MAYBE Initial complexity problem: 1: T: (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f999(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: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_0 >= 1 ] (Comp: 1, Cost: 1) f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 1 ] (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ] start location: koat_start leaf cost: 0 Applied AI with 'oct' on problem 2 to obtain the following invariants: For symbol f1: X_2 >= 0 /\ X_1 + X_2 - 1 >= 0 /\ X_1 - 1 >= 0 For symbol f2: X_2 >= 0 /\ X_1 + X_2 >= 0 /\ X_1 >= 0 This yielded the following problem: 3: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(koat_start) = 2*V_2 Pol(f999) = 2*V_2 Pol(f2) = 2*V_1 + 2*V_2 + 1 Pol(f1) = 2*V_1 + 2*V_2 orients all transitions weakly and the transition f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= 1 ] strictly and produces the following problem: 4: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= 1 ] (Comp: 2*Ar_1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= 1 ] start location: koat_start leaf cost: 0 A polynomial rank function with Pol(f2) = 1 Pol(f1) = 0 and size complexities S("f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-0) = ? S("f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_0 >= 1 ]", 0-1) = ? S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= 1 ]", 0-0) = ? S("f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_0 >= 1 ]", 0-1) = ? S("f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\\ Ar_0 = 0 ]", 0-0) = 1 S("f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\\ Ar_0 = 0 ]", 0-1) = Ar_1 S("f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = ? S("f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 - 1 >= 0 /\\ Ar_0 - 1 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = ? S("f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-0) = ? S("f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\\ Ar_0 + Ar_1 >= 0 /\\ Ar_0 >= 0 /\\ Ar_1 >= 1 ]", 0-1) = ? S("koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-0) = Ar_0 S("koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ]", 0-1) = Ar_1 orients the transitions f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= 1 ] f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] weakly and the transition f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] strictly and produces the following problem: 5: T: (Comp: 1, Cost: 0) koat_start(Ar_0, Ar_1) -> Com_1(f999(Ar_0, Ar_1)) [ 0 <= 0 ] (Comp: 2*Ar_1, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_1 >= 1 ] (Comp: ?, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f1(Ar_0 + 1, Ar_1 - 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_1 >= 1 ] (Comp: 1, Cost: 1) f999(Ar_0, Ar_1) -> Com_1(f1(1, Ar_1 - 1)) [ Ar_1 >= 1 /\ Ar_0 = 0 ] (Comp: ?, Cost: 1) f2(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1 + 1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 >= 0 /\ Ar_0 >= 0 /\ Ar_0 >= 1 ] (Comp: 2*Ar_1, Cost: 1) f1(Ar_0, Ar_1) -> Com_1(f2(Ar_0 - 1, Ar_1)) [ Ar_1 >= 0 /\ Ar_0 + Ar_1 - 1 >= 0 /\ Ar_0 - 1 >= 0 /\ Ar_0 >= 1 ] start location: koat_start leaf cost: 0 Complexity upper bound ? Time: 2.898 sec (SMT: 2.816 sec)