35.41/13.55	YES
35.41/13.56	We consider the system theBenchmark.
35.41/13.56	
35.41/13.56	  Alphabet:
35.41/13.56	
35.41/13.56	    !facplus : [a * a] --> a 
35.41/13.56	    !factimes : [a * a] --> a 
35.41/13.56	    cons : [c * d] --> d 
35.41/13.56	    false : [] --> b 
35.41/13.56	    filter : [c -> b * d] --> d 
35.41/13.56	    filter2 : [b * c -> b * c * d] --> d 
35.41/13.56	    map : [c -> c * d] --> d 
35.41/13.56	    nil : [] --> d 
35.41/13.56	    true : [] --> b 
35.41/13.56	
35.41/13.56	  Rules:
35.41/13.56	
35.41/13.56	    !factimes(x, !facplus(y, z)) => !facplus(!factimes(x, y), !factimes(x, z)) 
35.41/13.56	    !factimes(!facplus(x, y), z) => !facplus(!factimes(z, x), !factimes(z, y)) 
35.41/13.56	    !factimes(!factimes(x, y), z) => !factimes(x, !factimes(y, z)) 
35.41/13.56	    !facplus(!facplus(x, y), z) => !facplus(x, !facplus(y, z)) 
35.41/13.56	    map(f, nil) => nil 
35.41/13.56	    map(f, cons(x, y)) => cons(f x, map(f, y)) 
35.41/13.56	    filter(f, nil) => nil 
35.41/13.56	    filter(f, cons(x, y)) => filter2(f x, f, x, y) 
35.41/13.56	    filter2(true, f, x, y) => cons(x, filter(f, y)) 
35.41/13.56	    filter2(false, f, x, y) => filter(f, y) 
35.41/13.56	
35.41/13.56	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
35.41/13.56	
35.41/13.56	We observe that the rules contain a first-order subset:
35.41/13.56	
35.41/13.56	  !factimes(X, !facplus(Y, Z)) => !facplus(!factimes(X, Y), !factimes(X, Z)) 
35.41/13.56	  !factimes(!facplus(X, Y), Z) => !facplus(!factimes(Z, X), !factimes(Z, Y)) 
35.41/13.56	  !factimes(!factimes(X, Y), Z) => !factimes(X, !factimes(Y, Z)) 
35.41/13.56	  !facplus(!facplus(X, Y), Z) => !facplus(X, !facplus(Y, Z)) 
35.41/13.56	
35.41/13.56	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.
35.41/13.56	
35.41/13.56	According to the external first-order termination prover, this system is indeed Ce-terminating:
35.41/13.56	
35.41/13.56	 || proof of resources/system.trs
35.41/13.56	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || Termination w.r.t. Q of the given QTRS could be proven:
35.41/13.56	 || 
35.41/13.56	 || (0) QTRS
35.41/13.56	 || (1) DependencyPairsProof [EQUIVALENT]
35.41/13.56	 || (2) QDP
35.41/13.56	 || (3) DependencyGraphProof [EQUIVALENT]
35.41/13.56	 || (4) AND
35.41/13.56	 ||     (5) QDP
35.41/13.56	 ||         (6) UsableRulesProof [EQUIVALENT]
35.41/13.56	 ||         (7) QDP
35.41/13.56	 ||         (8) QDPSizeChangeProof [EQUIVALENT]
35.41/13.56	 ||         (9) YES
35.41/13.56	 ||     (10) QDP
35.41/13.56	 ||         (11) QDPOrderProof [EQUIVALENT]
35.41/13.56	 ||         (12) QDP
35.41/13.56	 ||         (13) QDPSizeChangeProof [EQUIVALENT]
35.41/13.56	 ||         (14) YES
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (0)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q restricted rewrite system:
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %X
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %Y
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (1) DependencyPairsProof (EQUIVALENT)
35.41/13.56	 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (2)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q DP problem:
35.41/13.56	 || The TRS P consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACPLUS(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Y)
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Z)
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACPLUS(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %X)
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %Y)
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%Y, %Z)
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%Y, %Z)
35.41/13.56	 || 
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %X
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %Y
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || We have to consider all minimal (P,Q,R)-chains.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (3) DependencyGraphProof (EQUIVALENT)
35.41/13.56	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (4)
35.41/13.56	 || Complex Obligation (AND)
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (5)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q DP problem:
35.41/13.56	 || The TRS P consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%Y, %Z)
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%X, !facplus(%Y, %Z))
35.41/13.56	 || 
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %X
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %Y
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || We have to consider all minimal (P,Q,R)-chains.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (6) UsableRulesProof (EQUIVALENT)
35.41/13.56	 || We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (7)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q DP problem:
35.41/13.56	 || The TRS P consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%Y, %Z)
35.41/13.56	 ||    !FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%X, !facplus(%Y, %Z))
35.41/13.56	 || 
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || We have to consider all minimal (P,Q,R)-chains.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (8) QDPSizeChangeProof (EQUIVALENT)
35.41/13.56	 || 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. 
35.41/13.56	 || 
35.41/13.56	 || From the DPs we obtained the following set of size-change graphs:
35.41/13.56	 || *!FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%Y, %Z)
35.41/13.56	 || The graph contains the following edges 1 > 1, 2 >= 2
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || *!FACPLUS(!facplus(%X, %Y), %Z) -> !FACPLUS(%X, !facplus(%Y, %Z))
35.41/13.56	 || The graph contains the following edges 1 > 1
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (9)
35.41/13.56	 || YES
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (10)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q DP problem:
35.41/13.56	 || The TRS P consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Z)
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Y)
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %X)
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %Y)
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%Y, %Z)
35.41/13.56	 || 
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %X
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %Y
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || We have to consider all minimal (P,Q,R)-chains.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (11) QDPOrderProof (EQUIVALENT)
35.41/13.56	 || We use the reduction pair processor [LPAR04,JAR06].
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || The following pairs can be oriented strictly and are deleted.
35.41/13.56	 || 
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %X)
35.41/13.56	 ||    !FACTIMES(!facplus(%X, %Y), %Z) -> !FACTIMES(%Z, %Y)
35.41/13.56	 || The remaining pairs can at least be oriented weakly.
35.41/13.56	 || Used ordering:  Polynomial interpretation [POLO]:
35.41/13.56	 || 
35.41/13.56	 ||    POL(!FACTIMES(x_1, x_2)) = x_1 + x_1*x_2
35.41/13.56	 ||    POL(!facplus(x_1, x_2)) = 1 + x_1 + x_2
35.41/13.56	 ||    POL(!factimes(x_1, x_2)) = x_1 + x_1*x_2 + x_2
35.41/13.56	 || 
35.41/13.56	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (12)
35.41/13.56	 || Obligation:
35.41/13.56	 || Q DP problem:
35.41/13.56	 || The TRS P consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Z)
35.41/13.56	 ||    !FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Y)
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%Y, %Z)
35.41/13.56	 || 
35.41/13.56	 || The TRS R consists of the following rules:
35.41/13.56	 || 
35.41/13.56	 ||    !factimes(%X, !facplus(%Y, %Z)) -> !facplus(!factimes(%X, %Y), !factimes(%X, %Z))
35.41/13.56	 ||    !factimes(!facplus(%X, %Y), %Z) -> !facplus(!factimes(%Z, %X), !factimes(%Z, %Y))
35.41/13.56	 ||    !factimes(!factimes(%X, %Y), %Z) -> !factimes(%X, !factimes(%Y, %Z))
35.41/13.56	 ||    !facplus(!facplus(%X, %Y), %Z) -> !facplus(%X, !facplus(%Y, %Z))
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %X
35.41/13.56	 ||    ~PAIR(%X, %Y) -> %Y
35.41/13.56	 || 
35.41/13.56	 || Q is empty.
35.41/13.56	 || We have to consider all minimal (P,Q,R)-chains.
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (13) QDPSizeChangeProof (EQUIVALENT)
35.41/13.56	 || 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. 
35.41/13.56	 || 
35.41/13.56	 || From the DPs we obtained the following set of size-change graphs:
35.41/13.56	 || *!FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Z)
35.41/13.56	 || The graph contains the following edges 1 >= 1, 2 > 2
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || *!FACTIMES(%X, !facplus(%Y, %Z)) -> !FACTIMES(%X, %Y)
35.41/13.56	 || The graph contains the following edges 1 >= 1, 2 > 2
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || *!FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%X, !factimes(%Y, %Z))
35.41/13.56	 || The graph contains the following edges 1 > 1
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || *!FACTIMES(!factimes(%X, %Y), %Z) -> !FACTIMES(%Y, %Z)
35.41/13.56	 || The graph contains the following edges 1 > 1, 2 >= 2
35.41/13.56	 || 
35.41/13.56	 || 
35.41/13.56	 || ----------------------------------------
35.41/13.56	 || 
35.41/13.56	 || (14)
35.41/13.56	 || YES
35.41/13.56	 || 
35.41/13.56	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]).
35.41/13.56	
35.41/13.56	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
35.41/13.56	
35.41/13.56	  Dependency Pairs P_0:
35.41/13.56	
35.41/13.56	    0] map#(F, cons(X, Y)) =#> map#(F, Y)   
35.41/13.56	    1] filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
35.41/13.56	    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	
35.41/13.56	  Rules R_0:
35.41/13.56	
35.41/13.56	    !factimes(X, !facplus(Y, Z)) => !facplus(!factimes(X, Y), !factimes(X, Z)) 
35.41/13.56	    !factimes(!facplus(X, Y), Z) => !facplus(!factimes(Z, X), !factimes(Z, Y)) 
35.41/13.56	    !factimes(!factimes(X, Y), Z) => !factimes(X, !factimes(Y, Z)) 
35.41/13.56	    !facplus(!facplus(X, Y), Z) => !facplus(X, !facplus(Y, Z)) 
35.41/13.56	    map(F, nil) => nil 
35.41/13.56	    map(F, cons(X, Y)) => cons(F X, map(F, Y)) 
35.41/13.56	    filter(F, nil) => nil 
35.41/13.56	    filter(F, cons(X, Y)) => filter2(F X, F, X, Y) 
35.41/13.56	    filter2(true, F, X, Y) => cons(X, filter(F, Y)) 
35.41/13.56	    filter2(false, F, X, Y) => filter(F, Y) 
35.41/13.56	
35.41/13.56	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
35.41/13.56	
35.41/13.56	We consider the dependency pair problem (P_0, R_0, static, formative).
35.41/13.56	
35.41/13.56	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
35.41/13.56	
35.41/13.56	    * 0 : 0 
35.41/13.56	    * 1 : 2, 3 
35.41/13.56	    * 2 : 1 
35.41/13.56	    * 3 : 1 
35.41/13.56	
35.41/13.56	This graph has the following strongly connected components:
35.41/13.56	
35.41/13.56	  P_1:
35.41/13.56	
35.41/13.56	    map#(F, cons(X, Y)) =#> map#(F, Y)   
35.41/13.56	
35.41/13.56	  P_2:
35.41/13.56	
35.41/13.56	    filter#(F, cons(X, Y)) =#> filter2#(F X, F, X, Y)   
35.41/13.56	    filter2#(true, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	    filter2#(false, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	
35.41/13.56	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).
35.41/13.56	
35.41/13.56	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.
35.41/13.56	
35.41/13.56	We consider the dependency pair problem (P_2, R_0, static, formative).
35.41/13.56	
35.41/13.56	We apply the subterm criterion with the following projection function:
35.41/13.56	
35.41/13.56	  nu(filter2#) = 4 
35.41/13.56	  nu(filter#) = 2 
35.41/13.56	
35.41/13.56	Thus, we can orient the dependency pairs as follows:
35.41/13.56	
35.41/13.56	  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
35.41/13.56	  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
35.41/13.56	  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
35.41/13.56	
35.41/13.56	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:
35.41/13.56	
35.41/13.56	  filter2#(true, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	  filter2#(false, F, X, Y) =#> filter#(F, Y)   
35.41/13.56	
35.41/13.56	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite.
35.41/13.56	
35.41/13.56	We consider the dependency pair problem (P_3, R_0, static, formative).
35.41/13.56	
35.41/13.56	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
35.41/13.56	
35.41/13.56	    * 0 :  
35.41/13.56	    * 1 :  
35.41/13.56	
35.41/13.56	This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.
35.41/13.56	
35.41/13.56	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
35.41/13.56	
35.41/13.56	We consider the dependency pair problem (P_1, R_0, static, formative).
35.41/13.56	
35.41/13.56	We apply the subterm criterion with the following projection function:
35.41/13.56	
35.41/13.56	  nu(map#) = 2 
35.41/13.56	
35.41/13.56	Thus, we can orient the dependency pairs as follows:
35.41/13.56	
35.41/13.56	  nu(map#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(map#(F, Y)) 
35.41/13.56	
35.41/13.56	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.
35.41/13.56	
35.41/13.56	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
35.41/13.56	
35.41/13.56	
35.41/13.56	+++ Citations +++
35.41/13.56	
35.41/13.56	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
35.41/13.56	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
35.41/13.56	[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.
35.41/13.56	EOF