4.59/2.16	YES
4.59/2.19	We consider the system theBenchmark.
4.59/2.19	
4.59/2.19	  Alphabet:
4.59/2.19	
4.59/2.19	    0 : [] --> b 
4.59/2.19	    add : [b * c] --> c 
4.59/2.19	    app : [c * c] --> c 
4.59/2.19	    eq : [b * b] --> a 
4.59/2.19	    false : [] --> a 
4.59/2.19	    filter : [b -> a * c] --> c 
4.59/2.19	    filter2 : [a * b -> a * b * c] --> c 
4.59/2.19	    if!fac6220min : [a * c] --> b 
4.59/2.19	    if!fac6220minsort : [a * c * c] --> c 
4.59/2.19	    if!fac6220rm : [a * b * c] --> c 
4.59/2.19	    le : [b * b] --> a 
4.59/2.19	    map : [b -> b * c] --> c 
4.59/2.19	    min : [c] --> b 
4.59/2.19	    minsort : [c * c] --> c 
4.59/2.19	    nil : [] --> c 
4.59/2.19	    rm : [b * c] --> c 
4.59/2.19	    s : [b] --> b 
4.59/2.19	    true : [] --> a 
4.59/2.19	
4.59/2.19	  Rules:
4.59/2.19	
4.59/2.19	    eq(0, 0) => true 
4.59/2.19	    eq(0, s(x)) => false 
4.59/2.19	    eq(s(x), 0) => false 
4.59/2.19	    eq(s(x), s(y)) => eq(x, y) 
4.59/2.19	    le(0, x) => true 
4.59/2.19	    le(s(x), 0) => false 
4.59/2.19	    le(s(x), s(y)) => le(x, y) 
4.59/2.19	    app(nil, x) => x 
4.59/2.19	    app(add(x, y), z) => add(x, app(y, z)) 
4.59/2.19	    min(add(x, nil)) => x 
4.59/2.19	    min(add(x, add(y, z))) => if!fac6220min(le(x, y), add(x, add(y, z))) 
4.59/2.19	    if!fac6220min(true, add(x, add(y, z))) => min(add(x, z)) 
4.59/2.19	    if!fac6220min(false, add(x, add(y, z))) => min(add(y, z)) 
4.59/2.19	    rm(x, nil) => nil 
4.59/2.19	    rm(x, add(y, z)) => if!fac6220rm(eq(x, y), x, add(y, z)) 
4.59/2.19	    if!fac6220rm(true, x, add(y, z)) => rm(x, z) 
4.59/2.19	    if!fac6220rm(false, x, add(y, z)) => add(y, rm(x, z)) 
4.59/2.19	    minsort(nil, nil) => nil 
4.59/2.19	    minsort(add(x, y), z) => if!fac6220minsort(eq(x, min(add(x, y))), add(x, y), z) 
4.59/2.19	    if!fac6220minsort(true, add(x, y), z) => add(x, minsort(app(rm(x, y), z), nil)) 
4.59/2.19	    if!fac6220minsort(false, add(x, y), z) => minsort(y, add(x, z)) 
4.59/2.19	    map(f, nil) => nil 
4.59/2.19	    map(f, add(x, y)) => add(f x, map(f, y)) 
4.59/2.19	    filter(f, nil) => nil 
4.59/2.19	    filter(f, add(x, y)) => filter2(f x, f, x, y) 
4.59/2.19	    filter2(true, f, x, y) => add(x, filter(f, y)) 
4.59/2.19	    filter2(false, f, x, y) => filter(f, y) 
4.59/2.19	
4.59/2.19	This AFS is converted to an AFSM simply by replacing all free variables by meta-variables (with arity 0).
4.59/2.19	
4.59/2.19	We observe that the rules contain a first-order subset:
4.59/2.19	
4.59/2.19	  eq(0, 0) => true 
4.59/2.19	  eq(0, s(X)) => false 
4.59/2.19	  eq(s(X), 0) => false 
4.59/2.19	  eq(s(X), s(Y)) => eq(X, Y) 
4.59/2.19	  le(0, X) => true 
4.59/2.19	  le(s(X), 0) => false 
4.59/2.19	  le(s(X), s(Y)) => le(X, Y) 
4.59/2.19	  app(nil, X) => X 
4.59/2.19	  app(add(X, Y), Z) => add(X, app(Y, Z)) 
4.59/2.19	  min(add(X, nil)) => X 
4.59/2.19	  min(add(X, add(Y, Z))) => if!fac6220min(le(X, Y), add(X, add(Y, Z))) 
4.59/2.19	  if!fac6220min(true, add(X, add(Y, Z))) => min(add(X, Z)) 
4.59/2.19	  if!fac6220min(false, add(X, add(Y, Z))) => min(add(Y, Z)) 
4.59/2.19	  rm(X, nil) => nil 
4.59/2.19	  rm(X, add(Y, Z)) => if!fac6220rm(eq(X, Y), X, add(Y, Z)) 
4.59/2.19	  if!fac6220rm(true, X, add(Y, Z)) => rm(X, Z) 
4.59/2.19	  if!fac6220rm(false, X, add(Y, Z)) => add(Y, rm(X, Z)) 
4.59/2.19	  minsort(nil, nil) => nil 
4.59/2.19	  minsort(add(X, Y), Z) => if!fac6220minsort(eq(X, min(add(X, Y))), add(X, Y), Z) 
4.59/2.19	  if!fac6220minsort(true, add(X, Y), Z) => add(X, minsort(app(rm(X, Y), Z), nil)) 
4.59/2.19	  if!fac6220minsort(false, add(X, Y), Z) => minsort(Y, add(X, Z)) 
4.59/2.19	
4.59/2.19	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.
4.59/2.19	
4.59/2.19	According to the external first-order termination prover, this system is indeed terminating:
4.59/2.19	
4.59/2.19	 || proof of resources/system.trs
4.59/2.19	 || # AProVE Commit ID: d84c10301d352dfd14de2104819581f4682260f5 fuhs 20130616
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || Termination w.r.t. Q of the given QTRS could be proven:
4.59/2.19	 || 
4.59/2.19	 || (0) QTRS
4.59/2.19	 || (1) Overlay + Local Confluence [EQUIVALENT]
4.59/2.19	 || (2) QTRS
4.59/2.19	 || (3) DependencyPairsProof [EQUIVALENT]
4.59/2.19	 || (4) QDP
4.59/2.19	 || (5) DependencyGraphProof [EQUIVALENT]
4.59/2.19	 || (6) AND
4.59/2.19	 ||     (7) QDP
4.59/2.19	 ||         (8) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (9) QDP
4.59/2.19	 ||         (10) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (11) QDP
4.59/2.19	 ||         (12) QDPSizeChangeProof [EQUIVALENT]
4.59/2.19	 ||         (13) YES
4.59/2.19	 ||     (14) QDP
4.59/2.19	 ||         (15) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (16) QDP
4.59/2.19	 ||         (17) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (18) QDP
4.59/2.19	 ||         (19) QDPSizeChangeProof [EQUIVALENT]
4.59/2.19	 ||         (20) YES
4.59/2.19	 ||     (21) QDP
4.59/2.19	 ||         (22) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (23) QDP
4.59/2.19	 ||         (24) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (25) QDP
4.59/2.19	 ||         (26) QDPOrderProof [EQUIVALENT]
4.59/2.19	 ||         (27) QDP
4.59/2.19	 ||         (28) DependencyGraphProof [EQUIVALENT]
4.59/2.19	 ||         (29) TRUE
4.59/2.19	 ||     (30) QDP
4.59/2.19	 ||         (31) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (32) QDP
4.59/2.19	 ||         (33) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (34) QDP
4.59/2.19	 ||         (35) QDPSizeChangeProof [EQUIVALENT]
4.59/2.19	 ||         (36) YES
4.59/2.19	 ||     (37) QDP
4.59/2.19	 ||         (38) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (39) QDP
4.59/2.19	 ||         (40) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (41) QDP
4.59/2.19	 ||         (42) QDPSizeChangeProof [EQUIVALENT]
4.59/2.19	 ||         (43) YES
4.59/2.19	 ||     (44) QDP
4.59/2.19	 ||         (45) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (46) QDP
4.59/2.19	 ||         (47) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (48) QDP
4.59/2.19	 ||         (49) QDPOrderProof [EQUIVALENT]
4.59/2.19	 ||         (50) QDP
4.59/2.19	 ||         (51) UsableRulesProof [EQUIVALENT]
4.59/2.19	 ||         (52) QDP
4.59/2.19	 ||         (53) QReductionProof [EQUIVALENT]
4.59/2.19	 ||         (54) QDP
4.59/2.19	 ||         (55) QDPSizeChangeProof [EQUIVALENT]
4.59/2.19	 ||         (56) YES
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (0)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q restricted rewrite system:
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || Q is empty.
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (1) Overlay + Local Confluence (EQUIVALENT)
4.59/2.19	 || The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (2)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q restricted rewrite system:
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (3) DependencyPairsProof (EQUIVALENT)
4.59/2.19	 || Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (4)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
4.59/2.19	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.59/2.19	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> IF!FAC6220MIN(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> LE(%X, %Y)
4.59/2.19	 ||    IF!FAC6220MIN(true, add(%X, add(%Y, %Z))) -> MIN(add(%X, %Z))
4.59/2.19	 ||    IF!FAC6220MIN(false, add(%X, add(%Y, %Z))) -> MIN(add(%Y, %Z))
4.59/2.19	 ||    RM(%X, add(%Y, %Z)) -> IF!FAC6220RM(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    RM(%X, add(%Y, %Z)) -> EQ(%X, %Y)
4.59/2.19	 ||    IF!FAC6220RM(true, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.19	 ||    IF!FAC6220RM(false, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.19	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    MINSORT(add(%X, %Y), %Z) -> EQ(%X, min(add(%X, %Y)))
4.59/2.19	 ||    MINSORT(add(%X, %Y), %Z) -> MIN(add(%X, %Y))
4.59/2.19	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> MINSORT(app(rm(%X, %Y), %Z), nil)
4.59/2.19	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> APP(rm(%X, %Y), %Z)
4.59/2.19	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> RM(%X, %Y)
4.59/2.19	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (5) DependencyGraphProof (EQUIVALENT)
4.59/2.19	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 6 less nodes.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (6)
4.59/2.19	 || Complex Obligation (AND)
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (7)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (8) UsableRulesProof (EQUIVALENT)
4.59/2.19	 || 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.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (9)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.59/2.19	 || 
4.59/2.19	 || R is empty.
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (10) QReductionProof (EQUIVALENT)
4.59/2.19	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (11)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.59/2.19	 || 
4.59/2.19	 || R is empty.
4.59/2.19	 || Q is empty.
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (12) QDPSizeChangeProof (EQUIVALENT)
4.59/2.19	 || 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. 
4.59/2.19	 || 
4.59/2.19	 || From the DPs we obtained the following set of size-change graphs:
4.59/2.19	 || *APP(add(%X, %Y), %Z) -> APP(%Y, %Z)
4.59/2.19	 || The graph contains the following edges 1 > 1, 2 >= 2
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (13)
4.59/2.19	 || YES
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (14)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (15) UsableRulesProof (EQUIVALENT)
4.59/2.19	 || 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.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (16)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || R is empty.
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (17) QReductionProof (EQUIVALENT)
4.59/2.19	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (18)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || R is empty.
4.59/2.19	 || Q is empty.
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (19) QDPSizeChangeProof (EQUIVALENT)
4.59/2.19	 || 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. 
4.59/2.19	 || 
4.59/2.19	 || From the DPs we obtained the following set of size-change graphs:
4.59/2.19	 || *LE(s(%X), s(%Y)) -> LE(%X, %Y)
4.59/2.19	 || The graph contains the following edges 1 > 1, 2 > 2
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (20)
4.59/2.19	 || YES
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (21)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> IF!FAC6220MIN(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    IF!FAC6220MIN(true, add(%X, add(%Y, %Z))) -> MIN(add(%X, %Z))
4.59/2.19	 ||    IF!FAC6220MIN(false, add(%X, add(%Y, %Z))) -> MIN(add(%Y, %Z))
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (22) UsableRulesProof (EQUIVALENT)
4.59/2.19	 || 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.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (23)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> IF!FAC6220MIN(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    IF!FAC6220MIN(true, add(%X, add(%Y, %Z))) -> MIN(add(%X, %Z))
4.59/2.19	 ||    IF!FAC6220MIN(false, add(%X, add(%Y, %Z))) -> MIN(add(%Y, %Z))
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (24) QReductionProof (EQUIVALENT)
4.59/2.19	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (25)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> IF!FAC6220MIN(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    IF!FAC6220MIN(true, add(%X, add(%Y, %Z))) -> MIN(add(%X, %Z))
4.59/2.19	 ||    IF!FAC6220MIN(false, add(%X, add(%Y, %Z))) -> MIN(add(%Y, %Z))
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (26) QDPOrderProof (EQUIVALENT)
4.59/2.19	 || We use the reduction pair processor [LPAR04,JAR06].
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || The following pairs can be oriented strictly and are deleted.
4.59/2.19	 || 
4.59/2.19	 ||    IF!FAC6220MIN(true, add(%X, add(%Y, %Z))) -> MIN(add(%X, %Z))
4.59/2.19	 ||    IF!FAC6220MIN(false, add(%X, add(%Y, %Z))) -> MIN(add(%Y, %Z))
4.59/2.19	 || The remaining pairs can at least be oriented weakly.
4.59/2.19	 || Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
4.59/2.19	 || 
4.59/2.19	 || POL( IF!FAC6220MIN_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 2}
4.59/2.19	 || POL( le_2(x_1, x_2) ) = 2
4.59/2.19	 || POL( 0 ) = 2
4.59/2.19	 || POL( true ) = 0
4.59/2.19	 || POL( s_1(x_1) ) = 2
4.59/2.19	 || POL( false ) = 0
4.59/2.19	 || POL( MIN_1(x_1) ) = 2x_1 + 2
4.59/2.19	 || POL( add_2(x_1, x_2) ) = x_1 + 2x_2 + 2
4.59/2.19	 || 
4.59/2.19	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (27)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    MIN(add(%X, add(%Y, %Z))) -> IF!FAC6220MIN(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (28) DependencyGraphProof (EQUIVALENT)
4.59/2.19	 || The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (29)
4.59/2.19	 || TRUE
4.59/2.19	 || 
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (30)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || The TRS R consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0) -> true
4.59/2.19	 ||    eq(0, s(%X)) -> false
4.59/2.19	 ||    eq(s(%X), 0) -> false
4.59/2.19	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.19	 ||    le(0, %X) -> true
4.59/2.19	 ||    le(s(%X), 0) -> false
4.59/2.19	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.19	 ||    app(nil, %X) -> %X
4.59/2.19	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.19	 ||    min(add(%X, nil)) -> %X
4.59/2.19	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.19	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.19	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.19	 ||    rm(%X, nil) -> nil
4.59/2.19	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.19	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.19	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.19	 ||    minsort(nil, nil) -> nil
4.59/2.19	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.19	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.19	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.19	 || 
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (31) UsableRulesProof (EQUIVALENT)
4.59/2.19	 || 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.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (32)
4.59/2.19	 || Obligation:
4.59/2.19	 || Q DP problem:
4.59/2.19	 || The TRS P consists of the following rules:
4.59/2.19	 || 
4.59/2.19	 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
4.59/2.19	 || 
4.59/2.19	 || R is empty.
4.59/2.19	 || The set Q consists of the following terms:
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.19	 ||    rm(x0, nil)
4.59/2.19	 ||    rm(x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.19	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.19	 ||    minsort(nil, nil)
4.59/2.19	 ||    minsort(add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.19	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.19	 || 
4.59/2.19	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.19	 || ----------------------------------------
4.59/2.19	 || 
4.59/2.19	 || (33) QReductionProof (EQUIVALENT)
4.59/2.19	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.19	 || 
4.59/2.19	 ||    eq(0, 0)
4.59/2.19	 ||    eq(0, s(x0))
4.59/2.19	 ||    eq(s(x0), 0)
4.59/2.19	 ||    eq(s(x0), s(x1))
4.59/2.19	 ||    le(0, x0)
4.59/2.19	 ||    le(s(x0), 0)
4.59/2.19	 ||    le(s(x0), s(x1))
4.59/2.19	 ||    app(nil, x0)
4.59/2.19	 ||    app(add(x0, x1), x2)
4.59/2.19	 ||    min(add(x0, nil))
4.59/2.19	 ||    min(add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.19	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (34)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
4.59/2.20	 || 
4.59/2.20	 || R is empty.
4.59/2.20	 || Q is empty.
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (35) QDPSizeChangeProof (EQUIVALENT)
4.59/2.20	 || 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. 
4.59/2.20	 || 
4.59/2.20	 || From the DPs we obtained the following set of size-change graphs:
4.59/2.20	 || *EQ(s(%X), s(%Y)) -> EQ(%X, %Y)
4.59/2.20	 || The graph contains the following edges 1 > 1, 2 > 2
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (36)
4.59/2.20	 || YES
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (37)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    RM(%X, add(%Y, %Z)) -> IF!FAC6220RM(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    IF!FAC6220RM(true, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 ||    IF!FAC6220RM(false, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 ||    minsort(nil, nil) -> nil
4.59/2.20	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.20	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (38) UsableRulesProof (EQUIVALENT)
4.59/2.20	 || 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.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (39)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    RM(%X, add(%Y, %Z)) -> IF!FAC6220RM(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    IF!FAC6220RM(true, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 ||    IF!FAC6220RM(false, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (40) QReductionProof (EQUIVALENT)
4.59/2.20	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.20	 || 
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (41)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    RM(%X, add(%Y, %Z)) -> IF!FAC6220RM(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    IF!FAC6220RM(true, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 ||    IF!FAC6220RM(false, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (42) QDPSizeChangeProof (EQUIVALENT)
4.59/2.20	 || 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. 
4.59/2.20	 || 
4.59/2.20	 || From the DPs we obtained the following set of size-change graphs:
4.59/2.20	 || *RM(%X, add(%Y, %Z)) -> IF!FAC6220RM(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 || The graph contains the following edges 1 >= 2, 2 >= 3
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || *IF!FAC6220RM(true, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 || The graph contains the following edges 2 >= 1, 3 > 2
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || *IF!FAC6220RM(false, %X, add(%Y, %Z)) -> RM(%X, %Z)
4.59/2.20	 || The graph contains the following edges 2 >= 1, 3 > 2
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (43)
4.59/2.20	 || YES
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (44)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> MINSORT(app(rm(%X, %Y), %Z), nil)
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 ||    minsort(nil, nil) -> nil
4.59/2.20	 ||    minsort(add(%X, %Y), %Z) -> if!fac6220minsort(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    if!fac6220minsort(true, add(%X, %Y), %Z) -> add(%X, minsort(app(rm(%X, %Y), %Z), nil))
4.59/2.20	 ||    if!fac6220minsort(false, add(%X, %Y), %Z) -> minsort(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (45) UsableRulesProof (EQUIVALENT)
4.59/2.20	 || 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.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (46)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> MINSORT(app(rm(%X, %Y), %Z), nil)
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (47) QReductionProof (EQUIVALENT)
4.59/2.20	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.20	 || 
4.59/2.20	 ||    minsort(nil, nil)
4.59/2.20	 ||    minsort(add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(true, add(x0, x1), x2)
4.59/2.20	 ||    if!fac6220minsort(false, add(x0, x1), x2)
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (48)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> MINSORT(app(rm(%X, %Y), %Z), nil)
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (49) QDPOrderProof (EQUIVALENT)
4.59/2.20	 || We use the reduction pair processor [LPAR04,JAR06].
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || The following pairs can be oriented strictly and are deleted.
4.59/2.20	 || 
4.59/2.20	 ||    IF!FAC6220MINSORT(true, add(%X, %Y), %Z) -> MINSORT(app(rm(%X, %Y), %Z), nil)
4.59/2.20	 || The remaining pairs can at least be oriented weakly.
4.59/2.20	 || Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
4.59/2.20	 || 
4.59/2.20	 || POL( MINSORT_2(x_1, x_2) ) = max{0, 2x_1 + 2x_2 - 1}
4.59/2.20	 || POL( app_2(x_1, x_2) ) = x_1 + x_2
4.59/2.20	 || POL( rm_2(x_1, x_2) ) = x_2
4.59/2.20	 || POL( nil ) = 0
4.59/2.20	 || POL( add_2(x_1, x_2) ) = x_1 + x_2 + 1
4.59/2.20	 || POL( if!fac6220rm_3(x_1, ..., x_3) ) = x_3
4.59/2.20	 || POL( eq_2(x_1, x_2) ) = max{0, x_2 - 2}
4.59/2.20	 || POL( true ) = 0
4.59/2.20	 || POL( IF!FAC6220MINSORT_3(x_1, ..., x_3) ) = max{0, 2x_2 + 2x_3 - 1}
4.59/2.20	 || POL( min_1(x_1) ) = max{0, -1}
4.59/2.20	 || POL( if!fac6220min_2(x_1, x_2) ) = 2x_1 + 2
4.59/2.20	 || POL( le_2(x_1, x_2) ) = 1
4.59/2.20	 || POL( false ) = 0
4.59/2.20	 || POL( 0 ) = 0
4.59/2.20	 || POL( s_1(x_1) ) = x_1
4.59/2.20	 || 
4.59/2.20	 || The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
4.59/2.20	 || 
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (50)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 ||    rm(%X, nil) -> nil
4.59/2.20	 ||    rm(%X, add(%Y, %Z)) -> if!fac6220rm(eq(%X, %Y), %X, add(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(true, %X, add(%Y, %Z)) -> rm(%X, %Z)
4.59/2.20	 ||    app(nil, %X) -> %X
4.59/2.20	 ||    app(add(%X, %Y), %Z) -> add(%X, app(%Y, %Z))
4.59/2.20	 ||    if!fac6220rm(false, %X, add(%Y, %Z)) -> add(%Y, rm(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (51) UsableRulesProof (EQUIVALENT)
4.59/2.20	 || 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.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (52)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (53) QReductionProof (EQUIVALENT)
4.59/2.20	 || We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
4.59/2.20	 || 
4.59/2.20	 ||    app(nil, x0)
4.59/2.20	 ||    app(add(x0, x1), x2)
4.59/2.20	 ||    rm(x0, nil)
4.59/2.20	 ||    rm(x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(true, x0, add(x1, x2))
4.59/2.20	 ||    if!fac6220rm(false, x0, add(x1, x2))
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (54)
4.59/2.20	 || Obligation:
4.59/2.20	 || Q DP problem:
4.59/2.20	 || The TRS P consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 ||    IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || 
4.59/2.20	 || The TRS R consists of the following rules:
4.59/2.20	 || 
4.59/2.20	 ||    min(add(%X, nil)) -> %X
4.59/2.20	 ||    min(add(%X, add(%Y, %Z))) -> if!fac6220min(le(%X, %Y), add(%X, add(%Y, %Z)))
4.59/2.20	 ||    if!fac6220min(true, add(%X, add(%Y, %Z))) -> min(add(%X, %Z))
4.59/2.20	 ||    if!fac6220min(false, add(%X, add(%Y, %Z))) -> min(add(%Y, %Z))
4.59/2.20	 ||    eq(0, 0) -> true
4.59/2.20	 ||    eq(0, s(%X)) -> false
4.59/2.20	 ||    eq(s(%X), 0) -> false
4.59/2.20	 ||    eq(s(%X), s(%Y)) -> eq(%X, %Y)
4.59/2.20	 ||    le(0, %X) -> true
4.59/2.20	 ||    le(s(%X), 0) -> false
4.59/2.20	 ||    le(s(%X), s(%Y)) -> le(%X, %Y)
4.59/2.20	 || 
4.59/2.20	 || The set Q consists of the following terms:
4.59/2.20	 || 
4.59/2.20	 ||    eq(0, 0)
4.59/2.20	 ||    eq(0, s(x0))
4.59/2.20	 ||    eq(s(x0), 0)
4.59/2.20	 ||    eq(s(x0), s(x1))
4.59/2.20	 ||    le(0, x0)
4.59/2.20	 ||    le(s(x0), 0)
4.59/2.20	 ||    le(s(x0), s(x1))
4.59/2.20	 ||    min(add(x0, nil))
4.59/2.20	 ||    min(add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(true, add(x0, add(x1, x2)))
4.59/2.20	 ||    if!fac6220min(false, add(x0, add(x1, x2)))
4.59/2.20	 || 
4.59/2.20	 || We have to consider all minimal (P,Q,R)-chains.
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (55) QDPSizeChangeProof (EQUIVALENT)
4.59/2.20	 || 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. 
4.59/2.20	 || 
4.59/2.20	 || From the DPs we obtained the following set of size-change graphs:
4.59/2.20	 || *IF!FAC6220MINSORT(false, add(%X, %Y), %Z) -> MINSORT(%Y, add(%X, %Z))
4.59/2.20	 || The graph contains the following edges 2 > 1
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || *MINSORT(add(%X, %Y), %Z) -> IF!FAC6220MINSORT(eq(%X, min(add(%X, %Y))), add(%X, %Y), %Z)
4.59/2.20	 || The graph contains the following edges 1 >= 2, 2 >= 3
4.59/2.20	 || 
4.59/2.20	 || 
4.59/2.20	 || ----------------------------------------
4.59/2.20	 || 
4.59/2.20	 || (56)
4.59/2.20	 || YES
4.59/2.20	 || 
4.59/2.20	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]).
4.59/2.20	
4.59/2.20	We thus obtain the following dependency pair problem (P_0, R_0, static, formative):
4.59/2.20	
4.59/2.20	  Dependency Pairs P_0:
4.59/2.20	
4.59/2.20	    0] map#(F, add(X, Y)) =#> map#(F, Y)   
4.59/2.20	    1] filter#(F, add(X, Y)) =#> filter2#(F X, F, X, Y)   
4.59/2.20	    2] filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	    3] filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	
4.59/2.20	  Rules R_0:
4.59/2.20	
4.59/2.20	    eq(0, 0) => true 
4.59/2.20	    eq(0, s(X)) => false 
4.59/2.20	    eq(s(X), 0) => false 
4.59/2.20	    eq(s(X), s(Y)) => eq(X, Y) 
4.59/2.20	    le(0, X) => true 
4.59/2.20	    le(s(X), 0) => false 
4.59/2.20	    le(s(X), s(Y)) => le(X, Y) 
4.59/2.20	    app(nil, X) => X 
4.59/2.20	    app(add(X, Y), Z) => add(X, app(Y, Z)) 
4.59/2.20	    min(add(X, nil)) => X 
4.59/2.20	    min(add(X, add(Y, Z))) => if!fac6220min(le(X, Y), add(X, add(Y, Z))) 
4.59/2.20	    if!fac6220min(true, add(X, add(Y, Z))) => min(add(X, Z)) 
4.59/2.20	    if!fac6220min(false, add(X, add(Y, Z))) => min(add(Y, Z)) 
4.59/2.20	    rm(X, nil) => nil 
4.59/2.20	    rm(X, add(Y, Z)) => if!fac6220rm(eq(X, Y), X, add(Y, Z)) 
4.59/2.20	    if!fac6220rm(true, X, add(Y, Z)) => rm(X, Z) 
4.59/2.20	    if!fac6220rm(false, X, add(Y, Z)) => add(Y, rm(X, Z)) 
4.59/2.20	    minsort(nil, nil) => nil 
4.59/2.20	    minsort(add(X, Y), Z) => if!fac6220minsort(eq(X, min(add(X, Y))), add(X, Y), Z) 
4.59/2.20	    if!fac6220minsort(true, add(X, Y), Z) => add(X, minsort(app(rm(X, Y), Z), nil)) 
4.59/2.20	    if!fac6220minsort(false, add(X, Y), Z) => minsort(Y, add(X, Z)) 
4.59/2.20	    map(F, nil) => nil 
4.59/2.20	    map(F, add(X, Y)) => add(F X, map(F, Y)) 
4.59/2.20	    filter(F, nil) => nil 
4.59/2.20	    filter(F, add(X, Y)) => filter2(F X, F, X, Y) 
4.59/2.20	    filter2(true, F, X, Y) => add(X, filter(F, Y)) 
4.59/2.20	    filter2(false, F, X, Y) => filter(F, Y) 
4.59/2.20	
4.59/2.20	Thus, the original system is terminating if (P_0, R_0, static, formative) is finite.
4.59/2.20	
4.59/2.20	We consider the dependency pair problem (P_0, R_0, static, formative).
4.59/2.20	
4.59/2.20	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
4.59/2.20	
4.59/2.20	    * 0 : 0 
4.59/2.20	    * 1 : 2, 3 
4.59/2.20	    * 2 : 1 
4.59/2.20	    * 3 : 1 
4.59/2.20	
4.59/2.20	This graph has the following strongly connected components:
4.59/2.20	
4.59/2.20	  P_1:
4.59/2.20	
4.59/2.20	    map#(F, add(X, Y)) =#> map#(F, Y)   
4.59/2.20	
4.59/2.20	  P_2:
4.59/2.20	
4.59/2.20	    filter#(F, add(X, Y)) =#> filter2#(F X, F, X, Y)   
4.59/2.20	    filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	    filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	
4.59/2.20	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).
4.59/2.20	
4.59/2.20	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_2, R_0, static, formative) is finite.
4.59/2.20	
4.59/2.20	We consider the dependency pair problem (P_2, R_0, static, formative).
4.59/2.20	
4.59/2.20	We apply the subterm criterion with the following projection function:
4.59/2.20	
4.59/2.20	  nu(filter2#) = 4 
4.59/2.20	  nu(filter#) = 2 
4.59/2.20	
4.59/2.20	Thus, we can orient the dependency pairs as follows:
4.59/2.20	
4.59/2.20	  nu(filter#(F, add(X, Y))) = add(X, Y) |> Y = nu(filter2#(F X, F, X, Y)) 
4.59/2.20	  nu(filter2#(true, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
4.59/2.20	  nu(filter2#(false, F, X, Y)) = Y = Y = nu(filter#(F, Y)) 
4.59/2.20	
4.59/2.20	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:
4.59/2.20	
4.59/2.20	  filter2#(true, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	  filter2#(false, F, X, Y) =#> filter#(F, Y)   
4.59/2.20	
4.59/2.20	Thus, the original system is terminating if each of (P_1, R_0, static, formative) and (P_3, R_0, static, formative) is finite.
4.59/2.20	
4.59/2.20	We consider the dependency pair problem (P_3, R_0, static, formative).
4.59/2.20	
4.59/2.20	We place the elements of P in a dependency graph approximation G (see e.g. [Kop12, Thm. 7.27, 7.29], as follows:
4.59/2.20	
4.59/2.20	    * 0 :  
4.59/2.20	    * 1 :  
4.59/2.20	
4.59/2.20	This graph has no strongly connected components.  By [Kop12, Thm. 7.31], this implies finiteness of the dependency pair problem.
4.59/2.20	
4.59/2.20	Thus, the original system is terminating if (P_1, R_0, static, formative) is finite.
4.59/2.20	
4.59/2.20	We consider the dependency pair problem (P_1, R_0, static, formative).
4.59/2.20	
4.59/2.20	We apply the subterm criterion with the following projection function:
4.59/2.20	
4.59/2.20	  nu(map#) = 2 
4.59/2.20	
4.59/2.20	Thus, we can orient the dependency pairs as follows:
4.59/2.20	
4.59/2.20	  nu(map#(F, add(X, Y))) = add(X, Y) |> Y = nu(map#(F, Y)) 
4.59/2.20	
4.59/2.20	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.
4.59/2.20	
4.59/2.20	As all dependency pair problems were succesfully simplified with sound (and complete) processors until nothing remained, we conclude termination.
4.59/2.20	
4.59/2.20	
4.59/2.20	+++ Citations +++
4.59/2.20	
4.59/2.20	[Kop12]  C. Kop.  Higher Order Termination.  PhD Thesis, 2012.
4.59/2.20	[Kop13]  C. Kop.  Static Dependency Pairs with Accessibility.  Unpublished manuscript, http://cl-informatik.uibk.ac.at/users/kop/static.pdf, 2013.
4.59/2.20	[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.
4.59/2.20	EOF