YES Problem: a__f(X) -> a__if(mark(X),c(),f(true())) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) mark(f(X)) -> a__f(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3) mark(c()) -> c() mark(true()) -> true() mark(false()) -> false() a__f(X) -> f(X) a__if(X1,X2,X3) -> if(X1,X2,X3) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 1 1]x1 + [0 1 1]x2 [0 0 0] [0 0 0] [0 0 0] , [1] [false] = [0] [0], [1 1 1] [1 0 1] [1 0 0] [a__if](x0, x1, x2) = [0 0 0]x0 + [0 1 1]x1 + [0 1 1]x2 [0 0 0] [0 0 0] [0 0 0] , [1 1 1] [0] [f](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [0] [true] = [0] [0], [0] [c] = [0] [0], [1 0 0] [mark](x0) = [0 1 1]x0 [0 0 0] , [1 1 1] [0] [a__f](x0) = [0 0 0]x0 + [1] [0 0 1] [0] orientation: [1 1 1] [0] [1 1 1] [0] a__f(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__if(mark(X),c(),f(true())) [0 0 1] [0] [0 0 0] [0] [1 0 1] [1 0 0] [1 0 0] a__if(true(),X,Y) = [0 1 1]X + [0 1 1]Y >= [0 1 1]X = mark(X) [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 0] a__if(false(),X,Y) = [0 1 1]X + [0 1 1]Y + [0] >= [0 1 1]Y = mark(Y) [0 0 0] [0 0 0] [0] [0 0 0] [1 1 1] [0] [1 1 1] [0] mark(f(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__f(mark(X)) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 1 1]X2 + [0 1 1]X3 >= [0 0 0]X1 + [0 1 1]X2 + [0 1 1]X3 = a__if(mark(X1),mark(X2),X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(c()) = [0] >= [0] = c() [0] [0] [0] [0] mark(true()) = [0] >= [0] = true() [0] [0] [1] [1] mark(false()) = [0] >= [0] = false() [0] [0] [1 1 1] [0] [1 1 1] [0] a__f(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = f(X) [0 0 1] [0] [0 0 0] [0] [1 1 1] [1 0 1] [1 0 0] [1 1 1] [1 0 0] [1 0 0] a__if(X1,X2,X3) = [0 0 0]X1 + [0 1 1]X2 + [0 1 1]X3 >= [0 0 0]X1 + [0 1 1]X2 + [0 1 1]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: a__f(X) -> a__if(mark(X),c(),f(true())) a__if(true(),X,Y) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3) mark(c()) -> c() mark(true()) -> true() mark(false()) -> false() a__f(X) -> f(X) a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [1 0 0] [if](x0, x1, x2) = x0 + [0 1 0]x1 + [0 0 0]x2 [0 1 0] [0 0 0] , [0] [false] = [0] [0], [1 0 1] [1 0 0] [a__if](x0, x1, x2) = x0 + [0 1 0]x1 + [0 0 0]x2 [0 1 0] [0 0 0] , [1 0 1] [0] [f](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [0] [true] = [0] [0], [0] [c] = [0] [0], [1 0 1] [mark](x0) = [0 1 0]x0 [0 1 0] , [1 0 1] [1] [a__f](x0) = [0 1 0]x0 + [1] [0 1 0] [1] orientation: [1 0 1] [1] [1 0 1] a__f(X) = [0 1 0]X + [1] >= [0 1 0]X = a__if(mark(X),c(),f(true())) [0 1 0] [1] [0 1 0] [1 0 1] [1 0 0] [1 0 1] a__if(true(),X,Y) = [0 1 0]X + [0 0 0]Y >= [0 1 0]X = mark(X) [0 1 0] [0 0 0] [0 1 0] [1 1 1] [1] [1 1 1] [1] mark(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__f(mark(X)) [0 1 0] [1] [0 1 0] [1] [1 0 1] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 0] mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 = a__if(mark(X1),mark(X2),X3) [0 1 0] [0 1 0] [0 0 0] [0 1 0] [0 1 0] [0 0 0] [0] [0] mark(c()) = [0] >= [0] = c() [0] [0] [0] [0] mark(true()) = [0] >= [0] = true() [0] [0] [0] [0] mark(false()) = [0] >= [0] = false() [0] [0] [1 0 1] [1] [1 0 1] [0] a__f(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(X) [0 1 0] [1] [0 1 0] [1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] a__if(X1,X2,X3) = X1 + [0 1 0]X2 + [0 0 0]X3 >= X1 + [0 1 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 1 0] [0 0 0] [0 1 0] [0 0 0] problem: a__if(true(),X,Y) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3) mark(c()) -> c() mark(true()) -> true() mark(false()) -> false() a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [if](x0, x1, x2) = x0 + 2x1 + 2x2 + 4, [false] = 2, [a__if](x0, x1, x2) = x0 + 2x1 + 4x2 + 4, [f](x0) = x0, [true] = 4, [c] = 0, [mark](x0) = 2x0 + 2, [a__f](x0) = x0 orientation: a__if(true(),X,Y) = 2X + 4Y + 8 >= 2X + 2 = mark(X) mark(f(X)) = 2X + 2 >= 2X + 2 = a__f(mark(X)) mark(if(X1,X2,X3)) = 2X1 + 4X2 + 4X3 + 10 >= 2X1 + 4X2 + 4X3 + 10 = a__if(mark(X1),mark(X2),X3) mark(c()) = 2 >= 0 = c() mark(true()) = 10 >= 4 = true() mark(false()) = 6 >= 2 = false() a__if(X1,X2,X3) = X1 + 2X2 + 4X3 + 4 >= X1 + 2X2 + 2X3 + 4 = if(X1,X2,X3) problem: mark(f(X)) -> a__f(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3) a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1 0 0] [0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 1 1] [0 0 0] [1], [1 0 0] [1 0 0] [1 0 0] [0] [a__if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 1] [0 1 1] [0 0 0] [1], [1 0 1] [0] [f](x0) = [1 0 1]x0 + [1] [0 0 0] [1], [1 0 1] [mark](x0) = [0 1 0]x0 [0 0 1] , [1 0 0] [0] [a__f](x0) = [1 0 0]x0 + [1] [0 0 0] [1] orientation: [1 0 1] [1] [1 0 1] [0] mark(f(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = a__f(mark(X)) [0 0 0] [1] [0 0 0] [1] [1 0 1] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 1] [1 0 0] [0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = a__if(mark(X1),mark(X2),X3) [0 0 1] [0 1 1] [0 0 0] [1] [0 0 1] [0 1 1] [0 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] a__if(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 1] [0 1 1] [0 0 0] [1] [0 0 1] [0 1 1] [0 0 0] [1] problem: a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1 0 0] [1] [a__if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 1] [0] orientation: [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] a__if(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 1] [0] [0 0 0] [0 0 0] [0 0 0] problem: Qed