4.49/1.95	YES
4.49/1.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.49/1.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.49/1.96	
4.49/1.96	
4.49/1.96	Termination w.r.t. Q of the given QTRS could be proven:
4.49/1.96	
4.49/1.96	(0) QTRS
4.49/1.96	(1) QTRSToCSRProof [SOUND, 0 ms]
4.49/1.96	(2) CSR
4.49/1.96	(3) CSRRRRProof [EQUIVALENT, 65 ms]
4.49/1.96	(4) CSR
4.49/1.96	(5) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(6) CSR
4.49/1.96	(7) CSRRRRProof [EQUIVALENT, 10 ms]
4.49/1.96	(8) CSR
4.49/1.96	(9) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(10) CSR
4.49/1.96	(11) CSRRRRProof [EQUIVALENT, 6 ms]
4.49/1.96	(12) CSR
4.49/1.96	(13) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(14) CSR
4.49/1.96	(15) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(16) CSR
4.49/1.96	(17) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(18) CSR
4.49/1.96	(19) CSRRRRProof [EQUIVALENT, 0 ms]
4.49/1.96	(20) CSR
4.49/1.96	(21) RisEmptyProof [EQUIVALENT, 0 ms]
4.49/1.96	(22) YES
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(0)
4.49/1.96	Obligation:
4.49/1.96	Q restricted rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
4.49/1.96	   active(U12(tt)) -> mark(tt)
4.49/1.96	   active(U21(tt)) -> mark(tt)
4.49/1.96	   active(U31(tt, N)) -> mark(N)
4.49/1.96	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
4.49/1.96	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
4.49/1.96	   active(isNat(0)) -> mark(tt)
4.49/1.96	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
4.49/1.96	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
4.49/1.96	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
4.49/1.96	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
4.49/1.96	   active(U11(X1, X2)) -> U11(active(X1), X2)
4.49/1.96	   active(U12(X)) -> U12(active(X))
4.49/1.96	   active(U21(X)) -> U21(active(X))
4.49/1.96	   active(U31(X1, X2)) -> U31(active(X1), X2)
4.49/1.96	   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
4.49/1.96	   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
4.49/1.96	   active(s(X)) -> s(active(X))
4.49/1.96	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.49/1.96	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.49/1.96	   U11(mark(X1), X2) -> mark(U11(X1, X2))
4.49/1.96	   U12(mark(X)) -> mark(U12(X))
4.49/1.96	   U21(mark(X)) -> mark(U21(X))
4.49/1.96	   U31(mark(X1), X2) -> mark(U31(X1, X2))
4.49/1.96	   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
4.49/1.96	   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
4.49/1.96	   s(mark(X)) -> mark(s(X))
4.49/1.96	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.49/1.96	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.49/1.96	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
4.49/1.96	   proper(tt) -> ok(tt)
4.49/1.96	   proper(U12(X)) -> U12(proper(X))
4.49/1.96	   proper(isNat(X)) -> isNat(proper(X))
4.49/1.96	   proper(U21(X)) -> U21(proper(X))
4.49/1.96	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
4.49/1.96	   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
4.49/1.96	   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
4.49/1.96	   proper(s(X)) -> s(proper(X))
4.49/1.96	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.49/1.96	   proper(0) -> ok(0)
4.49/1.96	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
4.49/1.96	   U12(ok(X)) -> ok(U12(X))
4.49/1.96	   isNat(ok(X)) -> ok(isNat(X))
4.49/1.96	   U21(ok(X)) -> ok(U21(X))
4.49/1.96	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
4.49/1.96	   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
4.49/1.96	   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
4.49/1.96	   s(ok(X)) -> ok(s(X))
4.49/1.96	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.49/1.96	   top(mark(X)) -> top(proper(X))
4.49/1.96	   top(ok(X)) -> top(active(X))
4.49/1.96	
4.49/1.96	The set Q consists of the following terms:
4.49/1.96	
4.49/1.96	   active(isNat(0))
4.49/1.96	   active(isNat(plus(x0, x1)))
4.49/1.96	   active(isNat(s(x0)))
4.49/1.96	   active(U11(x0, x1))
4.49/1.96	   active(U12(x0))
4.49/1.96	   active(U21(x0))
4.49/1.96	   active(U31(x0, x1))
4.49/1.96	   active(U41(x0, x1, x2))
4.49/1.96	   active(U42(x0, x1, x2))
4.49/1.96	   active(s(x0))
4.49/1.96	   active(plus(x0, x1))
4.49/1.96	   U11(mark(x0), x1)
4.49/1.96	   U12(mark(x0))
4.49/1.96	   U21(mark(x0))
4.49/1.96	   U31(mark(x0), x1)
4.49/1.96	   U41(mark(x0), x1, x2)
4.49/1.96	   U42(mark(x0), x1, x2)
4.49/1.96	   s(mark(x0))
4.49/1.96	   plus(mark(x0), x1)
4.49/1.96	   plus(x0, mark(x1))
4.49/1.96	   proper(U11(x0, x1))
4.49/1.96	   proper(tt)
4.49/1.96	   proper(U12(x0))
4.49/1.96	   proper(isNat(x0))
4.49/1.96	   proper(U21(x0))
4.49/1.96	   proper(U31(x0, x1))
4.49/1.96	   proper(U41(x0, x1, x2))
4.49/1.96	   proper(U42(x0, x1, x2))
4.49/1.96	   proper(s(x0))
4.49/1.96	   proper(plus(x0, x1))
4.49/1.96	   proper(0)
4.49/1.96	   U11(ok(x0), ok(x1))
4.49/1.96	   U12(ok(x0))
4.49/1.96	   isNat(ok(x0))
4.49/1.96	   U21(ok(x0))
4.49/1.96	   U31(ok(x0), ok(x1))
4.49/1.96	   U41(ok(x0), ok(x1), ok(x2))
4.49/1.96	   U42(ok(x0), ok(x1), ok(x2))
4.49/1.96	   s(ok(x0))
4.49/1.96	   plus(ok(x0), ok(x1))
4.49/1.96	   top(mark(x0))
4.49/1.96	   top(ok(x0))
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(1) QTRSToCSRProof (SOUND)
4.49/1.96	The following Q TRS is given: Q restricted rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   active(U11(tt, V2)) -> mark(U12(isNat(V2)))
4.49/1.96	   active(U12(tt)) -> mark(tt)
4.49/1.96	   active(U21(tt)) -> mark(tt)
4.49/1.96	   active(U31(tt, N)) -> mark(N)
4.49/1.96	   active(U41(tt, M, N)) -> mark(U42(isNat(N), M, N))
4.49/1.96	   active(U42(tt, M, N)) -> mark(s(plus(N, M)))
4.49/1.96	   active(isNat(0)) -> mark(tt)
4.49/1.96	   active(isNat(plus(V1, V2))) -> mark(U11(isNat(V1), V2))
4.49/1.96	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
4.49/1.96	   active(plus(N, 0)) -> mark(U31(isNat(N), N))
4.49/1.96	   active(plus(N, s(M))) -> mark(U41(isNat(M), M, N))
4.49/1.96	   active(U11(X1, X2)) -> U11(active(X1), X2)
4.49/1.96	   active(U12(X)) -> U12(active(X))
4.49/1.96	   active(U21(X)) -> U21(active(X))
4.49/1.96	   active(U31(X1, X2)) -> U31(active(X1), X2)
4.49/1.96	   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
4.49/1.96	   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
4.49/1.96	   active(s(X)) -> s(active(X))
4.49/1.96	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.49/1.96	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.49/1.96	   U11(mark(X1), X2) -> mark(U11(X1, X2))
4.49/1.96	   U12(mark(X)) -> mark(U12(X))
4.49/1.96	   U21(mark(X)) -> mark(U21(X))
4.49/1.96	   U31(mark(X1), X2) -> mark(U31(X1, X2))
4.49/1.96	   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
4.49/1.96	   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
4.49/1.96	   s(mark(X)) -> mark(s(X))
4.49/1.96	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.49/1.96	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.49/1.96	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
4.49/1.96	   proper(tt) -> ok(tt)
4.49/1.96	   proper(U12(X)) -> U12(proper(X))
4.49/1.96	   proper(isNat(X)) -> isNat(proper(X))
4.49/1.96	   proper(U21(X)) -> U21(proper(X))
4.49/1.96	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
4.49/1.96	   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
4.49/1.96	   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
4.49/1.96	   proper(s(X)) -> s(proper(X))
4.49/1.96	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.49/1.96	   proper(0) -> ok(0)
4.49/1.96	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
4.49/1.96	   U12(ok(X)) -> ok(U12(X))
4.49/1.96	   isNat(ok(X)) -> ok(isNat(X))
4.49/1.96	   U21(ok(X)) -> ok(U21(X))
4.49/1.96	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
4.49/1.96	   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
4.49/1.96	   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
4.49/1.96	   s(ok(X)) -> ok(s(X))
4.49/1.96	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.49/1.96	   top(mark(X)) -> top(proper(X))
4.49/1.96	   top(ok(X)) -> top(active(X))
4.49/1.96	
4.49/1.96	The set Q consists of the following terms:
4.49/1.96	
4.49/1.96	   active(isNat(0))
4.49/1.96	   active(isNat(plus(x0, x1)))
4.49/1.96	   active(isNat(s(x0)))
4.49/1.96	   active(U11(x0, x1))
4.49/1.96	   active(U12(x0))
4.49/1.96	   active(U21(x0))
4.49/1.96	   active(U31(x0, x1))
4.49/1.96	   active(U41(x0, x1, x2))
4.49/1.96	   active(U42(x0, x1, x2))
4.49/1.96	   active(s(x0))
4.49/1.96	   active(plus(x0, x1))
4.49/1.96	   U11(mark(x0), x1)
4.49/1.96	   U12(mark(x0))
4.49/1.96	   U21(mark(x0))
4.49/1.96	   U31(mark(x0), x1)
4.49/1.96	   U41(mark(x0), x1, x2)
4.49/1.96	   U42(mark(x0), x1, x2)
4.49/1.96	   s(mark(x0))
4.49/1.96	   plus(mark(x0), x1)
4.49/1.96	   plus(x0, mark(x1))
4.49/1.96	   proper(U11(x0, x1))
4.49/1.96	   proper(tt)
4.49/1.96	   proper(U12(x0))
4.49/1.96	   proper(isNat(x0))
4.49/1.96	   proper(U21(x0))
4.49/1.96	   proper(U31(x0, x1))
4.49/1.96	   proper(U41(x0, x1, x2))
4.49/1.96	   proper(U42(x0, x1, x2))
4.49/1.96	   proper(s(x0))
4.49/1.96	   proper(plus(x0, x1))
4.49/1.96	   proper(0)
4.49/1.96	   U11(ok(x0), ok(x1))
4.49/1.96	   U12(ok(x0))
4.49/1.96	   isNat(ok(x0))
4.49/1.96	   U21(ok(x0))
4.49/1.96	   U31(ok(x0), ok(x1))
4.49/1.96	   U41(ok(x0), ok(x1), ok(x2))
4.49/1.96	   U42(ok(x0), ok(x1), ok(x2))
4.49/1.96	   s(ok(x0))
4.49/1.96	   plus(ok(x0), ok(x1))
4.49/1.96	   top(mark(x0))
4.49/1.96	   top(ok(x0))
4.49/1.96	
4.49/1.96	Special symbols used for the transformation (see [GM04]):
4.49/1.96	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U31: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	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.
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(2)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U31(tt, N) -> N
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, 0) -> U31(isNat(N), N)
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U31: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(3) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U31(tt, N) -> N
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, 0) -> U31(isNat(N), N)
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U31: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(0) = 1
4.49/1.96	   POL(U11(x_1, x_2)) = 2*x_1
4.49/1.96	   POL(U12(x_1)) = 2*x_1
4.49/1.96	   POL(U21(x_1)) = 2*x_1
4.49/1.96	   POL(U31(x_1, x_2)) = 2*x_1 + 2*x_2
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3
4.49/1.96	   POL(isNat(x_1)) = 0
4.49/1.96	   POL(plus(x_1, x_2)) = 2*x_1 + x_2
4.49/1.96	   POL(s(x_1)) = x_1
4.49/1.96	   POL(tt) = 0
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   plus(N, 0) -> U31(isNat(N), N)
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(4)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U31(tt, N) -> N
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U31: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(5) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U31(tt, N) -> N
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U31: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(0) = 0
4.49/1.96	   POL(U11(x_1, x_2)) = x_1
4.49/1.96	   POL(U12(x_1)) = x_1
4.49/1.96	   POL(U21(x_1)) = x_1
4.49/1.96	   POL(U31(x_1, x_2)) = x_1 + x_2
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.49/1.96	   POL(isNat(x_1)) = 1
4.49/1.96	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
4.49/1.96	   POL(s(x_1)) = x_1
4.49/1.96	   POL(tt) = 1
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U31(tt, N) -> N
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(6)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(7) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(0) = 2
4.49/1.96	   POL(U11(x_1, x_2)) = x_1
4.49/1.96	   POL(U12(x_1)) = 2*x_1
4.49/1.96	   POL(U21(x_1)) = x_1
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3
4.49/1.96	   POL(isNat(x_1)) = 0
4.49/1.96	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
4.49/1.96	   POL(s(x_1)) = 2 + x_1
4.49/1.96	   POL(tt) = 0
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   plus(N, s(M)) -> U41(isNat(M), M, N)
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(8)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(9) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U21: {1}
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	0: empty set
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(0) = 1
4.49/1.96	   POL(U11(x_1, x_2)) = x_1 + 2*x_2
4.49/1.96	   POL(U12(x_1)) = 2*x_1
4.49/1.96	   POL(U21(x_1)) = 1 + x_1
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
4.49/1.96	   POL(isNat(x_1)) = x_1
4.49/1.96	   POL(plus(x_1, x_2)) = x_1 + 2*x_2
4.49/1.96	   POL(s(x_1)) = 2 + x_1
4.49/1.96	   POL(tt) = 0
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U21(tt) -> tt
4.49/1.96	   isNat(0) -> tt
4.49/1.96	   isNat(s(V1)) -> U21(isNat(V1))
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(10)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(11) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U12(tt) -> tt
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(U11(x_1, x_2)) = x_1
4.49/1.96	   POL(U12(x_1)) = 1 + x_1
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
4.49/1.96	   POL(isNat(x_1)) = 0
4.49/1.96	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
4.49/1.96	   POL(s(x_1)) = 1 + x_1
4.49/1.96	   POL(tt) = 1
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U12(tt) -> tt
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(12)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(13) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	s: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(U11(x_1, x_2)) = x_1
4.49/1.96	   POL(U12(x_1)) = x_1
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2 + x_3
4.49/1.96	   POL(isNat(x_1)) = 1
4.49/1.96	   POL(plus(x_1, x_2)) = x_1 + x_2
4.49/1.96	   POL(s(x_1)) = x_1
4.49/1.96	   POL(tt) = 1
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U42(tt, M, N) -> s(plus(N, M))
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(14)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(15) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	U41: {1}
4.49/1.96	U42: {1}
4.49/1.96	plus: {1, 2}
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(U11(x_1, x_2)) = 1 + x_1 + x_2
4.49/1.96	   POL(U12(x_1)) = 1 + x_1
4.49/1.96	   POL(U41(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
4.49/1.96	   POL(U42(x_1, x_2, x_3)) = x_1 + x_2
4.49/1.96	   POL(isNat(x_1)) = x_1
4.49/1.96	   POL(plus(x_1, x_2)) = 1 + x_1 + x_2
4.49/1.96	   POL(tt) = 0
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U41(tt, M, N) -> U42(isNat(N), M, N)
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(16)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	plus: {1, 2}
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(17) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	tt: empty set
4.49/1.96	U12: {1}
4.49/1.96	isNat: empty set
4.49/1.96	plus: {1, 2}
4.49/1.96	Used ordering:
4.49/1.96	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
4.49/1.96	
4.49/1.96	   <<<
4.49/1.96	 POL(U11(x_1, x_2)) =  	[[1], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
4.49/1.96	>>>
4.49/1.96	
4.49/1.96	   <<<
4.49/1.96	 POL(tt) =  	[[0], [1]]
4.49/1.96	>>>
4.49/1.96	
4.49/1.96	   <<<
4.49/1.96	 POL(U12(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
4.49/1.96	>>>
4.49/1.96	
4.49/1.96	   <<<
4.49/1.96	 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
4.49/1.96	>>>
4.49/1.96	
4.49/1.96	   <<<
4.49/1.96	 POL(plus(x_1, x_2)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
4.49/1.96	>>>
4.49/1.96	
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   U11(tt, V2) -> U12(isNat(V2))
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(18)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	isNat: empty set
4.49/1.96	plus: {1, 2}
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(19) CSRRRRProof (EQUIVALENT)
4.49/1.96	The following CSR is given: Context-sensitive rewrite system:
4.49/1.96	The TRS R consists of the following rules:
4.49/1.96	
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	The replacement map contains the following entries:
4.49/1.96	
4.49/1.96	U11: {1}
4.49/1.96	isNat: empty set
4.49/1.96	plus: {1, 2}
4.49/1.96	Used ordering:
4.49/1.96	Polynomial interpretation [POLO]:
4.49/1.96	
4.49/1.96	   POL(U11(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
4.49/1.96	   POL(isNat(x_1)) = 2*x_1
4.49/1.96	   POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2
4.49/1.96	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
4.49/1.96	
4.49/1.96	   isNat(plus(V1, V2)) -> U11(isNat(V1), V2)
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(20)
4.49/1.96	Obligation:
4.49/1.96	Context-sensitive rewrite system:
4.49/1.96	R is empty.
4.49/1.96	
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(21) RisEmptyProof (EQUIVALENT)
4.49/1.96	The CSR R is empty. Hence, termination is trivially proven.
4.49/1.96	----------------------------------------
4.49/1.96	
4.49/1.96	(22)
4.49/1.96	YES
4.65/1.99	EOF