YES Problem: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Proof: DP Processor: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) EDG Processor: DPs: a__and#(tt(),X) -> mark#(X) a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) 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__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) graph: a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,0()) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> mark#(M) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> mark#(N) a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) -> a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(M) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(M) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(M) -> mark#(s(X)) -> mark#(X) a__plus#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,s(M)) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__plus#(N,0()) -> mark#(N) -> mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X2) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,0()) -> mark#(N) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,0()) -> mark#(N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> mark#(M) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> mark#(N) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) -> a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(tt(),X) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X2) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> mark#(X1) a__and#(tt(),X) -> mark#(X) -> mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__and#(tt(),X) -> mark#(X) -> mark#(s(X)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 11 #arcs: 57/121 DPs: a__plus#(N,s(M)) -> a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) -> mark#(N) mark#(s(X)) -> mark#(X) mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) -> mark#(M) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(tt(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) a__plus#(N,0()) -> mark#(N) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) 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__and#](x0, x1) = x0 + x1 + 1, [plus](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1, [s](x0) = x0 + 1, [a__plus](x0, x1) = x0 + x1, [0] = 1, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [tt] = 1 orientation: a__plus#(N,s(M)) = M + N + 2 >= M + N + 1 = a__plus#(mark(N),mark(M)) a__plus#(N,s(M)) = M + N + 2 >= N + 1 = mark#(N) mark#(s(X)) = X + 2 >= X + 1 = mark#(X) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus#(mark(X1),mark(X2)) a__plus#(N,s(M)) = M + N + 2 >= M + 1 = mark#(M) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 + 1 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and#(mark(X1),X2) a__and#(tt(),X) = X + 2 >= X + 1 = mark#(X) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = mark#(X1) a__plus#(N,0()) = N + 2 >= N + 1 = mark#(N) a__and(tt(),X) = X + 1 >= X = mark(X) a__plus(N,0()) = N + 1 >= N = mark(N) a__plus(N,s(M)) = M + N + 1 >= M + N + 1 = s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 1 >= 1 = tt() mark(0()) = 1 >= 1 = 0() mark(s(X)) = X + 1 >= X + 1 = s(mark(X)) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = 1, [mark#](x0) = 1, [a__and#](x0, x1) = 0, [plus](x0, x1) = 0, [and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0 orientation: mark#(plus(X1,X2)) = 1 >= 1 = a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) = 1 >= 1 = mark#(X1) mark#(plus(X1,X2)) = 1 >= 1 = mark#(X2) mark#(and(X1,X2)) = 1 >= 0 = a__and#(mark(X1),X2) mark#(and(X1,X2)) = 1 >= 1 = mark#(X1) a__and(tt(),X) = 0 >= 0 = mark(X) a__plus(N,0()) = 0 >= 0 = mark(N) a__plus(N,s(M)) = 0 >= 0 = s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(mark(X)) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__plus(X1,X2) = 0 >= 0 = plus(X1,X2) problem: DPs: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = 0, [mark#](x0) = x0, [plus](x0, x1) = x0 + x1, [and](x0, x1) = x0 + x1 + 1, [s](x0) = x0, [a__plus](x0, x1) = x0 + x1, [0] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1 + 1, [tt] = 0 orientation: mark#(plus(X1,X2)) = X1 + X2 >= 0 = a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(and(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) a__and(tt(),X) = X + 1 >= X = mark(X) a__plus(N,0()) = N >= N = mark(N) a__plus(N,s(M)) = M + N >= M + N = s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 >= X1 + X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) a__and(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = and(X1,X2) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: DPs: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) -> mark#(X1) mark#(plus(X1,X2)) -> mark#(X2) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = x0 + 1, [mark#](x0) = x0, [plus](x0, x1) = x0 + x1 + 1, [and](x0, x1) = x1, [s](x0) = 0, [a__plus](x0, x1) = x0 + x1 + 1, [0] = 1, [mark](x0) = x0, [a__and](x0, x1) = x1, [tt] = 0 orientation: mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 + 1 = a__plus#(mark(X1),mark(X2)) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X1 = mark#(X1) mark#(plus(X1,X2)) = X1 + X2 + 1 >= X2 = mark#(X2) a__and(tt(),X) = X >= X = mark(X) a__plus(N,0()) = N + 2 >= N = mark(N) a__plus(N,s(M)) = N + 1 >= 0 = s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) = X2 >= X2 = a__and(mark(X1),X2) mark(plus(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 1 >= 1 = 0() mark(s(X)) = 0 >= 0 = s(mark(X)) a__and(X1,X2) = X2 >= X2 = and(X1,X2) a__plus(X1,X2) = X1 + X2 + 1 >= X1 + X2 + 1 = plus(X1,X2) problem: DPs: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dimension: 1 interpretation: [a__plus#](x0, x1) = 0, [mark#](x0) = 1, [plus](x0, x1) = 0, [and](x0, x1) = 0, [s](x0) = 0, [a__plus](x0, x1) = 0, [0] = 0, [mark](x0) = 0, [a__and](x0, x1) = 0, [tt] = 0 orientation: mark#(plus(X1,X2)) = 1 >= 0 = a__plus#(mark(X1),mark(X2)) a__and(tt(),X) = 0 >= 0 = mark(X) a__plus(N,0()) = 0 >= 0 = mark(N) a__plus(N,s(M)) = 0 >= 0 = s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) = 0 >= 0 = a__and(mark(X1),X2) mark(plus(X1,X2)) = 0 >= 0 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 0 >= 0 = 0() mark(s(X)) = 0 >= 0 = s(mark(X)) a__and(X1,X2) = 0 >= 0 = and(X1,X2) a__plus(X1,X2) = 0 >= 0 = plus(X1,X2) problem: DPs: TRS: a__and(tt(),X) -> mark(X) a__plus(N,0()) -> mark(N) a__plus(N,s(M)) -> s(a__plus(mark(N),mark(M))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Qed