WORST_CASE(?, O(n^2)) Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(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 1: f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ 0 >= Ar_0 + 1 /\ Ar_0 + Ar_1 >= 2 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ Ar_0 >= 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ 0 >= Ar_1 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ 0 >= Ar_1 + 1 ] We thus obtain the following problem: 2: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Repeatedly propagating knowledge in problem 2 produces the following problem: 3: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 + V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] strictly and produces the following problem: 4: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 + V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] strictly and produces the following problem: 5: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 + V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] strictly and produces the following problem: 6: T: (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -4*V_2 - 3 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] strictly and produces the following problem: 7: T: (Comp: 15*Ar_1 + 4*Ar_0^2 + 4*Ar_0*Ar_1 + 3*Ar_0 + 9, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 8: T: (Comp: 15*Ar_1 + 4*Ar_0^2 + 4*Ar_0*Ar_1 + 3*Ar_0 + 9, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: 2*Ar_0^2 + 2*Ar_0*Ar_1 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] strictly and produces the following problem: 9: T: (Comp: 15*Ar_1 + 4*Ar_0^2 + 4*Ar_0*Ar_1 + 3*Ar_0 + 9, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: 2*Ar_0^2 + 2*Ar_0*Ar_1 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = -V_1 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] strictly and produces the following problem: 10: T: (Comp: 15*Ar_1 + 4*Ar_0^2 + 4*Ar_0*Ar_1 + 3*Ar_0 + 9, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: 2*Ar_0^2 + 2*Ar_0*Ar_1 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = V_1 - V_2 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ Ar_1 >= 2 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\\ 0 >= Ar_1 /\\ Ar_0 >= 1 /\\ Ar_0 + Ar_1 >= 2 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ Ar_0 >= 2 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ Ar_1 >= 0 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-0) = Ar_0 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\\ Ar_1 >= Ar_0 + 1 /\\ 0 >= Ar_0 + 2 /\\ Ar_1 >= 1 ]", 0-1) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\\ Ar_0 >= Ar_1 + 1 /\\ 0 >= Ar_0 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\\ 0 >= Ar_1 + 2 /\\ 0 >= Ar_0 + 1 /\\ 0 >= Ar_1 + Ar_0 + 1 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 11: T: (Comp: 15*Ar_1 + 4*Ar_0^2 + 4*Ar_0*Ar_1 + 3*Ar_0 + 9, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ 0 >= Ar_1 + 2 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ 0 >= Ar_0 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0^2 + Ar_0*Ar_1 + Ar_0, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ 0 >= Ar_0 + 2 /\ Ar_1 >= 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 + 1)) [ Ar_1 >= Ar_0 /\ Ar_1 >= 0 /\ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + Ar_0 + 1 ] (Comp: 2*Ar_0^2 + 2*Ar_0*Ar_1 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ 1 >= Ar_0 + Ar_1 /\ Ar_0 >= Ar_1 + 1 /\ Ar_0 >= 2 /\ 0 >= Ar_1 + 1 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ 0 >= Ar_1 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 + 1, Ar_1)) [ Ar_1 + Ar_0 >= 0 /\ Ar_1 >= Ar_0 + 1 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: 2*Ar_0*Ar_1 + 2*Ar_1^2 + Ar_0 + Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1 - 1)) [ Ar_0 >= Ar_1 /\ Ar_1 >= 2 /\ Ar_0 >= 1 /\ Ar_0 + Ar_1 >= 2 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ 0 >= Ar_0 + 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f2(Ar_0, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound 21*Ar_1 + 7*Ar_0^2 + 9*Ar_0*Ar_1 + 10*Ar_0 + 2*Ar_1^2 + 13 Time: 6.279 sec (SMT: 6.060 sec)