0.00/0.21	YES
0.00/0.23	We consider the system theBenchmark.
0.00/0.23	
0.00/0.23	  Alphabet:
0.00/0.23	
0.00/0.23	    !facdot : [a * a] --> a 
0.00/0.23	    1 : [] --> a 
0.00/0.23	    cons : [c * d] --> d 
0.00/0.23	    false : [] --> b 
0.00/0.23	    filter : [c -> b * d] --> d 
0.00/0.23	    filter2 : [b * c -> b * c * d] --> d 
0.00/0.23	    i : [a] --> a 
0.00/0.23	    map : [c -> c * d] --> d 
0.00/0.23	    nil : [] --> d 
0.00/0.23	    true : [] --> b 
0.00/0.23	
0.00/0.23	  Rules:
0.00/0.23	
0.00/0.23	    !facdot(1, x) => x 
0.00/0.23	    !facdot(x, 1) => x 
0.00/0.23	    !facdot(i(x), x) => 1 
0.00/0.23	    !facdot(x, i(x)) => 1 
0.00/0.23	    !facdot(i(x), !facdot(x, y)) => y 
0.00/0.23	    !facdot(x, !facdot(i(x), y)) => y 
0.00/0.23	    !facdot(!facdot(x, y), z) => !facdot(x, !facdot(y, z)) 
0.00/0.23	    i(1) => 1 
0.00/0.23	    i(i(x)) => x 
0.00/0.23	    i(!facdot(x, y)) => !facdot(i(y), i(x)) 
0.00/0.23	    map(f, nil) => nil 
0.00/0.23	    map(f, cons(x, y)) => cons(f x, map(f, y)) 
0.00/0.23	    filter(f, nil) => nil 
0.00/0.23	    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
0.00/0.23	    filter2(true, f, x, y) => cons(x, filter(f, y)) 
0.00/0.23	    filter2(false, f, x, y) => filter(f, y) 
0.00/0.23	
0.00/0.23	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
0.00/0.23	
0.00/0.23	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.23	
0.00/0.23	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.23	
0.00/0.23	  !facdot(1, X) >? X 
0.00/0.23	  !facdot(X, 1) >? X 
0.00/0.23	  !facdot(i(X), X) >? 1 
0.00/0.23	  !facdot(X, i(X)) >? 1 
0.00/0.23	  !facdot(i(X), !facdot(X, Y)) >? Y 
0.00/0.23	  !facdot(X, !facdot(i(X), Y)) >? Y 
0.00/0.23	  !facdot(!facdot(X, Y), Z) >? !facdot(X, !facdot(Y, Z)) 
0.00/0.23	  i(1) >? 1 
0.00/0.23	  i(i(X)) >? X 
0.00/0.23	  i(!facdot(X, Y)) >? !facdot(i(Y), i(X)) 
0.00/0.23	  map(F, nil) >? nil 
0.00/0.23	  map(F, cons(X, Y)) >? cons(F X, map(F, Y)) 
0.00/0.23	  filter(F, nil) >? nil 
0.00/0.23	  filter(F, cons(X, Y)) >? filter2(F X, F, X, Y) 
0.00/0.23	  filter2(true, F, X, Y) >? cons(X, filter(F, Y)) 
0.00/0.23	  filter2(false, F, X, Y) >? filter(F, Y) 
0.00/0.23	
0.00/0.23	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.23	
0.00/0.23	The following interpretation satisfies the requirements:
0.00/0.23	
0.00/0.23	  !facdot = \y0y1.y0 + y1 
0.00/0.23	  1 = 0 
0.00/0.23	  cons = \y0y1.3 + y1 + 2y0 
0.00/0.23	  false = 3 
0.00/0.23	  filter = \G0y1.1 + 3y1 + G0(0) + 2y1G0(y1) 
0.00/0.23	  filter2 = \y0G1y2y3.1 + 2y0 + 2y2 + 3y3 + G1(0) + 2y3G1(y3) 
0.00/0.23	  i = \y0.y0 
0.00/0.23	  map = \G0y1.3y1 + G0(0) + 3y1G0(y1) 
0.00/0.23	  nil = 0 
0.00/0.23	  true = 3 
0.00/0.23	
0.00/0.23	Using this interpretation, the requirements translate to:
0.00/0.23	
0.00/0.23	  [[!facdot(1, _x0)]] = x0 >= x0 = [[_x0]] 
0.00/0.23	  [[!facdot(_x0, 1)]] = x0 >= x0 = [[_x0]] 
0.00/0.23	  [[!facdot(i(_x0), _x0)]] = 2x0 >= 0 = [[1]] 
0.00/0.23	  [[!facdot(_x0, i(_x0))]] = 2x0 >= 0 = [[1]] 
0.00/0.23	  [[!facdot(i(_x0), !facdot(_x0, _x1))]] = x1 + 2x0 >= x1 = [[_x1]] 
0.00/0.23	  [[!facdot(_x0, !facdot(i(_x0), _x1))]] = x1 + 2x0 >= x1 = [[_x1]] 
0.00/0.23	  [[!facdot(!facdot(_x0, _x1), _x2)]] = x0 + x1 + x2 >= x0 + x1 + x2 = [[!facdot(_x0, !facdot(_x1, _x2))]] 
0.00/0.23	  [[i(1)]] = 0 >= 0 = [[1]] 
0.00/0.23	  [[i(i(_x0))]] = x0 >= x0 = [[_x0]] 
0.00/0.23	  [[i(!facdot(_x0, _x1))]] = x0 + x1 >= x0 + x1 = [[!facdot(i(_x1), i(_x0))]] 
0.00/0.23	  [[map(_F0, nil)]] = F0(0) >= 0 = [[nil]] 
0.00/0.23	  [[map(_F0, cons(_x1, _x2))]] = 9 + 3x2 + 6x1 + F0(0) + 3x2F0(3 + x2 + 2x1) + 6x1F0(3 + x2 + 2x1) + 9F0(3 + x2 + 2x1) > 3 + 2x1 + 3x2 + F0(0) + 2F0(x1) + 3x2F0(x2) = [[cons(_F0 _x1, map(_F0, _x2))]] 
0.00/0.23	  [[filter(_F0, nil)]] = 1 + F0(0) > 0 = [[nil]] 
0.00/0.23	  [[filter(_F0, cons(_x1, _x2))]] = 10 + 3x2 + 6x1 + F0(0) + 2x2F0(3 + x2 + 2x1) + 4x1F0(3 + x2 + 2x1) + 6F0(3 + x2 + 2x1) > 1 + 3x2 + 4x1 + F0(0) + 2x2F0(x2) + 2F0(x1) = [[filter2(_F0 _x1, _F0, _x1, _x2)]] 
0.00/0.23	  [[filter2(true, _F0, _x1, _x2)]] = 7 + 2x1 + 3x2 + F0(0) + 2x2F0(x2) > 4 + 2x1 + 3x2 + F0(0) + 2x2F0(x2) = [[cons(_x1, filter(_F0, _x2))]] 
0.00/0.23	  [[filter2(false, _F0, _x1, _x2)]] = 7 + 2x1 + 3x2 + F0(0) + 2x2F0(x2) > 1 + 3x2 + F0(0) + 2x2F0(x2) = [[filter(_F0, _x2)]] 
0.00/0.23	
0.00/0.23	We can thus remove the following rules:
0.00/0.23	
0.00/0.23	  map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
0.00/0.23	  filter(F, nil) => nil 
0.00/0.23	  filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
0.00/0.23	  filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
0.00/0.23	  filter2(false, F, X, Y) => filter(F, Y) 
0.00/0.23	
0.00/0.23	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.23	
0.00/0.23	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.23	
0.00/0.23	  !facdot(1, X) >? X 
0.00/0.23	  !facdot(X, 1) >? X 
0.00/0.23	  !facdot(i(X), X) >? 1 
0.00/0.23	  !facdot(X, i(X)) >? 1 
0.00/0.23	  !facdot(i(X), !facdot(X, Y)) >? Y 
0.00/0.23	  !facdot(X, !facdot(i(X), Y)) >? Y 
0.00/0.23	  !facdot(!facdot(X, Y), Z) >? !facdot(X, !facdot(Y, Z)) 
0.00/0.23	  i(1) >? 1 
0.00/0.23	  i(i(X)) >? X 
0.00/0.23	  i(!facdot(X, Y)) >? !facdot(i(Y), i(X)) 
0.00/0.23	  map(F, nil) >? nil 
0.00/0.23	
0.00/0.23	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.23	
0.00/0.23	The following interpretation satisfies the requirements:
0.00/0.23	
0.00/0.23	  !facdot = \y0y1.3 + y0 + y1 
0.00/0.23	  1 = 1 
0.00/0.23	  i = \y0.2 + 2y0 
0.00/0.23	  map = \G0y1.3 + 3y1 + G0(0) 
0.00/0.23	  nil = 0 
0.00/0.23	
0.00/0.23	Using this interpretation, the requirements translate to:
0.00/0.23	
0.00/0.23	  [[!facdot(1, _x0)]] = 4 + x0 > x0 = [[_x0]] 
0.00/0.23	  [[!facdot(_x0, 1)]] = 4 + x0 > x0 = [[_x0]] 
0.00/0.23	  [[!facdot(i(_x0), _x0)]] = 5 + 3x0 > 1 = [[1]] 
0.00/0.23	  [[!facdot(_x0, i(_x0))]] = 5 + 3x0 > 1 = [[1]] 
0.00/0.23	  [[!facdot(i(_x0), !facdot(_x0, _x1))]] = 8 + x1 + 3x0 > x1 = [[_x1]] 
0.00/0.23	  [[!facdot(_x0, !facdot(i(_x0), _x1))]] = 8 + x1 + 3x0 > x1 = [[_x1]] 
0.00/0.23	  [[!facdot(!facdot(_x0, _x1), _x2)]] = 6 + x0 + x1 + x2 >= 6 + x0 + x1 + x2 = [[!facdot(_x0, !facdot(_x1, _x2))]] 
0.00/0.23	  [[i(1)]] = 4 > 1 = [[1]] 
0.00/0.23	  [[i(i(_x0))]] = 6 + 4x0 > x0 = [[_x0]] 
0.00/0.23	  [[i(!facdot(_x0, _x1))]] = 8 + 2x0 + 2x1 > 7 + 2x0 + 2x1 = [[!facdot(i(_x1), i(_x0))]] 
0.00/0.23	  [[map(_F0, nil)]] = 3 + F0(0) > 0 = [[nil]] 
0.00/0.23	
0.00/0.23	We can thus remove the following rules:
0.00/0.23	
0.00/0.23	  !facdot(1, X) => X 
0.00/0.23	  !facdot(X, 1) => X 
0.00/0.23	  !facdot(i(X), X) => 1 
0.00/0.23	  !facdot(X, i(X)) => 1 
0.00/0.23	  !facdot(i(X), !facdot(X, Y)) => Y 
0.00/0.23	  !facdot(X, !facdot(i(X), Y)) => Y 
0.00/0.23	  i(1) => 1 
0.00/0.23	  i(i(X)) => X 
0.00/0.23	  i(!facdot(X, Y)) => !facdot(i(Y), i(X)) 
0.00/0.23	  map(F, nil) => nil 
0.00/0.23	
0.00/0.23	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.23	
0.00/0.23	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.23	
0.00/0.23	  !facdot(!facdot(X, Y), Z) >? !facdot(X, !facdot(Y, Z)) 
0.00/0.23	
0.00/0.23	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.23	
0.00/0.23	The following interpretation satisfies the requirements:
0.00/0.23	
0.00/0.23	  !facdot = \y0y1.3 + y1 + 3y0 
0.00/0.23	
0.00/0.23	Using this interpretation, the requirements translate to:
0.00/0.23	
0.00/0.23	  [[!facdot(!facdot(_x0, _x1), _x2)]] = 12 + x2 + 3x1 + 9x0 > 6 + x2 + 3x0 + 3x1 = [[!facdot(_x0, !facdot(_x1, _x2))]] 
0.00/0.23	
0.00/0.23	We can thus remove the following rules:
0.00/0.23	
0.00/0.23	  !facdot(!facdot(X, Y), Z) => !facdot(X, !facdot(Y, Z)) 
0.00/0.23	
0.00/0.23	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.
0.00/0.23	
0.00/0.23	
0.00/0.23	+++ Citations +++
0.00/0.23	
0.00/0.23	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
0.00/0.23	EOF