4.18/4.20	YES
4.18/4.22	We consider the system theBenchmark.
4.18/4.22	
4.18/4.22	  Alphabet:
4.18/4.22	
4.18/4.22	    0 : [] --> nat 
4.18/4.22	    add : [nat * nat] --> nat 
4.18/4.22	    rec : [nat -> nat -> nat * nat * nat] --> nat 
4.18/4.22	    s : [nat] --> nat 
4.18/4.22	    succ : [] --> nat -> nat -> nat 
4.18/4.22	    xap : [nat -> nat -> nat * nat] --> nat -> nat 
4.18/4.22	    yap : [nat -> nat * nat] --> nat 
4.18/4.22	
4.18/4.22	  Rules:
4.18/4.22	
4.18/4.22	    rec(/\x./\y.yap(xap(f, x), y), z, 0) => z 
4.18/4.22	    rec(/\x./\y.yap(xap(f, x), y), z, s(u)) => yap(xap(f, u), rec(/\v./\w.yap(xap(f, v), w), z, u)) 
4.18/4.22	    succ x y => s(y) 
4.18/4.22	    add(x, y) => rec(/\z./\u.yap(xap(succ, z), u), x, y) 
4.18/4.22	    add(x, 0) => x 
4.18/4.22	    add(x, s(y)) => s(add(x, y)) 
4.18/4.22	    xap(f, x) => f x 
4.18/4.22	    yap(f, x) => f x 
4.18/4.22	
4.18/4.22	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.18/4.22	
4.18/4.22	Symbol xap is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:
4.18/4.22	
4.18/4.22	  Alphabet:
4.18/4.22	
4.18/4.22	    0 : [] --> nat 
4.18/4.22	    add : [nat * nat] --> nat 
4.18/4.22	    rec : [nat -> nat -> nat * nat * nat] --> nat 
4.18/4.22	    s : [nat] --> nat 
4.18/4.22	    succ : [nat] --> nat -> nat 
4.18/4.22	    yap : [nat -> nat * nat] --> nat 
4.18/4.22	
4.18/4.22	  Rules:
4.18/4.22	
4.18/4.22	    rec(/\x./\y.yap(F(x), y), X, 0) => X 
4.18/4.22	    rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 
4.18/4.22	    succ(X) Y => s(Y) 
4.18/4.22	    add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	    add(X, 0) => X 
4.18/4.22	    add(X, s(Y)) => s(add(X, Y)) 
4.18/4.22	    yap(F, X) => F X 
4.18/4.22	
4.18/4.22	We use rule removal, following [Kop12, Theorem 2.23].
4.18/4.22	
4.18/4.22	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
4.18/4.22	
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, 0) >? X 
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, s(Y)) >? yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 
4.18/4.22	  succ(X) Y >? s(Y) 
4.18/4.22	  add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	  add(X, 0) >? X 
4.18/4.22	  add(X, s(Y)) >? s(add(X, Y)) 
4.18/4.22	  yap(F, X) >? F X 
4.18/4.22	
4.18/4.22	We use a recursive path ordering as defined in [Kop12, Chapter 5].
4.18/4.22	
4.18/4.22	We choose Lex = {} and Mul = {0, @_{o -> o}, add, rec, s, succ, yap}, and the following precedence: 0 > add > succ > s > rec > yap > @_{o -> o}
4.18/4.22	
4.18/4.22	With these choices, we have:
4.18/4.22	
4.18/4.22	  1] rec(/\x./\y.yap(F(x), y), X, 0) > X  because [2], by definition 
4.18/4.22	  2] rec*(/\x./\y.yap(F(x), y), X, 0) >= X  because [3], by (Select) 
4.18/4.22	  3] X >= X  by (Meta) 
4.18/4.22	
4.18/4.22	  4] rec(/\x./\y.yap(F(x), y), X, s(Y)) > yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y))  because [5], by definition 
4.18/4.22	  5] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= yap(F(Y), rec(/\x./\y.yap(F(x), y), X, Y))  because rec > yap, [6] and [13], by (Copy) 
4.18/4.22	  6] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= F(Y)  because [7], by (Select) 
4.18/4.22	  7] /\x.yap(F(rec*(/\y./\z.yap(F(y), z), X, s(Y))), x) >= F(Y)  because [8], by (Eta)[Kop13:2] 
4.18/4.22	  8] F(rec*(/\x./\y.yap(F(x), y), X, s(Y))) >= F(Y)  because [9], by (Meta) 
4.18/4.22	  9] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= Y  because [10], by (Select) 
4.18/4.22	  10] s(Y) >= Y  because [11], by (Star) 
4.18/4.22	  11] s*(Y) >= Y  because [12], by (Select) 
4.18/4.22	  12] Y >= Y  by (Meta) 
4.18/4.22	  13] rec*(/\x./\y.yap(F(x), y), X, s(Y)) >= rec(/\x./\y.yap(F(x), y), X, Y)  because rec in Mul, [14], [20] and [21], by (Stat) 
4.18/4.22	  14] /\x./\z.yap(F(x), z) >= /\x./\z.yap(F(x), z)  because [15], by (Abs) 
4.18/4.22	  15] /\z.yap(F(y), z) >= /\z.yap(F(y), z)  because [16], by (Abs) 
4.18/4.22	  16] yap(F(y), x) >= yap(F(y), x)  because yap in Mul, [17] and [19], by (Fun) 
4.18/4.22	  17] F(y) >= F(y)  because [18], by (Meta) 
4.18/4.22	  18] y >= y  by (Var) 
4.18/4.22	  19] x >= x  by (Var) 
4.18/4.22	  20] X >= X  by (Meta) 
4.18/4.22	  21] s(Y) > Y  because [22], by definition 
4.18/4.22	  22] s*(Y) >= Y  because [12], by (Select) 
4.18/4.22	
4.18/4.22	  23] @_{o -> o}(succ(X), Y) > s(Y)  because [24], by definition 
4.18/4.22	  24] @_{o -> o}*(succ(X), Y) >= s(Y)  because [25], by (Select) 
4.18/4.22	  25] succ(X) @_{o -> o}*(succ(X), Y) >= s(Y)  because [26] 
4.18/4.22	  26] succ*(X, @_{o -> o}*(succ(X), Y)) >= s(Y)  because succ > s and [27], by (Copy) 
4.18/4.22	  27] succ*(X, @_{o -> o}*(succ(X), Y)) >= Y  because [28], by (Select) 
4.18/4.22	  28] @_{o -> o}*(succ(X), Y) >= Y  because [29], by (Select) 
4.18/4.22	  29] Y >= Y  by (Meta) 
4.18/4.22	
4.18/4.22	  30] add(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y)  because [31], by (Star) 
4.18/4.22	  31] add*(X, Y) >= rec(/\x./\y.yap(succ(x), y), X, Y)  because add > rec, [32], [40] and [42], by (Copy) 
4.18/4.22	  32] add*(X, Y) >= /\y./\z.yap(succ(y), z)  because [33], by (F-Abs) 
4.18/4.22	  33] add*(X, Y, x) >= /\z.yap(succ(x), z)  because [34], by (F-Abs) 
4.18/4.22	  34] add*(X, Y, x, y) >= yap(succ(x), y)  because add > yap, [35] and [38], by (Copy) 
4.18/4.22	  35] add*(X, Y, x, y) >= succ(x)  because add > succ and [36], by (Copy) 
4.18/4.22	  36] add*(X, Y, x, y) >= x  because [37], by (Select) 
4.18/4.22	  37] x >= x  by (Var) 
4.18/4.22	  38] add*(X, Y, x, y) >= y  because [39], by (Select) 
4.18/4.22	  39] y >= y  by (Var) 
4.18/4.22	  40] add*(X, Y) >= X  because [41], by (Select) 
4.18/4.22	  41] X >= X  by (Meta) 
4.18/4.22	  42] add*(X, Y) >= Y  because [43], by (Select) 
4.18/4.22	  43] Y >= Y  by (Meta) 
4.18/4.22	
4.18/4.22	  44] add(X, 0) >= X  because [45], by (Star) 
4.18/4.22	  45] add*(X, 0) >= X  because [46], by (Select) 
4.18/4.22	  46] X >= X  by (Meta) 
4.18/4.22	
4.18/4.22	  47] add(X, s(Y)) >= s(add(X, Y))  because [48], by (Star) 
4.18/4.22	  48] add*(X, s(Y)) >= s(add(X, Y))  because add > s and [49], by (Copy) 
4.18/4.22	  49] add*(X, s(Y)) >= add(X, Y)  because add in Mul, [50] and [51], by (Stat) 
4.18/4.22	  50] X >= X  by (Meta) 
4.18/4.22	  51] s(Y) > Y  because [52], by definition 
4.18/4.22	  52] s*(Y) >= Y  because [53], by (Select) 
4.18/4.22	  53] Y >= Y  by (Meta) 
4.18/4.22	
4.18/4.22	  54] yap(F, X) >= @_{o -> o}(F, X)  because [55], by (Star) 
4.18/4.22	  55] yap*(F, X) >= @_{o -> o}(F, X)  because yap > @_{o -> o}, [56] and [58], by (Copy) 
4.18/4.22	  56] yap*(F, X) >= F  because [57], by (Select) 
4.18/4.22	  57] F >= F  by (Meta) 
4.18/4.22	  58] yap*(F, X) >= X  because [59], by (Select) 
4.18/4.22	  59] X >= X  by (Meta) 
4.18/4.22	
4.18/4.22	We can thus remove the following rules:
4.18/4.22	
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, 0) => X 
4.18/4.22	  rec(/\x./\y.yap(F(x), y), X, s(Y)) => yap(F(Y), rec(/\z./\u.yap(F(z), u), X, Y)) 
4.18/4.22	  succ(X) Y => s(Y) 
4.18/4.22	
4.18/4.22	We use rule removal, following [Kop12, Theorem 2.23].
4.18/4.22	
4.18/4.22	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
4.18/4.22	
4.18/4.22	  add(X, Y) >? rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	  add(X, 0) >? X 
4.18/4.22	  add(X, s(Y)) >? s(add(X, Y)) 
4.18/4.22	  yap(F, X) >? F X 
4.18/4.22	
4.18/4.22	We orient these requirements with a polynomial interpretation in the natural numbers.
4.18/4.22	
4.18/4.22	The following interpretation satisfies the requirements:
4.18/4.22	
4.18/4.22	  0 = 3 
4.18/4.22	  add = \y0y1.3 + 3y0 + 3y1 
4.18/4.22	  rec = \G0y1y2.y1 + y2 + G0(0,0) 
4.18/4.22	  s = \y0.3 + y0 
4.18/4.22	  succ = \y0y1.y0 
4.18/4.22	  yap = \G0y1.y1 + G0(y1) 
4.18/4.22	
4.18/4.22	Using this interpretation, the requirements translate to:
4.18/4.22	
4.18/4.22	  [[add(_x0, _x1)]] = 3 + 3x0 + 3x1 > x0 + x1 = [[rec(/\x./\y.yap(succ(x), y), _x0, _x1)]] 
4.18/4.22	  [[add(_x0, 0)]] = 12 + 3x0 > x0 = [[_x0]] 
4.18/4.22	  [[add(_x0, s(_x1))]] = 12 + 3x0 + 3x1 > 6 + 3x0 + 3x1 = [[s(add(_x0, _x1))]] 
4.18/4.22	  [[yap(_F0, _x1)]] = x1 + F0(x1) >= x1 + F0(x1) = [[_F0 _x1]] 
4.18/4.22	
4.18/4.22	We can thus remove the following rules:
4.18/4.22	
4.18/4.22	  add(X, Y) => rec(/\x./\y.yap(succ(x), y), X, Y) 
4.18/4.22	  add(X, 0) => X 
4.18/4.22	  add(X, s(Y)) => s(add(X, Y)) 
4.18/4.22	
4.18/4.22	We use rule removal, following [Kop12, Theorem 2.23].
4.18/4.22	
4.18/4.22	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
4.18/4.22	
4.18/4.22	  yap(F, X) >? F X 
4.18/4.22	
4.18/4.22	We orient these requirements with a polynomial interpretation in the natural numbers.
4.18/4.22	
4.18/4.22	The following interpretation satisfies the requirements:
4.18/4.22	
4.18/4.22	  yap = \G0y1.1 + y1 + G0(y1) 
4.18/4.22	
4.18/4.22	Using this interpretation, the requirements translate to:
4.18/4.22	
4.18/4.22	  [[yap(_F0, _x1)]] = 1 + x1 + F0(x1) > x1 + F0(x1) = [[_F0 _x1]] 
4.18/4.22	
4.18/4.22	We can thus remove the following rules:
4.18/4.22	
4.18/4.22	  yap(F, X) => F X 
4.18/4.22	
4.18/4.22	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.
4.18/4.22	
4.18/4.22	
4.18/4.22	+++ Citations +++
4.18/4.22	
4.18/4.22	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
4.18/4.22	[Kop13:2]  C. Kop.  StarHorpo with an Eta-Rule.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/etahorpo.pdf, 2013.
4.18/4.22	EOF