YES(?,O(n^1)) * Step 1: UnsatRules WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (?,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (?,1) 2. eval(A,B,C) -> eval(A,B,-1 + C) [A >= 1 + B && B >= A && C >= 1 + B] (?,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (?,1) 4. eval(A,B,C) -> eval(A,B,C) [A >= 1 + B && B >= A && B >= C] (?,1) 5. eval(A,B,C) -> eval(A,B,C) [C >= 1 + B && B >= A && B >= C] (?,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,2,3,4,5},1->{0,1,2,3,4,5},2->{0,1,2,3,4,5},3->{0,1,2,3,4,5},4->{0,1,2,3,4,5},5->{0,1,2,3,4,5} ,6->{0,1,2,3,4,5}] + Applied Processor: UnsatRules + Details: Following transitions have unsatisfiable constraints and are removed: [2,4,5] * Step 2: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (?,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (?,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (?,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,3},1->{0,1,3},3->{0,1,3},6->{0,1,3}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(3,0),(3,1)] * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (?,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (?,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (?,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,3},1->{0,1,3},3->{3},6->{0,1,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = -1*x2 + x3 p(start) = -1*x2 + x3 Following rules are strictly oriented: [C >= 1 + B && B >= A] ==> eval(A,B,C) = -1*B + C > -1 + -1*B + C = eval(A,B,-1 + C) Following rules are weakly oriented: [A >= 1 + B] ==> eval(A,B,C) = -1*B + C >= -1*B + C = eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] ==> eval(A,B,C) = -1*B + C >= -1*B + C = eval(-1 + A,B,C) True ==> start(A,B,C) = -1*B + C >= -1*B + C = eval(A,B,C) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (?,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (?,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (B + C,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,3},1->{0,1,3},3->{3},6->{0,1,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x1 + -1*x2 p(start) = x1 + -1*x2 Following rules are strictly oriented: [C >= 1 + B && A >= 1 + B] ==> eval(A,B,C) = A + -1*B > -1 + A + -1*B = eval(-1 + A,B,C) Following rules are weakly oriented: [A >= 1 + B] ==> eval(A,B,C) = A + -1*B >= -1 + A + -1*B = eval(-1 + A,B,C) [C >= 1 + B && B >= A] ==> eval(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,-1 + C) True ==> start(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,C) * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (?,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (A + B,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (B + C,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,3},1->{0,1,3},3->{3},6->{0,1,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x1 + -1*x2 p(start) = x1 + -1*x2 Following rules are strictly oriented: [A >= 1 + B] ==> eval(A,B,C) = A + -1*B > -1 + A + -1*B = eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] ==> eval(A,B,C) = A + -1*B > -1 + A + -1*B = eval(-1 + A,B,C) Following rules are weakly oriented: [C >= 1 + B && B >= A] ==> eval(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,-1 + C) True ==> start(A,B,C) = A + -1*B >= A + -1*B = eval(A,B,C) * Step 6: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C) -> eval(-1 + A,B,C) [A >= 1 + B] (A + B,1) 1. eval(A,B,C) -> eval(-1 + A,B,C) [C >= 1 + B && A >= 1 + B] (A + B,1) 3. eval(A,B,C) -> eval(A,B,-1 + C) [C >= 1 + B && B >= A] (B + C,1) 6. start(A,B,C) -> eval(A,B,C) True (1,1) Signature: {(eval,3);(start,3)} Flow Graph: [0->{0,1,3},1->{0,1,3},3->{3},6->{0,1,3}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))