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=1 interpretation: [f](x0) = 4x0, [mark](x0) = x0, [true] = 0, [a__if](x0, x1, x2) = 4x0 + x1 + x2, [a__f](x0) = 4x0, [if](x0, x1, x2) = 4x0 + x1 + x2, [false] = 4, [c] = 0 orientation: a__f(X) = 4X >= 4X = a__if(mark(X),c(),f(true())) a__if(true(),X,Y) = X + Y >= X = mark(X) a__if(false(),X,Y) = X + Y + 16 >= Y = mark(Y) mark(f(X)) = 4X >= 4X = a__f(mark(X)) mark(if(X1,X2,X3)) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = a__if(mark(X1),mark(X2),X3) mark(c()) = 0 >= 0 = c() mark(true()) = 0 >= 0 = true() mark(false()) = 4 >= 4 = false() a__f(X) = 4X >= 4X = f(X) a__if(X1,X2,X3) = 4X1 + X2 + X3 >= 4X1 + X2 + X3 = if(X1,X2,X3) 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=1 interpretation: [f](x0) = 4x0, [mark](x0) = 4x0, [true] = 0, [a__if](x0, x1, x2) = x0 + 5x1 + 4x2, [a__f](x0) = 4x0, [if](x0, x1, x2) = x0 + 5x1 + x2, [false] = 4, [c] = 0 orientation: a__f(X) = 4X >= 4X = a__if(mark(X),c(),f(true())) a__if(true(),X,Y) = 5X + 4Y >= 4X = mark(X) mark(f(X)) = 16X >= 16X = a__f(mark(X)) mark(if(X1,X2,X3)) = 4X1 + 20X2 + 4X3 >= 4X1 + 20X2 + 4X3 = a__if(mark(X1),mark(X2),X3) mark(c()) = 0 >= 0 = c() mark(true()) = 0 >= 0 = true() mark(false()) = 16 >= 4 = false() a__f(X) = 4X >= 4X = f(X) a__if(X1,X2,X3) = X1 + 5X2 + 4X3 >= X1 + 5X2 + X3 = if(X1,X2,X3) 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() a__f(X) -> f(X) a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [f](x0) = [0 1 0]x0 + [1] [0 1 0] [0], [1 0 1] [mark](x0) = [0 1 0]x0 [0 1 0] , [0] [true] = [0] [0], [1 0 1] [1 0 0] [0] [a__if](x0, x1, x2) = x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 1 0] [0 0 0] [1], [1 0 1] [0] [a__f](x0) = [0 1 0]x0 + [1] [0 1 0] [1], [1 0 1] [1 0 0] [0] [if](x0, x1, x2) = x0 + [0 1 0]x1 + [0 0 0]x2 + [1] [0 1 0] [0 0 0] [1], [0] [c] = [0] [0] orientation: [1 0 1] [0] [1 0 1] [0] a__f(X) = [0 1 0]X + [1] >= [0 1 0]X + [1] = a__if(mark(X),c(),f(true())) [0 1 0] [1] [0 1 0] [1] [1 0 1] [1 0 0] [0] [1 0 1] a__if(true(),X,Y) = [0 1 0]X + [0 0 0]Y + [1] >= [0 1 0]X = mark(X) [0 1 0] [0 0 0] [1] [0 1 0] [1 1 1] [0] [1 1 1] [0] 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] [1 0 1] [1 1 1] [1 0 0] [0] mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = a__if(mark(X1),mark(X2),X3) [0 1 0] [0 1 0] [0 0 0] [1] [0 1 0] [0 1 0] [0 0 0] [1] [0] [0] mark(c()) = [0] >= [0] = c() [0] [0] [0] [0] mark(true()) = [0] >= [0] = true() [0] [0] [1 0 1] [0] [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] [0] [1 0 1] [1 0 0] [0] [1 0 1] [1 0 0] [0] a__if(X1,X2,X3) = X1 + [0 1 0]X2 + [0 0 0]X3 + [1] >= X1 + [0 1 0]X2 + [0 0 0]X3 + [1] = if(X1,X2,X3) [0 1 0] [0 0 0] [1] [0 1 0] [0 0 0] [1] 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(c()) -> c() mark(true()) -> true() a__f(X) -> f(X) a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=1 interpretation: [f](x0) = 2x0 + 2, [mark](x0) = 2x0, [true] = 0, [a__if](x0, x1, x2) = x0 + 2x1 + x2, [a__f](x0) = 2x0 + 3, [if](x0, x1, x2) = x0 + x1 + x2, [c] = 0 orientation: a__f(X) = 2X + 3 >= 2X + 2 = a__if(mark(X),c(),f(true())) a__if(true(),X,Y) = 2X + Y >= 2X = mark(X) mark(f(X)) = 4X + 4 >= 4X + 3 = a__f(mark(X)) mark(c()) = 0 >= 0 = c() mark(true()) = 0 >= 0 = true() a__f(X) = 2X + 3 >= 2X + 2 = f(X) a__if(X1,X2,X3) = X1 + 2X2 + X3 >= X1 + X2 + X3 = if(X1,X2,X3) problem: a__if(true(),X,Y) -> mark(X) mark(c()) -> c() mark(true()) -> true() a__if(X1,X2,X3) -> if(X1,X2,X3) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 1] , [0] [true] = [0] [1], [1 0 1] [1 0 1] [1 0 0] [a__if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [0 0 0] , [1 0 1] [1 0 1] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 1] [0 0 0] , [0] [c] = [0] [1] orientation: [1 0 1] [1 0 0] [1] [1 0 1] a__if(true(),X,Y) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X = mark(X) [0 0 1] [0 0 0] [0] [0 0 1] [1] [0] mark(c()) = [0] >= [0] = c() [1] [1] [1] [0] mark(true()) = [0] >= [0] = true() [1] [1] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] a__if(X1,X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 1] [0 0 0] [0 0 0] [0 0 1] [0 0 0] 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] [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], [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] 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