YES * Step 1: 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) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] (?,1) 2. f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] 3. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] 4. f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] 5. f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 (?,1) && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] 6. f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] (?,1) Signature: {(f0,4);(f19,4);(f4,4);(f7,4)} Flow Graph: [0->{1,6},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6},6->{}] + Applied Processor: FromIts + Details: () * Step 2: Decompose YES + Considered Problem: Rules: f0(A,B,C,D) -> f4(0,B,C,D) True f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] Signature: {(f0,4);(f19,4);(f4,4);(f7,4)} Rule Graph: [0->{1,6},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6},6->{}] + Applied Processor: Decompose NoGreedy + Details: We construct a looptree: P: [0,1,2,3,4,5,6] | `- p:[1,5,2,3,4] c: [1,5] | `- p:[2,3,4] c: [4] | `- p:[2,3] c: [3] | `- p:[2] c: [2] * Step 3: CloseWith YES + Considered Problem: (Rules: f0(A,B,C,D) -> f4(0,B,C,D) True f4(A,B,C,D) -> f7(A,B,1 + A,D) [A >= 0 && B >= 1 + A] f7(A,B,C,D) -> f7(A,B,1 + C,0) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C] f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && 0 >= 1 + E] f7(A,B,C,D) -> f7(A,-1 + B,C,E) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && B >= 1 + C && E >= 1] f7(A,B,C,D) -> f4(1 + A,B,C,D) [B + -1*C >= 0 && -1 + C >= 0 && -2 + B + C >= 0 && -1 + A + C >= 0 && -1 + -1*A + C >= 0 && -1 + B >= 0 && -1 + A + B >= 0 && -1 + -1*A + B >= 0 && A >= 0 && C >= B] f4(A,B,C,D) -> f19(A,B,C,D) [A >= 0 && A >= B] Signature: {(f0,4);(f19,4);(f4,4);(f7,4)} Rule Graph: [0->{1,6},1->{2,3,4,5},2->{2,3,4,5},3->{2,3,4,5},4->{2,3,4,5},5->{1,6},6->{}] ,We construct a looptree: P: [0,1,2,3,4,5,6] | `- p:[1,5,2,3,4] c: [1,5] | `- p:[2,3,4] c: [4] | `- p:[2,3] c: [3] | `- p:[2] c: [2]) + Applied Processor: CloseWith True + Details: () YES