1.91/1.05	YES
1.91/1.06	We consider the system theBenchmark.
1.91/1.06	
1.91/1.06	  Alphabet:
1.91/1.06	
1.91/1.06	    0 : [] --> a 
1.91/1.06	    cons : [c * c] --> c 
1.91/1.06	    copy : [a * c * c] --> c 
1.91/1.06	    f : [c] --> c 
1.91/1.06	    false : [] --> b 
1.91/1.06	    filter : [c -> b * c] --> c 
1.91/1.06	    filter2 : [b * c -> b * c * c] --> c 
1.91/1.06	    map : [c -> c * c] --> c 
1.91/1.06	    n : [] --> a 
1.91/1.06	    nil : [] --> c 
1.91/1.06	    s : [a] --> a 
1.91/1.06	    true : [] --> b 
1.91/1.06	
1.91/1.06	  Rules:
1.91/1.06	
1.91/1.06	    f(cons(nil, x)) => x 
1.91/1.06	    f(cons(f(cons(nil, x)), y)) => copy(n, x, y) 
1.91/1.06	    copy(0, x, y) => f(y) 
1.91/1.06	    copy(s(x), y, z) => copy(x, y, cons(f(y), z)) 
1.91/1.06	    map(g, nil) => nil 
1.91/1.06	    map(g, cons(x, y)) => cons(g x, map(g, y)) 
1.91/1.06	    filter(g, nil) => nil 
1.91/1.06	    filter(g, cons(x, y)) => filter2(g x, g, x, y) 
1.91/1.06	    filter2(true, g, x, y) => cons(x, filter(g, y)) 
1.91/1.06	    filter2(false, g, x, y) => filter(g, y) 
1.91/1.06	
1.91/1.06	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
1.91/1.06	
1.91/1.06	We observe that the rules contain a first-order subset:
1.91/1.06	
1.91/1.06	  f(cons(nil, X)) => X 
1.91/1.06	  f(cons(f(cons(nil, X)), Y)) => copy(n, X, Y) 
1.91/1.06	  copy(0, X, Y) => f(Y) 
1.91/1.06	  copy(s(X), Y, Z) => copy(X, Y, cons(f(Y), Z)) 
1.91/1.06	
1.91/1.06	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.
1.91/1.06	
1.91/1.06	According to the external first-order termination prover, this system is indeed Ce-terminating:
1.91/1.06	
1.91/1.06	 || proof of resources/system.trs
1.91/1.06	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
1.91/1.06	 || 
1.91/1.06	 || 
1.91/1.06	 || Termination w.r.t. Q of the given QTRS could be proven:
1.91/1.06	 || 
1.91/1.06	 || (0) QTRS
1.91/1.06	 || (1) DependencyPairsProof [EQUIVALENT]
1.91/1.06	 || (2) QDP
1.91/1.06	 || (3) DependencyGraphProof [EQUIVALENT]
1.91/1.06	 || (4) QDP
1.91/1.06	 || (5) QDPSizeChangeProof [EQUIVALENT]
1.91/1.06	 || (6) YES
1.91/1.06	 || 
1.91/1.06	 || 
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (0)
1.91/1.06	 || Obligation:
1.91/1.06	 || Q restricted rewrite system:
1.91/1.06	 || The TRS R consists of the following rules:
1.91/1.06	 || 
1.91/1.06	 ||    f(cons(nil, %X)) -> %X
1.91/1.06	 ||    f(cons(f(cons(nil, %X)), %Y)) -> copy(n, %X, %Y)
1.91/1.06	 ||    copy(0, %X, %Y) -> f(%Y)
1.91/1.06	 ||    copy(s(%X), %Y, %Z) -> copy(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %X
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %Y
1.91/1.06	 || 
1.91/1.06	 || Q is empty.
1.91/1.06	 || 
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (1) DependencyPairsProof (EQUIVALENT)
1.91/1.06	 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (2)
1.91/1.06	 || Obligation:
1.91/1.06	 || Q DP problem:
1.91/1.06	 || The TRS P consists of the following rules:
1.91/1.06	 || 
1.91/1.06	 ||    F(cons(f(cons(nil, %X)), %Y)) -> COPY(n, %X, %Y)
1.91/1.06	 ||    COPY(0, %X, %Y) -> F(%Y)
1.91/1.06	 ||    COPY(s(%X), %Y, %Z) -> COPY(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 ||    COPY(s(%X), %Y, %Z) -> F(%Y)
1.91/1.06	 || 
1.91/1.06	 || The TRS R consists of the following rules:
1.91/1.06	 || 
1.91/1.06	 ||    f(cons(nil, %X)) -> %X
1.91/1.06	 ||    f(cons(f(cons(nil, %X)), %Y)) -> copy(n, %X, %Y)
1.91/1.06	 ||    copy(0, %X, %Y) -> f(%Y)
1.91/1.06	 ||    copy(s(%X), %Y, %Z) -> copy(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %X
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %Y
1.91/1.06	 || 
1.91/1.06	 || Q is empty.
1.91/1.06	 || We have to consider all minimal (P,Q,R)-chains.
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (3) DependencyGraphProof (EQUIVALENT)
1.91/1.06	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (4)
1.91/1.06	 || Obligation:
1.91/1.06	 || Q DP problem:
1.91/1.06	 || The TRS P consists of the following rules:
1.91/1.06	 || 
1.91/1.06	 ||    COPY(s(%X), %Y, %Z) -> COPY(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 || 
1.91/1.06	 || The TRS R consists of the following rules:
1.91/1.06	 || 
1.91/1.06	 ||    f(cons(nil, %X)) -> %X
1.91/1.06	 ||    f(cons(f(cons(nil, %X)), %Y)) -> copy(n, %X, %Y)
1.91/1.06	 ||    copy(0, %X, %Y) -> f(%Y)
1.91/1.06	 ||    copy(s(%X), %Y, %Z) -> copy(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %X
1.91/1.06	 ||    ~PAIR(%X, %Y) -> %Y
1.91/1.06	 || 
1.91/1.06	 || Q is empty.
1.91/1.06	 || We have to consider all minimal (P,Q,R)-chains.
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (5) QDPSizeChangeProof (EQUIVALENT)
1.91/1.06	 || By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
1.91/1.06	 || 
1.91/1.06	 || From the DPs we obtained the following set of size-change graphs:
1.91/1.06	 || *COPY(s(%X), %Y, %Z) -> COPY(%X, %Y, cons(f(%Y), %Z))
1.91/1.06	 || The graph contains the following edges 1 > 1, 2 >= 2
1.91/1.06	 || 
1.91/1.06	 || 
1.91/1.06	 || ----------------------------------------
1.91/1.06	 || 
1.91/1.06	 || (6)
1.91/1.06	 || YES
1.91/1.06	 || 
1.91/1.06	We use the dependency pair framework as described in [Kop12, Ch. 6/7], with static dependency pairs (see [KusIsoSakBla09] and the adaptation for AFSMs and accessible arguments in [Kop13]).
1.91/1.06	
1.91/1.06	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
1.91/1.06	
1.91/1.06	  Dependency Pairs P_0:
1.91/1.06	
1.91/1.06	    0] map#(F, cons(X, Y)) =#> map#(F, Y)   
1.91/1.06	    1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
1.91/1.06	    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	
1.91/1.06	  Rules R_0:
1.91/1.06	
1.91/1.06	    f(cons(nil, X)) => X 
1.91/1.06	    f(cons(f(cons(nil, X)), Y)) => copy(n, X, Y) 
1.91/1.06	    copy(0, X, Y) => f(Y) 
1.91/1.06	    copy(s(X), Y, Z) => copy(X, Y, cons(f(Y), Z)) 
1.91/1.06	    map(F, nil) => nil 
1.91/1.06	    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
1.91/1.06	    filter(F, nil) => nil 
1.91/1.06	    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
1.91/1.06	    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
1.91/1.06	    filter2(false, F, X, Y) => filter(F, Y) 
1.91/1.06	
1.91/1.06	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
1.91/1.06	
1.91/1.06	We consider the dependency pair problem (P_0, R_0, static, formative).
1.91/1.06	
1.91/1.06	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
1.91/1.06	
1.91/1.06	    * 0 : 0 
1.91/1.06	    * 1 : 2, 3 
1.91/1.06	    * 2 : 1 
1.91/1.06	    * 3 : 1 
1.91/1.06	
1.91/1.06	This graph has the following strongly connected components:
1.91/1.06	
1.91/1.06	  P_1:
1.91/1.06	
1.91/1.06	    map#(F, cons(X, Y)) =#> map#(F, Y)   
1.91/1.06	
1.91/1.06	  P_2:
1.91/1.06	
1.91/1.06	    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
1.91/1.06	    filter2#(true, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	    filter2#(false, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	
1.91/1.06	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) and (P_2, R_0, m, f).
1.91/1.06	
1.91/1.06	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.
1.91/1.06	
1.91/1.06	We consider the dependency pair problem (P_2, R_0, static, formative).
1.91/1.06	
1.91/1.06	We apply the subterm criterion with the following projection function:
1.91/1.06	
1.91/1.06	  nu(filter2#) = 4 
1.91/1.06	  nu(filter#) = 2 
1.91/1.06	
1.91/1.06	Thus, we can orient the dependency pairs as follows:
1.91/1.06	
1.91/1.06	  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
1.91/1.06	  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
1.91/1.06	  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
1.91/1.06	
1.91/1.06	By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_2, R_0, static, f) by (P_3, R_0, static, f), where P_3 contains:
1.91/1.06	
1.91/1.06	  filter2#(true, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	  filter2#(false, F, X, Y) =#> filter#(F, Y)   
1.91/1.06	
1.91/1.06	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite.
1.91/1.06	
1.91/1.06	We consider the dependency pair problem (P_3, R_0, static, formative).
1.91/1.06	
1.91/1.06	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
1.91/1.06	
1.91/1.06	    * 0 :  
1.91/1.06	    * 1 :  
1.91/1.06	
1.91/1.06	This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.
1.91/1.06	
1.91/1.06	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
1.91/1.06	
1.91/1.06	We consider the dependency pair problem (P_1, R_0, static, formative).
1.91/1.06	
1.91/1.06	We apply the subterm criterion with the following projection function:
1.91/1.06	
1.91/1.06	  nu(map#) = 2 
1.91/1.06	
1.91/1.06	Thus, we can orient the dependency pairs as follows:
1.91/1.06	
1.91/1.06	  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 
1.91/1.06	
1.91/1.06	By [Kop12, Thm. 7.35] and [Kop13, Thm. 5], we may replace a dependency pair problem (P_1, R_0, static, f) by ({}, R_0, static, f).  By the empty set processor [Kop12, Thm. 7.15] this problem may be immediately removed.
1.91/1.06	
1.91/1.06	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
1.91/1.06	
1.91/1.06	
1.91/1.06	+++ Citations +++
1.91/1.06	
1.91/1.06	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
1.91/1.06	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
1.91/1.06	[KusIsoSakBla09]  K. Kusakari, Y. Isogai, M. Sakai, and F. Blanqui.  Static Dependency Pair Method Based On Strong Computability for Higher-Order Rewrite Systems.  In volume 92(10) of IEICE Transactions on Information and Systems.  2007--2015, 2009.
1.91/1.06	EOF