1.65/0.76	YES
1.65/0.77	We consider the system theBenchmark.
1.65/0.77	
1.65/0.77	  Alphabet:
1.65/0.77	
1.65/0.77	    dom : [N * N * N] --> N 
1.65/0.77	    eval : [N * N] --> N 
1.65/0.77	    fun : [N -> N * N * N] --> N 
1.65/0.77	    o : [] --> N 
1.65/0.77	    s : [N] --> N 
1.65/0.77	
1.65/0.77	  Rules:
1.65/0.77	
1.65/0.77	    eval(fun(f, x, y), z) => f dom(x, y, z) 
1.65/0.77	    dom(s(x), s(y), s(z)) => s(dom(x, y, z)) 
1.65/0.77	    dom(o, s(x), s(y)) => s(dom(o, x, y)) 
1.65/0.77	    dom(x, y, o) => x 
1.65/0.77	    dom(o, o, x) => o 
1.65/0.77	
1.65/0.77	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
1.65/0.77	
1.65/0.77	We observe that the rules contain a first-order subset:
1.65/0.77	
1.65/0.77	  dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 
1.65/0.77	  dom(o, s(X), s(Y)) => s(dom(o, X, Y)) 
1.65/0.77	  dom(X, Y, o) => X 
1.65/0.77	  dom(o, o, X) => o 
1.65/0.77	
1.65/0.77	Moreover, the system is finitely branching.  Thus, by [Kop12, Thm. 7.55], we may omit all first-order dependency pairs from the dependency pair problem (DP(R), R) if this first-order part is Ce-terminating when seen as a many-sorted first-order TRS.
1.65/0.77	
1.65/0.77	According to the external first-order termination prover, this system is indeed Ce-terminating:
1.65/0.77	
1.65/0.77	 || proof of resources/system.trs
1.65/0.77	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
1.65/0.77	 || 
1.65/0.77	 || 
1.65/0.77	 || Termination w.r.t. Q of the given QTRS could be proven:
1.65/0.77	 || 
1.65/0.77	 || (0) QTRS
1.65/0.77	 || (1) QTRSRRRProof [EQUIVALENT]
1.65/0.77	 || (2) QTRS
1.65/0.77	 || (3) RisEmptyProof [EQUIVALENT]
1.65/0.77	 || (4) YES
1.65/0.77	 || 
1.65/0.77	 || 
1.65/0.77	 || ----------------------------------------
1.65/0.77	 || 
1.65/0.77	 || (0)
1.65/0.77	 || Obligation:
1.65/0.77	 || Q restricted rewrite system:
1.65/0.77	 || The TRS R consists of the following rules:
1.65/0.77	 || 
1.65/0.77	 ||    dom(s(%X), s(%Y), s(%Z)) -> s(dom(%X, %Y, %Z))
1.65/0.77	 ||    dom(o, s(%X), s(%Y)) -> s(dom(o, %X, %Y))
1.65/0.77	 ||    dom(%X, %Y, o) -> %X
1.65/0.77	 ||    dom(o, o, %X) -> o
1.65/0.77	 ||    ~PAIR(%X, %Y) -> %X
1.65/0.77	 ||    ~PAIR(%X, %Y) -> %Y
1.65/0.77	 || 
1.65/0.77	 || Q is empty.
1.65/0.77	 || 
1.65/0.77	 || ----------------------------------------
1.65/0.77	 || 
1.65/0.77	 || (1) QTRSRRRProof (EQUIVALENT)
1.65/0.77	 || Used ordering:
1.65/0.77	 || Polynomial interpretation [POLO]:
1.65/0.77	 || 
1.65/0.77	 ||    POL(dom(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3
1.65/0.77	 ||    POL(o) = 2
1.65/0.77	 ||    POL(s(x_1)) = 1 + x_1
1.65/0.77	 ||    POL(~PAIR(x_1, x_2)) = 2 + x_1 + x_2
1.65/0.77	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
1.65/0.77	 || 
1.65/0.77	 ||    dom(s(%X), s(%Y), s(%Z)) -> s(dom(%X, %Y, %Z))
1.65/0.77	 ||    dom(o, s(%X), s(%Y)) -> s(dom(o, %X, %Y))
1.65/0.77	 ||    dom(%X, %Y, o) -> %X
1.65/0.77	 ||    dom(o, o, %X) -> o
1.65/0.77	 ||    ~PAIR(%X, %Y) -> %X
1.65/0.77	 ||    ~PAIR(%X, %Y) -> %Y
1.65/0.77	 || 
1.65/0.77	 || 
1.65/0.77	 || 
1.65/0.77	 || 
1.65/0.77	 || ----------------------------------------
1.65/0.77	 || 
1.65/0.77	 || (2)
1.65/0.77	 || Obligation:
1.65/0.77	 || Q restricted rewrite system:
1.65/0.77	 || R is empty.
1.65/0.77	 || Q is empty.
1.65/0.77	 || 
1.65/0.77	 || ----------------------------------------
1.65/0.77	 || 
1.65/0.77	 || (3) RisEmptyProof (EQUIVALENT)
1.65/0.77	 || The TRS R is empty. Hence, termination is trivially proven.
1.65/0.77	 || ----------------------------------------
1.65/0.77	 || 
1.65/0.77	 || (4)
1.65/0.77	 || YES
1.65/0.77	 || 
1.65/0.77	We use the dependency pair framework as described in [Kop12, Ch. 6/7], with dynamic dependency pairs.
1.65/0.77	
1.65/0.77	After applying [Kop12, Thm. 7.22] to denote collapsing dependency pairs in an extended form, we thus obtain the following dependency pair problem (P_0, R_0, minimal, formative):
1.65/0.77	
1.65/0.77	  Dependency Pairs P_0:
1.65/0.77	
1.65/0.77	    0] eval#(fun(F, X, Y), Z) =#> F(dom(X, Y, Z))   
1.65/0.77	    1] eval#(fun(F, X, Y), Z) =#> dom#(X, Y, Z)   
1.65/0.77	
1.65/0.77	  Rules R_0:
1.65/0.77	
1.65/0.77	    eval(fun(F, X, Y), Z) => F dom(X, Y, Z) 
1.65/0.77	    dom(s(X), s(Y), s(Z)) => s(dom(X, Y, Z)) 
1.65/0.77	    dom(o, s(X), s(Y)) => s(dom(o, X, Y)) 
1.65/0.77	    dom(X, Y, o) => X 
1.65/0.77	    dom(o, o, X) => o 
1.65/0.77	
1.65/0.77	Thus, the original system is terminating if (P_0, R_0, minimal, formative) is finite.
1.65/0.77	
1.65/0.77	We consider the dependency pair problem (P_0, R_0, minimal, formative).
1.65/0.77	
1.65/0.77	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
1.65/0.77	
1.65/0.77	    * 0 : 0, 1 
1.65/0.77	    * 1 :  
1.65/0.77	
1.65/0.77	This graph has the following strongly connected components:
1.65/0.77	
1.65/0.77	  P_1:
1.65/0.77	
1.65/0.77	    eval#(fun(F, X, Y), Z) =#> F(dom(X, Y, Z))   
1.65/0.77	
1.65/0.77	By [Kop12, Thm. 7.31], we may replace any dependency pair problem (P_0, R_0, m, f) by (P_1, R_0, m, f).
1.65/0.77	
1.65/0.77	Thus, the original system is terminating if (P_1, R_0, minimal, formative) is finite.
1.65/0.77	
1.65/0.77	We consider the dependency pair problem (P_1, R_0, minimal, formative).
1.65/0.77	
1.65/0.77	The formative rules of (P_1, R_0) are R_1 ::=
1.65/0.77	
1.65/0.77	  eval(fun(F, X, Y), Z) => F dom(X, Y, Z) 
1.65/0.77	  dom(X, Y, o) => X 
1.65/0.77	  dom(o, o, X) => o 
1.65/0.77	
1.65/0.77	By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, minimal, formative) by (P_1, R_1, minimal, formative).
1.65/0.77	
1.65/0.77	Thus, the original system is terminating if (P_1, R_1, minimal, formative) is finite.
1.65/0.77	
1.65/0.77	We consider the dependency pair problem (P_1, R_1, minimal, formative).
1.65/0.77	
1.65/0.77	We will use the reduction pair processor [Kop12, Thm. 7.16].  As the system is abstraction-simple and the formative flag is set, it suffices to find a tagged reduction pair [Kop12, Def. 6.70].  Thus, we must orient:
1.65/0.77	
1.65/0.77	  eval#(fun(F, X, Y), Z) >? F(dom(X, Y, Z)) 
1.65/0.77	  eval(fun(F, X, Y), Z) >= F dom(X, Y, Z) 
1.65/0.77	  dom(X, Y, o) >= X 
1.65/0.77	  dom(o, o, X) >= o 
1.65/0.77	
1.65/0.77	We orient these requirements with a polynomial interpretation in the natural numbers.
1.65/0.77	
1.65/0.77	The following interpretation satisfies the requirements:
1.65/0.77	
1.65/0.77	  dom = \y0y1y2.y0 
1.65/0.77	  eval = \y0y1.3 + 3y0 
1.65/0.77	  eval# = \y0y1.3 + y0 
1.65/0.77	  fun = \G0y1y2.3 + y1 + G0(y1) 
1.65/0.77	  o = 0 
1.65/0.77	
1.65/0.77	Using this interpretation, the requirements translate to:
1.65/0.77	
1.65/0.77	  [[eval#(fun(_F0, _x1, _x2), _x3)]] = 6 + x1 + F0(x1) > F0(x1) = [[_F0(dom(_x1, _x2, _x3))]] 
1.65/0.77	  [[eval(fun(_F0, _x1, _x2), _x3)]] = 12 + 3x1 + 3F0(x1) >= max(x1, F0(x1)) = [[_F0 dom(_x1, _x2, _x3)]] 
1.65/0.77	  [[dom(_x0, _x1, o)]] = x0 >= x0 = [[_x0]] 
1.65/0.77	  [[dom(o, o, _x0)]] = 0 >= 0 = [[o]] 
1.65/0.77	
1.65/0.77	By the observations in [Kop12, Sec. 6.6], this reduction pair suffices; we may thus replace a dependency pair problem (P_1, R_1) by ({}, R_1).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.
1.65/0.77	
1.65/0.77	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
1.65/0.77	
1.65/0.77	
1.65/0.77	+++ Citations +++
1.65/0.77	
1.65/0.77	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
1.65/0.77	EOF