2.26/1.35 WORST_CASE(?, O(n^2)) 2.26/1.36 proof of /export/starexec/sandbox/output/output_files/bench.koat 2.26/1.36 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 2.26/1.36 2.26/1.36 2.26/1.36 The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2). 2.26/1.36 2.26/1.36 (0) CpxIntTrs 2.26/1.36 (1) Koat Proof [FINISHED, 76 ms] 2.26/1.36 (2) BOUNDS(1, n^2) 2.26/1.36 2.26/1.36 2.26/1.36 ---------------------------------------- 2.26/1.36 2.26/1.36 (0) 2.26/1.36 Obligation: 2.26/1.36 Complexity Int TRS consisting of the following rules: 2.26/1.36 eval_foo_start(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb0_in(v_.01, v_.02, v_c, v_i, v_j)) :|: TRUE 2.26/1.36 eval_foo_bb0_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb1_in(v_i, v_.02, v_c, v_i, v_j)) :|: TRUE 2.26/1.36 eval_foo_bb1_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb2_in(v_.01, 0, v_c, v_i, v_j)) :|: v_.01 >= 0 2.26/1.36 eval_foo_bb1_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb5_in(v_.01, v_.02, v_c, v_i, v_j)) :|: v_.01 < 0 2.26/1.36 eval_foo_bb2_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb3_in(v_.01, v_.02, v_c, v_i, v_j)) :|: v_.02 <= v_.01 - 1 2.26/1.36 eval_foo_bb2_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb4_in(v_.01, v_.02, v_c, v_i, v_j)) :|: v_.02 > v_.01 - 1 2.26/1.36 eval_foo_bb3_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb2_in(v_.01, v_.02 + 1, v_c, v_i, v_j)) :|: TRUE 2.26/1.36 eval_foo_bb4_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_bb1_in(v_.01 - 1, v_.02, v_c, v_i, v_j)) :|: TRUE 2.26/1.36 eval_foo_bb5_in(v_.01, v_.02, v_c, v_i, v_j) -> Com_1(eval_foo_stop(v_.01, v_.02, v_c, v_i, v_j)) :|: TRUE 2.26/1.36 2.26/1.36 The start-symbols are:[eval_foo_start_5] 2.26/1.36 2.26/1.36 2.26/1.36 ---------------------------------------- 2.26/1.36 2.26/1.36 (1) Koat Proof (FINISHED) 2.26/1.36 YES(?, 49*ar_1 + 6*ar_1^2 + 49) 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 Initial complexity problem: 2.26/1.36 2.26/1.36 1: T: 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.36 2.26/1.36 start location: koat_start 2.26/1.36 2.26/1.36 leaf cost: 0 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 Repeatedly propagating knowledge in problem 1 produces the following problem: 2.26/1.36 2.26/1.36 2: T: 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.36 2.26/1.36 start location: koat_start 2.26/1.36 2.26/1.36 leaf cost: 0 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 A polynomial rank function with 2.26/1.36 2.26/1.36 Pol(evalfoostart) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb0in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb1in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb2in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb5in) = 1 2.26/1.36 2.26/1.36 Pol(evalfoobb3in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb4in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoostop) = 0 2.26/1.36 2.26/1.36 Pol(koat_start) = 2 2.26/1.36 2.26/1.36 orients all transitions weakly and the transitions 2.26/1.36 2.26/1.36 evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 strictly and produces the following problem: 2.26/1.36 2.26/1.36 3: T: 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.36 2.26/1.36 start location: koat_start 2.26/1.36 2.26/1.36 leaf cost: 0 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 A polynomial rank function with 2.26/1.36 2.26/1.36 Pol(evalfoostart) = V_2 + 1 2.26/1.36 2.26/1.36 Pol(evalfoobb0in) = V_2 + 1 2.26/1.36 2.26/1.36 Pol(evalfoobb1in) = V_1 + 1 2.26/1.36 2.26/1.36 Pol(evalfoobb2in) = V_1 2.26/1.36 2.26/1.36 Pol(evalfoobb5in) = V_1 2.26/1.36 2.26/1.36 Pol(evalfoobb3in) = V_1 2.26/1.36 2.26/1.36 Pol(evalfoobb4in) = V_1 2.26/1.36 2.26/1.36 Pol(evalfoostop) = V_1 2.26/1.36 2.26/1.36 Pol(koat_start) = V_2 + 1 2.26/1.36 2.26/1.36 orients all transitions weakly and the transition 2.26/1.36 2.26/1.36 evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 strictly and produces the following problem: 2.26/1.36 2.26/1.36 4: T: 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ar_1 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.36 2.26/1.36 start location: koat_start 2.26/1.36 2.26/1.36 leaf cost: 0 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 A polynomial rank function with 2.26/1.36 2.26/1.36 Pol(evalfoobb4in) = 1 2.26/1.36 2.26/1.36 Pol(evalfoobb1in) = 0 2.26/1.36 2.26/1.36 Pol(evalfoobb3in) = 2 2.26/1.36 2.26/1.36 Pol(evalfoobb2in) = 2 2.26/1.36 2.26/1.36 and size complexities 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = ? 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = ? 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = ? 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-0) = ? 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-0) = ? 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-0) = ? 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-0) = ? 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-2) = 0 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 2.26/1.36 2.26/1.36 orients the transitions 2.26/1.36 2.26/1.36 evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 weakly and the transitions 2.26/1.36 2.26/1.36 evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ] 2.26/1.36 2.26/1.36 strictly and produces the following problem: 2.26/1.36 2.26/1.36 5: T: 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: ar_1 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.36 2.26/1.36 (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 ] 2.26/1.36 2.26/1.36 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 (Comp: 2*ar_1 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.36 2.26/1.36 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.36 2.26/1.36 start location: koat_start 2.26/1.36 2.26/1.36 leaf cost: 0 2.26/1.36 2.26/1.36 2.26/1.36 2.26/1.36 A polynomial rank function with 2.26/1.36 2.26/1.36 Pol(evalfoobb3in) = V_1 - V_3 2.26/1.36 2.26/1.36 Pol(evalfoobb2in) = V_1 - V_3 + 1 2.26/1.36 2.26/1.36 and size complexities 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-0) = ar_0 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ]", 0-2) = ar_2 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-0) = 3*ar_1 + 162 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-0) = 3*ar_1 + 18 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-0) = 3*ar_1 + 18 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1))", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-0) = 3*ar_1 + 18 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb4in(ar_0, ar_1, ar_2)) [ ar_2 >= ar_0 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-0) = 3*ar_1 + 18 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-0) = 3*ar_1 + 54 2.26/1.36 2.26/1.36 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 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ]", 0-2) = ? 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-0) = 3*ar_1 + 18 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ]", 0-2) = 0 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-0) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2))", 0-2) = ar_2 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-0) = ar_0 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-1) = ar_1 2.26/1.36 2.26/1.36 S("evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2))", 0-2) = ar_2 2.26/1.36 2.26/1.36 orients the transitions 2.26/1.36 2.26/1.36 evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.36 2.26/1.36 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.37 2.26/1.37 weakly and the transition 2.26/1.37 2.26/1.37 evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.37 2.26/1.37 strictly and produces the following problem: 2.26/1.37 2.26/1.37 6: T: 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: ar_1 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.37 2.26/1.37 (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.37 2.26/1.37 (Comp: 3*ar_1^2 + 22*ar_1 + 19, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.37 2.26/1.37 (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 ] 2.26/1.37 2.26/1.37 (Comp: ?, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.37 2.26/1.37 (Comp: 2*ar_1 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.37 2.26/1.37 start location: koat_start 2.26/1.37 2.26/1.37 leaf cost: 0 2.26/1.37 2.26/1.37 2.26/1.37 2.26/1.37 Repeatedly propagating knowledge in problem 6 produces the following problem: 2.26/1.37 2.26/1.37 7: T: 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 1) evalfoostart(ar_0, ar_1, ar_2) -> Com_1(evalfoobb0in(ar_0, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 1) evalfoobb0in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_1, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: ar_1 + 1, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, 0)) [ ar_0 >= 0 ] 2.26/1.37 2.26/1.37 (Comp: 2, Cost: 1) evalfoobb1in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb5in(ar_0, ar_1, ar_2)) [ 0 >= ar_0 + 1 ] 2.26/1.37 2.26/1.37 (Comp: 3*ar_1^2 + 22*ar_1 + 19, Cost: 1) evalfoobb2in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb3in(ar_0, ar_1, ar_2)) [ ar_0 >= ar_2 + 1 ] 2.26/1.37 2.26/1.37 (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 ] 2.26/1.37 2.26/1.37 (Comp: 3*ar_1^2 + 22*ar_1 + 19, Cost: 1) evalfoobb3in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb2in(ar_0, ar_1, ar_2 + 1)) 2.26/1.37 2.26/1.37 (Comp: 2*ar_1 + 2, Cost: 1) evalfoobb4in(ar_0, ar_1, ar_2) -> Com_1(evalfoobb1in(ar_0 - 1, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 2, Cost: 1) evalfoobb5in(ar_0, ar_1, ar_2) -> Com_1(evalfoostop(ar_0, ar_1, ar_2)) 2.26/1.37 2.26/1.37 (Comp: 1, Cost: 0) koat_start(ar_0, ar_1, ar_2) -> Com_1(evalfoostart(ar_0, ar_1, ar_2)) [ 0 <= 0 ] 2.26/1.37 2.26/1.37 start location: koat_start 2.26/1.37 2.26/1.37 leaf cost: 0 2.26/1.37 2.26/1.37 2.26/1.37 2.26/1.37 Complexity upper bound 49*ar_1 + 6*ar_1^2 + 49 2.26/1.37 2.26/1.37 2.26/1.37 2.26/1.37 Time: 0.090 sec (SMT: 0.078 sec) 2.26/1.37 2.26/1.37 2.26/1.37 ---------------------------------------- 2.26/1.37 2.26/1.37 (2) 2.26/1.37 BOUNDS(1, n^2) 2.26/1.39 EOF