YES * Step 1: UnsatPaths YES + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f8(1 + A,B,0,D) [A >= 0 && B >= 1 + A] (?,1) 2. f8(A,B,C,D) -> f16(A,B,C,0) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] 3. f8(A,B,C,D) -> f16(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] 4. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] 5. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] 6. f16(A,B,C,D) -> f4(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] 7. f16(A,B,C,D) -> f4(-1 + A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] 8. f4(A,B,C,D) -> f20(A,B,C,D) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Flow Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1,8},8->{}] + Applied Processor: UnsatPaths + Details: We remove following edges from the transition graph: [(7,8)] * Step 2: FromIts YES + Considered Problem: Rules: 0. f0(A,B,C,D) -> f4(0,B,C,D) True (1,1) 1. f4(A,B,C,D) -> f8(1 + A,B,0,D) [A >= 0 && B >= 1 + A] (?,1) 2. f8(A,B,C,D) -> f16(A,B,C,0) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] 3. f8(A,B,C,D) -> f16(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] 4. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] 5. f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] 6. f16(A,B,C,D) -> f4(A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] 7. f16(A,B,C,D) -> f4(-1 + A,B,C,D) [-1 + B + -1*C >= 0 (?,1) && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] 8. f4(A,B,C,D) -> f20(A,B,C,D) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Flow Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{}] + Applied Processor: FromIts + Details: () * Step 3: Decompose YES + Considered Problem: Rules: f0(A,B,C,D) -> f4(0,B,C,D) True f4(A,B,C,D) -> f8(1 + A,B,0,D) [A >= 0 && B >= 1 + A] f8(A,B,C,D) -> f16(A,B,C,0) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] f8(A,B,C,D) -> f16(A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] f16(A,B,C,D) -> f4(A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] f16(A,B,C,D) -> f4(-1 + A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] f4(A,B,C,D) -> f20(A,B,C,D) [A >= 0 && A >= B] Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Rule Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{}] + Applied Processor: Decompose NoGreedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6,7,8] | `- p:[1,6,2,4,5,3,7] c: [7] | `- p:[1,6,2,4,5,3] c: [1,2,3,6] | `- p:[4,5] c: [5] | `- p:[4] c: [4] * Step 4: CloseWith YES + Considered Problem: (Rules: f0(A,B,C,D) -> f4(0,B,C,D) True f4(A,B,C,D) -> f8(1 + A,B,0,D) [A >= 0 && B >= 1 + A] f8(A,B,C,D) -> f16(A,B,C,0) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A] f8(A,B,C,D) -> f16(A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && A >= B] f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && 0 >= 1 + E] f8(A,B,C,D) -> f8(1 + A,B,1 + C,E) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && B >= 1 + A && E >= 1] f16(A,B,C,D) -> f4(A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && 0 >= C] f16(A,B,C,D) -> f4(-1 + A,B,C,D) [-1 + B + -1*C >= 0 && -1 + A + -1*C >= 0 && C >= 0 && -1 + B + C >= 0 && -1 + A + C >= 0 && -1 + B >= 0 && -2 + A + B >= 0 && -1*A + B >= 0 && -1 + A >= 0 && C >= 1] f4(A,B,C,D) -> f20(A,B,C,D) [A >= 0 && A >= B] Signature: {(f0,4);(f16,4);(f20,4);(f4,4);(f8,4)} Rule Graph: [0->{1,8},1->{2,3,4,5},2->{6,7},3->{6,7},4->{2,3,4,5},5->{2,3,4,5},6->{1,8},7->{1},8->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6,7,8] | `- p:[1,6,2,4,5,3,7] c: [7] | `- p:[1,6,2,4,5,3] c: [1,2,3,6] | `- p:[4,5] c: [5] | `- p:[4] c: [4]) + Applied Processor: CloseWith True + Details: () YES