WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Testing for reachability in the complexity graph removes the following transition from problem 1: evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: ?, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxreturnin) = 1 Pol(evalaxstop) = 0 Pol(evalaxbb3in) = 2 Pol(evalaxbbin) = 2 Pol(evalaxbb1in) = 2 Pol(evalaxbb2in) = 2 Pol(evalaxentryin) = 2 Pol(evalaxstart) = 2 Pol(koat_start) = 2 orients all transitions weakly and the transitions evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] strictly and produces the following problem: 4: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxreturnin) = -4*V_1 + 4*V_3 - 11 Pol(evalaxstop) = -4*V_1 + 4*V_3 - 11 Pol(evalaxbb3in) = -4*V_1 + 4*V_3 - 11 Pol(evalaxbbin) = -4*V_1 + 4*V_3 - 9 Pol(evalaxbb1in) = -4*V_1 + 4*V_3 - 10 Pol(evalaxbb2in) = -4*V_1 + 4*V_3 - 10 Pol(evalaxentryin) = 4*V_3 Pol(evalaxstart) = 4*V_3 Pol(koat_start) = 4*V_3 orients all transitions weakly and the transition evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] strictly and produces the following problem: 5: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: 4*Ar_2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: ?, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 5 produces the following problem: 6: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: 4*Ar_2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxbb2in) = 1 Pol(evalaxbb3in) = 0 Pol(evalaxbb1in) = 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-0) = 0 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-0) = 4*Ar_2 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-1) = 0 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = 4*Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = 4*Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-0) = 4*Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-1) = ? S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-0) = 4*Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-0) = 4*Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-0) = 4*Ar_2 S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) weakly and the transition evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] strictly and produces the following problem: 7: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: 4*Ar_2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: ?, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(evalaxbb2in) = -2*V_2 + 2*V_3 + 2 Pol(evalaxbb1in) = -2*V_2 + 2*V_3 + 1 and size complexities S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-1) = Ar_1 S("koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ]", 0-2) = Ar_2 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-0) = Ar_0 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-0) = 0 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-1) = Ar_1 S("evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2))", 0-2) = Ar_2 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-0) = 4*Ar_2 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-1) = 0 S("evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2))", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-0) = 4*Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ]", 0-2) = Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-0) = 4*Ar_2 S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-1) = ? S("evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-0) = 4*Ar_2 S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-1) = ? S("evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2))", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-0) = 4*Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\\ Ar_2 >= Ar_0 + 3 ]", 0-2) = Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-0) = 4*Ar_2 S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-1) = ? S("evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ]", 0-2) = Ar_2 S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-0) = 4*Ar_2 S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-1) = ? S("evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2))", 0-2) = Ar_2 orients the transitions evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) weakly and the transition evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] strictly and produces the following problem: 8: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: 4*Ar_2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: ?, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: 8*Ar_2^2 + 10*Ar_2 + 2, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 8 produces the following problem: 9: T: (Comp: 2, Cost: 1) evalaxreturnin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstop(Ar_0, Ar_1, Ar_2)) (Comp: 2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxreturnin(Ar_0, Ar_1, Ar_2)) [ Ar_0 + 2 >= Ar_2 ] (Comp: 4*Ar_2, Cost: 1) evalaxbb3in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(Ar_0 + 1, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 /\ Ar_2 >= Ar_0 + 3 ] (Comp: 8*Ar_2^2 + 10*Ar_2 + 2, Cost: 1) evalaxbb1in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, Ar_1 + 1, Ar_2)) (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb3in(Ar_0, Ar_1, Ar_2)) [ Ar_1 + 1 >= Ar_2 ] (Comp: 8*Ar_2^2 + 10*Ar_2 + 2, Cost: 1) evalaxbb2in(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb1in(Ar_0, Ar_1, Ar_2)) [ Ar_2 >= Ar_1 + 2 ] (Comp: 4*Ar_2 + 1, Cost: 1) evalaxbbin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbb2in(Ar_0, 0, Ar_2)) (Comp: 1, Cost: 1) evalaxentryin(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxbbin(0, Ar_1, Ar_2)) (Comp: 1, Cost: 1) evalaxstart(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxentryin(Ar_0, Ar_1, Ar_2)) (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1, Ar_2) -> Com_1(evalaxstart(Ar_0, Ar_1, Ar_2)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 32*Ar_2 + 16*Ar_2^2 + 12 Time: 0.779 sec (SMT: 0.748 sec)