YES(?,O(n^1)) * Step 1: UnsatPaths WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (?,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (?,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2,3},1->{0,1,2,3},2->{0,1,2,3},3->{0,1,2,3},4->{0,1,2,3}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(0,3),(1,2),(2,0),(2,1),(2,3),(3,0),(3,1),(3,2)] * Step 2: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (?,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (?,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2},1->{0,1,3},2->{2},3->{3},4->{0,1,2,3}] + 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 && D >= C] ==> eval(A,B,C,D,E) = A + -1*B > -1 + A + -1*B = eval(A,1 + B,C,D,1 + E) Following rules are weakly oriented: [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B >= -1 + A + -1*B = eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,1 + D,1 + E) True ==> start(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,D,E) * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (?,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (A + B,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2},1->{0,1,3},2->{2},3->{3},4->{0,1,2,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x3 + -1*x4 p(start) = x3 + -1*x4 Following rules are strictly oriented: [B >= A && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D > -1 + C + -1*D = eval(A,B,C,1 + D,1 + E) Following rules are weakly oriented: [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D >= -1 + C + -1*D = eval(A,B,C,1 + D,1 + E) [A >= 1 + B && D >= C] ==> eval(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,1 + B,C,D,1 + E) True ==> start(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,B,C,D,E) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (C + D,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (A + B,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2},1->{0,1,3},2->{2},3->{3},4->{0,1,2,3}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(eval) = x3 + -1*x4 p(start) = x3 + -1*x4 Following rules are strictly oriented: [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D > -1 + C + -1*D = eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D > -1 + C + -1*D = eval(A,B,C,1 + D,1 + E) Following rules are weakly oriented: [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] ==> eval(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,1 + B,C,D,1 + E) True ==> start(A,B,C,D,E) = C + -1*D >= C + -1*D = eval(A,B,C,D,E) * Step 5: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (?,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (C + D,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (C + D,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (A + B,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2},1->{0,1,3},2->{2},3->{3},4->{0,1,2,3}] + 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 && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B > -1 + A + -1*B = eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] ==> eval(A,B,C,D,E) = A + -1*B > -1 + A + -1*B = eval(A,1 + B,C,D,1 + E) Following rules are weakly oriented: [A >= 1 + B && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] ==> eval(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,1 + D,1 + E) True ==> start(A,B,C,D,E) = A + -1*B >= A + -1*B = eval(A,B,C,D,E) * Step 6: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && C >= 1 + D] (A + B,1) 1. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [A >= 1 + B && C >= 1 + D] (C + D,1) 2. eval(A,B,C,D,E) -> eval(A,B,C,1 + D,1 + E) [B >= A && C >= 1 + D] (C + D,1) 3. eval(A,B,C,D,E) -> eval(A,1 + B,C,D,1 + E) [A >= 1 + B && D >= C] (A + B,1) 4. start(A,B,C,D,E) -> eval(A,B,C,D,E) True (1,1) Signature: {(eval,5);(start,5)} Flow Graph: [0->{0,1,2},1->{0,1,3},2->{2},3->{3},4->{0,1,2,3}] + Applied Processor: KnowledgePropagation + Details: The problem is already solved. YES(?,O(n^1))