YES Problem: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Proof: DP Processor: DPs: a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) a__U12#(tt(),M,N) -> mark#(M) a__U12#(tt(),M,N) -> mark#(N) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) TDG Processor: DPs: a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) a__U12#(tt(),M,N) -> mark#(M) a__U12#(tt(),M,N) -> mark#(N) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) graph: a__plus#(N,0()) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,0()) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) a__plus#(N,0()) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__plus#(N,0()) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) -> a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__U11#(tt(),M,N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U12(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) -> a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) -> a__U12#(tt(),M,N) -> mark#(N) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) -> a__U12#(tt(),M,N) -> mark#(M) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(U11(X1,X2,X3)) -> mark#(X1) -> mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) -> a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__U11#(tt(),M,N) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> mark#(N) a__U12#(tt(),M,N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__U12#(tt(),M,N) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__U12#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(N) -> mark#(U12(X1,X2,X3)) -> mark#(X1) a__U12#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(N) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U12#(tt(),M,N) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__U12#(tt(),M,N) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__U12#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(M) -> mark#(U12(X1,X2,X3)) -> mark#(X1) a__U12#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(M) -> mark#(U11(X1,X2,X3)) -> mark#(X1) a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__U12#(tt(),M,N) -> mark#(N) a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) -> a__U12#(tt(),M,N) -> mark#(M) SCC Processor: #sccs: 1 #rules: 14 #arcs: 76/196 DPs: a__plus#(N,0()) -> mark#(N) mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) a__U12#(tt(),M,N) -> mark#(M) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) mark#(s(X)) -> mark#(X) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = x0 + x1, [mark#](x0) = x0, [a__U12#](x0, x1, x2) = x1 + x2, [a__U11#](x0, x1, x2) = x1 + x2, [plus](x0, x1) = x0 + x1, [U12](x0, x1, x2) = x0 + x1 + x2, [U11](x0, x1, x2) = x0 + x1 + x2, [0] = 1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1, [mark](x0) = x0, [a__U12](x0, x1, x2) = x0 + x1 + x2, [a__U11](x0, x1, x2) = x0 + x1 + x2, [tt] = 0 orientation: a__plus#(N,0()) = N + 1 >= N = mark#(N) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 >= X2 + X3 = a__U11#(mark(X1),X2,X3) a__U11#(tt(),M,N) = M + N >= M + N = a__U12#(tt(),M,N) a__U12#(tt(),M,N) = M + N >= M = mark#(M) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 >= X2 + X3 = a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) = M + N >= N = mark#(N) mark#(plus(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N >= M + N = a__U11#(tt(),M,N) mark#(s(X)) = X >= X = mark#(X) a__U12#(tt(),M,N) = M + N >= M + N = a__plus#(mark(N),mark(M)) a__U11(tt(),M,N) = M + N >= M + N = a__U12(tt(),M,N) a__U12(tt(),M,N) = M + N >= M + N = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = N + 1 >= N = mark(N) a__plus(N,s(M)) = M + N >= M + N = a__U11(tt(),M,N) mark(U11(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 1 >= 1 = 0() a__U11(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U12(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: mark#(U11(X1,X2,X3)) -> mark#(X1) mark#(U11(X1,X2,X3)) -> a__U11#(mark(X1),X2,X3) a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) a__U12#(tt(),M,N) -> mark#(M) mark#(U12(X1,X2,X3)) -> mark#(X1) mark#(U12(X1,X2,X3)) -> a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) -> mark#(N) mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) mark#(s(X)) -> mark#(X) a__U12#(tt(),M,N) -> a__plus#(mark(N),mark(M)) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = x0 + x1 + 1, [mark#](x0) = x0 + 1, [a__U12#](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U11#](x0, x1, x2) = x0 + x1 + x2 + 1, [plus](x0, x1) = x0 + x1 + 1, [U12](x0, x1, x2) = x0 + x1 + x2 + 1, [U11](x0, x1, x2) = x0 + x1 + x2 + 1, [0] = 0, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1 + 1, [mark](x0) = x0, [a__U12](x0, x1, x2) = x0 + x1 + x2 + 1, [a__U11](x0, x1, x2) = x0 + x1 + x2 + 1, [tt] = 1 orientation: mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1) mark#(U11(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 1 = a__U11#(mark(X1),X2,X3) a__U11#(tt(),M,N) = M + N + 2 >= M + N + 2 = a__U12#(tt(),M,N) a__U12#(tt(),M,N) = M + N + 2 >= M + 1 = mark#(M) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + 1 = mark#(X1) mark#(U12(X1,X2,X3)) = X1 + X2 + X3 + 2 >= X1 + X2 + X3 + 1 = a__U12#(mark(X1),X2,X3) a__U12#(tt(),M,N) = M + N + 2 >= N + 1 = mark#(N) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X2 + 1 = mark#(X2) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + 1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 1 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N + 2 >= M + N + 2 = a__U11#(tt(),M,N) mark#(s(X)) = X + 2 >= X + 1 = mark#(X) a__U12#(tt(),M,N) = M + N + 2 >= M + N + 1 = a__plus#(mark(N),mark(M)) a__U11(tt(),M,N) = M + N + 2 >= M + N + 2 = a__U12(tt(),M,N) a__U12(tt(),M,N) = M + N + 2 >= M + N + 2 = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = N + 1 >= N = mark(N) a__plus(N,s(M)) = M + N + 2 >= M + N + 2 = a__U11(tt(),M,N) mark(U11(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(tt()) = 1 >= 1 = tt() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U11(X1,X2,X3) a__U12(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 + 1 = U12(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) a__plus#(N,s(M)) -> a__U11#(tt(),M,N) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = 1, [a__U12#](x0, x1, x2) = 0, [a__U11#](x0, x1, x2) = 0, [plus](x0, x1) = 0, [U12](x0, x1, x2) = 0, [U11](x0, x1, x2) = 0, [0] = 0, [s](x0) = 0, [a__plus](x0, x1) = 0, [mark](x0) = 0, [a__U12](x0, x1, x2) = 0, [a__U11](x0, x1, x2) = 0, [tt] = 0 orientation: a__U11#(tt(),M,N) = 0 >= 0 = a__U12#(tt(),M,N) a__plus#(N,s(M)) = 1 >= 0 = a__U11#(tt(),M,N) a__U11(tt(),M,N) = 0 >= 0 = a__U12(tt(),M,N) a__U12(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = 0 >= 0 = mark(N) a__plus(N,s(M)) = 0 >= 0 = a__U11(tt(),M,N) mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3) a__plus(X1,X2) = 0 >= 0 = plus(X1,X2) problem: DPs: a__U11#(tt(),M,N) -> a__U12#(tt(),M,N) TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__U12#](x0, x1, x2) = 0, [a__U11#](x0, x1, x2) = 1, [plus](x0, x1) = 0, [U12](x0, x1, x2) = 0, [U11](x0, x1, x2) = 0, [0] = 0, [s](x0) = 0, [a__plus](x0, x1) = 0, [mark](x0) = 0, [a__U12](x0, x1, x2) = 0, [a__U11](x0, x1, x2) = 0, [tt] = 0 orientation: a__U11#(tt(),M,N) = 1 >= 0 = a__U12#(tt(),M,N) a__U11(tt(),M,N) = 0 >= 0 = a__U12(tt(),M,N) a__U12(tt(),M,N) = 0 >= 0 = s(a__plus(mark(N),mark(M))) a__plus(N,0()) = 0 >= 0 = mark(N) a__plus(N,s(M)) = 0 >= 0 = a__U11(tt(),M,N) mark(U11(X1,X2,X3)) = 0 >= 0 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = 0 >= 0 = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(s(X)) = 0 >= 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__U11(X1,X2,X3) = 0 >= 0 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 0 >= 0 = U12(X1,X2,X3) a__plus(X1,X2) = 0 >= 0 = plus(X1,X2) problem: DPs: TRS: a__U11(tt(),M,N) -> a__U12(tt(),M,N) a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> a__U11(tt(),M,N) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) Qed