3.62/2.14	YES
3.62/2.15	We consider the system theBenchmark.
3.62/2.15	
3.62/2.15	  Alphabet:
3.62/2.15	
3.62/2.15	    0 : [] --> nat 
3.62/2.15	    app : [natlist * natlist] --> natlist 
3.62/2.15	    apply2 : [pair -> nat -> pair * pair * nat] --> pair 
3.62/2.15	    cons : [nat * natlist] --> natlist 
3.62/2.15	    fst : [pair] --> natlist 
3.62/2.15	    nil : [] --> natlist 
3.62/2.15	    p : [natlist * natlist] --> pair 
3.62/2.15	    pcons : [pair * plist] --> plist 
3.62/2.15	    pnil : [] --> plist 
3.62/2.15	    pps : [natlist] --> plist 
3.62/2.15	    prefixshuffle : [pair * natlist] --> plist 
3.62/2.15	    pshuffle : [natlist] --> pair 
3.62/2.15	    reverse : [natlist] --> natlist 
3.62/2.15	    s : [nat] --> nat 
3.62/2.15	    shuffle : [natlist] --> natlist 
3.62/2.15	
3.62/2.15	  Rules:
3.62/2.15	
3.62/2.15	    app(nil, x) => x 
3.62/2.15	    app(cons(x, y), z) => cons(x, app(y, z)) 
3.62/2.15	    reverse(nil) => nil 
3.62/2.15	    reverse(cons(x, y)) => app(reverse(y), cons(x, nil)) 
3.62/2.15	    shuffle(nil) => nil 
3.62/2.15	    shuffle(cons(x, y)) => cons(x, shuffle(reverse(y))) 
3.62/2.15	    fst(p(x, y)) => x 
3.62/2.15	    pshuffle(x) => p(x, shuffle(x)) 
3.62/2.15	    prefixshuffle(x, nil) => pcons(x, pnil) 
3.62/2.15	    prefixshuffle(x, cons(y, z)) => pcons(x, prefixshuffle(apply2(/\u./\v.pshuffle(app(fst(u), cons(v, nil))), x, y), reverse(z))) 
3.62/2.16	    apply2(f, x, 0) => x 
3.62/2.16	    apply2(f, x, s(y)) => f x s(y) 
3.62/2.16	    pps(x) => prefixshuffle(p(nil, nil), x) 
3.62/2.16	
3.62/2.16	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
3.62/2.16	
3.62/2.16	We observe that the rules contain a first-order subset:
3.62/2.16	
3.62/2.16	  app(nil, X) => X 
3.62/2.16	  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
3.62/2.16	  reverse(nil) => nil 
3.62/2.16	  reverse(cons(X, Y)) => app(reverse(Y), cons(X, nil)) 
3.62/2.16	  shuffle(nil) => nil 
3.62/2.16	  shuffle(cons(X, Y)) => cons(X, shuffle(reverse(Y))) 
3.62/2.16	  fst(p(X, Y)) => X 
3.62/2.16	  pshuffle(X) => p(X, shuffle(X)) 
3.62/2.16	
3.62/2.16	Moreover, the system is orthogonal.  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 terminating when seen as a many-sorted first-order TRS.
3.62/2.16	
3.62/2.16	According to the external first-order termination prover, this system is indeed terminating:
3.62/2.16	
3.62/2.16	 || proof of resources/system.trs
3.62/2.16	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || Termination w.r.t. Q of the given QTRS could be proven:
3.62/2.16	 || 
3.62/2.16	 || (0) QTRS
3.62/2.16	 || (1) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (2) QTRS
3.62/2.16	 || (3) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (4) QTRS
3.62/2.16	 || (5) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (6) QTRS
3.62/2.16	 || (7) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (8) QTRS
3.62/2.16	 || (9) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (10) QTRS
3.62/2.16	 || (11) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (12) QTRS
3.62/2.16	 || (13) QTRSRRRProof [EQUIVALENT]
3.62/2.16	 || (14) QTRS
3.62/2.16	 || (15) RisEmptyProof [EQUIVALENT]
3.62/2.16	 || (16) YES
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (0)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 ||    shuffle(nil) -> nil
3.62/2.16	 ||    shuffle(cons(%X, %Y)) -> cons(%X, shuffle(reverse(%Y)))
3.62/2.16	 ||    fst(p(%X, %Y)) -> %X
3.62/2.16	 ||    pshuffle(%X) -> p(%X, shuffle(%X))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (1) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(fst(x_1)) = 2 + x_1
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(p(x_1, x_2)) = 2 + x_1 + x_2
3.62/2.16	 ||    POL(pshuffle(x_1)) = 2 + 2*x_1
3.62/2.16	 ||    POL(reverse(x_1)) = x_1
3.62/2.16	 ||    POL(shuffle(x_1)) = x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    fst(p(%X, %Y)) -> %X
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (2)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 ||    shuffle(nil) -> nil
3.62/2.16	 ||    shuffle(cons(%X, %Y)) -> cons(%X, shuffle(reverse(%Y)))
3.62/2.16	 ||    pshuffle(%X) -> p(%X, shuffle(%X))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (3) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = 2*x_1 + x_2
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(p(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(pshuffle(x_1)) = 1 + 2*x_1
3.62/2.16	 ||    POL(reverse(x_1)) = x_1
3.62/2.16	 ||    POL(shuffle(x_1)) = x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    pshuffle(%X) -> p(%X, shuffle(%X))
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (4)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 ||    shuffle(nil) -> nil
3.62/2.16	 ||    shuffle(cons(%X, %Y)) -> cons(%X, shuffle(reverse(%Y)))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (5) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = 2 + 2*x_1 + x_2
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(reverse(x_1)) = x_1
3.62/2.16	 ||    POL(shuffle(x_1)) = 2*x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    shuffle(cons(%X, %Y)) -> cons(%X, shuffle(reverse(%Y)))
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (6)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 ||    shuffle(nil) -> nil
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (7) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + 2*x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(reverse(x_1)) = 2*x_1
3.62/2.16	 ||    POL(shuffle(x_1)) = 1 + 2*x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    shuffle(nil) -> nil
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (8)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (9) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(reverse(x_1)) = 1 + x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    reverse(nil) -> nil
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (10)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (11) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Polynomial interpretation [POLO]:
3.62/2.16	 || 
3.62/2.16	 ||    POL(app(x_1, x_2)) = x_1 + x_2
3.62/2.16	 ||    POL(cons(x_1, x_2)) = 1 + 2*x_1 + x_2
3.62/2.16	 ||    POL(nil) = 0
3.62/2.16	 ||    POL(reverse(x_1)) = 2*x_1
3.62/2.16	 || With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    reverse(cons(%X, %Y)) -> app(reverse(%Y), cons(%X, nil))
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (12)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || The TRS R consists of the following rules:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 || 
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (13) QTRSRRRProof (EQUIVALENT)
3.62/2.16	 || Used ordering:
3.62/2.16	 || Knuth-Bendix order [KBO] with precedence:app_2 > cons_2 > nil
3.62/2.16	 || 
3.62/2.16	 || and weight map:
3.62/2.16	 || 
3.62/2.16	 ||    nil=1
3.62/2.16	 ||    app_2=0
3.62/2.16	 ||    cons_2=0
3.62/2.16	 || 
3.62/2.16	 || The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/2.16	 || 
3.62/2.16	 ||    app(nil, %X) -> %X
3.62/2.16	 ||    app(cons(%X, %Y), %Z) -> cons(%X, app(%Y, %Z))
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (14)
3.62/2.16	 || Obligation:
3.62/2.16	 || Q restricted rewrite system:
3.62/2.16	 || R is empty.
3.62/2.16	 || Q is empty.
3.62/2.16	 || 
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (15) RisEmptyProof (EQUIVALENT)
3.62/2.16	 || The TRS R is empty. Hence, termination is trivially proven.
3.62/2.16	 || ----------------------------------------
3.62/2.16	 || 
3.62/2.16	 || (16)
3.62/2.16	 || YES
3.62/2.16	 || 
3.62/2.16	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]).
3.62/2.16	
3.62/2.16	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
3.62/2.16	
3.62/2.16	  Dependency Pairs P_0:
3.62/2.16	
3.62/2.16	    0] prefixshuffle#(X, cons(Y, Z)) =#> prefixshuffle#(apply2(/\x./\y.pshuffle(app(fst(x), cons(y, nil))), X, Y), reverse(Z))   
3.62/2.16	    1] prefixshuffle#(X, cons(Y, Z)) =#> apply2#(/\x./\y.pshuffle(app(fst(x), cons(y, nil))), X, Y)   
3.62/2.16	    2] prefixshuffle#(X, cons(Y, Z)) =#> pshuffle#(app(fst(U), cons(V, nil)))   
3.62/2.16	    3] prefixshuffle#(X, cons(Y, Z)) =#> app#(fst(U), cons(V, nil))   
3.62/2.16	    4] prefixshuffle#(X, cons(Y, Z)) =#> fst#(U)   
3.62/2.16	    5] prefixshuffle#(X, cons(Y, Z)) =#> reverse#(Z)   
3.62/2.16	    6] pps#(X) =#> prefixshuffle#(p(nil, nil), X)   
3.62/2.16	
3.62/2.16	  Rules R_0:
3.62/2.16	
3.62/2.16	    app(nil, X) => X 
3.62/2.16	    app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
3.62/2.16	    reverse(nil) => nil 
3.62/2.16	    reverse(cons(X, Y)) => app(reverse(Y), cons(X, nil)) 
3.62/2.16	    shuffle(nil) => nil 
3.62/2.16	    shuffle(cons(X, Y)) => cons(X, shuffle(reverse(Y))) 
3.62/2.16	    fst(p(X, Y)) => X 
3.62/2.16	    pshuffle(X) => p(X, shuffle(X)) 
3.62/2.16	    prefixshuffle(X, nil) => pcons(X, pnil) 
3.62/2.16	    prefixshuffle(X, cons(Y, Z)) => pcons(X, prefixshuffle(apply2(/\x./\y.pshuffle(app(fst(x), cons(y, nil))), X, Y), reverse(Z))) 
3.62/2.16	    apply2(F, X, 0) => X 
3.62/2.16	    apply2(F, X, s(Y)) => F X s(Y) 
3.62/2.16	    pps(X) => prefixshuffle(p(nil, nil), X) 
3.62/2.16	
3.62/2.16	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
3.62/2.16	
3.62/2.16	We consider the dependency pair problem (P_0, R_0, static, formative).
3.62/2.16	
3.62/2.16	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
3.62/2.16	
3.62/2.16	    * 0 : 0, 1, 2, 3, 4, 5 
3.62/2.16	    * 1 :  
3.62/2.16	    * 2 :  
3.62/2.16	    * 3 :  
3.62/2.16	    * 4 :  
3.62/2.16	    * 5 :  
3.62/2.16	    * 6 : 0, 1, 2, 3, 4, 5 
3.62/2.16	
3.62/2.16	This graph has the following strongly connected components:
3.62/2.16	
3.62/2.16	  P_1:
3.62/2.16	
3.62/2.16	    prefixshuffle#(X, cons(Y, Z)) =#> prefixshuffle#(apply2(/\x./\y.pshuffle(app(fst(x), cons(y, nil))), X, Y), reverse(Z))   
3.62/2.16	
3.62/2.16	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).
3.62/2.16	
3.62/2.16	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
3.62/2.16	
3.62/2.16	We consider the dependency pair problem (P_1, R_0, static, formative).
3.62/2.16	
3.62/2.16	The formative rules of (P_1, R_0) are R_1 ::=
3.62/2.16	
3.62/2.16	  app(nil, X) => X 
3.62/2.16	  app(cons(X, Y), Z) => cons(X, app(Y, Z)) 
3.62/2.16	  reverse(nil) => nil 
3.62/2.16	  reverse(cons(X, Y)) => app(reverse(Y), cons(X, nil)) 
3.62/2.16	  shuffle(nil) => nil 
3.62/2.16	  shuffle(cons(X, Y)) => cons(X, shuffle(reverse(Y))) 
3.62/2.16	  fst(p(X, Y)) => X 
3.62/2.16	  pshuffle(X) => p(X, shuffle(X)) 
3.62/2.16	  apply2(F, X, 0) => X 
3.62/2.16	  apply2(F, X, s(Y)) => F X s(Y) 
3.62/2.16	
3.62/2.16	By [Kop12, Thm. 7.17], we may replace the dependency pair problem (P_1, R_0, static, formative) by (P_1, R_1, static, formative).
3.62/2.16	
3.62/2.16	Thus, the original system is terminating if (P_1, R_1, static, formative) is finite.
3.62/2.16	
3.62/2.16	We consider the dependency pair problem (P_1, R_1, static, formative).
3.62/2.16	
3.62/2.16	We will use the reduction pair processor [Kop12, Thm. 7.16].  It suffices to find a standard reduction pair [Kop12, Def. 6.69].  Thus, we must orient:
3.62/2.16	
3.62/2.16	  prefixshuffle#(X, cons(Y, Z)) >? prefixshuffle#(apply2(/\x./\y.pshuffle(app(fst(x), cons(y, nil))), X, Y), reverse(Z)) 
3.62/2.16	  app(nil, X) >= X 
3.62/2.16	  app(cons(X, Y), Z) >= cons(X, app(Y, Z)) 
3.62/2.16	  reverse(nil) >= nil 
3.62/2.16	  reverse(cons(X, Y)) >= app(reverse(Y), cons(X, nil)) 
3.62/2.16	  shuffle(nil) >= nil 
3.62/2.16	  shuffle(cons(X, Y)) >= cons(X, shuffle(reverse(Y))) 
3.62/2.16	  fst(p(X, Y)) >= X 
3.62/2.16	  pshuffle(X) >= p(X, shuffle(X)) 
3.62/2.16	  apply2(F, X, 0) >= X 
3.62/2.16	  apply2(F, X, s(Y)) >= F X s(Y) 
3.62/2.16	
3.62/2.16	We apply [Kop12, Thm. 6.75] and use the following argument functions:
3.62/2.16	
3.62/2.16	  pi( pshuffle(X) ) = #argfun-pshuffle#(p(X, shuffle(X))) 
3.62/2.16	
3.62/2.16	We orient these requirements with a polynomial interpretation in the natural numbers.
3.62/2.16	
3.62/2.16	The following interpretation satisfies the requirements:
3.62/2.16	
3.62/2.16	  #argfun-pshuffle# = \y0.y0 
3.62/2.16	  0 = 0 
3.62/2.16	  app = \y0y1.y0 + y1 
3.62/2.16	  apply2 = \G0y1y2.y1 + G0(y1,y1) 
3.62/2.16	  cons = \y0y1.1 + y1 
3.62/2.16	  fst = \y0.y0 
3.62/2.16	  nil = 0 
3.62/2.16	  p = \y0y1.y0 
3.62/2.16	  prefixshuffle# = \y0y1.2y1 
3.62/2.16	  pshuffle = \y0.0 
3.62/2.16	  reverse = \y0.y0 
3.62/2.16	  s = \y0.0 
3.62/2.16	  shuffle = \y0.2y0 
3.62/2.16	
3.62/2.16	Using this interpretation, the requirements translate to:
3.62/2.16	
3.62/2.16	  [[prefixshuffle#(_x0, cons(_x1, _x2))]] = 2 + 2x2 > 2x2 = [[prefixshuffle#(apply2(/\x./\y.#argfun-pshuffle#(p(app(fst(x), cons(y, nil)), shuffle(app(fst(x), cons(y, nil))))), _x0, _x1), reverse(_x2))]] 
3.62/2.16	  [[app(nil, _x0)]] = x0 >= x0 = [[_x0]] 
3.62/2.16	  [[app(cons(_x0, _x1), _x2)]] = 1 + x1 + x2 >= 1 + x1 + x2 = [[cons(_x0, app(_x1, _x2))]] 
3.62/2.16	  [[reverse(nil)]] = 0 >= 0 = [[nil]] 
3.62/2.16	  [[reverse(cons(_x0, _x1))]] = 1 + x1 >= 1 + x1 = [[app(reverse(_x1), cons(_x0, nil))]] 
3.62/2.16	  [[shuffle(nil)]] = 0 >= 0 = [[nil]] 
3.62/2.16	  [[shuffle(cons(_x0, _x1))]] = 2 + 2x1 >= 1 + 2x1 = [[cons(_x0, shuffle(reverse(_x1)))]] 
3.62/2.16	  [[fst(p(_x0, _x1))]] = x0 >= x0 = [[_x0]] 
3.62/2.16	  [[#argfun-pshuffle#(p(_x0, shuffle(_x0)))]] = x0 >= x0 = [[p(_x0, shuffle(_x0))]] 
3.62/2.16	  [[apply2(_F0, _x1, 0)]] = x1 + F0(x1,x1) >= x1 = [[_x1]] 
3.62/2.16	  [[apply2(_F0, _x1, s(_x2))]] = x1 + F0(x1,x1) >= F0(x1,0) = [[_F0 _x1 s(_x2)]] 
3.62/2.16	
3.62/2.16	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.
3.62/2.16	
3.62/2.16	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
3.62/2.16	
3.62/2.16	
3.62/2.16	+++ Citations +++
3.62/2.16	
3.62/2.16	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
3.62/2.16	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
3.62/2.16	[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.
3.62/2.16	EOF