YES Problem: g(A()) -> A() g(B()) -> A() g(B()) -> B() g(C()) -> A() g(C()) -> B() g(C()) -> C() foldf(x,nil()) -> x foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),A()) -> f''(foldf(triple(cons(A(),a),nil(),c),b)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [f''](x0) = [0 0 0]x0 + [1] [1 0 0] [0], [1 0 0] [1 0 0] [1 0 0] [1] [triple](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [0], [1 0 0] [1 0 0] [0] [f'](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [1 0 0] [0 0 0] [0], [1 0 0] [1 0 1] [0] [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [1 0 0] [0 0 0] [0], [1 0 1] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [0] [foldf](x0, x1) = [0 1 0]x0 + [0 0 0]x1 + [1] [1 0 1] [1 0 0] [0], [0] [nil] = [0] [0], [0] [C] = [0] [0], [0] [B] = [0] [1], [1 0 1] [0] [g](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [0] [A] = [0] [0] orientation: [0] [0] g(A()) = [1] >= [0] = A() [1] [0] [1] [0] g(B()) = [1] >= [0] = A() [1] [0] [1] [0] g(B()) = [1] >= [0] = B() [1] [1] [0] [0] g(C()) = [1] >= [0] = A() [1] [0] [0] [0] g(C()) = [1] >= [0] = B() [1] [1] [0] [0] g(C()) = [1] >= [0] = C() [1] [0] [1 0 0] [0] foldf(x,nil()) = [0 1 0]x + [1] >= x = x [1 0 1] [0] [1 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 1] [1 0 0] [0] foldf(x,cons(y,z)) = [0 1 0]x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] = f(foldf(x,z),y) [1 0 1] [1 0 1] [1 0 0] [0] [1 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 1] [0] f(t,x) = [0 0 0]t + [0 0 0]x + [1] >= [0 0 0]t + [0 0 0]x + [1] = f'(t,g(x)) [1 0 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] f'(triple(a,b,c),C()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [0] = triple(a,b,cons(C(),c)) [1 0 0] [1 0 0] [1 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] f'(triple(a,b,c),B()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] = f(triple(a,b,c),A()) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] f'(triple(a,b,c),A()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] = f''(foldf(triple(cons(A(),a),nil(),c),b)) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] f''(triple(a,b,c)) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] = foldf(triple(a,b,nil()),c) [1 0 0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] problem: g(A()) -> A() g(C()) -> A() g(C()) -> B() g(C()) -> C() foldf(x,nil()) -> x foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f'(triple(a,b,c),B()) -> f(triple(a,b,c),A()) f'(triple(a,b,c),A()) -> f''(foldf(triple(cons(A(),a),nil(),c),b)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=1 interpretation: [f''](x0) = x0, [triple](x0, x1, x2) = 4x0 + 4x1 + 4x2 + 4, [f'](x0, x1) = x0 + 4x1, [f](x0, x1) = x0 + 4x1, [cons](x0, x1) = x0 + x1, [foldf](x0, x1) = x0 + 4x1, [nil] = 0, [C] = 4, [B] = 2, [g](x0) = x0, [A] = 1 orientation: g(A()) = 1 >= 1 = A() g(C()) = 4 >= 1 = A() g(C()) = 4 >= 2 = B() g(C()) = 4 >= 4 = C() foldf(x,nil()) = x >= x = x foldf(x,cons(y,z)) = x + 4y + 4z >= x + 4y + 4z = f(foldf(x,z),y) f(t,x) = t + 4x >= t + 4x = f'(t,g(x)) f'(triple(a,b,c),C()) = 4a + 4b + 4c + 20 >= 4a + 4b + 4c + 20 = triple(a,b,cons(C(),c)) f'(triple(a,b,c),B()) = 4a + 4b + 4c + 12 >= 4a + 4b + 4c + 8 = f(triple(a,b,c),A()) f'(triple(a,b,c),A()) = 4a + 4b + 4c + 8 >= 4a + 4b + 4c + 8 = f''(foldf(triple(cons(A(),a),nil(),c),b)) f''(triple(a,b,c)) = 4a + 4b + 4c + 4 >= 4a + 4b + 4c + 4 = foldf(triple(a,b,nil()),c) problem: g(A()) -> A() g(C()) -> C() foldf(x,nil()) -> x foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f'(triple(a,b,c),A()) -> f''(foldf(triple(cons(A(),a),nil(),c),b)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [f''](x0) = [0 0 0]x0 + [1] [0 0 0] [1], [1 0 0] [1 1 1] [1 1 0] [triple](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] [0] [f'](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [1 1 0] [0] [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 0 0] [1 0 0] [cons](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [1 0 0] [0 0 0] , [1 1 0] [0] [foldf](x0, x1) = x0 + [0 0 0]x1 + [1] [0 0 0] [1], [0] [nil] = [0] [0], [1] [C] = [0] [0], [1 1 0] [0] [g](x0) = [0 0 1]x0 + [0] [0 0 0] [1], [0] [A] = [1] [1] orientation: [1] [0] g(A()) = [1] >= [1] = A() [1] [1] [1] [1] g(C()) = [0] >= [0] = C() [1] [0] [0] foldf(x,nil()) = x + [1] >= x = x [1] [1 1 0] [1 1 0] [0] [1 0 0] [1 1 0] [1 1 0] [0] foldf(x,cons(y,z)) = x + [0 0 0]y + [0 0 0]z + [1] >= [0 0 0]x + [0 0 0]y + [0 0 0]z + [1] = f(foldf(x,z),y) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] f(t,x) = [0 0 0]t + [0 0 0]x + [1] >= [0 0 0]t + [0 0 0]x + [1] = f'(t,g(x)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [1 1 0] [1] [1 0 0] [1 1 1] [1 1 0] [1] f'(triple(a,b,c),C()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [0] = triple(a,b,cons(C(),c)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 1 1] [1 1 0] [0] [1 0 0] [1 1 0] [1 1 0] [0] f'(triple(a,b,c),A()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] = f''(foldf(triple(cons(A(),a),nil(),c),b)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 1] [1 1 0] [0] [1 0 0] [1 1 1] [1 1 0] [0] f''(triple(a,b,c)) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [1] = foldf(triple(a,b,nil()),c) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] problem: g(C()) -> C() foldf(x,nil()) -> x foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f'(triple(a,b,c),A()) -> f''(foldf(triple(cons(A(),a),nil(),c),b)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [f''](x0) = [0 0 1]x0 [0 0 1] , [1 0 0] [1 1 0] [1 1 0] [triple](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 1] [0 1 0] [0 1 0] , [1 0 0] [1 0 1] [f'](x0, x1) = [0 0 1]x0 + [0 0 1]x1 [0 0 1] [0 0 1] , [1 0 0] [1 0 1] [f](x0, x1) = [0 0 1]x0 + [0 0 1]x1 [0 0 1] [0 0 1] , [1 0 0] [cons](x0, x1) = [0 0 1]x0 + x1 [0 0 1] , [1 0 0] [1 1 0] [foldf](x0, x1) = [0 1 1]x0 + [0 1 0]x1 [0 0 1] [0 1 0] , [0] [nil] = [0] [0], [0] [C] = [0] [1], [1 0 0] [g](x0) = [0 0 1]x0 [0 0 1] , [0] [A] = [0] [1] orientation: [0] [0] g(C()) = [1] >= [0] = C() [1] [1] [1 0 0] foldf(x,nil()) = [0 1 1]x >= x = x [0 0 1] [1 0 0] [1 0 1] [1 1 0] [1 0 0] [1 0 1] [1 1 0] foldf(x,cons(y,z)) = [0 1 1]x + [0 0 1]y + [0 1 0]z >= [0 0 1]x + [0 0 1]y + [0 1 0]z = f(foldf(x,z),y) [0 0 1] [0 0 1] [0 1 0] [0 0 1] [0 0 1] [0 1 0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] f(t,x) = [0 0 1]t + [0 0 1]x >= [0 0 1]t + [0 0 1]x = f'(t,g(x)) [0 0 1] [0 0 1] [0 0 1] [0 0 1] [1 0 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] f'(triple(a,b,c),C()) = [0 0 1]a + [0 1 0]b + [0 1 0]c + [1] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [0] = triple(a,b,cons(C(),c)) [0 0 1] [0 1 0] [0 1 0] [1] [0 0 1] [0 1 0] [0 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [0] f'(triple(a,b,c),A()) = [0 0 1]a + [0 1 0]b + [0 1 0]c + [1] >= [0 0 1]a + [0 1 0]b + [0 1 0]c + [1] = f''(foldf(triple(cons(A(),a),nil(),c),b)) [0 0 1] [0 1 0] [0 1 0] [1] [0 0 1] [0 1 0] [0 1 0] [1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] f''(triple(a,b,c)) = [0 0 1]a + [0 1 0]b + [0 1 0]c >= [0 0 1]a + [0 1 0]b + [0 1 0]c = foldf(triple(a,b,nil()),c) [0 0 1] [0 1 0] [0 1 0] [0 0 1] [0 1 0] [0 1 0] problem: g(C()) -> C() foldf(x,nil()) -> x foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1] [f''](x0) = [1 0 0]x0 + [0] [1 0 0] [0], [1 0 1] [1 0 1] [1 1 0] [triple](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 1] [f'](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 0] [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 1 0] [1 0 0] , [1 1 0] [1 1 0] [cons](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [0 1 0] [0 0 0] , [1 0 0] [1 0 0] [foldf](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 1 1] [1 0 0] , [1] [nil] = [0] [0], [0] [C] = [1] [1], [1 0 0] [0] [g](x0) = [0 0 0]x0 + [1] [0 1 0] [0] orientation: [0] [0] g(C()) = [1] >= [1] = C() [1] [1] [1 0 0] [1] foldf(x,nil()) = [0 1 0]x + [0] >= x = x [0 1 1] [1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] foldf(x,cons(y,z)) = [0 1 0]x + [1 0 0]y + [0 0 0]z >= [0 0 0]x + [0 0 0]y + [0 0 0]z = f(foldf(x,z),y) [0 1 1] [1 1 0] [1 1 0] [0 1 0] [1 0 0] [0 1 0] [1 0 0] [1 1 0] [1 0 0] [1 1 0] f(t,x) = [0 0 0]t + [0 0 0]x >= [0 0 0]t + [0 0 0]x = f'(t,g(x)) [0 1 0] [1 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 1 0] [1] f'(triple(a,b,c),C()) = [0 0 0]a + [0 0 0]b + [0 0 0]c + [0] >= [0 0 0]a + [0 0 0]b + [0 0 0]c + [0] = triple(a,b,cons(C(),c)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 1] [1 1 0] [1] [1 0 1] [1 0 1] [1 0 0] [1] f''(triple(a,b,c)) = [1 0 1]a + [1 0 1]b + [1 1 0]c + [0] >= [0 0 0]a + [0 0 0]b + [0 1 0]c + [0] = foldf(triple(a,b,nil()),c) [1 0 1] [1 0 1] [1 1 0] [0] [0 0 0] [0 0 0] [1 0 0] [0] problem: g(C()) -> C() foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [0] [f''](x0) = [1 0 0]x0 + [0] [0 1 1] [1], [1 0 0] [1 0 0] [1 1 0] [1] [triple](x0, x1, x2) = [1 0 0]x0 + [1 0 0]x1 + [1 0 0]x2 + [0] [0 0 0] [0 0 1] [0 0 1] [0], [1 0 0] [0] [f'](x0, x1) = x0 + [0 0 0]x1 + [0] [0 0 0] [1], [1 1 1] [1] [f](x0, x1) = x0 + [0 0 0]x1 + [0] [0 0 0] [1], [1 1 1] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 1] [1], [1 0 0] [1 0 1] [foldf](x0, x1) = [1 0 0]x0 + [0 0 0]x1 [1 0 0] [0 0 1] , [0] [nil] = [0] [0], [0] [C] = [0] [0], [1 1 0] [1] [g](x0) = [0 0 0]x0 + [1] [0 0 0] [1] orientation: [1] [0] g(C()) = [1] >= [0] = C() [1] [0] [1 0 0] [1 1 1] [1 0 1] [1] [1 0 0] [1 1 1] [1 0 1] [1] foldf(x,cons(y,z)) = [1 0 0]x + [0 0 0]y + [0 0 0]z + [0] >= [1 0 0]x + [0 0 0]y + [0 0 0]z + [0] = f(foldf(x,z),y) [1 0 0] [0 0 0] [0 0 1] [1] [1 0 0] [0 0 0] [0 0 1] [1] [1 1 1] [1] [1 1 0] [1] f(t,x) = t + [0 0 0]x + [0] >= t + [0 0 0]x + [0] = f'(t,g(x)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1 0 0] [1] f'(triple(a,b,c),C()) = [1 0 0]a + [1 0 0]b + [1 0 0]c + [0] >= [1 0 0]a + [1 0 0]b + [1 0 0]c + [0] = triple(a,b,cons(C(),c)) [0 0 0] [0 0 1] [0 0 1] [1] [0 0 0] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 1] [1 1 1] [1] [1 0 0] [1 0 0] [1 0 1] [1] f''(triple(a,b,c)) = [1 0 0]a + [1 0 0]b + [1 1 0]c + [1] >= [1 0 0]a + [1 0 0]b + [0 0 0]c + [1] = foldf(triple(a,b,nil()),c) [1 0 0] [1 0 1] [1 0 1] [1] [1 0 0] [1 0 0] [0 0 1] [1] problem: foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) f''(triple(a,b,c)) -> foldf(triple(a,b,nil()),c) Matrix Interpretation Processor: dim=1 interpretation: [f''](x0) = 2x0 + 5, [triple](x0, x1, x2) = 4x0 + x1 + x2 + 2, [f'](x0, x1) = x0 + x1, [f](x0, x1) = x0 + 4x1, [cons](x0, x1) = 4x0 + x1, [foldf](x0, x1) = x0 + 2x1 + 6, [nil] = 0, [C] = 0, [g](x0) = 4x0 orientation: foldf(x,cons(y,z)) = x + 8y + 2z + 6 >= x + 4y + 2z + 6 = f(foldf(x,z),y) f(t,x) = t + 4x >= t + 4x = f'(t,g(x)) f'(triple(a,b,c),C()) = 4a + b + c + 2 >= 4a + b + c + 2 = triple(a,b,cons(C(),c)) f''(triple(a,b,c)) = 8a + 2b + 2c + 9 >= 4a + b + 2c + 8 = foldf(triple(a,b,nil()),c) problem: foldf(x,cons(y,z)) -> f(foldf(x,z),y) f(t,x) -> f'(t,g(x)) f'(triple(a,b,c),C()) -> triple(a,b,cons(C(),c)) Matrix Interpretation Processor: dim=1 interpretation: [triple](x0, x1, x2) = 2x0 + x1 + x2 + 4, [f'](x0, x1) = x0 + 6x1 + 1, [f](x0, x1) = x0 + 7x1 + 1, [cons](x0, x1) = 4x0 + x1 + 4, [foldf](x0, x1) = 4x0 + 4x1 + 4, [C] = 3, [g](x0) = x0 orientation: foldf(x,cons(y,z)) = 4x + 16y + 4z + 20 >= 4x + 7y + 4z + 5 = f(foldf(x,z),y) f(t,x) = t + 7x + 1 >= t + 6x + 1 = f'(t,g(x)) f'(triple(a,b,c),C()) = 2a + b + c + 23 >= 2a + b + c + 20 = triple(a,b,cons(C(),c)) problem: f(t,x) -> f'(t,g(x)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [1 0 0] [f'](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [f](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [g](x0) = [0 0 0]x0 [0 0 0] orientation: [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] f(t,x) = [0 0 0]t + [0 0 0]x + [0] >= [0 0 0]t + [0 0 0]x = f'(t,g(x)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] problem: Qed