4.07/1.95	YES
4.07/1.98	We consider the system theBenchmark.
4.07/1.98	
4.07/1.98	  Alphabet:
4.07/1.98	
4.07/1.98	    0 : [] --> b 
4.07/1.98	    add : [b * c] --> c 
4.07/1.98	    app : [c * c] --> c 
4.07/1.98	    false : [] --> a 
4.07/1.98	    filter : [b -> a * c] --> c 
4.07/1.98	    filter2 : [a * b -> a * b * c] --> c 
4.07/1.98	    high : [b * c] --> c 
4.07/1.98	    if!fac6220high : [a * b * c] --> c 
4.07/1.98	    if!fac6220low : [a * b * c] --> c 
4.07/1.98	    le : [b * b] --> a 
4.07/1.98	    low : [b * c] --> c 
4.07/1.98	    map : [b -> b * c] --> c 
4.07/1.98	    minus : [b * b] --> b 
4.07/1.98	    nil : [] --> c 
4.07/1.98	    quicksort : [c] --> c 
4.07/1.98	    quot : [b * b] --> b 
4.07/1.98	    s : [b] --> b 
4.07/1.98	    true : [] --> a 
4.07/1.98	
4.07/1.98	  Rules:
4.07/1.98	
4.07/1.98	    minus(x, 0) => x 
4.07/1.98	    minus(s(x), s(y)) => minus(x, y) 
4.07/1.98	    quot(0, s(x)) => 0 
4.07/1.98	    quot(s(x), s(y)) => s(quot(minus(x, y), s(y))) 
4.07/1.98	    le(0, x) => true 
4.07/1.98	    le(s(x), 0) => false 
4.07/1.98	    le(s(x), s(y)) => le(x, y) 
4.07/1.98	    app(nil, x) => x 
4.07/1.98	    app(add(x, y), z) => add(x, app(y, z)) 
4.07/1.98	    low(x, nil) => nil 
4.07/1.98	    low(x, add(y, z)) => if!fac6220low(le(y, x), x, add(y, z)) 
4.07/1.98	    if!fac6220low(true, x, add(y, z)) => add(y, low(x, z)) 
4.07/1.98	    if!fac6220low(false, x, add(y, z)) => low(x, z) 
4.07/1.98	    high(x, nil) => nil 
4.07/1.98	    high(x, add(y, z)) => if!fac6220high(le(y, x), x, add(y, z)) 
4.07/1.98	    if!fac6220high(true, x, add(y, z)) => high(x, z) 
4.07/1.98	    if!fac6220high(false, x, add(y, z)) => add(y, high(x, z)) 
4.07/1.98	    quicksort(nil) => nil 
4.07/1.98	    quicksort(add(x, y)) => app(quicksort(low(x, y)), add(x, quicksort(high(x, y)))) 
4.07/1.98	    map(f, nil) => nil 
4.07/1.98	    map(f, add(x, y)) => add(f x, map(f, y)) 
4.07/1.98	    filter(f, nil) => nil 
4.07/1.98	    filter(f, add(x, y)) => filter2(f x, f, x, y) 
4.07/1.98	    filter2(true, f, x, y) => add(x, filter(f, y)) 
4.07/1.98	    filter2(false, f, x, y) => filter(f, y) 
4.07/1.98	
4.07/1.98	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.07/1.98	
4.07/1.98	We observe that the rules contain a first-order subset:
4.07/1.98	
4.07/1.98	  minus(X, 0) => X 
4.07/1.98	  minus(s(X), s(Y)) => minus(X, Y) 
4.07/1.98	  quot(0, s(X)) => 0 
4.07/1.98	  quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
4.07/1.98	  le(0, X) => true 
4.07/1.98	  le(s(X), 0) => false 
4.07/1.98	  le(s(X), s(Y)) => le(X, Y) 
4.07/1.98	  app(nil, X) => X 
4.07/1.98	  app(add(X, Y), Z) => add(X, app(Y, Z)) 
4.07/1.98	  low(X, nil) => nil 
4.07/1.98	  low(X, add(Y, Z)) => if!fac6220low(le(Y, X), X, add(Y, Z)) 
4.07/1.98	  if!fac6220low(true, X, add(Y, Z)) => add(Y, low(X, Z)) 
4.07/1.98	  if!fac6220low(false, X, add(Y, Z)) => low(X, Z) 
4.07/1.98	  high(X, nil) => nil 
4.07/1.98	  high(X, add(Y, Z)) => if!fac6220high(le(Y, X), X, add(Y, Z)) 
4.07/1.98	  if!fac6220high(true, X, add(Y, Z)) => high(X, Z) 
4.07/1.98	  if!fac6220high(false, X, add(Y, Z)) => add(Y, high(X, Z)) 
4.07/1.98	  quicksort(nil) => nil 
4.07/1.98	  quicksort(add(X, Y)) => app(quicksort(low(X, Y)), add(X, quicksort(high(X, Y)))) 
4.07/1.98	
4.07/1.98	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.
4.07/1.98	
4.07/1.98	According to the external first-order termination prover, this system is indeed terminating:
4.07/1.98	
4.07/1.98	 || proof of resources/system.trs
4.07/1.98	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || Termination w.r.t. Q of the given QTRS could be proven:
4.07/1.98	 || 
4.07/1.98	 || (0) QTRS
4.07/1.98	 || (1) Overlay + Local Confluence [EQUIVALENT]
4.07/1.98	 || (2) QTRS
4.07/1.98	 || (3) DependencyPairsProof [EQUIVALENT]
4.07/1.98	 || (4) QDP
4.07/1.98	 || (5) DependencyGraphProof [EQUIVALENT]
4.07/1.98	 || (6) AND
4.07/1.98	 ||     (7) QDP
4.07/1.98	 ||         (8) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (9) QDP
4.07/1.98	 ||         (10) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (11) QDP
4.07/1.98	 ||         (12) QDPSizeChangeProof [EQUIVALENT]
4.07/1.98	 ||         (13) YES
4.07/1.98	 ||     (14) QDP
4.07/1.98	 ||         (15) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (16) QDP
4.07/1.98	 ||         (17) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (18) QDP
4.07/1.98	 ||         (19) QDPSizeChangeProof [EQUIVALENT]
4.07/1.98	 ||         (20) YES
4.07/1.98	 ||     (21) QDP
4.07/1.98	 ||         (22) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (23) QDP
4.07/1.98	 ||         (24) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (25) QDP
4.07/1.98	 ||         (26) QDPSizeChangeProof [EQUIVALENT]
4.07/1.98	 ||         (27) YES
4.07/1.98	 ||     (28) QDP
4.07/1.98	 ||         (29) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (30) QDP
4.07/1.98	 ||         (31) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (32) QDP
4.07/1.98	 ||         (33) QDPSizeChangeProof [EQUIVALENT]
4.07/1.98	 ||         (34) YES
4.07/1.98	 ||     (35) QDP
4.07/1.98	 ||         (36) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (37) QDP
4.07/1.98	 ||         (38) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (39) QDP
4.07/1.98	 ||         (40) QDPOrderProof [EQUIVALENT]
4.07/1.98	 ||         (41) QDP
4.07/1.98	 ||         (42) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (43) QDP
4.07/1.98	 ||         (44) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (45) QDP
4.07/1.98	 ||         (46) QDPOrderProof [EQUIVALENT]
4.07/1.98	 ||         (47) QDP
4.07/1.98	 ||         (48) PisEmptyProof [EQUIVALENT]
4.07/1.98	 ||         (49) YES
4.07/1.98	 ||     (50) QDP
4.07/1.98	 ||         (51) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (52) QDP
4.07/1.98	 ||         (53) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (54) QDP
4.07/1.98	 ||         (55) QDPSizeChangeProof [EQUIVALENT]
4.07/1.98	 ||         (56) YES
4.07/1.98	 ||     (57) QDP
4.07/1.98	 ||         (58) UsableRulesProof [EQUIVALENT]
4.07/1.98	 ||         (59) QDP
4.07/1.98	 ||         (60) QReductionProof [EQUIVALENT]
4.07/1.98	 ||         (61) QDP
4.07/1.98	 ||         (62) QDPOrderProof [EQUIVALENT]
4.07/1.98	 ||         (63) QDP
4.07/1.98	 ||         (64) PisEmptyProof [EQUIVALENT]
4.07/1.98	 ||         (65) YES
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (0)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q restricted rewrite system:
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || Q is empty.
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (1) Overlay + Local Confluence (EQUIVALENT)
4.07/1.98	 || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (2)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q restricted rewrite system:
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (3) DependencyPairsProof (EQUIVALENT)
4.07/1.98	 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (4)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    MINUS(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> QUOT(minus(%X, %Y), s(%Y))
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.07/1.98	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.07/1.98	 ||    LOW(%X, add(%Y, %Z)) -> IF!FAC6220LOW(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    LOW(%X, add(%Y, %Z)) -> LE(%Y, %X)
4.07/1.98	 ||    IF!FAC6220LOW(true, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 ||    IF!FAC6220LOW(false, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 ||    HIGH(%X, add(%Y, %Z)) -> IF!FAC6220HIGH(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    HIGH(%X, add(%Y, %Z)) -> LE(%Y, %X)
4.07/1.98	 ||    IF!FAC6220HIGH(true, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 ||    IF!FAC6220HIGH(false, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> APP(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> LOW(%X, %Y)
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(high(%X, %Y))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> HIGH(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (5) DependencyGraphProof (EQUIVALENT)
4.07/1.98	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 6 less nodes.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (6)
4.07/1.98	 || Complex Obligation (AND)
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (7)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (8) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (9)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (10) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (11)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || Q is empty.
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (12) QDPSizeChangeProof (EQUIVALENT)
4.07/1.98	 || 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. 
4.07/1.98	 || 
4.07/1.98	 || From the DPs we obtained the following set of size-change graphs:
4.07/1.98	 || *APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.07/1.98	 || The graph contains the following edges 1 > 1, 2 >= 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (13)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (14)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (15) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (16)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (17) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (18)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || Q is empty.
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (19) QDPSizeChangeProof (EQUIVALENT)
4.07/1.98	 || 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. 
4.07/1.98	 || 
4.07/1.98	 || From the DPs we obtained the following set of size-change graphs:
4.07/1.98	 || *LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.07/1.98	 || The graph contains the following edges 1 > 1, 2 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (20)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (21)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    HIGH(%X, add(%Y, %Z)) -> IF!FAC6220HIGH(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220HIGH(true, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 ||    IF!FAC6220HIGH(false, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (22) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (23)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    HIGH(%X, add(%Y, %Z)) -> IF!FAC6220HIGH(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220HIGH(true, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 ||    IF!FAC6220HIGH(false, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (24) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (25)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    HIGH(%X, add(%Y, %Z)) -> IF!FAC6220HIGH(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220HIGH(true, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 ||    IF!FAC6220HIGH(false, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (26) QDPSizeChangeProof (EQUIVALENT)
4.07/1.98	 || 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. 
4.07/1.98	 || 
4.07/1.98	 || From the DPs we obtained the following set of size-change graphs:
4.07/1.98	 || *HIGH(%X, add(%Y, %Z)) -> IF!FAC6220HIGH(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 || The graph contains the following edges 1 >= 2, 2 >= 3
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || *IF!FAC6220HIGH(true, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 || The graph contains the following edges 2 >= 1, 3 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || *IF!FAC6220HIGH(false, %X, add(%Y, %Z)) -> HIGH(%X, %Z)
4.07/1.98	 || The graph contains the following edges 2 >= 1, 3 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (27)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (28)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LOW(%X, add(%Y, %Z)) -> IF!FAC6220LOW(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220LOW(true, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 ||    IF!FAC6220LOW(false, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (29) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (30)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LOW(%X, add(%Y, %Z)) -> IF!FAC6220LOW(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220LOW(true, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 ||    IF!FAC6220LOW(false, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (31) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (32)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    LOW(%X, add(%Y, %Z)) -> IF!FAC6220LOW(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    IF!FAC6220LOW(true, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 ||    IF!FAC6220LOW(false, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (33) QDPSizeChangeProof (EQUIVALENT)
4.07/1.98	 || 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. 
4.07/1.98	 || 
4.07/1.98	 || From the DPs we obtained the following set of size-change graphs:
4.07/1.98	 || *LOW(%X, add(%Y, %Z)) -> IF!FAC6220LOW(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 || The graph contains the following edges 1 >= 2, 2 >= 3
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || *IF!FAC6220LOW(true, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 || The graph contains the following edges 2 >= 1, 3 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || *IF!FAC6220LOW(false, %X, add(%Y, %Z)) -> LOW(%X, %Z)
4.07/1.98	 || The graph contains the following edges 2 >= 1, 3 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (34)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (35)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(high(%X, %Y))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (36) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (37)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(high(%X, %Y))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (38) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (39)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(high(%X, %Y))
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (40) QDPOrderProof (EQUIVALENT)
4.07/1.98	 || We use the reduction pair processor [LPAR04,JAR06].
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || The following pairs can be oriented strictly and are deleted.
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(high(%X, %Y))
4.07/1.98	 || The remaining pairs can at least be oriented weakly.
4.07/1.98	 || Used ordering:  Polynomial interpretation [POLO]:
4.07/1.98	 || 
4.07/1.98	 ||    POL(0) = 0
4.07/1.98	 ||    POL(QUICKSORT(x_1)) = x_1
4.07/1.98	 ||    POL(add(x_1, x_2)) = 1 + x_2
4.07/1.98	 ||    POL(false) = 0
4.07/1.98	 ||    POL(high(x_1, x_2)) = x_2
4.07/1.98	 ||    POL(if!fac6220high(x_1, x_2, x_3)) = x_3
4.07/1.98	 ||    POL(if!fac6220low(x_1, x_2, x_3)) = 1 + x_3
4.07/1.98	 ||    POL(le(x_1, x_2)) = 0
4.07/1.98	 ||    POL(low(x_1, x_2)) = 1 + x_2
4.07/1.98	 ||    POL(nil) = 0
4.07/1.98	 ||    POL(s(x_1)) = 0
4.07/1.98	 ||    POL(true) = 0
4.07/1.98	 || 
4.07/1.98	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4.07/1.98	 || 
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (41)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (42) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (43)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (44) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (45)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (46) QDPOrderProof (EQUIVALENT)
4.07/1.98	 || We use the reduction pair processor [LPAR04,JAR06].
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || The following pairs can be oriented strictly and are deleted.
4.07/1.98	 || 
4.07/1.98	 ||    QUICKSORT(add(%X, %Y)) -> QUICKSORT(low(%X, %Y))
4.07/1.98	 || The remaining pairs can at least be oriented weakly.
4.07/1.98	 || Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
4.07/1.98	 || 
4.07/1.98	 || POL( QUICKSORT_1(x_1) ) = max{0, 2x_1 - 1}
4.07/1.98	 || POL( low_2(x_1, x_2) ) = x_2
4.07/1.98	 || POL( nil ) = 1
4.07/1.98	 || POL( add_2(x_1, x_2) ) = 2x_2 + 2
4.07/1.98	 || POL( if!fac6220low_3(x_1, ..., x_3) ) = max{0, x_1 + x_3 - 2}
4.07/1.98	 || POL( le_2(x_1, x_2) ) = 2
4.07/1.98	 || POL( false ) = 0
4.07/1.98	 || POL( 0 ) = 2
4.07/1.98	 || POL( true ) = 2
4.07/1.98	 || POL( s_1(x_1) ) = 2x_1 + 2
4.07/1.98	 || 
4.07/1.98	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (47)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || P is empty.
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (48) PisEmptyProof (EQUIVALENT)
4.07/1.98	 || The TRS P is empty. Hence, there is no (P,Q,R) chain.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (49)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (50)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    MINUS(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (51) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (52)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    MINUS(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (53) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (54)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    MINUS(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || R is empty.
4.07/1.98	 || Q is empty.
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (55) QDPSizeChangeProof (EQUIVALENT)
4.07/1.98	 || 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. 
4.07/1.98	 || 
4.07/1.98	 || From the DPs we obtained the following set of size-change graphs:
4.07/1.98	 || *MINUS(s(%X), s(%Y)) -> MINUS(%X, %Y)
4.07/1.98	 || The graph contains the following edges 1 > 1, 2 > 2
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (56)
4.07/1.98	 || YES
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (57)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> QUOT(minus(%X, %Y), s(%Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 ||    quot(0, s(%X)) -> 0
4.07/1.98	 ||    quot(s(%X), s(%Y)) -> s(quot(minus(%X, %Y), s(%Y)))
4.07/1.98	 ||    le(0, %X) -> true
4.07/1.98	 ||    le(s(%X), 0) -> false
4.07/1.98	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.07/1.98	 ||    app(nil, %X) -> %X
4.07/1.98	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.07/1.98	 ||    low(%X, nil) -> nil
4.07/1.98	 ||    low(%X, add(%Y, %Z)) -> if!fac6220low(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220low(true, %X, add(%Y, %Z)) -> add(%Y, low(%X, %Z))
4.07/1.98	 ||    if!fac6220low(false, %X, add(%Y, %Z)) -> low(%X, %Z)
4.07/1.98	 ||    high(%X, nil) -> nil
4.07/1.98	 ||    high(%X, add(%Y, %Z)) -> if!fac6220high(le(%Y, %X), %X, add(%Y, %Z))
4.07/1.98	 ||    if!fac6220high(true, %X, add(%Y, %Z)) -> high(%X, %Z)
4.07/1.98	 ||    if!fac6220high(false, %X, add(%Y, %Z)) -> add(%Y, high(%X, %Z))
4.07/1.98	 ||    quicksort(nil) -> nil
4.07/1.98	 ||    quicksort(add(%X, %Y)) -> app(quicksort(low(%X, %Y)), add(%X, quicksort(high(%X, %Y))))
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (58) UsableRulesProof (EQUIVALENT)
4.07/1.98	 || 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.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (59)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> QUOT(minus(%X, %Y), s(%Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (60) QReductionProof (EQUIVALENT)
4.07/1.98	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.07/1.98	 || 
4.07/1.98	 ||    quot(0, s(x0))
4.07/1.98	 ||    quot(s(x0), s(x1))
4.07/1.98	 ||    le(0, x0)
4.07/1.98	 ||    le(s(x0), 0)
4.07/1.98	 ||    le(s(x0), s(x1))
4.07/1.98	 ||    app(nil, x0)
4.07/1.98	 ||    app(add(x0, x1), x2)
4.07/1.98	 ||    low(x0, nil)
4.07/1.98	 ||    low(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220low(false, x0, add(x1, x2))
4.07/1.98	 ||    high(x0, nil)
4.07/1.98	 ||    high(x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(true, x0, add(x1, x2))
4.07/1.98	 ||    if!fac6220high(false, x0, add(x1, x2))
4.07/1.98	 ||    quicksort(nil)
4.07/1.98	 ||    quicksort(add(x0, x1))
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (61)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || The TRS P consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> QUOT(minus(%X, %Y), s(%Y))
4.07/1.98	 || 
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (62) QDPOrderProof (EQUIVALENT)
4.07/1.98	 || We use the reduction pair processor [LPAR04,JAR06].
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || The following pairs can be oriented strictly and are deleted.
4.07/1.98	 || 
4.07/1.98	 ||    QUOT(s(%X), s(%Y)) -> QUOT(minus(%X, %Y), s(%Y))
4.07/1.98	 || The remaining pairs can at least be oriented weakly.
4.07/1.98	 || Used ordering:  Combined order from the following AFS and order.
4.07/1.98	 || QUOT(x1, x2)  =  QUOT(x1)
4.07/1.98	 || 
4.07/1.98	 || s(x1)  =  s(x1)
4.07/1.98	 || 
4.07/1.98	 || minus(x1, x2)  =  x1
4.07/1.98	 || 
4.07/1.98	 || 0  =  0
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || Recursive path order with status [RPO].
4.07/1.98	 || Quasi-Precedence: [QUOT_1, s_1]
4.07/1.98	 || 
4.07/1.98	 || Status: QUOT_1: multiset status
4.07/1.98	 || s_1: multiset status
4.07/1.98	 || 0: multiset status
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || 
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (63)
4.07/1.98	 || Obligation:
4.07/1.98	 || Q DP problem:
4.07/1.98	 || P is empty.
4.07/1.98	 || The TRS R consists of the following rules:
4.07/1.98	 || 
4.07/1.98	 ||    minus(%X, 0) -> %X
4.07/1.98	 ||    minus(s(%X), s(%Y)) -> minus(%X, %Y)
4.07/1.98	 || 
4.07/1.98	 || The set Q consists of the following terms:
4.07/1.98	 || 
4.07/1.98	 ||    minus(x0, 0)
4.07/1.98	 ||    minus(s(x0), s(x1))
4.07/1.98	 || 
4.07/1.98	 || We have to consider all minimal (P,Q,R)-chains.
4.07/1.98	 || ----------------------------------------
4.07/1.98	 || 
4.07/1.98	 || (64) PisEmptyProof (EQUIVALENT)
4.07/1.98	 || The TRS P is empty. Hence, there is no (P,Q,R) chain.
4.07/1.99	 || ----------------------------------------
4.07/1.99	 || 
4.07/1.99	 || (65)
4.07/1.99	 || YES
4.07/1.99	 || 
4.07/1.99	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]).
4.07/1.99	
4.07/1.99	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
4.07/1.99	
4.07/1.99	  Dependency Pairs P_0:
4.07/1.99	
4.07/1.99	    0] map#(F, add(X, Y)) =#> map#(F, Y)   
4.07/1.99	    1] filter#(F, add(X, Y)) =#> filter2#(F X, F, X, Y)   
4.07/1.99	    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	
4.07/1.99	  Rules R_0:
4.07/1.99	
4.07/1.99	    minus(X, 0) => X 
4.07/1.99	    minus(s(X), s(Y)) => minus(X, Y) 
4.07/1.99	    quot(0, s(X)) => 0 
4.07/1.99	    quot(s(X), s(Y)) => s(quot(minus(X, Y), s(Y))) 
4.07/1.99	    le(0, X) => true 
4.07/1.99	    le(s(X), 0) => false 
4.07/1.99	    le(s(X), s(Y)) => le(X, Y) 
4.07/1.99	    app(nil, X) => X 
4.07/1.99	    app(add(X, Y), Z) => add(X, app(Y, Z)) 
4.07/1.99	    low(X, nil) => nil 
4.07/1.99	    low(X, add(Y, Z)) => if!fac6220low(le(Y, X), X, add(Y, Z)) 
4.07/1.99	    if!fac6220low(true, X, add(Y, Z)) => add(Y, low(X, Z)) 
4.07/1.99	    if!fac6220low(false, X, add(Y, Z)) => low(X, Z) 
4.07/1.99	    high(X, nil) => nil 
4.07/1.99	    high(X, add(Y, Z)) => if!fac6220high(le(Y, X), X, add(Y, Z)) 
4.07/1.99	    if!fac6220high(true, X, add(Y, Z)) => high(X, Z) 
4.07/1.99	    if!fac6220high(false, X, add(Y, Z)) => add(Y, high(X, Z)) 
4.07/1.99	    quicksort(nil) => nil 
4.07/1.99	    quicksort(add(X, Y)) => app(quicksort(low(X, Y)), add(X, quicksort(high(X, Y)))) 
4.07/1.99	    map(F, nil) => nil 
4.07/1.99	    map(F, add(X, Y)) => add(F X, map(F, Y)) 
4.07/1.99	    filter(F, nil) => nil 
4.07/1.99	    filter(F, add(X, Y)) => filter2(F X, F, X, Y) 
4.07/1.99	    filter2(true, F, X, Y) => add(X, filter(F, Y)) 
4.07/1.99	    filter2(false, F, X, Y) => filter(F, Y) 
4.07/1.99	
4.07/1.99	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
4.07/1.99	
4.07/1.99	We consider the dependency pair problem (P_0, R_0, static, formative).
4.07/1.99	
4.07/1.99	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
4.07/1.99	
4.07/1.99	    * 0 : 0 
4.07/1.99	    * 1 : 2, 3 
4.07/1.99	    * 2 : 1 
4.07/1.99	    * 3 : 1 
4.07/1.99	
4.07/1.99	This graph has the following strongly connected components:
4.07/1.99	
4.07/1.99	  P_1:
4.07/1.99	
4.07/1.99	    map#(F, add(X, Y)) =#> map#(F, Y)   
4.07/1.99	
4.07/1.99	  P_2:
4.07/1.99	
4.07/1.99	    filter#(F, add(X, Y)) =#> filter2#(F X, F, X, Y)   
4.07/1.99	    filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	    filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	
4.07/1.99	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).
4.07/1.99	
4.07/1.99	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.
4.07/1.99	
4.07/1.99	We consider the dependency pair problem (P_2, R_0, static, formative).
4.07/1.99	
4.07/1.99	We apply the subterm criterion with the following projection function:
4.07/1.99	
4.07/1.99	  nu(filter2#) = 4 
4.07/1.99	  nu(filter#) = 2 
4.07/1.99	
4.07/1.99	Thus, we can orient the dependency pairs as follows:
4.07/1.99	
4.07/1.99	  nu(filter#(F, add(X, Y))) = add(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
4.07/1.99	  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
4.07/1.99	  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
4.07/1.99	
4.07/1.99	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:
4.07/1.99	
4.07/1.99	  filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	  filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.07/1.99	
4.07/1.99	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite.
4.07/1.99	
4.07/1.99	We consider the dependency pair problem (P_3, R_0, static, formative).
4.07/1.99	
4.07/1.99	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
4.07/1.99	
4.07/1.99	    * 0 :  
4.07/1.99	    * 1 :  
4.07/1.99	
4.07/1.99	This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.
4.07/1.99	
4.07/1.99	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
4.07/1.99	
4.07/1.99	We consider the dependency pair problem (P_1, R_0, static, formative).
4.07/1.99	
4.07/1.99	We apply the subterm criterion with the following projection function:
4.07/1.99	
4.07/1.99	  nu(map#) = 2 
4.07/1.99	
4.07/1.99	Thus, we can orient the dependency pairs as follows:
4.07/1.99	
4.07/1.99	  nu(map#(F, add(X, Y))) = add(X, Y) |> Y = nu(map#(F, Y)) 
4.07/1.99	
4.07/1.99	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.
4.07/1.99	
4.07/1.99	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
4.07/1.99	
4.07/1.99	
4.07/1.99	+++ Citations +++
4.07/1.99	
4.07/1.99	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
4.07/1.99	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
4.07/1.99	[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.
4.07/1.99	EOF