MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(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) f1(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f0) = V_1 and size complexities S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-0) = Ar_0 + Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\\ 0 >= Ar_1 + 1 ]", 0-1) = Ar_1 S("f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-0) = ? S("f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\\ Ar_1 >= 1 ]", 0-1) = Ar_1 S("f1(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1))", 0-0) = Ar_0 S("f1(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1))", 0-1) = Ar_1 orients the transitions f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] weakly and the transition f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] strictly and produces the following problem: 3: T: (Comp: 1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f0(Ar_0, Ar_1)) (Comp: ?, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ Ar_1 >= 1 ] (Comp: Ar_0, Cost: 1) f0(Ar_0, Ar_1) -> Com_1(f0(Ar_0 + Ar_1, Ar_1)) [ Ar_0 >= 1 /\ 0 >= Ar_1 + 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f1(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 0.939 sec (SMT: 0.900 sec)