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: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 1 0] [0] [a__plus](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [1 0 0] [a__and](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 1] [0 0 0] , [1] [0] = [0] [1], [0] [s](x0) = x0 + [1] [0], [0] [tt] = [1] [1], [1 0 0] [1 1 0] [0] [plus](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 1 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 1 0]x1 [0 0 1] [0 0 0] , [1 0 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 0] [1] orientation: [1 0 0] [1] [1 0 0] [0] a__and(tt(),X) = [0 1 0]X + [0] >= [0 1 0]X + [0] = mark(X) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [0] a__plus(N,0()) = [0 1 0]N + [0] >= [0 1 0]N + [0] = mark(N) [0 0 0] [1] [0 0 0] [1] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] a__plus(N,s(M)) = [0 1 0]M + [0 1 0]N + [1] >= [0 1 0]M + [0 1 0]N + [1] = s(a__plus(mark(N),mark(M))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 1 0] [1 0 0] [0] mark(and(X1,X2)) = [0 0 0]X1 + [0 1 0]X2 + [0] >= [0 0 0]X1 + [0 1 0]X2 + [0] = a__and(mark(X1),X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] mark(plus(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = a__plus(mark(X1),mark(X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [0] [0] mark(tt()) = [1] >= [1] = tt() [1] [1] [1] [1] mark(0()) = [0] >= [0] = 0() [1] [1] [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] [1] [0 0 0] [1] [1 1 0] [1 0 0] [1 1 0] [1 0 0] a__and(X1,X2) = [0 0 0]X1 + [0 1 0]X2 >= [0 0 0]X1 + [0 1 0]X2 = and(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] a__plus(X1,X2) = [0 1 0]X1 + [0 1 0]X2 + [0] >= [0 1 0]X1 + [0 1 0]X2 + [0] = plus(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] problem: 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: dim=1 interpretation: [a__plus](x0, x1) = 3x0 + 4x1, [a__and](x0, x1) = x0 + 6x1, [0] = 1, [s](x0) = 3x0, [tt] = 0, [plus](x0, x1) = 3x0 + 4x1, [and](x0, x1) = x0 + 2x1, [mark](x0) = 3x0 orientation: mark(and(X1,X2)) = 3X1 + 6X2 >= 3X1 + 6X2 = a__and(mark(X1),X2) mark(plus(X1,X2)) = 9X1 + 12X2 >= 9X1 + 12X2 = a__plus(mark(X1),mark(X2)) mark(tt()) = 0 >= 0 = tt() mark(0()) = 3 >= 1 = 0() mark(s(X)) = 9X >= 9X = s(mark(X)) a__and(X1,X2) = X1 + 6X2 >= X1 + 2X2 = and(X1,X2) a__plus(X1,X2) = 3X1 + 4X2 >= 3X1 + 4X2 = plus(X1,X2) problem: mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__plus](x0, x1) = 2x0 + 2x1 + 5, [a__and](x0, x1) = x0 + 4x1, [s](x0) = 2x0 + 4, [tt] = 0, [plus](x0, x1) = 2x0 + 2x1 + 5, [and](x0, x1) = x0 + x1, [mark](x0) = 4x0 + 5 orientation: mark(and(X1,X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = a__and(mark(X1),X2) mark(plus(X1,X2)) = 8X1 + 8X2 + 25 >= 8X1 + 8X2 + 25 = a__plus(mark(X1),mark(X2)) mark(tt()) = 5 >= 0 = tt() mark(s(X)) = 8X + 21 >= 8X + 14 = s(mark(X)) a__and(X1,X2) = X1 + 4X2 >= X1 + X2 = and(X1,X2) a__plus(X1,X2) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = plus(X1,X2) problem: mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) a__and(X1,X2) -> and(X1,X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [a__plus](x0, x1) = 3x0 + 4x1 + 4, [a__and](x0, x1) = 4x0 + 7x1 + 6, [plus](x0, x1) = 3x0 + 4x1 + 4, [and](x0, x1) = 4x0 + 7x1 + 3, [mark](x0) = 3x0 + 1 orientation: mark(and(X1,X2)) = 12X1 + 21X2 + 10 >= 12X1 + 7X2 + 10 = a__and(mark(X1),X2) mark(plus(X1,X2)) = 9X1 + 12X2 + 13 >= 9X1 + 12X2 + 11 = a__plus(mark(X1),mark(X2)) a__and(X1,X2) = 4X1 + 7X2 + 6 >= 4X1 + 7X2 + 3 = and(X1,X2) a__plus(X1,X2) = 3X1 + 4X2 + 4 >= 3X1 + 4X2 + 4 = plus(X1,X2) problem: mark(and(X1,X2)) -> a__and(mark(X1),X2) a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [a__plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [a__and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [mark](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = a__and(mark(X1),X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] a__plus(X1,X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: a__plus(X1,X2) -> plus(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1] [a__plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [0], [1 0 0] [1 0 0] [plus](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] a__plus(X1,X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = plus(X1,X2) [0 0 0] [0 0 1] [0] [0 0 0] [0 0 0] problem: Qed