/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?, O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(1, n^2).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, 132 ms] (2) BOUNDS(1, n^2) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: evalloopsstart(A, B) -> Com_1(evalloopsentryin(A, B)) :|: TRUE evalloopsentryin(A, B) -> Com_1(evalloopsbb6in(A, B)) :|: A >= 0 evalloopsentryin(A, B) -> Com_1(evalloopsreturnin(A, B)) :|: 0 >= A + 1 evalloopsbb6in(A, B) -> Com_1(evalloopsbb1in(A, B)) :|: A >= 0 evalloopsbb6in(A, B) -> Com_1(evalloopsreturnin(A, B)) :|: 0 >= A + 1 evalloopsbb1in(A, B) -> Com_1(evalloopsbb4in(A, 1)) :|: A >= 2 evalloopsbb1in(A, B) -> Com_1(evalloopsbb5in(A, C)) :|: 1 >= A evalloopsbb4in(A, B) -> Com_1(evalloopsbb3in(A, B)) :|: A >= B + 1 evalloopsbb4in(A, B) -> Com_1(evalloopsbb5in(A, B)) :|: B >= A evalloopsbb3in(A, B) -> Com_1(evalloopsbb4in(A, 2 * B)) :|: TRUE evalloopsbb5in(A, B) -> Com_1(evalloopsbb6in(A - 1, B)) :|: TRUE evalloopsreturnin(A, B) -> Com_1(evalloopsstop(A, B)) :|: TRUE The start-symbols are:[evalloopsstart_2] ---------------------------------------- (1) Koat Proof (FINISHED) YES(?, 54*Ar_0 + 8*Ar_0^2 + 53) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: ?, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: ?, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(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) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: ?, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsstart) = 2 Pol(evalloopsentryin) = 2 Pol(evalloopsbb6in) = 2 Pol(evalloopsreturnin) = 1 Pol(evalloopsbb1in) = 2 Pol(evalloopsbb4in) = 2 Pol(evalloopsbb5in) = 2 Pol(evalloopsbb3in) = 2 Pol(evalloopsstop) = 0 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsstart) = 2*V_1 + 2 Pol(evalloopsentryin) = 2*V_1 + 2 Pol(evalloopsbb6in) = 2*V_1 + 2 Pol(evalloopsreturnin) = 2*V_1 Pol(evalloopsbb1in) = 2*V_1 + 1 Pol(evalloopsbb4in) = 2*V_1 Pol(evalloopsbb5in) = 2*V_1 Pol(evalloopsbb3in) = 2*V_1 Pol(evalloopsstop) = 2*V_1 Pol(koat_start) = 2*V_1 + 2 orients all transitions weakly and the transitions evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: ?, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 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) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: ?, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsbb5in) = 1 Pol(evalloopsbb6in) = 0 Pol(evalloopsbb4in) = 2 Pol(evalloopsbb3in) = 2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1))", 0-0) = ? S("evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1))", 0-1) = ? S("evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1))", 0-0) = ? S("evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1))", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1))", 0-0) = ? S("evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1))", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ]", 0-0) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ]", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ]", 0-0) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ]", 0-0) = ? S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ]", 0-0) = ? S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ]", 0-1) = 1 S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-0) = ? S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-1) = ? S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-0) = ? S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-1) = ? S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1))", 0-0) = Ar_0 S("evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1))", 0-1) = Ar_1 orients the transitions evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) weakly and the transitions evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] strictly and produces the following problem: 6: T: (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 2 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ 1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_0 >= Ar_1 + 1 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 6 to obtain the following invariants: For symbol evalloopsbb1in: X_1 >= 0 For symbol evalloopsbb3in: X_1 - X_2 - 1 >= 0 /\ X_2 - 1 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ X_1 - 2 >= 0 For symbol evalloopsbb4in: X_2 - 1 >= 0 /\ X_1 + X_2 - 3 >= 0 /\ X_1 - 2 >= 0 For symbol evalloopsbb5in: X_1 >= 0 For symbol evalloopsreturnin: -X_1 - 1 >= 0 This yielded the following problem: 7: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) [ -Ar_0 - 1 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) [ Ar_0 >= 0 ] (Comp: ?, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: ?, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ Ar_0 >= 0 /\ 1 >= Ar_0 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 0 /\ Ar_0 >= 2 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalloopsbb4in) = 2*V_1 - 2*V_2 + 1 Pol(evalloopsbb3in) = 2*V_1 - 2*V_2 and size complexities S("evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1))", 0-0) = Ar_0 S("evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1))", 0-1) = Ar_1 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-1) = Ar_1 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 S("evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-1) = Ar_1 S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 + 2 S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ]", 0-1) = ? S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-0) = Ar_0 + 2 S("evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ]", 0-1) = ? S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 0 /\\ Ar_0 >= 2 ]", 0-0) = Ar_0 + 2 S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 0 /\\ Ar_0 >= 2 ]", 0-1) = 1 S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ Ar_0 >= 0 /\\ 1 >= Ar_0 ]", 0-0) = 1 S("evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ Ar_0 >= 0 /\\ 1 >= Ar_0 ]", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-0) = Ar_0 + 2 S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_0 >= Ar_1 + 1 ]", 0-1) = ? S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-0) = Ar_0 + 2 S("evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 /\\ Ar_1 >= Ar_0 ]", 0-1) = ? S("evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-0) = Ar_0 + 2 S("evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\\ Ar_1 - 1 >= 0 /\\ Ar_0 + Ar_1 - 3 >= 0 /\\ Ar_0 - 2 >= 0 ]", 0-1) = ? S("evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) [ Ar_0 >= 0 ]", 0-0) = Ar_0 + 2 S("evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) [ Ar_0 >= 0 ]", 0-1) = ? S("evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) [ -Ar_0 - 1 >= 0 ]", 0-0) = Ar_0 + 2 S("evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) [ -Ar_0 - 1 >= 0 ]", 0-1) = ? S("koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 orients the transitions evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_0 >= Ar_1 + 1 ] evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 ] weakly and the transitions evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_0 >= Ar_1 + 1 ] evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 ] strictly and produces the following problem: 8: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(evalloopsstart(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 2, Cost: 1) evalloopsreturnin(Ar_0, Ar_1) -> Com_1(evalloopsstop(Ar_0, Ar_1)) [ -Ar_0 - 1 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb5in(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0 - 1, Ar_1)) [ Ar_0 >= 0 ] (Comp: 4*Ar_0^2 + 18*Ar_0 + 14, Cost: 1) evalloopsbb3in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 2*Ar_1)) [ Ar_0 - Ar_1 - 1 >= 0 /\ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 ] (Comp: 6*Ar_0 + 6, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_1 >= Ar_0 ] (Comp: 4*Ar_0^2 + 18*Ar_0 + 14, Cost: 1) evalloopsbb4in(Ar_0, Ar_1) -> Com_1(evalloopsbb3in(Ar_0, Ar_1)) [ Ar_1 - 1 >= 0 /\ Ar_0 + Ar_1 - 3 >= 0 /\ Ar_0 - 2 >= 0 /\ Ar_0 >= Ar_1 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb5in(Ar_0, Fresh_0)) [ Ar_0 >= 0 /\ 1 >= Ar_0 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb1in(Ar_0, Ar_1) -> Com_1(evalloopsbb4in(Ar_0, 1)) [ Ar_0 >= 0 /\ Ar_0 >= 2 ] (Comp: 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 2*Ar_0 + 2, Cost: 1) evalloopsbb6in(Ar_0, Ar_1) -> Com_1(evalloopsbb1in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsreturnin(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 ] (Comp: 1, Cost: 1) evalloopsentryin(Ar_0, Ar_1) -> Com_1(evalloopsbb6in(Ar_0, Ar_1)) [ Ar_0 >= 0 ] (Comp: 1, Cost: 1) evalloopsstart(Ar_0, Ar_1) -> Com_1(evalloopsentryin(Ar_0, Ar_1)) start location: koat_start leaf cost: 0 Complexity upper bound 54*Ar_0 + 8*Ar_0^2 + 53 Time: 0.198 sec (SMT: 0.171 sec) ---------------------------------------- (2) BOUNDS(1, n^2)