pol(evalfoobb5in) = 2 Pol(evalfoostop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb6in(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(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] strictly and produces the following problem: 3: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, 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_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 A polynomial rank function with Pol(evalfoostart) = 2*V_2 + 1 Pol(evalfoobb0in) = 2*V_2 + 1 Pol(evalfoobb1in) = 2*V_1 + 1 Pol(evalfoobb2in) = 2*V_1 Pol(evalfoobb6in) = 2*V_1 Pol(evalfoobb3in) = 2*V_4 - 2 Pol(evalfoobb4in) = 2*V_4 - 2 Pol(evalfoobb5in) = 2*V_4 - 2 Pol(evalfoostop) = 2*V_1 Pol(koat_start) = 2*V_2 + 1 orients all transitions weakly and the transition evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] strictly and produces the following problem: 4: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 Repeatedly propagating knowledge in problem 4 produces the following problem: 5: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: ?, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 A polynomial rank function with Pol(evalfoobb5in) = 1 Pol(evalfoobb1in) = 0 Pol(evalfoobb4in) = 2 Pol(evalfoobb3in) = 2 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("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-0) = ? S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-3) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-0) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-1) = Ar_1 S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-2) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-3) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-0) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-3) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-0) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-3) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-0) = ? S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-2) = 1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-3) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-0) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-3) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ]", 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 >= 0 ]", 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 >= 0 ]", 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 >= 0 ]", 0-3) = ? S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 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 orients the transitions evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] weakly and the transitions evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] strictly and produces the following problem: 6: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 A polynomial rank function with Pol(evalfoobb4in) = -V_3 + V_4 Pol(evalfoobb3in) = -V_3 + V_4 + 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("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-0) = 11*Ar_1 + 73205 S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoostop(Ar_0, Ar_1, Ar_2, Ar_3))", 0-3) = 11*Ar_1 + 11*Ar_3 + 805255 S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-0) = 11*Ar_1 + 605 S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-2) = ? S("evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3))", 0-3) = 11*Ar_1 + 6655 S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-0) = 11*Ar_1 + 73205 S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-1) = Ar_1 S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-2) = ? S("evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3))", 0-3) = 11*Ar_1 + 605 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-0) = 11*Ar_1 + 805255 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ]", 0-3) = 11*Ar_1 + 605 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-0) = 11*Ar_1 + 73205 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-2) = ? S("evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ]", 0-3) = 11*Ar_1 + 605 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-0) = 11*Ar_1 + 6655 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-2) = 1 S("evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1))", 0-3) = 11*Ar_1 + 605 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-0) = 11*Ar_1 + 6655 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-2) = ? S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ]", 0-3) = 11*Ar_1 + 11*Ar_3 + 73205 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ]", 0-0) = 11*Ar_1 + 605 S("evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ]", 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 >= 0 ]", 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 >= 0 ]", 0-3) = 11*Ar_1 + 11*Ar_3 + 73205 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-0) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 0-2) = Ar_2 S("evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3))", 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 orients the transitions evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] weakly and the transition evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] strictly and produces the following problem: 7: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: 22*Ar_1^2 + 1225*Ar_1 + 607, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: ?, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 Repeatedly propagating knowledge in problem 7 produces the following problem: 8: T: (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: 1) evalfoobb0in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_1, Ar_1, Ar_2, Ar_3)) (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb6in(Ar_0, Ar_1, Ar_2, Ar_3)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_1 + 1, Cost: 1) evalfoobb2in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, 1, Ar_0 + 1)) (Comp: 22*Ar_1^2 + 1225*Ar_1 + 607, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_3 >= Ar_2 ] (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3)) [ Ar_2 >= Ar_3 + 1 ] (Comp: 22*Ar_1^2 + 1225*Ar_1 + 607, Cost: 1) evalfoobb4in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb3in(Ar_0, Ar_1, Ar_2 + 1, Ar_3)) (Comp: 4*Ar_1 + 2, Cost: 1) evalfoobb5in(Ar_0, Ar_1, Ar_2, Ar_3) -> Com_1(evalfoobb1in(Ar_3 - 2, Ar_1, Ar_2, Ar_3)) (Comp: 2, Cost: 1) evalfoobb6in(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 Complexity upper bound 2462*Ar_1 + 44*Ar_1^2 + 1226 Time: 0.091 sec (SMT: 0.069 sec) ---------------------------------------- (2) BOUNDS(1, n^2)