/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^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^1). (0) CpxIntTrs (1) Koat Proof [FINISHED, 184 ms] (2) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_foo_start(v_.0, v_x) -> Com_1(eval_foo_bb0_in(v_.0, v_x)) :|: TRUE eval_foo_bb0_in(v_.0, v_x) -> Com_1(eval_foo_bb1_in(v_x, v_x)) :|: v_x > 0 eval_foo_bb0_in(v_.0, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_x)) :|: v_x <= 0 eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_x)) :|: v_.0 < 0 eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb2_in(v_.0, v_x)) :|: v_.0 > 0 eval_foo_bb1_in(v_.0, v_x) -> Com_1(eval_foo_bb3_in(v_.0, v_x)) :|: v_.0 >= 0 && v_.0 <= 0 eval_foo_bb2_in(v_.0, v_x) -> Com_1(eval_foo_bb1_in(v_.0 - 1, v_x)) :|: TRUE eval_foo_bb3_in(v_.0, v_x) -> Com_1(eval_foo_stop(v_.0, v_x)) :|: TRUE The start-symbols are:[eval_foo_start_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 2*Ar_0 + 9) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 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) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) (Comp: ?, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoostart) = 2 Pol(evalfoobb0in) = 2 Pol(evalfoobb1in) = 2 Pol(evalfoobb3in) = 1 Pol(evalfoobb2in) = 2 Pol(evalfoostop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalfoobb2in) = V_2 - 1 Pol(evalfoobb1in) = V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1))", 0-0) = Ar_0 S("evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1))", 0-1) = Ar_1 S("evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1))", 0-0) = Ar_0 S("evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1))", 0-1) = ? S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ]", 0-0) = Ar_0 S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ]", 0-1) = 0 S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ]", 0-0) = Ar_0 S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ]", 0-1) = ? S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ]", 0-1) = ? S("evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ]", 0-0) = Ar_0 S("evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ]", 0-1) = Ar_1 S("evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-0) = Ar_0 S("evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ]", 0-1) = Ar_0 S("evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1))", 0-0) = Ar_0 S("evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1))", 0-1) = Ar_1 orients the transitions evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] weakly and the transition evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ 0 >= Ar_1 + 1 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 4 to obtain the following invariants: For symbol evalfoobb1in: X_1 - X_2 >= 0 /\ X_1 - 1 >= 0 For symbol evalfoobb2in: X_1 - X_2 >= 0 /\ X_1 - 1 >= 0 This yielded the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) start location: koat_start leaf cost: 0 By chaining the transition koat_start(Ar_0, Ar_1) -> Com_1(evalfoostart(Ar_0, Ar_1)) [ 0 <= 0 ] with all transitions in problem 5, the following new transition is obtained: koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] We thus obtain the following problem: 6: T: (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 6: evalfoostart(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) We thus obtain the following problem: 7: T: (Comp: ?, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] with all transitions in problem 7, the following new transition is obtained: evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] We thus obtain the following problem: 8: T: (Comp: ?, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: ?, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 8 produces the following problem: 9: T: (Comp: ?, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: 2, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: Ar_0, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] with all transitions in problem 9, the following new transition is obtained: evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] We thus obtain the following problem: 10: T: (Comp: 2, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: ?, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: Ar_0, Cost: 1) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 By chaining the transition evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb2in(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] with all transitions in problem 10, the following new transition is obtained: evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] We thus obtain the following problem: 11: T: (Comp: Ar_0, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: ?, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0, Cost: 1) evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transitions from problem 11: evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ 0 >= Ar_1 + 1 ] evalfoobb2in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 ] We thus obtain the following problem: 12: T: (Comp: 2, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 = 0 ] (Comp: Ar_0, Cost: 2) evalfoobb1in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_1 - 1)) [ Ar_0 - Ar_1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 2, Cost: 1) evalfoobb3in(Ar_0, Ar_1) -> Com_1(evalfoostop(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb1in(Ar_0, Ar_0)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) evalfoobb0in(Ar_0, Ar_1) -> Com_1(evalfoobb3in(Ar_0, Ar_1)) [ 0 >= Ar_0 ] (Comp: 1, Cost: 1) koat_start(Ar_0, Ar_1) -> Com_1(evalfoobb0in(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 2*Ar_0 + 9 Time: 0.174 sec (SMT: 0.149 sec) ---------------------------------------- (2) BOUNDS(1, n^1)