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: Matrix Interpretation Processor: dim=1 interpretation: [a__plus](x0, x1) = x0 + x1, [U12](x0, x1, x2) = x0 + x1 + x2, [a__U11](x0, x1, x2) = 4x0 + x1 + x2, [mark](x0) = x0, [s](x0) = x0, [tt] = 0, [U11](x0, x1, x2) = 4x0 + x1 + x2, [plus](x0, x1) = x0 + x1, [0] = 2, [a__U12](x0, x1, x2) = x0 + x1 + x2 orientation: 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 + 2 >= N = mark(N) a__plus(N,s(M)) = M + N >= M + N = a__U11(tt(),M,N) mark(U11(X1,X2,X3)) = 4X1 + X2 + X3 >= 4X1 + 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()) = 2 >= 2 = 0() a__U11(X1,X2,X3) = 4X1 + X2 + X3 >= 4X1 + 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: 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,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: dim=3 interpretation: [1 0 0] [1 1 0] [a__plus](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 0] [1 0 0] [0] [U12](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 0] [1 0 0] [1] [a__U11](x0, x1, x2) = [0 0 1]x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 0 1] [0 0 0] [0 0 0] [0], [1 0 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [s](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [0] [tt] = [0] [0], [1 0 0] [1 1 0] [1 0 0] [1] [U11](x0, x1, x2) = [0 0 1]x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0], [1 0 0] [1 1 0] [plus](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 0 1] [1 1 0] [1 0 0] [0] [a__U12](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 0 0] [0 0 0] [0 0 0] [0] orientation: [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] a__U11(tt(),M,N) = [0 1 0]M + [0 0 0]N + [1] >= [0 1 0]M + [0 0 0]N + [1] = a__U12(tt(),M,N) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] a__U12(tt(),M,N) = [0 1 0]M + [0 0 0]N + [1] >= [0 1 0]M + [0 0 0]N + [1] = s(a__plus(mark(N),mark(M))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] a__plus(N,s(M)) = [0 1 0]M + [0 0 0]N + [1] >= [0 1 0]M + [0 0 0]N + [1] = a__U11(tt(),M,N) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 1 0] [1 0 0] [1] mark(U11(X1,X2,X3)) = [0 0 1]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = a__U11(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [0] mark(U12(X1,X2,X3)) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = a__U12(mark(X1),X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] mark(plus(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = a__plus(mark(X1),mark(X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(tt()) = [0] >= [0] = tt() [0] [0] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(mark(X)) [0 0 0] [0] [0 0 0] [0] [0] [0] mark(0()) = [0] >= [0] = 0() [0] [0] [1 0 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 1 0] [1 0 0] [1] a__U11(X1,X2,X3) = [0 0 1]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= [0 0 1]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = U11(X1,X2,X3) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 1] [1 1 0] [1 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [0] a__U12(X1,X2,X3) = [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = U12(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] a__plus(X1,X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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: dim=1 interpretation: [a__plus](x0, x1) = x0 + x1, [U12](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 1, [a__U11](x0, x1, x2) = x0 + x1 + x2, [mark](x0) = x0, [s](x0) = 4x0 + 1, [tt] = 0, [U11](x0, x1, x2) = x0 + x1 + x2, [plus](x0, x1) = x0 + x1, [0] = 2, [a__U12](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 1 orientation: a__U12(tt(),M,N) = 4M + 4N + 1 >= 4M + 4N + 1 = s(a__plus(mark(N),mark(M))) a__plus(N,s(M)) = 4M + N + 1 >= 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)) = 4X1 + 4X2 + 4X3 + 1 >= 4X1 + 4X2 + 4X3 + 1 = 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)) = 4X + 1 >= 4X + 1 = s(mark(X)) mark(0()) = 2 >= 2 = 0() a__U11(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 4X1 + 4X2 + 4X3 + 1 >= 4X1 + 4X2 + 4X3 + 1 = U12(X1,X2,X3) a__plus(X1,X2) = X1 + X2 >= X1 + X2 = plus(X1,X2) problem: a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) 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: dim=1 interpretation: [a__plus](x0, x1) = x0 + x1 + 3, [U12](x0, x1, x2) = 4x0 + x1 + 2x2 + 3, [a__U11](x0, x1, x2) = 7x0 + 4x1 + 4x2 + 6, [mark](x0) = 4x0 + 3, [s](x0) = x0 + 2, [tt] = 2, [U11](x0, x1, x2) = 7x0 + x1 + x2 + 6, [plus](x0, x1) = x0 + x1 + 3, [0] = 0, [a__U12](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 3 orientation: a__U12(tt(),M,N) = 4M + 4N + 11 >= 4M + 4N + 11 = s(a__plus(mark(N),mark(M))) mark(U11(X1,X2,X3)) = 28X1 + 4X2 + 4X3 + 27 >= 28X1 + 4X2 + 4X3 + 27 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = 16X1 + 4X2 + 8X3 + 15 >= 16X1 + 4X2 + 4X3 + 15 = a__U12(mark(X1),X2,X3) mark(plus(X1,X2)) = 4X1 + 4X2 + 15 >= 4X1 + 4X2 + 9 = a__plus(mark(X1),mark(X2)) mark(tt()) = 11 >= 2 = tt() mark(s(X)) = 4X + 11 >= 4X + 5 = s(mark(X)) mark(0()) = 3 >= 0 = 0() a__U11(X1,X2,X3) = 7X1 + 4X2 + 4X3 + 6 >= 7X1 + X2 + X3 + 6 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 4X1 + 4X2 + 4X3 + 3 >= 4X1 + X2 + 2X3 + 3 = U12(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 3 >= X1 + X2 + 3 = plus(X1,X2) problem: a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) 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: dim=1 interpretation: [a__plus](x0, x1) = x0 + x1 + 7, [U12](x0, x1, x2) = x0 + x1 + x2 + 4, [a__U11](x0, x1, x2) = x0 + x1 + x2, [mark](x0) = x0, [s](x0) = x0, [tt] = 3, [U11](x0, x1, x2) = x0 + x1 + x2, [plus](x0, x1) = x0 + x1, [a__U12](x0, x1, x2) = x0 + x1 + x2 + 4 orientation: a__U12(tt(),M,N) = M + N + 7 >= M + N + 7 = s(a__plus(mark(N),mark(M))) 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 + 4 >= X1 + X2 + X3 + 4 = a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = U12(X1,X2,X3) a__plus(X1,X2) = X1 + X2 + 7 >= X1 + X2 = plus(X1,X2) problem: a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M))) mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [a__plus](x0, x1) = x0 + 4x1, [U12](x0, x1, x2) = 2x0 + 4x1 + x2, [a__U11](x0, x1, x2) = 2x0 + x1 + 2x2, [mark](x0) = x0, [s](x0) = x0, [tt] = 1, [U11](x0, x1, x2) = 2x0 + x1 + 2x2, [a__U12](x0, x1, x2) = 2x0 + 4x1 + x2 orientation: a__U12(tt(),M,N) = 4M + N + 2 >= 4M + N = s(a__plus(mark(N),mark(M))) mark(U11(X1,X2,X3)) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) = 2X1 + 4X2 + X3 >= 2X1 + 4X2 + X3 = a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 2X1 + 4X2 + X3 >= 2X1 + 4X2 + X3 = U12(X1,X2,X3) problem: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3) mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1 0 0] [U12](x0, x1, x2) = [0 1 1]x0 + [0 0 1]x1 + [0 0 1]x2 [0 1 1] [0 0 1] [0 0 1] , [1 1 0] [1 0 0] [1 0 0] [0] [a__U11](x0, x1, x2) = [0 0 1]x0 + [0 0 1]x1 + [0 0 0]x2 + [1] [0 1 0] [0 0 1] [0 0 0] [1], [1 1 0] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 1 0] [1 0 0] [1 0 0] [0] [U11](x0, x1, x2) = [0 0 1]x0 + [0 0 1]x1 + [0 0 0]x2 + [1] [0 1 0] [0 0 1] [0 0 0] [1], [1 1 0] [1 0 0] [1 0 1] [a__U12](x0, x1, x2) = [0 1 1]x0 + [0 0 1]x1 + [0 0 1]x2 [0 1 1] [0 0 1] [0 0 1] orientation: [1 1 1] [1 0 1] [1 0 0] [1] [1 1 1] [1 0 0] [1 0 0] [0] mark(U11(X1,X2,X3)) = [0 1 0]X1 + [0 0 1]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 1]X2 + [0 0 0]X3 + [1] = a__U11(mark(X1),X2,X3) [0 0 1] [0 0 1] [0 0 0] [1] [0 0 1] [0 0 1] [0 0 0] [1] [1 1 1] [1 0 1] [1 0 1] [1 1 1] [1 0 0] [1 0 1] mark(U12(X1,X2,X3)) = [0 1 1]X1 + [0 0 1]X2 + [0 0 1]X3 >= [0 1 1]X1 + [0 0 1]X2 + [0 0 1]X3 = a__U12(mark(X1),X2,X3) [0 1 1] [0 0 1] [0 0 1] [0 1 1] [0 0 1] [0 0 1] [1 1 0] [1 0 0] [1 0 0] [0] [1 1 0] [1 0 0] [1 0 0] [0] a__U11(X1,X2,X3) = [0 0 1]X1 + [0 0 1]X2 + [0 0 0]X3 + [1] >= [0 0 1]X1 + [0 0 1]X2 + [0 0 0]X3 + [1] = U11(X1,X2,X3) [0 1 0] [0 0 1] [0 0 0] [1] [0 1 0] [0 0 1] [0 0 0] [1] [1 1 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] a__U12(X1,X2,X3) = [0 1 1]X1 + [0 0 1]X2 + [0 0 1]X3 >= [0 1 1]X1 + [0 0 1]X2 + [0 0 1]X3 = U12(X1,X2,X3) [0 1 1] [0 0 1] [0 0 1] [0 1 1] [0 0 1] [0 0 1] problem: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) -> U11(X1,X2,X3) a__U12(X1,X2,X3) -> U12(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [U12](x0, x1, x2) = 2x0 + 2x1 + 2x2 + 4, [a__U11](x0, x1, x2) = x0 + x1 + x2, [mark](x0) = 2x0 + 1, [U11](x0, x1, x2) = x0 + x1 + x2, [a__U12](x0, x1, x2) = 2x0 + 4x1 + 4x2 + 7 orientation: mark(U12(X1,X2,X3)) = 4X1 + 4X2 + 4X3 + 9 >= 4X1 + 4X2 + 4X3 + 9 = a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = U11(X1,X2,X3) a__U12(X1,X2,X3) = 2X1 + 4X2 + 4X3 + 7 >= 2X1 + 2X2 + 2X3 + 4 = U12(X1,X2,X3) problem: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) -> U11(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [U12](x0, x1, x2) = x0 + x1 + x2, [a__U11](x0, x1, x2) = x0 + x1 + x2 + 1, [mark](x0) = x0 + 4, [U11](x0, x1, x2) = x0 + x1 + x2, [a__U12](x0, x1, x2) = x0 + x1 + x2 orientation: mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 4 >= X1 + X2 + X3 + 4 = a__U12(mark(X1),X2,X3) a__U11(X1,X2,X3) = X1 + X2 + X3 + 1 >= X1 + X2 + X3 = U11(X1,X2,X3) problem: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [U12](x0, x1, x2) = x0 + x1 + x2 + 1, [mark](x0) = x0 + 2, [a__U12](x0, x1, x2) = x0 + x1 + x2 orientation: mark(U12(X1,X2,X3)) = X1 + X2 + X3 + 3 >= X1 + X2 + X3 + 2 = a__U12(mark(X1),X2,X3) problem: Qed