1.82/1.91	YES
2.22/2.24	We consider the system theBenchmark.
2.22/2.24	
2.22/2.24	  Alphabet:
2.22/2.24	
2.22/2.24	    0 : [] --> R 
2.22/2.24	    1 : [] --> R 
2.22/2.24	    cos : [] --> R -> R 
2.22/2.24	    d : [] --> (R -> R) -> R -> R 
2.22/2.24	    minus : [] --> R -> R 
2.22/2.24	    mul : [] --> R -> R -> R 
2.22/2.24	    pls : [] --> R -> R -> R 
2.22/2.24	    sin : [] --> R -> R 
2.22/2.24	
2.22/2.24	  Rules:
2.22/2.24	
2.22/2.24	    d (/\x.y) z => 0 
2.22/2.24	    d (/\x.x) y => 1 
2.22/2.24	    d (/\x.minus (f x)) y => minus (d (/\z.f z) y) 
2.22/2.24	    d (/\x.pls (f x) (g x)) y => pls (d (/\z.f z) y) (d (/\u.g u) y) 
2.22/2.24	    d (/\x.mul (f x) (g x)) y => pls (mul (d (/\z.f z) y) (g y)) (mul (f y) (d (/\u.g u) y)) 
2.22/2.24	    d (/\x.sin (f x)) y => mul (cos y) (d (/\z.f z) y) 
2.22/2.24	    d (/\x.cos (f x)) y => mul (minus (sin y)) (d (/\z.f z) y) 
2.22/2.24	    minus 0 => 0 
2.22/2.24	    mul 0 x => 0 
2.22/2.24	    mul x 0 => 0 
2.22/2.24	    pls 0 x => x 
2.22/2.24	
2.22/2.24	Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.
2.22/2.24	
2.22/2.24	We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:
2.22/2.24	
2.22/2.24	  Alphabet:
2.22/2.24	
2.22/2.24	    0 : [] --> R 
2.22/2.24	    1 : [] --> R 
2.22/2.24	    cos : [R] --> R 
2.22/2.24	    d : [R -> R * R] --> R 
2.22/2.24	    minus : [R] --> R 
2.22/2.24	    mul : [R * R] --> R 
2.22/2.24	    pls : [R * R] --> R 
2.22/2.24	    sin : [R] --> R 
2.22/2.24	    ~AP1 : [R -> R * R] --> R 
2.22/2.24	
2.22/2.24	  Rules:
2.22/2.24	
2.22/2.24	    d(/\x.X, Y) => 0 
2.22/2.24	    d(/\x.x, X) => 1 
2.22/2.24	    d(/\x.minus(~AP1(F, x)), X) => minus(d(/\y.~AP1(F, y), X)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), ~AP1(G, x)), X) => pls(d(/\y.~AP1(F, y), X), d(/\z.~AP1(G, z), X)) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), ~AP1(G, x)), X) => pls(mul(d(/\y.~AP1(F, y), X), ~AP1(G, X)), mul(~AP1(F, X), d(/\z.~AP1(G, z), X))) 
2.22/2.24	    d(/\x.sin(~AP1(F, x)), X) => mul(cos(X), d(/\y.~AP1(F, y), X)) 
2.22/2.24	    d(/\x.cos(~AP1(F, x)), X) => mul(minus(sin(X)), d(/\y.~AP1(F, y), X)) 
2.22/2.24	    minus(0) => 0 
2.22/2.24	    mul(0, X) => 0 
2.22/2.24	    mul(X, 0) => 0 
2.22/2.24	    pls(0, X) => X 
2.22/2.24	    d(/\x.minus(cos(x)), X) => minus(d(/\y.cos(y), X)) 
2.22/2.24	    d(/\x.minus(d(F, x)), X) => minus(d(/\y.d(F, y), X)) 
2.22/2.24	    d(/\x.minus(minus(x)), X) => minus(d(/\y.minus(y), X)) 
2.22/2.24	    d(/\x.minus(mul(X, x)), Y) => minus(d(/\y.mul(X, y), Y)) 
2.22/2.24	    d(/\x.minus(pls(X, x)), Y) => minus(d(/\y.pls(X, y), Y)) 
2.22/2.24	    d(/\x.minus(sin(x)), X) => minus(d(/\y.sin(y), X)) 
2.22/2.24	    d(/\x.pls(cos(x), ~AP1(F, x)), X) => pls(d(/\y.cos(y), X), d(/\z.~AP1(F, z), X)) 
2.22/2.24	    d(/\x.pls(d(F, x), ~AP1(G, x)), X) => pls(d(/\y.d(F, y), X), d(/\z.~AP1(G, z), X)) 
2.22/2.24	    d(/\x.pls(minus(x), ~AP1(F, x)), X) => pls(d(/\y.minus(y), X), d(/\z.~AP1(F, z), X)) 
2.22/2.24	    d(/\x.pls(mul(X, x), ~AP1(F, x)), Y) => pls(d(/\y.mul(X, y), Y), d(/\z.~AP1(F, z), Y)) 
2.22/2.24	    d(/\x.pls(pls(X, x), ~AP1(F, x)), Y) => pls(d(/\y.pls(X, y), Y), d(/\z.~AP1(F, z), Y)) 
2.22/2.24	    d(/\x.pls(sin(x), ~AP1(F, x)), X) => pls(d(/\y.sin(y), X), d(/\z.~AP1(F, z), X)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), cos(x)), X) => pls(d(/\y.~AP1(F, y), X), d(/\z.cos(z), X)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), d(G, x)), X) => pls(d(/\y.~AP1(F, y), X), d(/\z.d(G, z), X)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), minus(x)), X) => pls(d(/\y.~AP1(F, y), X), d(/\z.minus(z), X)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), mul(X, x)), Y) => pls(d(/\y.~AP1(F, y), Y), d(/\z.mul(X, z), Y)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), pls(X, x)), Y) => pls(d(/\y.~AP1(F, y), Y), d(/\z.pls(X, z), Y)) 
2.22/2.24	    d(/\x.pls(~AP1(F, x), sin(x)), X) => pls(d(/\y.~AP1(F, y), X), d(/\z.sin(z), X)) 
2.22/2.24	    d(/\x.mul(cos(x), ~AP1(F, x)), X) => pls(mul(d(/\y.cos(y), X), ~AP1(F, X)), mul(cos(X), d(/\z.~AP1(F, z), X))) 
2.22/2.24	    d(/\x.mul(d(F, x), ~AP1(G, x)), X) => pls(mul(d(/\y.d(F, y), X), ~AP1(G, X)), mul(d(F, X), d(/\z.~AP1(G, z), X))) 
2.22/2.24	    d(/\x.mul(minus(x), ~AP1(F, x)), X) => pls(mul(d(/\y.minus(y), X), ~AP1(F, X)), mul(minus(X), d(/\z.~AP1(F, z), X))) 
2.22/2.24	    d(/\x.mul(mul(X, x), ~AP1(F, x)), Y) => pls(mul(d(/\y.mul(X, y), Y), ~AP1(F, Y)), mul(mul(X, Y), d(/\z.~AP1(F, z), Y))) 
2.22/2.24	    d(/\x.mul(pls(X, x), ~AP1(F, x)), Y) => pls(mul(d(/\y.pls(X, y), Y), ~AP1(F, Y)), mul(pls(X, Y), d(/\z.~AP1(F, z), Y))) 
2.22/2.24	    d(/\x.mul(sin(x), ~AP1(F, x)), X) => pls(mul(d(/\y.sin(y), X), ~AP1(F, X)), mul(sin(X), d(/\z.~AP1(F, z), X))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), cos(x)), X) => pls(mul(d(/\y.~AP1(F, y), X), cos(X)), mul(~AP1(F, X), d(/\z.cos(z), X))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), d(G, x)), X) => pls(mul(d(/\y.~AP1(F, y), X), d(G, X)), mul(~AP1(F, X), d(/\z.d(G, z), X))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), minus(x)), X) => pls(mul(d(/\y.~AP1(F, y), X), minus(X)), mul(~AP1(F, X), d(/\z.minus(z), X))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), mul(X, x)), Y) => pls(mul(d(/\y.~AP1(F, y), Y), mul(X, Y)), mul(~AP1(F, Y), d(/\z.mul(X, z), Y))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), pls(X, x)), Y) => pls(mul(d(/\y.~AP1(F, y), Y), pls(X, Y)), mul(~AP1(F, Y), d(/\z.pls(X, z), Y))) 
2.22/2.24	    d(/\x.mul(~AP1(F, x), sin(x)), X) => pls(mul(d(/\y.~AP1(F, y), X), sin(X)), mul(~AP1(F, X), d(/\z.sin(z), X))) 
2.22/2.24	    d(/\x.sin(cos(x)), X) => mul(cos(X), d(/\y.cos(y), X)) 
2.22/2.24	    d(/\x.sin(d(F, x)), X) => mul(cos(X), d(/\y.d(F, y), X)) 
2.22/2.24	    d(/\x.sin(minus(x)), X) => mul(cos(X), d(/\y.minus(y), X)) 
2.22/2.24	    d(/\x.sin(mul(X, x)), Y) => mul(cos(Y), d(/\y.mul(X, y), Y)) 
2.22/2.24	    d(/\x.sin(pls(X, x)), Y) => mul(cos(Y), d(/\y.pls(X, y), Y)) 
2.22/2.24	    d(/\x.sin(sin(x)), X) => mul(cos(X), d(/\y.sin(y), X)) 
2.22/2.24	    d(/\x.cos(cos(x)), X) => mul(minus(sin(X)), d(/\y.cos(y), X)) 
2.22/2.24	    d(/\x.cos(d(F, x)), X) => mul(minus(sin(X)), d(/\y.d(F, y), X)) 
2.22/2.24	    d(/\x.cos(minus(x)), X) => mul(minus(sin(X)), d(/\y.minus(y), X)) 
2.22/2.24	    d(/\x.cos(mul(X, x)), Y) => mul(minus(sin(Y)), d(/\y.mul(X, y), Y)) 
2.22/2.24	    d(/\x.cos(pls(X, x)), Y) => mul(minus(sin(Y)), d(/\y.pls(X, y), Y)) 
2.22/2.24	    d(/\x.cos(sin(x)), X) => mul(minus(sin(X)), d(/\y.sin(y), X)) 
2.22/2.24	    d(/\x.pls(cos(x), cos(x)), X) => pls(d(/\y.cos(y), X), d(/\z.cos(z), X)) 
2.22/2.24	    d(/\x.pls(cos(x), d(F, x)), X) => pls(d(/\y.cos(y), X), d(/\z.d(F, z), X)) 
2.22/2.24	    d(/\x.pls(cos(x), minus(x)), X) => pls(d(/\y.cos(y), X), d(/\z.minus(z), X)) 
2.22/2.24	    d(/\x.pls(cos(x), mul(X, x)), Y) => pls(d(/\y.cos(y), Y), d(/\z.mul(X, z), Y)) 
2.22/2.24	    d(/\x.pls(cos(x), pls(X, x)), Y) => pls(d(/\y.cos(y), Y), d(/\z.pls(X, z), Y)) 
2.22/2.24	    d(/\x.pls(cos(x), sin(x)), X) => pls(d(/\y.cos(y), X), d(/\z.sin(z), X)) 
2.22/2.24	    d(/\x.pls(d(F, x), cos(x)), X) => pls(d(/\y.d(F, y), X), d(/\z.cos(z), X)) 
2.22/2.24	    d(/\x.pls(d(F, x), d(G, x)), X) => pls(d(/\y.d(F, y), X), d(/\z.d(G, z), X)) 
2.22/2.24	    d(/\x.pls(d(F, x), minus(x)), X) => pls(d(/\y.d(F, y), X), d(/\z.minus(z), X)) 
2.22/2.24	    d(/\x.pls(d(F, x), mul(X, x)), Y) => pls(d(/\y.d(F, y), Y), d(/\z.mul(X, z), Y)) 
2.22/2.24	    d(/\x.pls(d(F, x), pls(X, x)), Y) => pls(d(/\y.d(F, y), Y), d(/\z.pls(X, z), Y)) 
2.22/2.24	    d(/\x.pls(d(F, x), sin(x)), X) => pls(d(/\y.d(F, y), X), d(/\z.sin(z), X)) 
2.22/2.24	    d(/\x.pls(minus(x), cos(x)), X) => pls(d(/\y.minus(y), X), d(/\z.cos(z), X)) 
2.22/2.24	    d(/\x.pls(minus(x), d(F, x)), X) => pls(d(/\y.minus(y), X), d(/\z.d(F, z), X)) 
2.22/2.24	    d(/\x.pls(minus(x), minus(x)), X) => pls(d(/\y.minus(y), X), d(/\z.minus(z), X)) 
2.22/2.24	    d(/\x.pls(minus(x), mul(X, x)), Y) => pls(d(/\y.minus(y), Y), d(/\z.mul(X, z), Y)) 
2.22/2.24	    d(/\x.pls(minus(x), pls(X, x)), Y) => pls(d(/\y.minus(y), Y), d(/\z.pls(X, z), Y)) 
2.22/2.24	    d(/\x.pls(minus(x), sin(x)), X) => pls(d(/\y.minus(y), X), d(/\z.sin(z), X)) 
2.22/2.24	    d(/\x.pls(mul(X, x), cos(x)), Y) => pls(d(/\y.mul(X, y), Y), d(/\z.cos(z), Y)) 
2.22/2.24	    d(/\x.pls(mul(X, x), d(F, x)), Y) => pls(d(/\y.mul(X, y), Y), d(/\z.d(F, z), Y)) 
2.22/2.25	    d(/\x.pls(mul(X, x), minus(x)), Y) => pls(d(/\y.mul(X, y), Y), d(/\z.minus(z), Y)) 
2.22/2.25	    d(/\x.pls(mul(X, x), mul(Y, x)), Z) => pls(d(/\y.mul(X, y), Z), d(/\z.mul(Y, z), Z)) 
2.22/2.25	    d(/\x.pls(mul(X, x), pls(Y, x)), Z) => pls(d(/\y.mul(X, y), Z), d(/\z.pls(Y, z), Z)) 
2.22/2.25	    d(/\x.pls(mul(X, x), sin(x)), Y) => pls(d(/\y.mul(X, y), Y), d(/\z.sin(z), Y)) 
2.22/2.25	    d(/\x.pls(pls(X, x), cos(x)), Y) => pls(d(/\y.pls(X, y), Y), d(/\z.cos(z), Y)) 
2.22/2.25	    d(/\x.pls(pls(X, x), d(F, x)), Y) => pls(d(/\y.pls(X, y), Y), d(/\z.d(F, z), Y)) 
2.22/2.25	    d(/\x.pls(pls(X, x), minus(x)), Y) => pls(d(/\y.pls(X, y), Y), d(/\z.minus(z), Y)) 
2.22/2.25	    d(/\x.pls(pls(X, x), mul(Y, x)), Z) => pls(d(/\y.pls(X, y), Z), d(/\z.mul(Y, z), Z)) 
2.22/2.25	    d(/\x.pls(pls(X, x), pls(Y, x)), Z) => pls(d(/\y.pls(X, y), Z), d(/\z.pls(Y, z), Z)) 
2.22/2.25	    d(/\x.pls(pls(X, x), sin(x)), Y) => pls(d(/\y.pls(X, y), Y), d(/\z.sin(z), Y)) 
2.22/2.25	    d(/\x.pls(sin(x), cos(x)), X) => pls(d(/\y.sin(y), X), d(/\z.cos(z), X)) 
2.22/2.25	    d(/\x.pls(sin(x), d(F, x)), X) => pls(d(/\y.sin(y), X), d(/\z.d(F, z), X)) 
2.22/2.25	    d(/\x.pls(sin(x), minus(x)), X) => pls(d(/\y.sin(y), X), d(/\z.minus(z), X)) 
2.22/2.25	    d(/\x.pls(sin(x), mul(X, x)), Y) => pls(d(/\y.sin(y), Y), d(/\z.mul(X, z), Y)) 
2.22/2.25	    d(/\x.pls(sin(x), pls(X, x)), Y) => pls(d(/\y.sin(y), Y), d(/\z.pls(X, z), Y)) 
2.22/2.25	    d(/\x.pls(sin(x), sin(x)), X) => pls(d(/\y.sin(y), X), d(/\z.sin(z), X)) 
2.22/2.25	    d(/\x.mul(cos(x), cos(x)), X) => pls(mul(d(/\y.cos(y), X), cos(X)), mul(cos(X), d(/\z.cos(z), X))) 
2.22/2.25	    d(/\x.mul(cos(x), d(F, x)), X) => pls(mul(d(/\y.cos(y), X), d(F, X)), mul(cos(X), d(/\z.d(F, z), X))) 
2.22/2.25	    d(/\x.mul(cos(x), minus(x)), X) => pls(mul(d(/\y.cos(y), X), minus(X)), mul(cos(X), d(/\z.minus(z), X))) 
2.22/2.25	    d(/\x.mul(cos(x), mul(X, x)), Y) => pls(mul(d(/\y.cos(y), Y), mul(X, Y)), mul(cos(Y), d(/\z.mul(X, z), Y))) 
2.22/2.25	    d(/\x.mul(cos(x), pls(X, x)), Y) => pls(mul(d(/\y.cos(y), Y), pls(X, Y)), mul(cos(Y), d(/\z.pls(X, z), Y))) 
2.22/2.25	    d(/\x.mul(cos(x), sin(x)), X) => pls(mul(d(/\y.cos(y), X), sin(X)), mul(cos(X), d(/\z.sin(z), X))) 
2.22/2.25	    d(/\x.mul(d(F, x), cos(x)), X) => pls(mul(d(/\y.d(F, y), X), cos(X)), mul(d(F, X), d(/\z.cos(z), X))) 
2.22/2.25	    d(/\x.mul(d(F, x), d(G, x)), X) => pls(mul(d(/\y.d(F, y), X), d(G, X)), mul(d(F, X), d(/\z.d(G, z), X))) 
2.22/2.25	    d(/\x.mul(d(F, x), minus(x)), X) => pls(mul(d(/\y.d(F, y), X), minus(X)), mul(d(F, X), d(/\z.minus(z), X))) 
2.22/2.25	    d(/\x.mul(d(F, x), mul(X, x)), Y) => pls(mul(d(/\y.d(F, y), Y), mul(X, Y)), mul(d(F, Y), d(/\z.mul(X, z), Y))) 
2.22/2.25	    d(/\x.mul(d(F, x), pls(X, x)), Y) => pls(mul(d(/\y.d(F, y), Y), pls(X, Y)), mul(d(F, Y), d(/\z.pls(X, z), Y))) 
2.22/2.25	    d(/\x.mul(d(F, x), sin(x)), X) => pls(mul(d(/\y.d(F, y), X), sin(X)), mul(d(F, X), d(/\z.sin(z), X))) 
2.22/2.25	    d(/\x.mul(minus(x), cos(x)), X) => pls(mul(d(/\y.minus(y), X), cos(X)), mul(minus(X), d(/\z.cos(z), X))) 
2.22/2.25	    d(/\x.mul(minus(x), d(F, x)), X) => pls(mul(d(/\y.minus(y), X), d(F, X)), mul(minus(X), d(/\z.d(F, z), X))) 
2.22/2.25	    d(/\x.mul(minus(x), minus(x)), X) => pls(mul(d(/\y.minus(y), X), minus(X)), mul(minus(X), d(/\z.minus(z), X))) 
2.22/2.25	    d(/\x.mul(minus(x), mul(X, x)), Y) => pls(mul(d(/\y.minus(y), Y), mul(X, Y)), mul(minus(Y), d(/\z.mul(X, z), Y))) 
2.22/2.25	    d(/\x.mul(minus(x), pls(X, x)), Y) => pls(mul(d(/\y.minus(y), Y), pls(X, Y)), mul(minus(Y), d(/\z.pls(X, z), Y))) 
2.22/2.25	    d(/\x.mul(minus(x), sin(x)), X) => pls(mul(d(/\y.minus(y), X), sin(X)), mul(minus(X), d(/\z.sin(z), X))) 
2.22/2.25	    d(/\x.mul(mul(X, x), cos(x)), Y) => pls(mul(d(/\y.mul(X, y), Y), cos(Y)), mul(mul(X, Y), d(/\z.cos(z), Y))) 
2.22/2.25	    d(/\x.mul(mul(X, x), d(F, x)), Y) => pls(mul(d(/\y.mul(X, y), Y), d(F, Y)), mul(mul(X, Y), d(/\z.d(F, z), Y))) 
2.22/2.25	    d(/\x.mul(mul(X, x), minus(x)), Y) => pls(mul(d(/\y.mul(X, y), Y), minus(Y)), mul(mul(X, Y), d(/\z.minus(z), Y))) 
2.22/2.25	    d(/\x.mul(mul(X, x), mul(Y, x)), Z) => pls(mul(d(/\y.mul(X, y), Z), mul(Y, Z)), mul(mul(X, Z), d(/\z.mul(Y, z), Z))) 
2.22/2.25	    d(/\x.mul(mul(X, x), pls(Y, x)), Z) => pls(mul(d(/\y.mul(X, y), Z), pls(Y, Z)), mul(mul(X, Z), d(/\z.pls(Y, z), Z))) 
2.22/2.25	    d(/\x.mul(mul(X, x), sin(x)), Y) => pls(mul(d(/\y.mul(X, y), Y), sin(Y)), mul(mul(X, Y), d(/\z.sin(z), Y))) 
2.22/2.25	    d(/\x.mul(pls(X, x), cos(x)), Y) => pls(mul(d(/\y.pls(X, y), Y), cos(Y)), mul(pls(X, Y), d(/\z.cos(z), Y))) 
2.22/2.25	    d(/\x.mul(pls(X, x), d(F, x)), Y) => pls(mul(d(/\y.pls(X, y), Y), d(F, Y)), mul(pls(X, Y), d(/\z.d(F, z), Y))) 
2.22/2.25	    d(/\x.mul(pls(X, x), minus(x)), Y) => pls(mul(d(/\y.pls(X, y), Y), minus(Y)), mul(pls(X, Y), d(/\z.minus(z), Y))) 
2.22/2.25	    d(/\x.mul(pls(X, x), mul(Y, x)), Z) => pls(mul(d(/\y.pls(X, y), Z), mul(Y, Z)), mul(pls(X, Z), d(/\z.mul(Y, z), Z))) 
2.22/2.25	    d(/\x.mul(pls(X, x), pls(Y, x)), Z) => pls(mul(d(/\y.pls(X, y), Z), pls(Y, Z)), mul(pls(X, Z), d(/\z.pls(Y, z), Z))) 
2.22/2.25	    d(/\x.mul(pls(X, x), sin(x)), Y) => pls(mul(d(/\y.pls(X, y), Y), sin(Y)), mul(pls(X, Y), d(/\z.sin(z), Y))) 
2.22/2.25	    d(/\x.mul(sin(x), cos(x)), X) => pls(mul(d(/\y.sin(y), X), cos(X)), mul(sin(X), d(/\z.cos(z), X))) 
2.22/2.25	    d(/\x.mul(sin(x), d(F, x)), X) => pls(mul(d(/\y.sin(y), X), d(F, X)), mul(sin(X), d(/\z.d(F, z), X))) 
2.22/2.25	    d(/\x.mul(sin(x), minus(x)), X) => pls(mul(d(/\y.sin(y), X), minus(X)), mul(sin(X), d(/\z.minus(z), X))) 
2.22/2.25	    d(/\x.mul(sin(x), mul(X, x)), Y) => pls(mul(d(/\y.sin(y), Y), mul(X, Y)), mul(sin(Y), d(/\z.mul(X, z), Y))) 
2.22/2.25	    d(/\x.mul(sin(x), pls(X, x)), Y) => pls(mul(d(/\y.sin(y), Y), pls(X, Y)), mul(sin(Y), d(/\z.pls(X, z), Y))) 
2.22/2.25	    d(/\x.mul(sin(x), sin(x)), X) => pls(mul(d(/\y.sin(y), X), sin(X)), mul(sin(X), d(/\z.sin(z), X))) 
2.22/2.25	    ~AP1(F, X) => F X 
2.22/2.25	
2.22/2.25	Symbol ~AP1 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  Additionally, we can remove some (now-)redundant rules.  This gives:
2.22/2.25	
2.22/2.25	  Alphabet:
2.22/2.25	
2.22/2.25	    0 : [] --> R 
2.22/2.25	    1 : [] --> R 
2.22/2.25	    cos : [R] --> R 
2.22/2.25	    d : [R -> R * R] --> R 
2.22/2.25	    minus : [R] --> R 
2.22/2.25	    mul : [R * R] --> R 
2.22/2.25	    pls : [R * R] --> R 
2.22/2.25	    sin : [R] --> R 
2.22/2.25	
2.22/2.25	  Rules:
2.22/2.25	
2.22/2.25	    d(/\x.X, Y) => 0 
2.22/2.25	    d(/\x.x, X) => 1 
2.22/2.25	    d(/\x.minus(X(x)), Y) => minus(d(/\y.X(y), Y)) 
2.22/2.25	    d(/\x.pls(X(x), Y(x)), Z) => pls(d(/\y.X(y), Z), d(/\z.Y(z), Z)) 
2.22/2.25	    d(/\x.mul(X(x), Y(x)), Z) => pls(mul(d(/\y.X(y), Z), Y(Z)), mul(X(Z), d(/\z.Y(z), Z))) 
2.22/2.25	    d(/\x.sin(X(x)), Y) => mul(cos(Y), d(/\y.X(y), Y)) 
2.22/2.25	    d(/\x.cos(X(x)), Y) => mul(minus(sin(Y)), d(/\y.X(y), Y)) 
2.22/2.25	    minus(0) => 0 
2.22/2.25	    mul(0, X) => 0 
2.22/2.25	    mul(X, 0) => 0 
2.22/2.25	    pls(0, X) => X 
2.22/2.25	
2.22/2.25	We use rule removal, following [Kop12, Theorem 2.23].
2.22/2.25	
2.22/2.25	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) >? 0 
2.22/2.25	  d(/\x.x, X) >? 1 
2.22/2.25	  d(/\x.minus(X(x)), Y) >? minus(d(/\y.X(y), Y)) 
2.22/2.25	  d(/\x.pls(X(x), Y(x)), Z) >? pls(d(/\y.X(y), Z), d(/\z.Y(z), Z)) 
2.22/2.25	  d(/\x.mul(X(x), Y(x)), Z) >? pls(mul(d(/\y.X(y), Z), Y(Z)), mul(X(Z), d(/\z.Y(z), Z))) 
2.22/2.25	  d(/\x.sin(X(x)), Y) >? mul(cos(Y), d(/\y.X(y), Y)) 
2.22/2.25	  d(/\x.cos(X(x)), Y) >? mul(minus(sin(Y)), d(/\y.X(y), Y)) 
2.22/2.25	  minus(0) >? 0 
2.22/2.25	  mul(0, X) >? 0 
2.22/2.25	  mul(X, 0) >? 0 
2.22/2.25	  pls(0, X) >? X 
2.22/2.25	
2.22/2.25	We use a recursive path ordering as defined in [Kop12, Chapter 5].
2.22/2.25	
2.22/2.25	Argument functions:
2.22/2.25	
2.22/2.25	  [[0]] = _|_ 
2.22/2.25	  [[1]] = _|_ 
2.22/2.25	
2.22/2.25	We choose Lex = {} and Mul = {cos, d, minus, mul, pls, sin}, and the following precedence: d > cos > minus > mul > pls > sin
2.22/2.25	
2.22/2.25	Taking the argument function into account, and fixing the greater / greater equal choices, the constraints can be denoted as follows:
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) >= _|_ 
2.22/2.25	  d(/\x.x, X) >= _|_ 
2.22/2.25	  d(/\x.minus(X(x)), Y) > minus(d(/\x.X(x), Y)) 
2.22/2.25	  d(/\x.pls(X(x), Y(x)), Z) >= pls(d(/\x.X(x), Z), d(/\y.Y(y), Z)) 
2.22/2.25	  d(/\x.mul(X(x), Y(x)), Z) > pls(mul(d(/\x.X(x), Z), Y(Z)), mul(X(Z), d(/\y.Y(y), Z))) 
2.22/2.25	  d(/\x.sin(X(x)), Y) >= mul(cos(Y), d(/\x.X(x), Y)) 
2.22/2.25	  d(/\x.cos(X(x)), Y) >= mul(minus(sin(Y)), d(/\x.X(x), Y)) 
2.22/2.25	  minus(_|_) >= _|_ 
2.22/2.25	  mul(_|_, X) >= _|_ 
2.22/2.25	  mul(X, _|_) >= _|_ 
2.22/2.25	  pls(_|_, X) >= X 
2.22/2.25	
2.22/2.25	With these choices, we have:
2.22/2.25	
2.22/2.25	  1] d(/\x.X, Y) >= _|_  by (Bot) 
2.22/2.25	
2.22/2.25	  2] d(/\x.x, X) >= _|_  by (Bot) 
2.22/2.25	
2.22/2.25	  3] d(/\x.minus(X(x)), Y) > minus(d(/\x.X(x), Y))  because [4], by definition 
2.22/2.25	  4] d*(/\x.minus(X(x)), Y) >= minus(d(/\x.X(x), Y))  because d > minus and [5], by (Copy) 
2.22/2.25	  5] d*(/\x.minus(X(x)), Y) >= d(/\x.X(x), Y)  because d in Mul, [6] and [11], by (Stat) 
2.22/2.25	  6] /\x.minus(X(x)) > /\x.X(x)  because [7], by definition 
2.22/2.25	  7] /\y.minus*(X(y)) >= /\y.X(y)  because [8], by (Abs) 
2.22/2.25	  8] minus*(X(x)) >= X(x)  because [9], by (Select) 
2.22/2.25	  9] X(x) >= X(x)  because [10], by (Meta) 
2.22/2.25	  10] x >= x  by (Var) 
2.22/2.25	  11] Y >= Y  by (Meta) 
2.22/2.25	
2.22/2.25	  12] d(/\x.pls(X(x), Y(x)), Z) >= pls(d(/\x.X(x), Z), d(/\y.Y(y), Z))  because [13], by (Star) 
2.22/2.25	  13] d*(/\x.pls(X(x), Y(x)), Z) >= pls(d(/\x.X(x), Z), d(/\y.Y(y), Z))  because d > pls, [14] and [21], by (Copy) 
2.22/2.25	  14] d*(/\x.pls(X(x), Y(x)), Z) >= d(/\x.X(x), Z)  because d in Mul, [15] and [20], by (Stat) 
2.22/2.25	  15] /\x.pls(X(x), Y(x)) > /\x.X(x)  because [16], by definition 
2.22/2.25	  16] /\y.pls*(X(y), Y(y)) >= /\y.X(y)  because [17], by (Abs) 
2.22/2.25	  17] pls*(X(x), Y(x)) >= X(x)  because [18], by (Select) 
2.22/2.25	  18] X(x) >= X(x)  because [19], by (Meta) 
2.22/2.25	  19] x >= x  by (Var) 
2.22/2.25	  20] Z >= Z  by (Meta) 
2.22/2.25	  21] d*(/\y.pls(X(y), Y(y)), Z) >= d(/\y.Y(y), Z)  because d in Mul, [22] and [20], by (Stat) 
2.22/2.25	  22] /\y.pls(X(y), Y(y)) > /\y.Y(y)  because [23], by definition 
2.22/2.25	  23] /\z.pls*(X(z), Y(z)) >= /\z.Y(z)  because [24], by (Abs) 
2.22/2.25	  24] pls*(X(y), Y(y)) >= Y(y)  because [25], by (Select) 
2.22/2.25	  25] Y(y) >= Y(y)  because [26], by (Meta) 
2.22/2.25	  26] y >= y  by (Var) 
2.22/2.25	
2.22/2.25	  27] d(/\x.mul(X(x), Y(x)), Z) > pls(mul(d(/\x.X(x), Z), Y(Z)), mul(X(Z), d(/\y.Y(y), Z)))  because [28], by definition 
2.22/2.25	  28] d*(/\x.mul(X(x), Y(x)), Z) >= pls(mul(d(/\x.X(x), Z), Y(Z)), mul(X(Z), d(/\y.Y(y), Z)))  because d > pls, [29] and [42], by (Copy) 
2.22/2.25	  29] d*(/\x.mul(X(x), Y(x)), Z) >= mul(d(/\x.X(x), Z), Y(Z))  because d > mul, [30] and [37], by (Copy) 
2.22/2.25	  30] d*(/\x.mul(X(x), Y(x)), Z) >= d(/\x.X(x), Z)  because d in Mul, [31] and [36], by (Stat) 
2.22/2.25	  31] /\x.mul(X(x), Y(x)) > /\x.X(x)  because [32], by definition 
2.22/2.25	  32] /\y.mul*(X(y), Y(y)) >= /\y.X(y)  because [33], by (Abs) 
2.22/2.25	  33] mul*(X(x), Y(x)) >= X(x)  because [34], by (Select) 
2.22/2.25	  34] X(x) >= X(x)  because [35], by (Meta) 
2.22/2.25	  35] x >= x  by (Var) 
2.22/2.25	  36] Z >= Z  by (Meta) 
2.22/2.25	  37] d*(/\y.mul(X(y), Y(y)), Z) >= Y(Z)  because [38], by (Select) 
2.22/2.25	  38] mul(X(d*(/\y.mul(X(y), Y(y)), Z)), Y(d*(/\z.mul(X(z), Y(z)), Z))) >= Y(Z)  because [39], by (Star) 
2.22/2.25	  39] mul*(X(d*(/\y.mul(X(y), Y(y)), Z)), Y(d*(/\z.mul(X(z), Y(z)), Z))) >= Y(Z)  because [40], by (Select) 
2.22/2.25	  40] Y(d*(/\y.mul(X(y), Y(y)), Z)) >= Y(Z)  because [41], by (Meta) 
2.22/2.25	  41] d*(/\y.mul(X(y), Y(y)), Z) >= Z  because [36], by (Select) 
2.22/2.25	  42] d*(/\y.mul(X(y), Y(y)), Z) >= mul(X(Z), d(/\y.Y(y), Z))  because d > mul, [43] and [47], by (Copy) 
2.22/2.25	  43] d*(/\y.mul(X(y), Y(y)), Z) >= X(Z)  because [44], by (Select) 
2.22/2.25	  44] mul(X(d*(/\y.mul(X(y), Y(y)), Z)), Y(d*(/\z.mul(X(z), Y(z)), Z))) >= X(Z)  because [45], by (Star) 
2.22/2.25	  45] mul*(X(d*(/\y.mul(X(y), Y(y)), Z)), Y(d*(/\z.mul(X(z), Y(z)), Z))) >= X(Z)  because [46], by (Select) 
2.22/2.25	  46] X(d*(/\y.mul(X(y), Y(y)), Z)) >= X(Z)  because [41], by (Meta) 
2.22/2.25	  47] d*(/\y.mul(X(y), Y(y)), Z) >= d(/\y.Y(y), Z)  because d in Mul, [48] and [36], by (Stat) 
2.22/2.25	  48] /\y.mul(X(y), Y(y)) > /\y.Y(y)  because [49], by definition 
2.22/2.25	  49] /\z.mul*(X(z), Y(z)) >= /\z.Y(z)  because [50], by (Abs) 
2.22/2.25	  50] mul*(X(y), Y(y)) >= Y(y)  because [51], by (Select) 
2.22/2.25	  51] Y(y) >= Y(y)  because [52], by (Meta) 
2.22/2.25	  52] y >= y  by (Var) 
2.22/2.25	
2.22/2.25	  53] d(/\x.sin(X(x)), Y) >= mul(cos(Y), d(/\x.X(x), Y))  because [54], by (Star) 
2.22/2.25	  54] d*(/\x.sin(X(x)), Y) >= mul(cos(Y), d(/\x.X(x), Y))  because d > mul, [55] and [58], by (Copy) 
2.22/2.25	  55] d*(/\x.sin(X(x)), Y) >= cos(Y)  because d > cos and [56], by (Copy) 
2.22/2.25	  56] d*(/\x.sin(X(x)), Y) >= Y  because [57], by (Select) 
2.22/2.25	  57] Y >= Y  by (Meta) 
2.22/2.25	  58] d*(/\x.sin(X(x)), Y) >= d(/\x.X(x), Y)  because d in Mul, [59] and [64], by (Stat) 
2.22/2.25	  59] /\x.sin(X(x)) > /\x.X(x)  because [60], by definition 
2.22/2.25	  60] /\y.sin*(X(y)) >= /\y.X(y)  because [61], by (Abs) 
2.22/2.25	  61] sin*(X(x)) >= X(x)  because [62], by (Select) 
2.22/2.25	  62] X(x) >= X(x)  because [63], by (Meta) 
2.22/2.25	  63] x >= x  by (Var) 
2.22/2.25	  64] Y >= Y  by (Meta) 
2.22/2.25	
2.22/2.25	  65] d(/\x.cos(X(x)), Y) >= mul(minus(sin(Y)), d(/\x.X(x), Y))  because [66], by (Star) 
2.22/2.25	  66] d*(/\x.cos(X(x)), Y) >= mul(minus(sin(Y)), d(/\x.X(x), Y))  because d > mul, [67] and [71], by (Copy) 
2.22/2.25	  67] d*(/\x.cos(X(x)), Y) >= minus(sin(Y))  because d > minus and [68], by (Copy) 
2.22/2.25	  68] d*(/\x.cos(X(x)), Y) >= sin(Y)  because d > sin and [69], by (Copy) 
2.22/2.25	  69] d*(/\x.cos(X(x)), Y) >= Y  because [70], by (Select) 
2.22/2.25	  70] Y >= Y  by (Meta) 
2.22/2.25	  71] d*(/\x.cos(X(x)), Y) >= d(/\x.X(x), Y)  because d in Mul, [72] and [77], by (Stat) 
2.22/2.25	  72] /\x.cos(X(x)) > /\x.X(x)  because [73], by definition 
2.22/2.25	  73] /\y.cos*(X(y)) >= /\y.X(y)  because [74], by (Abs) 
2.22/2.25	  74] cos*(X(x)) >= X(x)  because [75], by (Select) 
2.22/2.25	  75] X(x) >= X(x)  because [76], by (Meta) 
2.22/2.25	  76] x >= x  by (Var) 
2.22/2.25	  77] Y >= Y  by (Meta) 
2.22/2.25	
2.22/2.25	  78] minus(_|_) >= _|_  by (Bot) 
2.22/2.25	
2.22/2.25	  79] mul(_|_, X) >= _|_  by (Bot) 
2.22/2.25	
2.22/2.25	  80] mul(X, _|_) >= _|_  by (Bot) 
2.22/2.25	
2.22/2.25	  81] pls(_|_, X) >= X  because [82], by (Star) 
2.22/2.25	  82] pls*(_|_, X) >= X  because [83], by (Select) 
2.22/2.25	  83] X >= X  by (Meta) 
2.22/2.25	
2.22/2.25	We can thus remove the following rules:
2.22/2.25	
2.22/2.25	  d(/\x.minus(X(x)), Y) => minus(d(/\y.X(y), Y)) 
2.22/2.25	  d(/\x.mul(X(x), Y(x)), Z) => pls(mul(d(/\y.X(y), Z), Y(Z)), mul(X(Z), d(/\z.Y(z), Z))) 
2.22/2.25	
2.22/2.25	We use rule removal, following [Kop12, Theorem 2.23].
2.22/2.25	
2.22/2.25	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) >? 0 
2.22/2.25	  d(/\x.x, X) >? 1 
2.22/2.25	  d(/\x.pls(X(x), Y(x)), Z) >? pls(d(/\y.X(y), Z), d(/\z.Y(z), Z)) 
2.22/2.25	  d(/\x.sin(X(x)), Y) >? mul(cos(Y), d(/\y.X(y), Y)) 
2.22/2.25	  d(/\x.cos(X(x)), Y) >? mul(minus(sin(Y)), d(/\y.X(y), Y)) 
2.22/2.25	  minus(0) >? 0 
2.22/2.25	  mul(0, X) >? 0 
2.22/2.25	  mul(X, 0) >? 0 
2.22/2.25	  pls(0, X) >? X 
2.22/2.25	
2.22/2.25	We orient these requirements with a polynomial interpretation in the natural numbers.
2.22/2.25	
2.22/2.25	The following interpretation satisfies the requirements:
2.22/2.25	
2.22/2.25	  0 = 0 
2.22/2.25	  1 = 0 
2.22/2.25	  cos = \y0.1 + y0 
2.22/2.25	  d = \G0y1.y1 + G0(0) + G0(y1) + y1G0(y1) 
2.22/2.25	  minus = \y0.y0 
2.22/2.25	  mul = \y0y1.y0 + y1 
2.22/2.25	  pls = \y0y1.1 + y0 + y1 
2.22/2.25	  sin = \y0.2 + y0 
2.22/2.25	
2.22/2.25	Using this interpretation, the requirements translate to:
2.22/2.25	
2.22/2.25	  [[d(/\x._x0, _x1)]] = x1 + 2x0 + x0x1 >= 0 = [[0]] 
2.22/2.25	  [[d(/\x.x, _x0)]] = 2x0 + x0x0 >= 0 = [[1]] 
2.22/2.25	  [[d(/\x.pls(_x0(x), _x1(x)), _x2)]] = 2 + 2x2 + F0(0) + F0(x2) + F1(0) + F1(x2) + x2F0(x2) + x2F1(x2) > 1 + 2x2 + F0(0) + F0(x2) + F1(0) + F1(x2) + x2F0(x2) + x2F1(x2) = [[pls(d(/\x._x0(x), _x2), d(/\y._x1(y), _x2))]] 
2.22/2.25	  [[d(/\x.sin(_x0(x)), _x1)]] = 4 + 3x1 + F0(0) + F0(x1) + x1F0(x1) > 1 + 2x1 + F0(0) + F0(x1) + x1F0(x1) = [[mul(cos(_x1), d(/\x._x0(x), _x1))]] 
2.22/2.25	  [[d(/\x.cos(_x0(x)), _x1)]] = 2 + 2x1 + F0(0) + F0(x1) + x1F0(x1) >= 2 + 2x1 + F0(0) + F0(x1) + x1F0(x1) = [[mul(minus(sin(_x1)), d(/\x._x0(x), _x1))]] 
2.22/2.25	  [[minus(0)]] = 0 >= 0 = [[0]] 
2.22/2.25	  [[mul(0, _x0)]] = x0 >= 0 = [[0]] 
2.22/2.25	  [[mul(_x0, 0)]] = x0 >= 0 = [[0]] 
2.22/2.25	  [[pls(0, _x0)]] = 1 + x0 > x0 = [[_x0]] 
2.22/2.25	
2.22/2.25	We can thus remove the following rules:
2.22/2.25	
2.22/2.25	  d(/\x.pls(X(x), Y(x)), Z) => pls(d(/\y.X(y), Z), d(/\z.Y(z), Z)) 
2.22/2.25	  d(/\x.sin(X(x)), Y) => mul(cos(Y), d(/\y.X(y), Y)) 
2.22/2.25	  pls(0, X) => X 
2.22/2.25	
2.22/2.25	We use rule removal, following [Kop12, Theorem 2.23].
2.22/2.25	
2.22/2.25	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) >? 0 
2.22/2.25	  d(/\x.x, X) >? 1 
2.22/2.25	  d(/\x.cos(X(x)), Y) >? mul(minus(sin(Y)), d(/\y.X(y), Y)) 
2.22/2.25	  minus(0) >? 0 
2.22/2.25	  mul(0, X) >? 0 
2.22/2.25	  mul(X, 0) >? 0 
2.22/2.25	
2.22/2.25	We orient these requirements with a polynomial interpretation in the natural numbers.
2.22/2.25	
2.22/2.25	The following interpretation satisfies the requirements:
2.22/2.25	
2.22/2.25	  0 = 0 
2.22/2.25	  1 = 0 
2.22/2.25	  cos = \y0.3 + 3y0 
2.22/2.25	  d = \G0y1.2y1 + 3y1G0(y1) + 3G0(0) + 3G0(y1) 
2.22/2.25	  minus = \y0.1 + y0 
2.22/2.25	  mul = \y0y1.2 + y0 + y1 
2.22/2.25	  sin = \y0.y0 
2.22/2.25	
2.22/2.25	Using this interpretation, the requirements translate to:
2.22/2.25	
2.22/2.25	  [[d(/\x._x0, _x1)]] = 2x1 + 3x0x1 + 6x0 >= 0 = [[0]] 
2.22/2.25	  [[d(/\x.x, _x0)]] = 3x0x0 + 5x0 >= 0 = [[1]] 
2.22/2.25	  [[d(/\x.cos(_x0(x)), _x1)]] = 18 + 11x1 + 9x1F0(x1) + 9F0(0) + 9F0(x1) > 3 + 3x1 + 3x1F0(x1) + 3F0(0) + 3F0(x1) = [[mul(minus(sin(_x1)), d(/\x._x0(x), _x1))]] 
2.22/2.25	  [[minus(0)]] = 1 > 0 = [[0]] 
2.22/2.25	  [[mul(0, _x0)]] = 2 + x0 > 0 = [[0]] 
2.22/2.25	  [[mul(_x0, 0)]] = 2 + x0 > 0 = [[0]] 
2.22/2.25	
2.22/2.25	We can thus remove the following rules:
2.22/2.25	
2.22/2.25	  d(/\x.cos(X(x)), Y) => mul(minus(sin(Y)), d(/\y.X(y), Y)) 
2.22/2.25	  minus(0) => 0 
2.22/2.25	  mul(0, X) => 0 
2.22/2.25	  mul(X, 0) => 0 
2.22/2.25	
2.22/2.25	We use rule removal, following [Kop12, Theorem 2.23].
2.22/2.25	
2.22/2.25	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) >? 0 
2.22/2.25	  d(/\x.x, X) >? 1 
2.22/2.25	
2.22/2.25	We orient these requirements with a polynomial interpretation in the natural numbers.
2.22/2.25	
2.22/2.25	The following interpretation satisfies the requirements:
2.22/2.25	
2.22/2.25	  0 = 0 
2.22/2.25	  1 = 0 
2.22/2.25	  d = \G0y1.3 + y1 + G0(0) 
2.22/2.25	
2.22/2.25	Using this interpretation, the requirements translate to:
2.22/2.25	
2.22/2.25	  [[d(/\x._x0, _x1)]] = 3 + x0 + x1 > 0 = [[0]] 
2.22/2.25	  [[d(/\x.x, _x0)]] = 3 + x0 > 0 = [[1]] 
2.22/2.25	
2.22/2.25	We can thus remove the following rules:
2.22/2.25	
2.22/2.25	  d(/\x.X, Y) => 0 
2.22/2.25	  d(/\x.x, X) => 1 
2.22/2.25	
2.22/2.25	All rules were succesfully removed.  Thus, termination of the original system has been reduced to termination of the beta-rule, which is well-known to hold.
2.22/2.25	
2.22/2.25	
2.22/2.25	+++ Citations +++
2.22/2.25	
2.22/2.25	[Kop11]  C. Kop.  Simplifying Algebraic Functional Systems.  In Proceedings of CAI 2011, volume 6742 of LNCS.  201--215, Springer, 2011.
2.22/2.25	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
2.22/2.25	EOF