YES(?,O(n^1)) * Step 1: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval1(A,B,C) -> eval2(A,B,C) [A >= 1 + B] (?,1) 1. eval2(A,B,C) -> eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] (?,1) 2. eval2(A,B,C) -> eval1(A,B,1 + C) [-1 + A + -1*B >= 0 && A >= 1 + C] (?,1) 3. eval2(A,B,C) -> eval1(-1 + A,B,C) [-1 + A + -1*B >= 0 && C >= A] (?,1) 4. start(A,B,C) -> eval1(A,B,C) True (1,1) Signature: {(eval1,3);(eval2,3);(start,3)} Flow Graph: [0->{1,2,3},1->{0},2->{0},3->{0},4->{0}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval1) = x1 + -1*x2 p(eval2) = x1 + -1*x2 p(start) = x1 + -1*x2 Following rules are strictly oriented: [-1 + A + -1*B >= 0 && C >= A] ==> eval2(A,B,C) = A + -1*B > -1 + A + -1*B = eval1(-1 + A,B,C) Following rules are weakly oriented: [A >= 1 + B] ==> eval1(A,B,C) = A + -1*B >= A + -1*B = eval2(A,B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*B >= -1 + A + -1*B = eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*B >= A + -1*B = eval1(A,B,1 + C) True ==> start(A,B,C) = A + -1*B >= A + -1*B = eval1(A,B,C) * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval1(A,B,C) -> eval2(A,B,C) [A >= 1 + B] (?,1) 1. eval2(A,B,C) -> eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] (?,1) 2. eval2(A,B,C) -> eval1(A,B,1 + C) [-1 + A + -1*B >= 0 && A >= 1 + C] (?,1) 3. eval2(A,B,C) -> eval1(-1 + A,B,C) [-1 + A + -1*B >= 0 && C >= A] (A + B,1) 4. start(A,B,C) -> eval1(A,B,C) True (1,1) Signature: {(eval1,3);(eval2,3);(start,3)} Flow Graph: [0->{1,2,3},1->{0},2->{0},3->{0},4->{0}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval1) = x1 + -1*x3 p(eval2) = x1 + -1*x3 p(start) = x1 + -1*x3 Following rules are strictly oriented: [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*C > -1 + A + -1*C = eval1(A,B,1 + C) Following rules are weakly oriented: [A >= 1 + B] ==> eval1(A,B,C) = A + -1*C >= A + -1*C = eval2(A,B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*C >= A + -1*C = eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && C >= A] ==> eval2(A,B,C) = A + -1*C >= -1 + A + -1*C = eval1(-1 + A,B,C) True ==> start(A,B,C) = A + -1*C >= A + -1*C = eval1(A,B,C) * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval1(A,B,C) -> eval2(A,B,C) [A >= 1 + B] (?,1) 1. eval2(A,B,C) -> eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] (?,1) 2. eval2(A,B,C) -> eval1(A,B,1 + C) [-1 + A + -1*B >= 0 && A >= 1 + C] (A + C,1) 3. eval2(A,B,C) -> eval1(-1 + A,B,C) [-1 + A + -1*B >= 0 && C >= A] (A + B,1) 4. start(A,B,C) -> eval1(A,B,C) True (1,1) Signature: {(eval1,3);(eval2,3);(start,3)} Flow Graph: [0->{1,2,3},1->{0},2->{0},3->{0},4->{0}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval1) = x1 + -1*x2 p(eval2) = x1 + -1*x2 p(start) = x1 + -1*x2 Following rules are strictly oriented: [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*B > -1 + A + -1*B = eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && C >= A] ==> eval2(A,B,C) = A + -1*B > -1 + A + -1*B = eval1(-1 + A,B,C) Following rules are weakly oriented: [A >= 1 + B] ==> eval1(A,B,C) = A + -1*B >= A + -1*B = eval2(A,B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] ==> eval2(A,B,C) = A + -1*B >= A + -1*B = eval1(A,B,1 + C) True ==> start(A,B,C) = A + -1*B >= A + -1*B = eval1(A,B,C) * Step 4: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval1(A,B,C) -> eval2(A,B,C) [A >= 1 + B] (?,1) 1. eval2(A,B,C) -> eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] (A + B,1) 2. eval2(A,B,C) -> eval1(A,B,1 + C) [-1 + A + -1*B >= 0 && A >= 1 + C] (A + C,1) 3. eval2(A,B,C) -> eval1(-1 + A,B,C) [-1 + A + -1*B >= 0 && C >= A] (A + B,1) 4. start(A,B,C) -> eval1(A,B,C) True (1,1) Signature: {(eval1,3);(eval2,3);(start,3)} Flow Graph: [0->{1,2,3},1->{0},2->{0},3->{0},4->{0}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval1(A,B,C) -> eval2(A,B,C) [A >= 1 + B] (1 + 3*A + 2*B + C,1) 1. eval2(A,B,C) -> eval1(A,1 + B,C) [-1 + A + -1*B >= 0 && A >= 1 + C] (A + B,1) 2. eval2(A,B,C) -> eval1(A,B,1 + C) [-1 + A + -1*B >= 0 && A >= 1 + C] (A + C,1) 3. eval2(A,B,C) -> eval1(-1 + A,B,C) [-1 + A + -1*B >= 0 && C >= A] (A + B,1) 4. start(A,B,C) -> eval1(A,B,C) True (1,1) Signature: {(eval1,3);(eval2,3);(start,3)} Flow Graph: [0->{1,2,3},1->{0},2->{0},3->{0},4->{0}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))