YES(?,O(n^1)) * Step 1: ArgumentFilter WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) True (1,1) 1. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f17(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [B >= 1 + A] (?,1) 2. f5(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f8(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [A >= B] (?,1) 3. f17(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f27(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [-1 + -1*A + B >= 0 && B >= 1 + A] (?,1) 4. f8(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f5(A,1 + B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [A + -1*B >= 0 && C >= 1 + A] (?,1) 5. f8(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f8(A,B,1 + C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [A + -1*B >= 0 && A >= C] (?,1) 6. f27(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [-1 + -1*A + B >= 0 && E >= 51] (?,1) 7. f27(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f31(A,B,C,D,E,0,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [-1 + -1*A + B >= 0 && 50 >= E] (?,1) 8. f31(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) -> f1(A,B,C,D,E,0,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,A1) [-1*F >= 0 (?,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: ArgumentFilter [3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] + Details: We remove following argument positions: [3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. * Step 2: TrivialSCCs WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,E,F) -> f5(A,B,C,E,F) True (1,1) 1. f5(A,B,C,E,F) -> f17(A,B,C,E,F) [B >= 1 + A] (?,1) 2. f5(A,B,C,E,F) -> f8(A,B,C,E,F) [A >= B] (?,1) 3. f17(A,B,C,E,F) -> f27(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] (?,1) 4. f8(A,B,C,E,F) -> f5(A,1 + B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] (?,1) 5. f8(A,B,C,E,F) -> f8(A,B,1 + C,E,F) [A + -1*B >= 0 && A >= C] (?,1) 6. f27(A,B,C,E,F) -> f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] (?,1) 7. f27(A,B,C,E,F) -> f31(A,B,C,E,0) [-1 + -1*A + B >= 0 && 50 >= E] (?,1) 8. f31(A,B,C,E,F) -> f1(A,B,C,E,0) [-1*F >= 0 (?,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: TrivialSCCs + Details: All trivial SCCs of the transition graph admit timebound 1. * Step 3: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,E,F) -> f5(A,B,C,E,F) True (1,1) 1. f5(A,B,C,E,F) -> f17(A,B,C,E,F) [B >= 1 + A] (1,1) 2. f5(A,B,C,E,F) -> f8(A,B,C,E,F) [A >= B] (?,1) 3. f17(A,B,C,E,F) -> f27(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] (1,1) 4. f8(A,B,C,E,F) -> f5(A,1 + B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] (?,1) 5. f8(A,B,C,E,F) -> f8(A,B,1 + C,E,F) [A + -1*B >= 0 && A >= C] (?,1) 6. f27(A,B,C,E,F) -> f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] (1,1) 7. f27(A,B,C,E,F) -> f31(A,B,C,E,0) [-1 + -1*A + B >= 0 && 50 >= E] (1,1) 8. f31(A,B,C,E,F) -> f1(A,B,C,E,0) [-1*F >= 0 (1,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x3 p(f17) = x1 + -1*x3 p(f2) = x1 + -1*x3 + x27 p(f27) = x1 + -1*x3 p(f31) = x1 + -1*x3 p(f5) = x1 + -1*x3 + x27 p(f8) = x1 + -1*x3 + x27 Following rules are strictly oriented: [A + -1*B >= 0 && A >= C] ==> f8(A,B,C,E,F) = 1 + A + -1*C > A + -1*C = f8(A,B,1 + C,E,F) Following rules are weakly oriented: True ==> f2(A,B,C,E,F) = 1 + A + -1*C >= 1 + A + -1*C = f5(A,B,C,E,F) [B >= 1 + A] ==> f5(A,B,C,E,F) = 1 + A + -1*C >= A + -1*C = f17(A,B,C,E,F) [A >= B] ==> f5(A,B,C,E,F) = 1 + A + -1*C >= 1 + A + -1*C = f8(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] ==> f17(A,B,C,E,F) = A + -1*C >= A + -1*C = f27(A,B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] ==> f8(A,B,C,E,F) = 1 + A + -1*C >= 1 + A + -1*C = f5(A,1 + B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] ==> f27(A,B,C,E,F) = A + -1*C >= A + -1*C = f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && 50 >= E] ==> f27(A,B,C,E,F) = A + -1*C >= A + -1*C = f31(A,B,C,E,0) [-1*F >= 0 ==> && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] f31(A,B,C,E,F) = A + -1*C >= A + -1*C = f1(A,B,C,E,0) * Step 4: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,E,F) -> f5(A,B,C,E,F) True (1,1) 1. f5(A,B,C,E,F) -> f17(A,B,C,E,F) [B >= 1 + A] (1,1) 2. f5(A,B,C,E,F) -> f8(A,B,C,E,F) [A >= B] (?,1) 3. f17(A,B,C,E,F) -> f27(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] (1,1) 4. f8(A,B,C,E,F) -> f5(A,1 + B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] (?,1) 5. f8(A,B,C,E,F) -> f8(A,B,1 + C,E,F) [A + -1*B >= 0 && A >= C] (1 + A + C,1) 6. f27(A,B,C,E,F) -> f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] (1,1) 7. f27(A,B,C,E,F) -> f31(A,B,C,E,0) [-1 + -1*A + B >= 0 && 50 >= E] (1,1) 8. f31(A,B,C,E,F) -> f1(A,B,C,E,0) [-1*F >= 0 (1,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: We apply a polynomial interpretation of shape linear: p(f1) = x1 + -1*x2 p(f17) = x1 + -1*x2 + x27 p(f2) = x1 + -1*x2 + x27 p(f27) = x1 + -1*x2 p(f31) = x1 + -1*x2 p(f5) = x1 + -1*x2 + x27 p(f8) = x1 + -1*x2 + x27 Following rules are strictly oriented: [A + -1*B >= 0 && C >= 1 + A] ==> f8(A,B,C,E,F) = 1 + A + -1*B > A + -1*B = f5(A,1 + B,C,E,F) Following rules are weakly oriented: True ==> f2(A,B,C,E,F) = 1 + A + -1*B >= 1 + A + -1*B = f5(A,B,C,E,F) [B >= 1 + A] ==> f5(A,B,C,E,F) = 1 + A + -1*B >= 1 + A + -1*B = f17(A,B,C,E,F) [A >= B] ==> f5(A,B,C,E,F) = 1 + A + -1*B >= 1 + A + -1*B = f8(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] ==> f17(A,B,C,E,F) = 1 + A + -1*B >= A + -1*B = f27(A,B,C,E,F) [A + -1*B >= 0 && A >= C] ==> f8(A,B,C,E,F) = 1 + A + -1*B >= 1 + A + -1*B = f8(A,B,1 + C,E,F) [-1 + -1*A + B >= 0 && E >= 51] ==> f27(A,B,C,E,F) = A + -1*B >= A + -1*B = f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && 50 >= E] ==> f27(A,B,C,E,F) = A + -1*B >= A + -1*B = f31(A,B,C,E,0) [-1*F >= 0 ==> && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] f31(A,B,C,E,F) = A + -1*B >= A + -1*B = f1(A,B,C,E,0) * Step 5: KnowledgePropagation WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,E,F) -> f5(A,B,C,E,F) True (1,1) 1. f5(A,B,C,E,F) -> f17(A,B,C,E,F) [B >= 1 + A] (1,1) 2. f5(A,B,C,E,F) -> f8(A,B,C,E,F) [A >= B] (?,1) 3. f17(A,B,C,E,F) -> f27(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] (1,1) 4. f8(A,B,C,E,F) -> f5(A,1 + B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] (1 + A + B,1) 5. f8(A,B,C,E,F) -> f8(A,B,1 + C,E,F) [A + -1*B >= 0 && A >= C] (1 + A + C,1) 6. f27(A,B,C,E,F) -> f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] (1,1) 7. f27(A,B,C,E,F) -> f31(A,B,C,E,0) [-1 + -1*A + B >= 0 && 50 >= E] (1,1) 8. f31(A,B,C,E,F) -> f1(A,B,C,E,0) [-1*F >= 0 (1,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: KnowledgePropagation + Details: We propagate bounds from predecessors. * Step 6: PolyRank WORST_CASE(?,O(n^1)) + Considered Problem: Rules: 0. f2(A,B,C,E,F) -> f5(A,B,C,E,F) True (1,1) 1. f5(A,B,C,E,F) -> f17(A,B,C,E,F) [B >= 1 + A] (1,1) 2. f5(A,B,C,E,F) -> f8(A,B,C,E,F) [A >= B] (2 + A + B,1) 3. f17(A,B,C,E,F) -> f27(A,B,C,E,F) [-1 + -1*A + B >= 0 && B >= 1 + A] (1,1) 4. f8(A,B,C,E,F) -> f5(A,1 + B,C,E,F) [A + -1*B >= 0 && C >= 1 + A] (1 + A + B,1) 5. f8(A,B,C,E,F) -> f8(A,B,1 + C,E,F) [A + -1*B >= 0 && A >= C] (1 + A + C,1) 6. f27(A,B,C,E,F) -> f1(A,B,C,E,F) [-1 + -1*A + B >= 0 && E >= 51] (1,1) 7. f27(A,B,C,E,F) -> f31(A,B,C,E,0) [-1 + -1*A + B >= 0 && 50 >= E] (1,1) 8. f31(A,B,C,E,F) -> f1(A,B,C,E,0) [-1*F >= 0 (1,1) && 50 + -1*E + -1*F >= 0 && F >= 0 && 50 + -1*E + F >= 0 && 50 + -1*E >= 0 && -1 + -1*A + B >= 0 && B >= A && F = 0] Signature: {(f1,27);(f17,27);(f2,27);(f27,27);(f31,27);(f5,27);(f8,27)} Flow Graph: [0->{1,2},1->{3},2->{4,5},3->{6,7},4->{1,2},5->{4,5},6->{},7->{8},8->{}] + Applied Processor: PolyRank {useFarkas = True, withSizebounds = [], shape = Linear} + Details: The problem is already solved. YES(?,O(n^1))