2.18/1.13	YES
2.18/1.14	We consider the system theBenchmark.
2.18/1.14	
2.18/1.14	  Alphabet:
2.18/1.14	
2.18/1.14	    0 : [] --> a 
2.18/1.14	    cons : [b * c] --> c 
2.18/1.14	    d : [a * a] --> c 
2.18/1.14	    false : [] --> c 
2.18/1.14	    filter : [b -> c * c] --> c 
2.18/1.14	    gtr : [a * a] --> c 
2.18/1.14	    if : [c * c * c] --> c 
2.18/1.14	    len : [c] --> a 
2.18/1.14	    nil : [] --> c 
2.18/1.14	    s : [a] --> a 
2.18/1.14	    sub : [a * a] --> a 
2.18/1.14	    true : [] --> c 
2.18/1.14	
2.18/1.14	  Rules:
2.18/1.14	
2.18/1.14	    if(true, x, y) => x 
2.18/1.14	    if(false, x, y) => y 
2.18/1.14	    sub(x, 0) => x 
2.18/1.14	    sub(s(x), s(y)) => sub(x, y) 
2.18/1.14	    gtr(0, x) => false 
2.18/1.14	    gtr(s(x), 0) => true 
2.18/1.14	    gtr(s(x), s(y)) => gtr(x, y) 
2.18/1.14	    d(x, 0) => true 
2.18/1.14	    d(s(x), s(y)) => if(gtr(x, y), false, d(s(x), sub(y, x))) 
2.18/1.14	    len(nil) => 0 
2.18/1.14	    len(cons(x, y)) => s(len(y)) 
2.18/1.14	    filter(f, nil) => nil 
2.18/1.14	    filter(f, cons(x, y)) => if(f x, cons(x, filter(f, y)), filter(f, y)) 
2.18/1.14	
2.18/1.14	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
2.18/1.14	
2.18/1.14	We observe that the rules contain a first-order subset:
2.18/1.14	
2.18/1.14	  if(true, X, Y) => X 
2.18/1.14	  if(false, X, Y) => Y 
2.18/1.14	  sub(X, 0) => X 
2.18/1.14	  sub(s(X), s(Y)) => sub(X, Y) 
2.18/1.14	  gtr(0, X) => false 
2.18/1.14	  gtr(s(X), 0) => true 
2.18/1.14	  gtr(s(X), s(Y)) => gtr(X, Y) 
2.18/1.14	  d(X, 0) => true 
2.18/1.14	  d(s(X), s(Y)) => if(gtr(X, Y), false, d(s(X), sub(Y, X))) 
2.18/1.14	  len(nil) => 0 
2.18/1.14	  len(cons(X, Y)) => s(len(Y)) 
2.18/1.14	
2.18/1.14	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.
2.18/1.14	
2.18/1.14	According to the external first-order termination prover, this system is indeed terminating:
2.18/1.14	
2.18/1.14	 || proof of resources/system.trs
2.18/1.14	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || Termination w.r.t. Q of the given QTRS could be proven:
2.18/1.14	 || 
2.18/1.14	 || (0) QTRS
2.18/1.14	 || (1) Overlay + Local Confluence [EQUIVALENT]
2.18/1.14	 || (2) QTRS
2.18/1.14	 || (3) DependencyPairsProof [EQUIVALENT]
2.18/1.14	 || (4) QDP
2.18/1.14	 || (5) DependencyGraphProof [EQUIVALENT]
2.18/1.14	 || (6) AND
2.18/1.14	 ||     (7) QDP
2.18/1.14	 ||         (8) UsableRulesProof [EQUIVALENT]
2.18/1.14	 ||         (9) QDP
2.18/1.14	 ||         (10) QReductionProof [EQUIVALENT]
2.18/1.14	 ||         (11) QDP
2.18/1.14	 ||         (12) QDPSizeChangeProof [EQUIVALENT]
2.18/1.14	 ||         (13) YES
2.18/1.14	 ||     (14) QDP
2.18/1.14	 ||         (15) UsableRulesProof [EQUIVALENT]
2.18/1.14	 ||         (16) QDP
2.18/1.14	 ||         (17) QReductionProof [EQUIVALENT]
2.18/1.14	 ||         (18) QDP
2.18/1.14	 ||         (19) QDPSizeChangeProof [EQUIVALENT]
2.18/1.14	 ||         (20) YES
2.18/1.14	 ||     (21) QDP
2.18/1.14	 ||         (22) UsableRulesProof [EQUIVALENT]
2.18/1.14	 ||         (23) QDP
2.18/1.14	 ||         (24) QReductionProof [EQUIVALENT]
2.18/1.14	 ||         (25) QDP
2.18/1.14	 ||         (26) QDPSizeChangeProof [EQUIVALENT]
2.18/1.14	 ||         (27) YES
2.18/1.14	 ||     (28) QDP
2.18/1.14	 ||         (29) UsableRulesProof [EQUIVALENT]
2.18/1.14	 ||         (30) QDP
2.18/1.14	 ||         (31) QReductionProof [EQUIVALENT]
2.18/1.14	 ||         (32) QDP
2.18/1.14	 ||         (33) QDPOrderProof [EQUIVALENT]
2.18/1.14	 ||         (34) QDP
2.18/1.14	 ||         (35) PisEmptyProof [EQUIVALENT]
2.18/1.14	 ||         (36) YES
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (0)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q restricted rewrite system:
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || Q is empty.
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (1) Overlay + Local Confluence (EQUIVALENT)
2.18/1.14	 || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (2)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q restricted rewrite system:
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (3) DependencyPairsProof (EQUIVALENT)
2.18/1.14	 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (4)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    SUB(s(%X), s(%Y)) -> SUB(%X, %Y)
2.18/1.14	 ||    GTR(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 ||    D(s(%X), s(%Y)) -> IF(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    D(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 ||    D(s(%X), s(%Y)) -> D(s(%X), sub(%Y, %X))
2.18/1.14	 ||    D(s(%X), s(%Y)) -> SUB(%Y, %X)
2.18/1.14	 ||    LEN(cons(%X, %Y)) -> LEN(%Y)
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (5) DependencyGraphProof (EQUIVALENT)
2.18/1.14	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 3 less nodes.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (6)
2.18/1.14	 || Complex Obligation (AND)
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (7)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    LEN(cons(%X, %Y)) -> LEN(%Y)
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (8) UsableRulesProof (EQUIVALENT)
2.18/1.14	 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (9)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    LEN(cons(%X, %Y)) -> LEN(%Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (10) QReductionProof (EQUIVALENT)
2.18/1.14	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (11)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    LEN(cons(%X, %Y)) -> LEN(%Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || Q is empty.
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (12) QDPSizeChangeProof (EQUIVALENT)
2.18/1.14	 || 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. 
2.18/1.14	 || 
2.18/1.14	 || From the DPs we obtained the following set of size-change graphs:
2.18/1.14	 || *LEN(cons(%X, %Y)) -> LEN(%Y)
2.18/1.14	 || The graph contains the following edges 1 > 1
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (13)
2.18/1.14	 || YES
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (14)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    GTR(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (15) UsableRulesProof (EQUIVALENT)
2.18/1.14	 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (16)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    GTR(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (17) QReductionProof (EQUIVALENT)
2.18/1.14	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (18)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    GTR(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || Q is empty.
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (19) QDPSizeChangeProof (EQUIVALENT)
2.18/1.14	 || 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. 
2.18/1.14	 || 
2.18/1.14	 || From the DPs we obtained the following set of size-change graphs:
2.18/1.14	 || *GTR(s(%X), s(%Y)) -> GTR(%X, %Y)
2.18/1.14	 || The graph contains the following edges 1 > 1, 2 > 2
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (20)
2.18/1.14	 || YES
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (21)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    SUB(s(%X), s(%Y)) -> SUB(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (22) UsableRulesProof (EQUIVALENT)
2.18/1.14	 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (23)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    SUB(s(%X), s(%Y)) -> SUB(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (24) QReductionProof (EQUIVALENT)
2.18/1.14	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (25)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    SUB(s(%X), s(%Y)) -> SUB(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || R is empty.
2.18/1.14	 || Q is empty.
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (26) QDPSizeChangeProof (EQUIVALENT)
2.18/1.14	 || 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. 
2.18/1.14	 || 
2.18/1.14	 || From the DPs we obtained the following set of size-change graphs:
2.18/1.14	 || *SUB(s(%X), s(%Y)) -> SUB(%X, %Y)
2.18/1.14	 || The graph contains the following edges 1 > 1, 2 > 2
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (27)
2.18/1.14	 || YES
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (28)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    D(s(%X), s(%Y)) -> D(s(%X), sub(%Y, %X))
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, %X, %Y) -> %X
2.18/1.14	 ||    if(false, %X, %Y) -> %Y
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 ||    gtr(0, %X) -> false
2.18/1.14	 ||    gtr(s(%X), 0) -> true
2.18/1.14	 ||    gtr(s(%X), s(%Y)) -> gtr(%X, %Y)
2.18/1.14	 ||    d(%X, 0) -> true
2.18/1.14	 ||    d(s(%X), s(%Y)) -> if(gtr(%X, %Y), false, d(s(%X), sub(%Y, %X)))
2.18/1.14	 ||    len(nil) -> 0
2.18/1.14	 ||    len(cons(%X, %Y)) -> s(len(%Y))
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (29) UsableRulesProof (EQUIVALENT)
2.18/1.14	 || As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (30)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    D(s(%X), s(%Y)) -> D(s(%X), sub(%Y, %X))
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (31) QReductionProof (EQUIVALENT)
2.18/1.14	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
2.18/1.14	 || 
2.18/1.14	 ||    if(true, x0, x1)
2.18/1.14	 ||    if(false, x0, x1)
2.18/1.14	 ||    gtr(0, x0)
2.18/1.14	 ||    gtr(s(x0), 0)
2.18/1.14	 ||    gtr(s(x0), s(x1))
2.18/1.14	 ||    d(x0, 0)
2.18/1.14	 ||    d(s(x0), s(x1))
2.18/1.14	 ||    len(nil)
2.18/1.14	 ||    len(cons(x0, x1))
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (32)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || The TRS P consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    D(s(%X), s(%Y)) -> D(s(%X), sub(%Y, %X))
2.18/1.14	 || 
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (33) QDPOrderProof (EQUIVALENT)
2.18/1.14	 || We use the reduction pair processor [LPAR04,JAR06].
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || The following pairs can be oriented strictly and are deleted.
2.18/1.14	 || 
2.18/1.14	 ||    D(s(%X), s(%Y)) -> D(s(%X), sub(%Y, %X))
2.18/1.14	 || The remaining pairs can at least be oriented weakly.
2.18/1.14	 || Used ordering:  Polynomial interpretation [POLO]:
2.18/1.14	 || 
2.18/1.14	 ||    POL(0) = 0
2.18/1.14	 ||    POL(D(x_1, x_2)) = x_2
2.18/1.14	 ||    POL(s(x_1)) = 1 + x_1
2.18/1.14	 ||    POL(sub(x_1, x_2)) = x_1
2.18/1.14	 || 
2.18/1.14	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
2.18/1.14	 || 
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || 
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (34)
2.18/1.14	 || Obligation:
2.18/1.14	 || Q DP problem:
2.18/1.14	 || P is empty.
2.18/1.14	 || The TRS R consists of the following rules:
2.18/1.14	 || 
2.18/1.14	 ||    sub(%X, 0) -> %X
2.18/1.14	 ||    sub(s(%X), s(%Y)) -> sub(%X, %Y)
2.18/1.14	 || 
2.18/1.14	 || The set Q consists of the following terms:
2.18/1.14	 || 
2.18/1.14	 ||    sub(x0, 0)
2.18/1.14	 ||    sub(s(x0), s(x1))
2.18/1.14	 || 
2.18/1.14	 || We have to consider all minimal (P,Q,R)-chains.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (35) PisEmptyProof (EQUIVALENT)
2.18/1.14	 || The TRS P is empty. Hence, there is no (P,Q,R) chain.
2.18/1.14	 || ----------------------------------------
2.18/1.14	 || 
2.18/1.14	 || (36)
2.18/1.14	 || YES
2.18/1.14	 || 
2.18/1.14	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]).
2.18/1.14	
2.18/1.14	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
2.18/1.14	
2.18/1.14	  Dependency Pairs P_0:
2.18/1.14	
2.18/1.14	    0] filter#(F, cons(X, Y)) =#> if#(F X, cons(X, filter(F, Y)), filter(F, Y))   
2.18/1.14	    1] filter#(F, cons(X, Y)) =#> filter#(F, Y)   
2.18/1.14	    2] filter#(F, cons(X, Y)) =#> filter#(F, Y)   
2.18/1.14	
2.18/1.14	  Rules R_0:
2.18/1.14	
2.18/1.14	    if(true, X, Y) => X 
2.18/1.14	    if(false, X, Y) => Y 
2.18/1.14	    sub(X, 0) => X 
2.18/1.14	    sub(s(X), s(Y)) => sub(X, Y) 
2.18/1.14	    gtr(0, X) => false 
2.18/1.14	    gtr(s(X), 0) => true 
2.18/1.14	    gtr(s(X), s(Y)) => gtr(X, Y) 
2.18/1.14	    d(X, 0) => true 
2.18/1.14	    d(s(X), s(Y)) => if(gtr(X, Y), false, d(s(X), sub(Y, X))) 
2.18/1.14	    len(nil) => 0 
2.18/1.14	    len(cons(X, Y)) => s(len(Y)) 
2.18/1.14	    filter(F, nil) => nil 
2.18/1.14	    filter(F, cons(X, Y)) => if(F X, cons(X, filter(F, Y)), filter(F, Y)) 
2.18/1.14	
2.18/1.14	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
2.18/1.14	
2.18/1.14	We consider the dependency pair problem (P_0, R_0, static, formative).
2.18/1.14	
2.18/1.14	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
2.18/1.14	
2.18/1.14	    * 0 :  
2.18/1.14	    * 1 : 0, 1, 2 
2.18/1.14	    * 2 : 0, 1, 2 
2.18/1.14	
2.18/1.14	This graph has the following strongly connected components:
2.18/1.14	
2.18/1.14	  P_1:
2.18/1.14	
2.18/1.14	    filter#(F, cons(X, Y)) =#> filter#(F, Y)   
2.18/1.14	    filter#(F, cons(X, Y)) =#> filter#(F, Y)   
2.18/1.14	
2.18/1.14	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).
2.18/1.14	
2.18/1.14	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
2.18/1.14	
2.18/1.14	We consider the dependency pair problem (P_1, R_0, static, formative).
2.18/1.14	
2.18/1.14	We apply the subterm criterion with the following projection function:
2.18/1.14	
2.18/1.14	  nu(filter#) = 2 
2.18/1.14	
2.18/1.14	Thus, we can orient the dependency pairs as follows:
2.18/1.14	
2.18/1.14	  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 
2.18/1.14	  nu(filter#(F, cons(X, Y))) = cons(X, Y) |> Y = nu(filter#(F, Y)) 
2.18/1.14	
2.18/1.14	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.
2.18/1.14	
2.18/1.14	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
2.18/1.14	
2.18/1.14	
2.18/1.14	+++ Citations +++
2.18/1.14	
2.18/1.14	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
2.18/1.14	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
2.18/1.14	[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.
2.18/1.14	EOF