4.08/1.90	YES
4.08/1.92	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.08/1.92	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.08/1.92	
4.08/1.92	
4.08/1.92	Termination w.r.t. Q of the given QTRS could be proven:
4.08/1.92	
4.08/1.92	(0) QTRS
4.08/1.92	(1) QTRSToCSRProof [SOUND, 0 ms]
4.08/1.92	(2) CSR
4.08/1.92	(3) CSDependencyPairsProof [EQUIVALENT, 0 ms]
4.08/1.92	(4) QCSDP
4.08/1.92	(5) QCSDependencyGraphProof [EQUIVALENT, 3 ms]
4.08/1.92	(6) AND
4.08/1.92	    (7) QCSDP
4.08/1.92	        (8) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.08/1.92	        (9) QCSDP
4.08/1.92	        (10) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.08/1.92	        (11) TRUE
4.08/1.92	    (12) QCSDP
4.08/1.92	        (13) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.08/1.92	        (14) QCSDP
4.08/1.92	        (15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.08/1.92	        (16) TRUE
4.08/1.92	    (17) QCSDP
4.08/1.92	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.08/1.92	        (19) QCSDP
4.08/1.92	        (20) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.08/1.92	        (21) TRUE
4.08/1.92	
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(0)
4.08/1.92	Obligation:
4.08/1.92	Q restricted rewrite system:
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
4.08/1.92	   active(U12(tt)) -> mark(tt)
4.08/1.92	   active(U21(tt)) -> mark(tt)
4.08/1.92	   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
4.08/1.92	   active(U32(tt)) -> mark(tt)
4.08/1.92	   active(U41(tt, N)) -> mark(N)
4.08/1.92	   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
4.08/1.92	   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
4.08/1.92	   active(U61(tt)) -> mark(0)
4.08/1.92	   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
4.08/1.92	   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
4.08/1.92	   active(isNat(0)) -> mark(tt)
4.08/1.92	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
4.08/1.92	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
4.08/1.92	   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
4.08/1.92	   active(plus(N, 0)) -> mark(U41(isNat(N), N))
4.08/1.92	   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
4.08/1.92	   active(x(N, 0)) -> mark(U61(isNat(N)))
4.08/1.92	   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
4.08/1.92	   active(U11(X1, X2)) -> U11(active(X1), X2)
4.08/1.92	   active(U12(X)) -> U12(active(X))
4.08/1.92	   active(U21(X)) -> U21(active(X))
4.08/1.92	   active(U31(X1, X2)) -> U31(active(X1), X2)
4.08/1.92	   active(U32(X)) -> U32(active(X))
4.08/1.92	   active(U41(X1, X2)) -> U41(active(X1), X2)
4.08/1.92	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
4.08/1.92	   active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3)
4.08/1.92	   active(s(X)) -> s(active(X))
4.08/1.92	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.08/1.92	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.08/1.92	   active(U61(X)) -> U61(active(X))
4.08/1.92	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
4.08/1.92	   active(U72(X1, X2, X3)) -> U72(active(X1), X2, X3)
4.08/1.92	   active(x(X1, X2)) -> x(active(X1), X2)
4.08/1.92	   active(x(X1, X2)) -> x(X1, active(X2))
4.08/1.92	   U11(mark(X1), X2) -> mark(U11(X1, X2))
4.08/1.92	   U12(mark(X)) -> mark(U12(X))
4.08/1.92	   U21(mark(X)) -> mark(U21(X))
4.08/1.92	   U31(mark(X1), X2) -> mark(U31(X1, X2))
4.08/1.92	   U32(mark(X)) -> mark(U32(X))
4.08/1.92	   U41(mark(X1), X2) -> mark(U41(X1, X2))
4.08/1.92	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
4.08/1.92	   U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3))
4.08/1.92	   s(mark(X)) -> mark(s(X))
4.08/1.92	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.08/1.92	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.08/1.92	   U61(mark(X)) -> mark(U61(X))
4.08/1.92	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
4.08/1.92	   U72(mark(X1), X2, X3) -> mark(U72(X1, X2, X3))
4.08/1.92	   x(mark(X1), X2) -> mark(x(X1, X2))
4.08/1.92	   x(X1, mark(X2)) -> mark(x(X1, X2))
4.08/1.92	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
4.08/1.92	   proper(tt) -> ok(tt)
4.08/1.92	   proper(U12(X)) -> U12(proper(X))
4.08/1.92	   proper(isNat(X)) -> isNat(proper(X))
4.08/1.92	   proper(U21(X)) -> U21(proper(X))
4.08/1.92	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
4.08/1.92	   proper(U32(X)) -> U32(proper(X))
4.08/1.92	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
4.08/1.92	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(s(X)) -> s(proper(X))
4.08/1.92	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.08/1.92	   proper(U61(X)) -> U61(proper(X))
4.08/1.92	   proper(0) -> ok(0)
4.08/1.92	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(U72(X1, X2, X3)) -> U72(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
4.08/1.92	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
4.08/1.92	   U12(ok(X)) -> ok(U12(X))
4.08/1.92	   isNat(ok(X)) -> ok(isNat(X))
4.08/1.92	   U21(ok(X)) -> ok(U21(X))
4.08/1.92	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
4.08/1.92	   U32(ok(X)) -> ok(U32(X))
4.08/1.92	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
4.08/1.92	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
4.08/1.92	   U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3))
4.08/1.92	   s(ok(X)) -> ok(s(X))
4.08/1.92	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.08/1.92	   U61(ok(X)) -> ok(U61(X))
4.08/1.92	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
4.08/1.92	   U72(ok(X1), ok(X2), ok(X3)) -> ok(U72(X1, X2, X3))
4.08/1.92	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
4.08/1.92	   top(mark(X)) -> top(proper(X))
4.08/1.92	   top(ok(X)) -> top(active(X))
4.08/1.92	
4.08/1.92	The set Q consists of the following terms:
4.08/1.92	
4.08/1.92	   active(isNat(0))
4.08/1.92	   active(isNat(plus(x0, x1)))
4.08/1.92	   active(isNat(s(x0)))
4.08/1.92	   active(isNat(x(x0, x1)))
4.08/1.92	   active(U11(x0, x1))
4.08/1.92	   active(U12(x0))
4.08/1.92	   active(U21(x0))
4.08/1.92	   active(U31(x0, x1))
4.08/1.92	   active(U32(x0))
4.08/1.92	   active(U41(x0, x1))
4.08/1.92	   active(U51(x0, x1, x2))
4.08/1.92	   active(U52(x0, x1, x2))
4.08/1.92	   active(s(x0))
4.08/1.92	   active(plus(x0, x1))
4.08/1.92	   active(U61(x0))
4.08/1.92	   active(U71(x0, x1, x2))
4.08/1.92	   active(U72(x0, x1, x2))
4.08/1.92	   active(x(x0, x1))
4.08/1.92	   U11(mark(x0), x1)
4.08/1.92	   U12(mark(x0))
4.08/1.92	   U21(mark(x0))
4.08/1.92	   U31(mark(x0), x1)
4.08/1.92	   U32(mark(x0))
4.08/1.92	   U41(mark(x0), x1)
4.08/1.92	   U51(mark(x0), x1, x2)
4.08/1.92	   U52(mark(x0), x1, x2)
4.08/1.92	   s(mark(x0))
4.08/1.92	   plus(mark(x0), x1)
4.08/1.92	   plus(x0, mark(x1))
4.08/1.92	   U61(mark(x0))
4.08/1.92	   U71(mark(x0), x1, x2)
4.08/1.92	   U72(mark(x0), x1, x2)
4.08/1.92	   x(mark(x0), x1)
4.08/1.92	   x(x0, mark(x1))
4.08/1.92	   proper(U11(x0, x1))
4.08/1.92	   proper(tt)
4.08/1.92	   proper(U12(x0))
4.08/1.92	   proper(isNat(x0))
4.08/1.92	   proper(U21(x0))
4.08/1.92	   proper(U31(x0, x1))
4.08/1.92	   proper(U32(x0))
4.08/1.92	   proper(U41(x0, x1))
4.08/1.92	   proper(U51(x0, x1, x2))
4.08/1.92	   proper(U52(x0, x1, x2))
4.08/1.92	   proper(s(x0))
4.08/1.92	   proper(plus(x0, x1))
4.08/1.92	   proper(U61(x0))
4.08/1.92	   proper(0)
4.08/1.92	   proper(U71(x0, x1, x2))
4.08/1.92	   proper(U72(x0, x1, x2))
4.08/1.92	   proper(x(x0, x1))
4.08/1.92	   U11(ok(x0), ok(x1))
4.08/1.92	   U12(ok(x0))
4.08/1.92	   isNat(ok(x0))
4.08/1.92	   U21(ok(x0))
4.08/1.92	   U31(ok(x0), ok(x1))
4.08/1.92	   U32(ok(x0))
4.08/1.92	   U41(ok(x0), ok(x1))
4.08/1.92	   U51(ok(x0), ok(x1), ok(x2))
4.08/1.92	   U52(ok(x0), ok(x1), ok(x2))
4.08/1.92	   s(ok(x0))
4.08/1.92	   plus(ok(x0), ok(x1))
4.08/1.92	   U61(ok(x0))
4.08/1.92	   U71(ok(x0), ok(x1), ok(x2))
4.08/1.92	   U72(ok(x0), ok(x1), ok(x2))
4.08/1.92	   x(ok(x0), ok(x1))
4.08/1.92	   top(mark(x0))
4.08/1.92	   top(ok(x0))
4.08/1.92	
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(1) QTRSToCSRProof (SOUND)
4.08/1.92	The following Q TRS is given: Q restricted rewrite system:
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
4.08/1.92	   active(U12(tt)) -> mark(tt)
4.08/1.92	   active(U21(tt)) -> mark(tt)
4.08/1.92	   active(U31(tt, V2)) -> mark(U32(isNat(V2)))
4.08/1.92	   active(U32(tt)) -> mark(tt)
4.08/1.92	   active(U41(tt, N)) -> mark(N)
4.08/1.92	   active(U51(tt, M, N)) -> mark(U52(isNat(N), M, N))
4.08/1.92	   active(U52(tt, M, N)) -> mark(s(plus(N, M)))
4.08/1.92	   active(U61(tt)) -> mark(0)
4.08/1.92	   active(U71(tt, M, N)) -> mark(U72(isNat(N), M, N))
4.08/1.92	   active(U72(tt, M, N)) -> mark(plus(x(N, M), N))
4.08/1.92	   active(isNat(0)) -> mark(tt)
4.08/1.92	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
4.08/1.92	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
4.08/1.92	   active(isNat(x(V1, V2))) -> mark(U31(isNat(V1), V2))
4.08/1.92	   active(plus(N, 0)) -> mark(U41(isNat(N), N))
4.08/1.92	   active(plus(N, s(M))) -> mark(U51(isNat(M), M, N))
4.08/1.92	   active(x(N, 0)) -> mark(U61(isNat(N)))
4.08/1.92	   active(x(N, s(M))) -> mark(U71(isNat(M), M, N))
4.08/1.92	   active(U11(X1, X2)) -> U11(active(X1), X2)
4.08/1.92	   active(U12(X)) -> U12(active(X))
4.08/1.92	   active(U21(X)) -> U21(active(X))
4.08/1.92	   active(U31(X1, X2)) -> U31(active(X1), X2)
4.08/1.92	   active(U32(X)) -> U32(active(X))
4.08/1.92	   active(U41(X1, X2)) -> U41(active(X1), X2)
4.08/1.92	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
4.08/1.92	   active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3)
4.08/1.92	   active(s(X)) -> s(active(X))
4.08/1.92	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.08/1.92	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.08/1.92	   active(U61(X)) -> U61(active(X))
4.08/1.92	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
4.08/1.92	   active(U72(X1, X2, X3)) -> U72(active(X1), X2, X3)
4.08/1.92	   active(x(X1, X2)) -> x(active(X1), X2)
4.08/1.92	   active(x(X1, X2)) -> x(X1, active(X2))
4.08/1.92	   U11(mark(X1), X2) -> mark(U11(X1, X2))
4.08/1.92	   U12(mark(X)) -> mark(U12(X))
4.08/1.92	   U21(mark(X)) -> mark(U21(X))
4.08/1.92	   U31(mark(X1), X2) -> mark(U31(X1, X2))
4.08/1.92	   U32(mark(X)) -> mark(U32(X))
4.08/1.92	   U41(mark(X1), X2) -> mark(U41(X1, X2))
4.08/1.92	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
4.08/1.92	   U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3))
4.08/1.92	   s(mark(X)) -> mark(s(X))
4.08/1.92	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.08/1.92	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.08/1.92	   U61(mark(X)) -> mark(U61(X))
4.08/1.92	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
4.08/1.92	   U72(mark(X1), X2, X3) -> mark(U72(X1, X2, X3))
4.08/1.92	   x(mark(X1), X2) -> mark(x(X1, X2))
4.08/1.92	   x(X1, mark(X2)) -> mark(x(X1, X2))
4.08/1.92	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
4.08/1.92	   proper(tt) -> ok(tt)
4.08/1.92	   proper(U12(X)) -> U12(proper(X))
4.08/1.92	   proper(isNat(X)) -> isNat(proper(X))
4.08/1.92	   proper(U21(X)) -> U21(proper(X))
4.08/1.92	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
4.08/1.92	   proper(U32(X)) -> U32(proper(X))
4.08/1.92	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
4.08/1.92	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(s(X)) -> s(proper(X))
4.08/1.92	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.08/1.92	   proper(U61(X)) -> U61(proper(X))
4.08/1.92	   proper(0) -> ok(0)
4.08/1.92	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(U72(X1, X2, X3)) -> U72(proper(X1), proper(X2), proper(X3))
4.08/1.92	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
4.08/1.92	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
4.08/1.92	   U12(ok(X)) -> ok(U12(X))
4.08/1.92	   isNat(ok(X)) -> ok(isNat(X))
4.08/1.92	   U21(ok(X)) -> ok(U21(X))
4.08/1.92	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
4.08/1.92	   U32(ok(X)) -> ok(U32(X))
4.08/1.92	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
4.08/1.92	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
4.08/1.92	   U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3))
4.08/1.92	   s(ok(X)) -> ok(s(X))
4.08/1.92	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.08/1.92	   U61(ok(X)) -> ok(U61(X))
4.08/1.92	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
4.08/1.92	   U72(ok(X1), ok(X2), ok(X3)) -> ok(U72(X1, X2, X3))
4.08/1.92	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
4.08/1.92	   top(mark(X)) -> top(proper(X))
4.08/1.92	   top(ok(X)) -> top(active(X))
4.08/1.92	
4.08/1.92	The set Q consists of the following terms:
4.08/1.92	
4.08/1.92	   active(isNat(0))
4.08/1.92	   active(isNat(plus(x0, x1)))
4.08/1.92	   active(isNat(s(x0)))
4.08/1.92	   active(isNat(x(x0, x1)))
4.08/1.92	   active(U11(x0, x1))
4.08/1.92	   active(U12(x0))
4.08/1.92	   active(U21(x0))
4.08/1.92	   active(U31(x0, x1))
4.08/1.92	   active(U32(x0))
4.08/1.92	   active(U41(x0, x1))
4.08/1.92	   active(U51(x0, x1, x2))
4.08/1.92	   active(U52(x0, x1, x2))
4.08/1.92	   active(s(x0))
4.08/1.92	   active(plus(x0, x1))
4.08/1.92	   active(U61(x0))
4.08/1.92	   active(U71(x0, x1, x2))
4.08/1.92	   active(U72(x0, x1, x2))
4.08/1.92	   active(x(x0, x1))
4.08/1.92	   U11(mark(x0), x1)
4.08/1.92	   U12(mark(x0))
4.08/1.92	   U21(mark(x0))
4.08/1.92	   U31(mark(x0), x1)
4.08/1.92	   U32(mark(x0))
4.08/1.92	   U41(mark(x0), x1)
4.08/1.92	   U51(mark(x0), x1, x2)
4.08/1.92	   U52(mark(x0), x1, x2)
4.08/1.92	   s(mark(x0))
4.08/1.92	   plus(mark(x0), x1)
4.08/1.92	   plus(x0, mark(x1))
4.08/1.92	   U61(mark(x0))
4.08/1.92	   U71(mark(x0), x1, x2)
4.08/1.92	   U72(mark(x0), x1, x2)
4.08/1.92	   x(mark(x0), x1)
4.08/1.92	   x(x0, mark(x1))
4.08/1.92	   proper(U11(x0, x1))
4.08/1.92	   proper(tt)
4.08/1.92	   proper(U12(x0))
4.08/1.92	   proper(isNat(x0))
4.08/1.92	   proper(U21(x0))
4.08/1.92	   proper(U31(x0, x1))
4.08/1.92	   proper(U32(x0))
4.08/1.92	   proper(U41(x0, x1))
4.08/1.92	   proper(U51(x0, x1, x2))
4.08/1.92	   proper(U52(x0, x1, x2))
4.08/1.92	   proper(s(x0))
4.08/1.92	   proper(plus(x0, x1))
4.08/1.92	   proper(U61(x0))
4.08/1.92	   proper(0)
4.08/1.92	   proper(U71(x0, x1, x2))
4.08/1.92	   proper(U72(x0, x1, x2))
4.08/1.92	   proper(x(x0, x1))
4.08/1.92	   U11(ok(x0), ok(x1))
4.08/1.92	   U12(ok(x0))
4.08/1.92	   isNat(ok(x0))
4.08/1.92	   U21(ok(x0))
4.08/1.92	   U31(ok(x0), ok(x1))
4.08/1.92	   U32(ok(x0))
4.08/1.92	   U41(ok(x0), ok(x1))
4.08/1.92	   U51(ok(x0), ok(x1), ok(x2))
4.08/1.92	   U52(ok(x0), ok(x1), ok(x2))
4.08/1.92	   s(ok(x0))
4.08/1.92	   plus(ok(x0), ok(x1))
4.08/1.92	   U61(ok(x0))
4.08/1.92	   U71(ok(x0), ok(x1), ok(x2))
4.08/1.92	   U72(ok(x0), ok(x1), ok(x2))
4.08/1.92	   x(ok(x0), ok(x1))
4.08/1.92	   top(mark(x0))
4.08/1.92	   top(ok(x0))
4.08/1.92	
4.08/1.92	Special symbols used for the transformation (see [GM04]):
4.08/1.92	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.08/1.92	The replacement map contains the following entries:
4.08/1.92	
4.08/1.92	U11: {1}
4.08/1.92	tt: empty set
4.08/1.92	U12: {1}
4.08/1.92	isNat: empty set
4.08/1.92	U21: {1}
4.08/1.92	U31: {1}
4.08/1.92	U32: {1}
4.08/1.92	U41: {1}
4.08/1.92	U51: {1}
4.08/1.92	U52: {1}
4.08/1.92	s: {1}
4.08/1.92	plus: {1, 2}
4.08/1.92	U61: {1}
4.08/1.92	0: empty set
4.08/1.92	U71: {1}
4.08/1.92	U72: {1}
4.08/1.92	x: {1, 2}
4.08/1.92	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.
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(2)
4.08/1.92	Obligation:
4.08/1.92	Context-sensitive rewrite system:
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	The replacement map contains the following entries:
4.08/1.92	
4.08/1.92	U11: {1}
4.08/1.92	tt: empty set
4.08/1.92	U12: {1}
4.08/1.92	isNat: empty set
4.08/1.92	U21: {1}
4.08/1.92	U31: {1}
4.08/1.92	U32: {1}
4.08/1.92	U41: {1}
4.08/1.92	U51: {1}
4.08/1.92	U52: {1}
4.08/1.92	s: {1}
4.08/1.92	plus: {1, 2}
4.08/1.92	U61: {1}
4.08/1.92	0: empty set
4.08/1.92	U71: {1}
4.08/1.92	U72: {1}
4.08/1.92	x: {1, 2}
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(3) CSDependencyPairsProof (EQUIVALENT)
4.08/1.92	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(4)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, U12'_1, U32'_1, PLUS_2, X_2, U21'_1, U61'_1} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2, U52'_3, U51'_3, U72'_3, U71'_3, U41'_2} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1, ISNAT_1, U_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The ordinary context-sensitive dependency pairs DP_o are:
4.08/1.92	   U11'(tt, V2) -> U12'(isNat(V2))
4.08/1.92	   U11'(tt, V2) -> ISNAT(V2)
4.08/1.92	   U31'(tt, V2) -> U32'(isNat(V2))
4.08/1.92	   U31'(tt, V2) -> ISNAT(V2)
4.08/1.92	   U51'(tt, M, N) -> U52'(isNat(N), M, N)
4.08/1.92	   U51'(tt, M, N) -> ISNAT(N)
4.08/1.92	   U52'(tt, M, N) -> PLUS(N, M)
4.08/1.92	   U71'(tt, M, N) -> U72'(isNat(N), M, N)
4.08/1.92	   U71'(tt, M, N) -> ISNAT(N)
4.08/1.92	   U72'(tt, M, N) -> PLUS(x(N, M), N)
4.08/1.92	   U72'(tt, M, N) -> X(N, M)
4.08/1.92	   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
4.08/1.92	   ISNAT(plus(V1, V2)) -> ISNAT(V1)
4.08/1.92	   ISNAT(s(V1)) -> U21'(isNat(V1))
4.08/1.92	   ISNAT(s(V1)) -> ISNAT(V1)
4.08/1.92	   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
4.08/1.92	   ISNAT(x(V1, V2)) -> ISNAT(V1)
4.08/1.92	   PLUS(N, 0) -> U41'(isNat(N), N)
4.08/1.92	   PLUS(N, 0) -> ISNAT(N)
4.08/1.92	   PLUS(N, s(M)) -> U51'(isNat(M), M, N)
4.08/1.92	   PLUS(N, s(M)) -> ISNAT(M)
4.08/1.92	   X(N, 0) -> U61'(isNat(N))
4.08/1.92	   X(N, 0) -> ISNAT(N)
4.08/1.92	   X(N, s(M)) -> U71'(isNat(M), M, N)
4.08/1.92	   X(N, s(M)) -> ISNAT(M)
4.08/1.92	
4.08/1.92	The collapsing dependency pairs are DP_c:
4.08/1.92	   U41'(tt, N) -> N
4.08/1.92	   U52'(tt, M, N) -> N
4.08/1.92	   U52'(tt, M, N) -> M
4.08/1.92	   U72'(tt, M, N) -> N
4.08/1.92	   U72'(tt, M, N) -> M
4.08/1.92	
4.08/1.92	
4.08/1.92	The hidden terms of R are:
4.08/1.92	none
4.08/1.92	
4.08/1.92	Every hiding context is built from:none
4.08/1.92	
4.08/1.92	Hence, the new unhiding pairs DP_u are :
4.08/1.92	   U41'(tt, N) -> U(N)
4.08/1.92	   U52'(tt, M, N) -> U(N)
4.08/1.92	   U52'(tt, M, N) -> U(M)
4.08/1.92	   U72'(tt, M, N) -> U(N)
4.08/1.92	   U72'(tt, M, N) -> U(M)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(5) QCSDependencyGraphProof (EQUIVALENT)
4.08/1.92	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 17 less nodes.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(6)
4.08/1.92	Complex Obligation (AND)
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(7)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1, ISNAT_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
4.08/1.92	   U11'(tt, V2) -> ISNAT(V2)
4.08/1.92	   ISNAT(plus(V1, V2)) -> ISNAT(V1)
4.08/1.92	   ISNAT(s(V1)) -> ISNAT(V1)
4.08/1.92	   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
4.08/1.92	   U31'(tt, V2) -> ISNAT(V2)
4.08/1.92	   ISNAT(x(V1, V2)) -> ISNAT(V1)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(8) QCSDPSubtermProof (EQUIVALENT)
4.08/1.92	We use the subterm processor [DA_EMMES].
4.08/1.92	
4.08/1.92	
4.08/1.92	The following pairs can be oriented strictly and are deleted.
4.08/1.92	
4.08/1.92	   ISNAT(plus(V1, V2)) -> U11'(isNat(V1), V2)
4.08/1.92	   ISNAT(plus(V1, V2)) -> ISNAT(V1)
4.08/1.92	   ISNAT(s(V1)) -> ISNAT(V1)
4.08/1.92	   ISNAT(x(V1, V2)) -> U31'(isNat(V1), V2)
4.08/1.92	   ISNAT(x(V1, V2)) -> ISNAT(V1)
4.08/1.92	The remaining pairs can at least be oriented weakly.
4.08/1.92	
4.08/1.92	   U11'(tt, V2) -> ISNAT(V2)
4.08/1.92	   U31'(tt, V2) -> ISNAT(V2)
4.08/1.92	Used ordering:  Combined order from the following AFS and order.
4.08/1.92	U11'(x1, x2)  =  x2
4.08/1.92	
4.08/1.92	ISNAT(x1)  =  x1
4.08/1.92	
4.08/1.92	U31'(x1, x2)  =  x2
4.08/1.92	
4.08/1.92	
4.08/1.92	Subterm Order
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(9)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U11'_2, U31'_2} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1, ISNAT_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   U11'(tt, V2) -> ISNAT(V2)
4.08/1.92	   U31'(tt, V2) -> ISNAT(V2)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(10) QCSDependencyGraphProof (EQUIVALENT)
4.08/1.92	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(11)
4.08/1.92	TRUE
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(12)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U52'_3, U51'_3} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   U51'(tt, M, N) -> U52'(isNat(N), M, N)
4.08/1.92	   U52'(tt, M, N) -> PLUS(N, M)
4.08/1.92	   PLUS(N, s(M)) -> U51'(isNat(M), M, N)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(13) QCSDPSubtermProof (EQUIVALENT)
4.08/1.92	We use the subterm processor [DA_EMMES].
4.08/1.92	
4.08/1.92	
4.08/1.92	The following pairs can be oriented strictly and are deleted.
4.08/1.92	
4.08/1.92	   PLUS(N, s(M)) -> U51'(isNat(M), M, N)
4.08/1.92	The remaining pairs can at least be oriented weakly.
4.08/1.92	
4.08/1.92	   U51'(tt, M, N) -> U52'(isNat(N), M, N)
4.08/1.92	   U52'(tt, M, N) -> PLUS(N, M)
4.08/1.92	Used ordering:  Combined order from the following AFS and order.
4.08/1.92	U52'(x1, x2, x3)  =  x2
4.08/1.92	
4.08/1.92	U51'(x1, x2, x3)  =  x2
4.08/1.92	
4.08/1.92	PLUS(x1, x2)  =  x2
4.08/1.92	
4.08/1.92	
4.08/1.92	Subterm Order
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(14)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U52'_3, U51'_3} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   U51'(tt, M, N) -> U52'(isNat(N), M, N)
4.08/1.92	   U52'(tt, M, N) -> PLUS(N, M)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(15) QCSDependencyGraphProof (EQUIVALENT)
4.08/1.92	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(16)
4.08/1.92	TRUE
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(17)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U72'_3, U71'_3} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   U72'(tt, M, N) -> X(N, M)
4.08/1.92	   X(N, s(M)) -> U71'(isNat(M), M, N)
4.08/1.92	   U71'(tt, M, N) -> U72'(isNat(N), M, N)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(18) QCSDPSubtermProof (EQUIVALENT)
4.08/1.92	We use the subterm processor [DA_EMMES].
4.08/1.92	
4.08/1.92	
4.08/1.92	The following pairs can be oriented strictly and are deleted.
4.08/1.92	
4.08/1.92	   X(N, s(M)) -> U71'(isNat(M), M, N)
4.08/1.92	The remaining pairs can at least be oriented weakly.
4.08/1.92	
4.08/1.92	   U72'(tt, M, N) -> X(N, M)
4.08/1.92	   U71'(tt, M, N) -> U72'(isNat(N), M, N)
4.08/1.92	Used ordering:  Combined order from the following AFS and order.
4.08/1.92	X(x1, x2)  =  x2
4.08/1.92	
4.08/1.92	U72'(x1, x2, x3)  =  x2
4.08/1.92	
4.08/1.92	U71'(x1, x2, x3)  =  x2
4.08/1.92	
4.08/1.92	
4.08/1.92	Subterm Order
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(19)
4.08/1.92	Obligation:
4.08/1.92	Q-restricted context-sensitive dependency pair problem:
4.08/1.92	The symbols in {U12_1, U21_1, U32_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
4.08/1.92	For all symbols f in {U11_2, U31_2, U41_2, U51_3, U52_3, U71_3, U72_3, U72'_3, U71'_3} we have mu(f) = {1}.
4.08/1.92	The symbols in {isNat_1} are not replacing on any position.
4.08/1.92	
4.08/1.92	The TRS P consists of the following rules:
4.08/1.92	
4.08/1.92	   U72'(tt, M, N) -> X(N, M)
4.08/1.92	   U71'(tt, M, N) -> U72'(isNat(N), M, N)
4.08/1.92	
4.08/1.92	The TRS R consists of the following rules:
4.08/1.92	
4.08/1.92	   U11(tt, V2) -> U12(isNat(V2))
4.08/1.92	   U12(tt) -> tt
4.08/1.92	   U21(tt) -> tt
4.08/1.92	   U31(tt, V2) -> U32(isNat(V2))
4.08/1.92	   U32(tt) -> tt
4.08/1.92	   U41(tt, N) -> N
4.08/1.92	   U51(tt, M, N) -> U52(isNat(N), M, N)
4.08/1.92	   U52(tt, M, N) -> s(plus(N, M))
4.08/1.92	   U61(tt) -> 0
4.08/1.92	   U71(tt, M, N) -> U72(isNat(N), M, N)
4.08/1.92	   U72(tt, M, N) -> plus(x(N, M), N)
4.08/1.92	   isNat(0) -> tt
4.08/1.92	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.08/1.92	   isNat(s(V1)) -> U21(isNat(V1))
4.08/1.92	   isNat(x(V1, V2)) -> U31(isNat(V1), V2)
4.08/1.92	   plus(N, 0) -> U41(isNat(N), N)
4.08/1.92	   plus(N, s(M)) -> U51(isNat(M), M, N)
4.08/1.92	   x(N, 0) -> U61(isNat(N))
4.08/1.92	   x(N, s(M)) -> U71(isNat(M), M, N)
4.08/1.92	
4.08/1.92	Q is empty.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(20) QCSDependencyGraphProof (EQUIVALENT)
4.08/1.92	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
4.08/1.92	
4.08/1.92	----------------------------------------
4.08/1.92	
4.08/1.92	(21)
4.08/1.92	TRUE
4.40/1.95	EOF