11.09/3.68	YES
11.09/3.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.09/3.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.09/3.69	
11.09/3.69	
11.09/3.69	Termination w.r.t. Q of the given QTRS could be proven:
11.09/3.69	
11.09/3.69	(0) QTRS
11.09/3.69	(1) QTRSToCSRProof [SOUND, 0 ms]
11.09/3.69	(2) CSR
11.09/3.69	(3) CSDependencyPairsProof [EQUIVALENT, 14 ms]
11.09/3.69	(4) QCSDP
11.09/3.69	(5) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	(6) AND
11.09/3.69	    (7) QCSDP
11.09/3.69	        (8) QCSDPReductionPairProof [EQUIVALENT, 54 ms]
11.09/3.69	        (9) QCSDP
11.09/3.69	        (10) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	        (11) AND
11.09/3.69	            (12) QCSDP
11.09/3.69	                (13) QCSDPReductionPairProof [EQUIVALENT, 266 ms]
11.09/3.69	                (14) QCSDP
11.09/3.69	                (15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	                (16) TRUE
11.09/3.69	            (17) QCSDP
11.09/3.69	                (18) QCSDPReductionPairProof [EQUIVALENT, 335 ms]
11.09/3.69	                (19) QCSDP
11.09/3.69	                (20) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	                (21) TRUE
11.09/3.69	    (22) QCSDP
11.09/3.69	        (23) QCSDPSubtermProof [EQUIVALENT, 0 ms]
11.09/3.69	        (24) QCSDP
11.09/3.69	        (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	        (26) TRUE
11.09/3.69	    (27) QCSDP
11.09/3.69	        (28) QCSDPSubtermProof [EQUIVALENT, 0 ms]
11.09/3.69	        (29) QCSDP
11.09/3.69	        (30) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.69	        (31) TRUE
11.09/3.69	
11.09/3.69	
11.09/3.69	----------------------------------------
11.09/3.69	
11.09/3.69	(0)
11.09/3.69	Obligation:
11.09/3.69	Q restricted rewrite system:
11.09/3.69	The TRS R consists of the following rules:
11.09/3.69	
11.09/3.69	   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
11.09/3.69	   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
11.09/3.69	   active(U13(tt)) -> mark(tt)
11.09/3.69	   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
11.09/3.69	   active(U22(tt)) -> mark(tt)
11.09/3.69	   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
11.09/3.69	   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
11.09/3.69	   active(U33(tt)) -> mark(tt)
11.09/3.69	   active(U41(tt, N)) -> mark(N)
11.09/3.69	   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
11.09/3.69	   active(U61(tt)) -> mark(0)
11.09/3.69	   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
11.09/3.69	   active(and(tt, X)) -> mark(X)
11.09/3.69	   active(isNat(0)) -> mark(tt)
11.09/3.69	   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
11.09/3.69	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
11.09/3.69	   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
11.09/3.69	   active(isNatKind(0)) -> mark(tt)
11.09/3.69	   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
11.09/3.69	   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
11.09/3.69	   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
11.09/3.69	   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
11.09/3.69	   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
11.09/3.69	   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
11.09/3.69	   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
11.09/3.69	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
11.09/3.69	   active(U12(X1, X2)) -> U12(active(X1), X2)
11.09/3.69	   active(U13(X)) -> U13(active(X))
11.09/3.69	   active(U21(X1, X2)) -> U21(active(X1), X2)
11.09/3.69	   active(U22(X)) -> U22(active(X))
11.09/3.69	   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
11.09/3.69	   active(U32(X1, X2)) -> U32(active(X1), X2)
11.09/3.69	   active(U33(X)) -> U33(active(X))
11.09/3.69	   active(U41(X1, X2)) -> U41(active(X1), X2)
11.09/3.69	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
11.09/3.69	   active(s(X)) -> s(active(X))
11.09/3.69	   active(plus(X1, X2)) -> plus(active(X1), X2)
11.09/3.69	   active(plus(X1, X2)) -> plus(X1, active(X2))
11.09/3.69	   active(U61(X)) -> U61(active(X))
11.09/3.69	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
11.09/3.69	   active(x(X1, X2)) -> x(active(X1), X2)
11.09/3.69	   active(x(X1, X2)) -> x(X1, active(X2))
11.09/3.69	   active(and(X1, X2)) -> and(active(X1), X2)
11.09/3.69	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
11.09/3.69	   U12(mark(X1), X2) -> mark(U12(X1, X2))
11.09/3.69	   U13(mark(X)) -> mark(U13(X))
11.09/3.69	   U21(mark(X1), X2) -> mark(U21(X1, X2))
11.09/3.69	   U22(mark(X)) -> mark(U22(X))
11.09/3.69	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
11.09/3.69	   U32(mark(X1), X2) -> mark(U32(X1, X2))
11.09/3.69	   U33(mark(X)) -> mark(U33(X))
11.09/3.69	   U41(mark(X1), X2) -> mark(U41(X1, X2))
11.09/3.69	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
11.09/3.69	   s(mark(X)) -> mark(s(X))
11.09/3.69	   plus(mark(X1), X2) -> mark(plus(X1, X2))
11.09/3.69	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
11.09/3.69	   U61(mark(X)) -> mark(U61(X))
11.09/3.69	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
11.09/3.69	   x(mark(X1), X2) -> mark(x(X1, X2))
11.09/3.69	   x(X1, mark(X2)) -> mark(x(X1, X2))
11.09/3.69	   and(mark(X1), X2) -> mark(and(X1, X2))
11.09/3.69	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(tt) -> ok(tt)
11.09/3.69	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
11.09/3.69	   proper(isNat(X)) -> isNat(proper(X))
11.09/3.69	   proper(U13(X)) -> U13(proper(X))
11.09/3.69	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
11.09/3.69	   proper(U22(X)) -> U22(proper(X))
11.09/3.69	   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
11.09/3.69	   proper(U33(X)) -> U33(proper(X))
11.09/3.69	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
11.09/3.69	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(s(X)) -> s(proper(X))
11.09/3.69	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
11.09/3.69	   proper(U61(X)) -> U61(proper(X))
11.09/3.69	   proper(0) -> ok(0)
11.09/3.69	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
11.09/3.69	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
11.09/3.69	   proper(isNatKind(X)) -> isNatKind(proper(X))
11.09/3.69	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
11.09/3.69	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
11.09/3.69	   isNat(ok(X)) -> ok(isNat(X))
11.09/3.69	   U13(ok(X)) -> ok(U13(X))
11.09/3.69	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
11.09/3.69	   U22(ok(X)) -> ok(U22(X))
11.09/3.69	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
11.09/3.69	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
11.09/3.69	   U33(ok(X)) -> ok(U33(X))
11.09/3.69	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
11.09/3.69	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
11.09/3.69	   s(ok(X)) -> ok(s(X))
11.09/3.69	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
11.09/3.69	   U61(ok(X)) -> ok(U61(X))
11.09/3.69	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
11.09/3.69	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
11.09/3.69	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
11.09/3.69	   isNatKind(ok(X)) -> ok(isNatKind(X))
11.09/3.69	   top(mark(X)) -> top(proper(X))
11.09/3.69	   top(ok(X)) -> top(active(X))
11.09/3.69	
11.09/3.69	The set Q consists of the following terms:
11.09/3.69	
11.09/3.69	   active(isNat(0))
11.09/3.69	   active(isNat(plus(x0, x1)))
11.09/3.69	   active(isNat(s(x0)))
11.09/3.69	   active(isNat(x(x0, x1)))
11.09/3.69	   active(isNatKind(0))
11.09/3.69	   active(isNatKind(plus(x0, x1)))
11.09/3.69	   active(isNatKind(s(x0)))
11.09/3.69	   active(isNatKind(x(x0, x1)))
11.09/3.69	   active(U11(x0, x1, x2))
11.09/3.69	   active(U12(x0, x1))
11.09/3.69	   active(U13(x0))
11.09/3.69	   active(U21(x0, x1))
11.09/3.69	   active(U22(x0))
11.09/3.69	   active(U31(x0, x1, x2))
11.09/3.69	   active(U32(x0, x1))
11.09/3.69	   active(U33(x0))
11.09/3.69	   active(U41(x0, x1))
11.09/3.69	   active(U51(x0, x1, x2))
11.09/3.69	   active(s(x0))
11.09/3.69	   active(plus(x0, x1))
11.09/3.69	   active(U61(x0))
11.09/3.69	   active(U71(x0, x1, x2))
11.09/3.69	   active(x(x0, x1))
11.09/3.69	   active(and(x0, x1))
11.09/3.69	   U11(mark(x0), x1, x2)
11.09/3.69	   U12(mark(x0), x1)
11.09/3.69	   U13(mark(x0))
11.09/3.69	   U21(mark(x0), x1)
11.09/3.69	   U22(mark(x0))
11.09/3.69	   U31(mark(x0), x1, x2)
11.09/3.69	   U32(mark(x0), x1)
11.09/3.69	   U33(mark(x0))
11.09/3.69	   U41(mark(x0), x1)
11.09/3.69	   U51(mark(x0), x1, x2)
11.09/3.69	   s(mark(x0))
11.09/3.69	   plus(mark(x0), x1)
11.09/3.69	   plus(x0, mark(x1))
11.09/3.69	   U61(mark(x0))
11.09/3.69	   U71(mark(x0), x1, x2)
11.09/3.69	   x(mark(x0), x1)
11.09/3.69	   x(x0, mark(x1))
11.09/3.69	   and(mark(x0), x1)
11.09/3.69	   proper(U11(x0, x1, x2))
11.09/3.69	   proper(tt)
11.09/3.69	   proper(U12(x0, x1))
11.09/3.69	   proper(isNat(x0))
11.09/3.69	   proper(U13(x0))
11.09/3.69	   proper(U21(x0, x1))
11.09/3.69	   proper(U22(x0))
11.09/3.69	   proper(U31(x0, x1, x2))
11.09/3.69	   proper(U32(x0, x1))
11.09/3.69	   proper(U33(x0))
11.09/3.69	   proper(U41(x0, x1))
11.09/3.69	   proper(U51(x0, x1, x2))
11.09/3.69	   proper(s(x0))
11.09/3.69	   proper(plus(x0, x1))
11.09/3.69	   proper(U61(x0))
11.09/3.69	   proper(0)
11.09/3.69	   proper(U71(x0, x1, x2))
11.09/3.69	   proper(x(x0, x1))
11.09/3.69	   proper(and(x0, x1))
11.09/3.69	   proper(isNatKind(x0))
11.09/3.69	   U11(ok(x0), ok(x1), ok(x2))
11.09/3.69	   U12(ok(x0), ok(x1))
11.09/3.69	   isNat(ok(x0))
11.09/3.69	   U13(ok(x0))
11.09/3.69	   U21(ok(x0), ok(x1))
11.09/3.69	   U22(ok(x0))
11.09/3.69	   U31(ok(x0), ok(x1), ok(x2))
11.09/3.69	   U32(ok(x0), ok(x1))
11.09/3.69	   U33(ok(x0))
11.09/3.69	   U41(ok(x0), ok(x1))
11.09/3.69	   U51(ok(x0), ok(x1), ok(x2))
11.09/3.69	   s(ok(x0))
11.09/3.69	   plus(ok(x0), ok(x1))
11.09/3.69	   U61(ok(x0))
11.09/3.69	   U71(ok(x0), ok(x1), ok(x2))
11.09/3.69	   x(ok(x0), ok(x1))
11.09/3.69	   and(ok(x0), ok(x1))
11.09/3.69	   isNatKind(ok(x0))
11.09/3.69	   top(mark(x0))
11.09/3.69	   top(ok(x0))
11.09/3.69	
11.09/3.69	
11.09/3.69	----------------------------------------
11.09/3.69	
11.09/3.69	(1) QTRSToCSRProof (SOUND)
11.09/3.69	The following Q TRS is given: Q restricted rewrite system:
11.09/3.69	The TRS R consists of the following rules:
11.09/3.69	
11.09/3.69	   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
11.09/3.69	   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
11.09/3.69	   active(U13(tt)) -> mark(tt)
11.09/3.69	   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
11.09/3.69	   active(U22(tt)) -> mark(tt)
11.09/3.69	   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
11.09/3.69	   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
11.09/3.69	   active(U33(tt)) -> mark(tt)
11.09/3.69	   active(U41(tt, N)) -> mark(N)
11.09/3.69	   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
11.09/3.69	   active(U61(tt)) -> mark(0)
11.09/3.69	   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
11.09/3.69	   active(and(tt, X)) -> mark(X)
11.09/3.69	   active(isNat(0)) -> mark(tt)
11.09/3.69	   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
11.09/3.69	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
11.09/3.69	   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
11.09/3.69	   active(isNatKind(0)) -> mark(tt)
11.09/3.69	   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
11.09/3.69	   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
11.09/3.69	   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
11.09/3.69	   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
11.09/3.69	   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
11.09/3.69	   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
11.09/3.69	   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
11.09/3.69	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
11.09/3.69	   active(U12(X1, X2)) -> U12(active(X1), X2)
11.09/3.69	   active(U13(X)) -> U13(active(X))
11.09/3.69	   active(U21(X1, X2)) -> U21(active(X1), X2)
11.09/3.69	   active(U22(X)) -> U22(active(X))
11.09/3.69	   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
11.09/3.69	   active(U32(X1, X2)) -> U32(active(X1), X2)
11.09/3.69	   active(U33(X)) -> U33(active(X))
11.09/3.69	   active(U41(X1, X2)) -> U41(active(X1), X2)
11.09/3.69	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
11.09/3.69	   active(s(X)) -> s(active(X))
11.09/3.69	   active(plus(X1, X2)) -> plus(active(X1), X2)
11.09/3.69	   active(plus(X1, X2)) -> plus(X1, active(X2))
11.09/3.69	   active(U61(X)) -> U61(active(X))
11.09/3.69	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
11.09/3.69	   active(x(X1, X2)) -> x(active(X1), X2)
11.09/3.69	   active(x(X1, X2)) -> x(X1, active(X2))
11.09/3.69	   active(and(X1, X2)) -> and(active(X1), X2)
11.09/3.69	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
11.09/3.69	   U12(mark(X1), X2) -> mark(U12(X1, X2))
11.09/3.69	   U13(mark(X)) -> mark(U13(X))
11.09/3.69	   U21(mark(X1), X2) -> mark(U21(X1, X2))
11.09/3.69	   U22(mark(X)) -> mark(U22(X))
11.09/3.69	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
11.09/3.69	   U32(mark(X1), X2) -> mark(U32(X1, X2))
11.09/3.69	   U33(mark(X)) -> mark(U33(X))
11.09/3.69	   U41(mark(X1), X2) -> mark(U41(X1, X2))
11.09/3.69	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
11.09/3.69	   s(mark(X)) -> mark(s(X))
11.09/3.69	   plus(mark(X1), X2) -> mark(plus(X1, X2))
11.09/3.69	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
11.09/3.69	   U61(mark(X)) -> mark(U61(X))
11.09/3.69	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
11.09/3.69	   x(mark(X1), X2) -> mark(x(X1, X2))
11.09/3.69	   x(X1, mark(X2)) -> mark(x(X1, X2))
11.09/3.69	   and(mark(X1), X2) -> mark(and(X1, X2))
11.09/3.69	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(tt) -> ok(tt)
11.09/3.69	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
11.09/3.69	   proper(isNat(X)) -> isNat(proper(X))
11.09/3.69	   proper(U13(X)) -> U13(proper(X))
11.09/3.69	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
11.09/3.69	   proper(U22(X)) -> U22(proper(X))
11.09/3.69	   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
11.09/3.69	   proper(U33(X)) -> U33(proper(X))
11.09/3.69	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
11.09/3.69	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(s(X)) -> s(proper(X))
11.09/3.69	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
11.09/3.69	   proper(U61(X)) -> U61(proper(X))
11.09/3.69	   proper(0) -> ok(0)
11.09/3.69	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
11.09/3.69	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
11.09/3.69	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
11.09/3.69	   proper(isNatKind(X)) -> isNatKind(proper(X))
11.09/3.69	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
11.09/3.69	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
11.09/3.69	   isNat(ok(X)) -> ok(isNat(X))
11.09/3.69	   U13(ok(X)) -> ok(U13(X))
11.09/3.69	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
11.09/3.69	   U22(ok(X)) -> ok(U22(X))
11.09/3.69	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
11.09/3.69	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
11.09/3.69	   U33(ok(X)) -> ok(U33(X))
11.09/3.69	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
11.09/3.69	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
11.09/3.69	   s(ok(X)) -> ok(s(X))
11.09/3.69	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
11.09/3.69	   U61(ok(X)) -> ok(U61(X))
11.09/3.69	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
11.09/3.69	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
11.09/3.69	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
11.09/3.69	   isNatKind(ok(X)) -> ok(isNatKind(X))
11.09/3.69	   top(mark(X)) -> top(proper(X))
11.09/3.69	   top(ok(X)) -> top(active(X))
11.09/3.69	
11.09/3.69	The set Q consists of the following terms:
11.09/3.69	
11.09/3.69	   active(isNat(0))
11.09/3.69	   active(isNat(plus(x0, x1)))
11.09/3.69	   active(isNat(s(x0)))
11.09/3.69	   active(isNat(x(x0, x1)))
11.09/3.69	   active(isNatKind(0))
11.09/3.69	   active(isNatKind(plus(x0, x1)))
11.09/3.69	   active(isNatKind(s(x0)))
11.09/3.69	   active(isNatKind(x(x0, x1)))
11.09/3.69	   active(U11(x0, x1, x2))
11.09/3.69	   active(U12(x0, x1))
11.09/3.69	   active(U13(x0))
11.09/3.69	   active(U21(x0, x1))
11.09/3.69	   active(U22(x0))
11.09/3.69	   active(U31(x0, x1, x2))
11.09/3.69	   active(U32(x0, x1))
11.09/3.69	   active(U33(x0))
11.09/3.69	   active(U41(x0, x1))
11.09/3.69	   active(U51(x0, x1, x2))
11.09/3.69	   active(s(x0))
11.09/3.69	   active(plus(x0, x1))
11.09/3.69	   active(U61(x0))
11.09/3.69	   active(U71(x0, x1, x2))
11.09/3.69	   active(x(x0, x1))
11.09/3.69	   active(and(x0, x1))
11.09/3.69	   U11(mark(x0), x1, x2)
11.09/3.69	   U12(mark(x0), x1)
11.09/3.69	   U13(mark(x0))
11.09/3.69	   U21(mark(x0), x1)
11.09/3.69	   U22(mark(x0))
11.09/3.69	   U31(mark(x0), x1, x2)
11.09/3.69	   U32(mark(x0), x1)
11.09/3.69	   U33(mark(x0))
11.09/3.69	   U41(mark(x0), x1)
11.09/3.69	   U51(mark(x0), x1, x2)
11.09/3.69	   s(mark(x0))
11.09/3.69	   plus(mark(x0), x1)
11.09/3.69	   plus(x0, mark(x1))
11.09/3.69	   U61(mark(x0))
11.09/3.69	   U71(mark(x0), x1, x2)
11.09/3.69	   x(mark(x0), x1)
11.09/3.69	   x(x0, mark(x1))
11.09/3.69	   and(mark(x0), x1)
11.09/3.69	   proper(U11(x0, x1, x2))
11.09/3.69	   proper(tt)
11.09/3.69	   proper(U12(x0, x1))
11.09/3.69	   proper(isNat(x0))
11.09/3.69	   proper(U13(x0))
11.09/3.69	   proper(U21(x0, x1))
11.09/3.69	   proper(U22(x0))
11.09/3.69	   proper(U31(x0, x1, x2))
11.09/3.69	   proper(U32(x0, x1))
11.09/3.69	   proper(U33(x0))
11.09/3.69	   proper(U41(x0, x1))
11.09/3.69	   proper(U51(x0, x1, x2))
11.09/3.69	   proper(s(x0))
11.09/3.69	   proper(plus(x0, x1))
11.09/3.69	   proper(U61(x0))
11.09/3.69	   proper(0)
11.09/3.69	   proper(U71(x0, x1, x2))
11.09/3.69	   proper(x(x0, x1))
11.09/3.69	   proper(and(x0, x1))
11.09/3.69	   proper(isNatKind(x0))
11.09/3.69	   U11(ok(x0), ok(x1), ok(x2))
11.09/3.69	   U12(ok(x0), ok(x1))
11.09/3.69	   isNat(ok(x0))
11.09/3.69	   U13(ok(x0))
11.09/3.69	   U21(ok(x0), ok(x1))
11.09/3.69	   U22(ok(x0))
11.09/3.69	   U31(ok(x0), ok(x1), ok(x2))
11.09/3.69	   U32(ok(x0), ok(x1))
11.09/3.69	   U33(ok(x0))
11.09/3.69	   U41(ok(x0), ok(x1))
11.09/3.69	   U51(ok(x0), ok(x1), ok(x2))
11.09/3.69	   s(ok(x0))
11.09/3.69	   plus(ok(x0), ok(x1))
11.09/3.69	   U61(ok(x0))
11.09/3.69	   U71(ok(x0), ok(x1), ok(x2))
11.09/3.69	   x(ok(x0), ok(x1))
11.09/3.69	   and(ok(x0), ok(x1))
11.09/3.69	   isNatKind(ok(x0))
11.09/3.69	   top(mark(x0))
11.09/3.69	   top(ok(x0))
11.09/3.69	
11.09/3.69	Special symbols used for the transformation (see [GM04]):
11.09/3.69	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
11.09/3.69	The replacement map contains the following entries:
11.09/3.69	
11.09/3.69	U11: {1}
11.09/3.69	tt: empty set
11.09/3.69	U12: {1}
11.09/3.69	isNat: empty set
11.09/3.69	U13: {1}
11.09/3.69	U21: {1}
11.09/3.69	U22: {1}
11.09/3.69	U31: {1}
11.09/3.69	U32: {1}
11.09/3.69	U33: {1}
11.09/3.69	U41: {1}
11.09/3.69	U51: {1}
11.09/3.69	s: {1}
11.09/3.69	plus: {1, 2}
11.09/3.69	U61: {1}
11.09/3.69	0: empty set
11.09/3.69	U71: {1}
11.09/3.69	x: {1, 2}
11.09/3.69	and: {1}
11.09/3.69	isNatKind: empty set
11.09/3.69	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
11.09/3.69	----------------------------------------
11.09/3.69	
11.09/3.69	(2)
11.09/3.69	Obligation:
11.09/3.69	Context-sensitive rewrite system:
11.09/3.69	The TRS R consists of the following rules:
11.09/3.69	
11.09/3.69	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.69	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.69	   U13(tt) -> tt
11.09/3.69	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.69	   U22(tt) -> tt
11.09/3.69	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.69	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.69	   U33(tt) -> tt
11.09/3.69	   U41(tt, N) -> N
11.09/3.69	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.69	   U61(tt) -> 0
11.09/3.69	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.69	   and(tt, X) -> X
11.09/3.69	   isNat(0) -> tt
11.09/3.69	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.69	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.69	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.69	   isNatKind(0) -> tt
11.09/3.69	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.69	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.69	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.69	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.69	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.69	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.69	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.69	
11.09/3.69	The replacement map contains the following entries:
11.09/3.69	
11.09/3.69	U11: {1}
11.09/3.69	tt: empty set
11.09/3.69	U12: {1}
11.09/3.69	isNat: empty set
11.09/3.69	U13: {1}
11.09/3.69	U21: {1}
11.09/3.69	U22: {1}
11.09/3.69	U31: {1}
11.09/3.69	U32: {1}
11.09/3.69	U33: {1}
11.09/3.69	U41: {1}
11.09/3.69	U51: {1}
11.09/3.69	s: {1}
11.09/3.69	plus: {1, 2}
11.09/3.69	U61: {1}
11.09/3.69	0: empty set
11.09/3.69	U71: {1}
11.09/3.69	x: {1, 2}
11.09/3.69	and: {1}
11.09/3.69	isNatKind: empty set
11.09/3.69	
11.09/3.69	----------------------------------------
11.09/3.69	
11.09/3.69	(3) CSDependencyPairsProof (EQUIVALENT)
11.09/3.69	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
11.09/3.69	----------------------------------------
11.09/3.69	
11.09/3.69	(4)
11.09/3.69	Obligation:
11.09/3.69	Q-restricted context-sensitive dependency pair problem:
11.09/3.69	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, U13'_1, U22'_1, U33'_1, PLUS_2, X_2, U61'_1} are replacing on all positions.
11.09/3.69	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U12'_2, U11'_3, U21'_2, U32'_2, U31'_3, U51'_3, U71'_3, AND_2, U41'_2} we have mu(f) = {1}.
11.09/3.69	The symbols in {isNat_1, isNatKind_1, ISNAT_1, ISNATKIND_1, U_1} are not replacing on any position.
11.09/3.69	
11.09/3.69	The ordinary context-sensitive dependency pairs DP_o are:
11.09/3.69	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
11.09/3.69	   U11'(tt, V1, V2) -> ISNAT(V1)
11.09/3.69	   U12'(tt, V2) -> U13'(isNat(V2))
11.09/3.69	   U12'(tt, V2) -> ISNAT(V2)
11.09/3.69	   U21'(tt, V1) -> U22'(isNat(V1))
11.09/3.69	   U21'(tt, V1) -> ISNAT(V1)
11.09/3.69	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
11.09/3.69	   U31'(tt, V1, V2) -> ISNAT(V1)
11.09/3.69	   U32'(tt, V2) -> U33'(isNat(V2))
11.09/3.69	   U32'(tt, V2) -> ISNAT(V2)
11.09/3.69	   U51'(tt, M, N) -> PLUS(N, M)
11.09/3.69	   U71'(tt, M, N) -> PLUS(x(N, M), N)
11.09/3.69	   U71'(tt, M, N) -> X(N, M)
11.09/3.69	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.69	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.69	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.69	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.09/3.69	   ISNAT(s(V1)) -> ISNATKIND(V1)
11.09/3.69	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.69	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.69	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.69	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.69	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.69	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.09/3.69	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.69	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   PLUS(N, 0) -> U41'(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   PLUS(N, 0) -> AND(isNat(N), isNatKind(N))
11.09/3.70	   PLUS(N, 0) -> ISNAT(N)
11.09/3.70	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   PLUS(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
11.09/3.70	   PLUS(N, s(M)) -> AND(isNat(M), isNatKind(M))
11.09/3.70	   PLUS(N, s(M)) -> ISNAT(M)
11.09/3.70	   X(N, 0) -> U61'(and(isNat(N), isNatKind(N)))
11.09/3.70	   X(N, 0) -> AND(isNat(N), isNatKind(N))
11.09/3.70	   X(N, 0) -> ISNAT(N)
11.09/3.70	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   X(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
11.09/3.70	   X(N, s(M)) -> AND(isNat(M), isNatKind(M))
11.09/3.70	   X(N, s(M)) -> ISNAT(M)
11.09/3.70	
11.09/3.70	The collapsing dependency pairs are DP_c:
11.09/3.70	   U41'(tt, N) -> N
11.09/3.70	   U51'(tt, M, N) -> N
11.09/3.70	   U51'(tt, M, N) -> M
11.09/3.70	   U71'(tt, M, N) -> N
11.09/3.70	   U71'(tt, M, N) -> M
11.09/3.70	   AND(tt, X) -> X
11.09/3.70	
11.09/3.70	
11.09/3.70	The hidden terms of R are:
11.09/3.70	
11.09/3.70	   isNatKind(x0)
11.09/3.70	   and(isNat(x0), isNatKind(x0))
11.09/3.70	   isNat(x0)
11.09/3.70	
11.09/3.70	Every hiding context is built from:
11.09/3.70	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@bebcc6a
11.09/3.70	
11.09/3.70	Hence, the new unhiding pairs DP_u are :
11.09/3.70	   U41'(tt, N) -> U(N)
11.09/3.70	   U51'(tt, M, N) -> U(N)
11.09/3.70	   U51'(tt, M, N) -> U(M)
11.09/3.70	   U71'(tt, M, N) -> U(N)
11.09/3.70	   U71'(tt, M, N) -> U(M)
11.09/3.70	   AND(tt, X) -> U(X)
11.09/3.70	   U(and(x_0, x_1)) -> U(x_0)
11.09/3.70	   U(isNatKind(x0)) -> ISNATKIND(x0)
11.09/3.70	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
11.09/3.70	   U(isNat(x0)) -> ISNAT(x0)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(5) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 21 less nodes.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(6)
11.09/3.70	Complex Obligation (AND)
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(7)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U12'_2, U11'_3, AND_2, U21'_2, U31'_3, U32'_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, ISNAT_1, U_1, ISNATKIND_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U12'(tt, V2) -> ISNAT(V2)
11.09/3.70	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
11.09/3.70	   U11'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   AND(tt, X) -> U(X)
11.09/3.70	   U(and(x_0, x_1)) -> U(x_0)
11.09/3.70	   U(isNatKind(x0)) -> ISNATKIND(x0)
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
11.09/3.70	   U(isNat(x0)) -> ISNAT(x0)
11.09/3.70	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.09/3.70	   U21'(tt, V1) -> ISNAT(V1)
11.09/3.70	   ISNAT(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
11.09/3.70	   U32'(tt, V2) -> ISNAT(V2)
11.09/3.70	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   U31'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(8) QCSDPReductionPairProof (EQUIVALENT)
11.09/3.70	Using the order
11.09/3.70	
11.09/3.70	Polynomial interpretation [POLO]:
11.09/3.70	
11.09/3.70	   POL(0) = 0
11.09/3.70	   POL(AND(x_1, x_2)) = x_1 + 2*x_2
11.09/3.70	   POL(ISNAT(x_1)) = 0
11.09/3.70	   POL(ISNATKIND(x_1)) = 0
11.09/3.70	   POL(U(x_1)) = 2*x_1
11.09/3.70	   POL(U11(x_1, x_2, x_3)) = 0
11.09/3.70	   POL(U11'(x_1, x_2, x_3)) = 0
11.09/3.70	   POL(U12(x_1, x_2)) = 0
11.09/3.70	   POL(U12'(x_1, x_2)) = 0
11.09/3.70	   POL(U13(x_1)) = 0
11.09/3.70	   POL(U21(x_1, x_2)) = 0
11.09/3.70	   POL(U21'(x_1, x_2)) = 0
11.09/3.70	   POL(U22(x_1)) = 0
11.09/3.70	   POL(U31(x_1, x_2, x_3)) = 2
11.09/3.70	   POL(U31'(x_1, x_2, x_3)) = 0
11.09/3.70	   POL(U32(x_1, x_2)) = 0
11.09/3.70	   POL(U32'(x_1, x_2)) = 0
11.09/3.70	   POL(U33(x_1)) = 0
11.09/3.70	   POL(U41(x_1, x_2)) = x_2
11.09/3.70	   POL(U51(x_1, x_2, x_3)) = 0
11.09/3.70	   POL(U61(x_1)) = 0
11.09/3.70	   POL(U71(x_1, x_2, x_3)) = 0
11.09/3.70	   POL(and(x_1, x_2)) = x_1 + 2*x_2
11.09/3.70	   POL(isNat(x_1)) = 2
11.09/3.70	   POL(isNatKind(x_1)) = 0
11.09/3.70	   POL(plus(x_1, x_2)) = x_1
11.09/3.70	   POL(s(x_1)) = 0
11.09/3.70	   POL(tt) = 0
11.09/3.70	   POL(x(x_1, x_2)) = 0
11.09/3.70	
11.09/3.70	
11.09/3.70	the following usable rules
11.09/3.70	
11.09/3.70	
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	
11.09/3.70	
11.09/3.70	could all be oriented weakly.
11.09/3.70	
11.09/3.70	Furthermore, the pairs
11.09/3.70	
11.09/3.70	
11.09/3.70	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
11.09/3.70	   U(isNat(x0)) -> ISNAT(x0)
11.09/3.70	
11.09/3.70	
11.09/3.70	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
11.09/3.70	
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(9)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U12'_2, U11'_3, AND_2, U21'_2, U31'_3, U32'_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, ISNAT_1, U_1, ISNATKIND_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U12'(tt, V2) -> ISNAT(V2)
11.09/3.70	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
11.09/3.70	   U11'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   AND(tt, X) -> U(X)
11.09/3.70	   U(and(x_0, x_1)) -> U(x_0)
11.09/3.70	   U(isNatKind(x0)) -> ISNATKIND(x0)
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.09/3.70	   U21'(tt, V1) -> ISNAT(V1)
11.09/3.70	   ISNAT(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
11.09/3.70	   U32'(tt, V2) -> ISNAT(V2)
11.09/3.70	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   U31'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(10) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 5 less nodes.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(11)
11.09/3.70	Complex Obligation (AND)
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(12)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, AND_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, U_1, ISNATKIND_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U(and(x_0, x_1)) -> U(x_0)
11.09/3.70	   U(isNatKind(x0)) -> ISNATKIND(x0)
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   AND(tt, X) -> U(X)
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(13) QCSDPReductionPairProof (EQUIVALENT)
11.09/3.70	Using the order
11.09/3.70	
11.09/3.70	U/1(YES)
11.09/3.70	and/2(YES,YES)
11.09/3.70	isNatKind/1)YES(
11.09/3.70	ISNATKIND/1(YES)
11.09/3.70	plus/2(YES,YES)
11.09/3.70	AND/2(NO,YES)
11.09/3.70	tt/0)
11.09/3.70	s/1(YES)
11.09/3.70	x/2(YES,YES)
11.09/3.70	0/0)
11.09/3.70	U41/2)NO,YES(
11.09/3.70	isNat/1)YES(
11.09/3.70	U51/3(YES,YES,YES)
11.09/3.70	U11/3(NO,NO,NO)
11.09/3.70	U21/2(NO,YES)
11.09/3.70	U31/3(NO,NO,YES)
11.09/3.70	U12/2(NO,NO)
11.09/3.70	U13/1(NO)
11.09/3.70	U22/1)YES(
11.09/3.70	U61/1(YES)
11.09/3.70	U71/3(YES,YES,YES)
11.09/3.70	U32/2)NO,YES(
11.09/3.70	U33/1)YES(
11.09/3.70	
11.09/3.70	Quasi precedence:
11.09/3.70	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > [plus_2, U51_3, U11] > [and_2, tt, s_1, U21_1, U12, U13]
11.09/3.70	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > [0, U61_1] > [and_2, tt, s_1, U21_1, U12, U13]
11.09/3.70	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > U31_1
11.09/3.70	
11.09/3.70	
11.09/3.70	Status:
11.09/3.70	U_1: multiset status
11.09/3.70	and_2: [1,2]
11.09/3.70	ISNATKIND_1: multiset status
11.09/3.70	plus_2: [1,2]
11.09/3.70	AND_1: multiset status
11.09/3.70	tt: multiset status
11.09/3.70	s_1: [1]
11.09/3.70	x_2: [2,1]
11.09/3.70	0: multiset status
11.09/3.70	U51_3: [3,2,1]
11.09/3.70	U11: []
11.09/3.70	U21_1: [1]
11.09/3.70	U31_1: multiset status
11.09/3.70	U12: []
11.09/3.70	U13: []
11.09/3.70	U61_1: multiset status
11.09/3.70	U71_3: [2,3,1]
11.09/3.70	
11.09/3.70	
11.09/3.70	
11.09/3.70	the following usable rules
11.09/3.70	
11.09/3.70	
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	
11.09/3.70	
11.09/3.70	could all be oriented weakly.
11.09/3.70	
11.09/3.70	Furthermore, the pairs
11.09/3.70	
11.09/3.70	
11.09/3.70	   U(and(x_0, x_1)) -> U(x_0)
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.09/3.70	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
11.09/3.70	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
11.09/3.70	
11.09/3.70	
11.09/3.70	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
11.09/3.70	
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(14)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, AND_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, ISNATKIND_1, U_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U(isNatKind(x0)) -> ISNATKIND(x0)
11.09/3.70	   AND(tt, X) -> U(X)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(15) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(16)
11.09/3.70	TRUE
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(17)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U11'_3, U12'_2, U21'_2, U31'_3, U32'_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, ISNAT_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
11.09/3.70	   U12'(tt, V2) -> ISNAT(V2)
11.09/3.70	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.09/3.70	   U21'(tt, V1) -> ISNAT(V1)
11.09/3.70	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
11.09/3.70	   U32'(tt, V2) -> ISNAT(V2)
11.09/3.70	   U31'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	   U11'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(18) QCSDPReductionPairProof (EQUIVALENT)
11.09/3.70	Using the order
11.09/3.70	
11.09/3.70	ISNAT/1(YES)
11.09/3.70	plus/2(YES,YES)
11.09/3.70	U11'/3(NO,YES,YES)
11.09/3.70	and/2)NO,YES(
11.09/3.70	isNatKind/1(YES)
11.09/3.70	tt/0)
11.09/3.70	U12'/2(NO,YES)
11.09/3.70	isNat/1(NO)
11.09/3.70	s/1(YES)
11.09/3.70	U21'/2(NO,YES)
11.09/3.70	x/2(YES,YES)
11.09/3.70	U31'/3(YES,YES,YES)
11.09/3.70	U32'/2(NO,YES)
11.09/3.70	0/0)
11.09/3.70	U41/2(YES,YES)
11.09/3.70	U51/3(YES,YES,YES)
11.09/3.70	U11/3(NO,NO,NO)
11.09/3.70	U21/2(NO,NO)
11.09/3.70	U31/3(NO,NO,NO)
11.09/3.70	U12/2(NO,NO)
11.09/3.70	U13/1(NO)
11.09/3.70	U61/1(NO)
11.09/3.70	U71/3(YES,YES,YES)
11.09/3.70	U22/1)YES(
11.09/3.70	U32/2(NO,NO)
11.09/3.70	U33/1(NO)
11.09/3.70	
11.09/3.70	Quasi precedence:
11.09/3.70	[ISNAT_1, U11'_2, U12'_1, U21'_1, x_2, U31'_3, U32'_1, U71_3] > [plus_2, isNatKind_1, isNat, U51_3, U21] > s_1 > U41_2
11.09/3.70	[ISNAT_1, U11'_2, U12'_1, U21'_1, x_2, U31'_3, U32'_1, U71_3] > [plus_2, isNatKind_1, isNat, U51_3, U21] > U11 > U12 > U13 > [tt, 0, U61] > U41_2
11.09/3.70	[ISNAT_1, U11'_2, U12'_1, U21'_1, x_2, U31'_3, U32'_1, U71_3] > [plus_2, isNatKind_1, isNat, U51_3, U21] > U31 > U32 > U33 > [tt, 0, U61] > U41_2
11.09/3.70	
11.09/3.70	
11.09/3.70	Status:
11.09/3.70	ISNAT_1: multiset status
11.09/3.70	plus_2: [1,2]
11.09/3.70	U11'_2: multiset status
11.09/3.70	isNatKind_1: [1]
11.09/3.70	tt: multiset status
11.09/3.70	U12'_1: multiset status
11.09/3.70	isNat: multiset status
11.09/3.70	s_1: [1]
11.09/3.70	U21'_1: multiset status
11.09/3.70	x_2: [2,1]
11.09/3.70	U31'_3: multiset status
11.09/3.70	U32'_1: multiset status
11.09/3.70	0: multiset status
11.09/3.70	U41_2: [1,2]
11.09/3.70	U51_3: [3,2,1]
11.09/3.70	U11: multiset status
11.09/3.70	U21: multiset status
11.09/3.70	U31: []
11.09/3.70	U12: []
11.09/3.70	U13: multiset status
11.09/3.70	U61: multiset status
11.09/3.70	U71_3: [2,3,1]
11.09/3.70	U32: multiset status
11.09/3.70	U33: multiset status
11.09/3.70	
11.09/3.70	
11.09/3.70	
11.09/3.70	the following usable rules
11.09/3.70	
11.09/3.70	
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	
11.09/3.70	
11.09/3.70	could all be oriented weakly.
11.09/3.70	
11.09/3.70	Furthermore, the pairs
11.09/3.70	
11.09/3.70	
11.09/3.70	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
11.09/3.70	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.09/3.70	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
11.09/3.70	   U31'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	   U11'(tt, V1, V2) -> ISNAT(V1)
11.09/3.70	
11.09/3.70	
11.09/3.70	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
11.09/3.70	
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(19)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U12'_2, U21'_2, U32'_2} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1, ISNAT_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U12'(tt, V2) -> ISNAT(V2)
11.09/3.70	   U21'(tt, V1) -> ISNAT(V1)
11.09/3.70	   U32'(tt, V2) -> ISNAT(V2)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(20) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(21)
11.09/3.70	TRUE
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(22)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U51'_3} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   U51'(tt, M, N) -> PLUS(N, M)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(23) QCSDPSubtermProof (EQUIVALENT)
11.09/3.70	We use the subterm processor [DA_EMMES].
11.09/3.70	
11.09/3.70	
11.09/3.70	The following pairs can be oriented strictly and are deleted.
11.09/3.70	
11.09/3.70	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	The remaining pairs can at least be oriented weakly.
11.09/3.70	
11.09/3.70	   U51'(tt, M, N) -> PLUS(N, M)
11.09/3.70	Used ordering:  Combined order from the following AFS and order.
11.09/3.70	U51'(x1, x2, x3)  =  x2
11.09/3.70	
11.09/3.70	PLUS(x1, x2)  =  x2
11.09/3.70	
11.09/3.70	
11.09/3.70	Subterm Order
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(24)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U51'_3} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U51'(tt, M, N) -> PLUS(N, M)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(25) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(26)
11.09/3.70	TRUE
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(27)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U71'_3} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U71'(tt, M, N) -> X(N, M)
11.09/3.70	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(28) QCSDPSubtermProof (EQUIVALENT)
11.09/3.70	We use the subterm processor [DA_EMMES].
11.09/3.70	
11.09/3.70	
11.09/3.70	The following pairs can be oriented strictly and are deleted.
11.09/3.70	
11.09/3.70	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	The remaining pairs can at least be oriented weakly.
11.09/3.70	
11.09/3.70	   U71'(tt, M, N) -> X(N, M)
11.09/3.70	Used ordering:  Combined order from the following AFS and order.
11.09/3.70	X(x1, x2)  =  x2
11.09/3.70	
11.09/3.70	U71'(x1, x2, x3)  =  x2
11.09/3.70	
11.09/3.70	
11.09/3.70	Subterm Order
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(29)
11.09/3.70	Obligation:
11.09/3.70	Q-restricted context-sensitive dependency pair problem:
11.09/3.70	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
11.09/3.70	For all symbols f in {U11_3, U12_2, U21_2, U31_3, U32_2, U41_2, U51_3, U71_3, and_2, U71'_3} we have mu(f) = {1}.
11.09/3.70	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
11.09/3.70	
11.09/3.70	The TRS P consists of the following rules:
11.09/3.70	
11.09/3.70	   U71'(tt, M, N) -> X(N, M)
11.09/3.70	
11.09/3.70	The TRS R consists of the following rules:
11.09/3.70	
11.09/3.70	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
11.09/3.70	   U12(tt, V2) -> U13(isNat(V2))
11.09/3.70	   U13(tt) -> tt
11.09/3.70	   U21(tt, V1) -> U22(isNat(V1))
11.09/3.70	   U22(tt) -> tt
11.09/3.70	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
11.09/3.70	   U32(tt, V2) -> U33(isNat(V2))
11.09/3.70	   U33(tt) -> tt
11.09/3.70	   U41(tt, N) -> N
11.09/3.70	   U51(tt, M, N) -> s(plus(N, M))
11.09/3.70	   U61(tt) -> 0
11.09/3.70	   U71(tt, M, N) -> plus(x(N, M), N)
11.09/3.70	   and(tt, X) -> X
11.09/3.70	   isNat(0) -> tt
11.09/3.70	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.09/3.70	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
11.09/3.70	   isNatKind(0) -> tt
11.09/3.70	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   isNatKind(s(V1)) -> isNatKind(V1)
11.09/3.70	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
11.09/3.70	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
11.09/3.70	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
11.09/3.70	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
11.09/3.70	
11.09/3.70	Q is empty.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(30) QCSDependencyGraphProof (EQUIVALENT)
11.09/3.70	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
11.09/3.70	
11.09/3.70	----------------------------------------
11.09/3.70	
11.09/3.70	(31)
11.09/3.70	TRUE
11.44/3.77	EOF