12.12/3.76	YES
12.12/3.78	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
12.12/3.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.12/3.78	
12.12/3.78	
12.12/3.78	Termination of the given CSR could be proven:
12.12/3.78	
12.12/3.78	(0) CSR
12.12/3.78	(1) CSDependencyPairsProof [EQUIVALENT, 78 ms]
12.12/3.78	(2) QCSDP
12.12/3.78	(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	(4) AND
12.12/3.78	    (5) QCSDP
12.12/3.78	        (6) QCSDPReductionPairProof [EQUIVALENT, 75 ms]
12.12/3.78	        (7) QCSDP
12.12/3.78	        (8) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	        (9) AND
12.12/3.78	            (10) QCSDP
12.12/3.78	                (11) QCSDPReductionPairProof [EQUIVALENT, 256 ms]
12.12/3.78	                (12) QCSDP
12.12/3.78	                (13) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	                (14) TRUE
12.12/3.78	            (15) QCSDP
12.12/3.78	                (16) QCSDPReductionPairProof [EQUIVALENT, 218 ms]
12.12/3.78	                (17) QCSDP
12.12/3.78	                (18) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	                (19) TRUE
12.12/3.78	    (20) QCSDP
12.12/3.78	        (21) QCSDPSubtermProof [EQUIVALENT, 0 ms]
12.12/3.78	        (22) QCSDP
12.12/3.78	        (23) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	        (24) TRUE
12.12/3.78	    (25) QCSDP
12.12/3.78	        (26) QCSDPSubtermProof [EQUIVALENT, 0 ms]
12.12/3.78	        (27) QCSDP
12.12/3.78	        (28) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
12.12/3.78	        (29) TRUE
12.12/3.78	
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(0)
12.12/3.78	Obligation:
12.12/3.78	Context-sensitive rewrite system:
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	The replacement map contains the following entries:
12.12/3.78	
12.12/3.78	U11: {1}
12.12/3.78	tt: empty set
12.12/3.78	U12: {1}
12.12/3.78	isNat: empty set
12.12/3.78	U13: {1}
12.12/3.78	U21: {1}
12.12/3.78	U22: {1}
12.12/3.78	U31: {1}
12.12/3.78	U32: {1}
12.12/3.78	U33: {1}
12.12/3.78	U41: {1}
12.12/3.78	U51: {1}
12.12/3.78	s: {1}
12.12/3.78	plus: {1, 2}
12.12/3.78	U61: {1}
12.12/3.78	0: empty set
12.12/3.78	U71: {1}
12.12/3.78	x: {1, 2}
12.12/3.78	and: {1}
12.12/3.78	isNatKind: empty set
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(1) CSDependencyPairsProof (EQUIVALENT)
12.12/3.78	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(2)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	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.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNAT_1, ISNATKIND_1, U_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The ordinary context-sensitive dependency pairs DP_o are:
12.12/3.78	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
12.12/3.78	   U11'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   U12'(tt, V2) -> U13'(isNat(V2))
12.12/3.78	   U12'(tt, V2) -> ISNAT(V2)
12.12/3.78	   U21'(tt, V1) -> U22'(isNat(V1))
12.12/3.78	   U21'(tt, V1) -> ISNAT(V1)
12.12/3.78	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
12.12/3.78	   U31'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   U32'(tt, V2) -> U33'(isNat(V2))
12.12/3.78	   U32'(tt, V2) -> ISNAT(V2)
12.12/3.78	   U51'(tt, M, N) -> PLUS(N, M)
12.12/3.78	   U71'(tt, M, N) -> PLUS(x(N, M), N)
12.12/3.78	   U71'(tt, M, N) -> X(N, M)
12.12/3.78	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
12.12/3.78	   ISNAT(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   PLUS(N, 0) -> U41'(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   PLUS(N, 0) -> AND(isNat(N), isNatKind(N))
12.12/3.78	   PLUS(N, 0) -> ISNAT(N)
12.12/3.78	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   PLUS(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
12.12/3.78	   PLUS(N, s(M)) -> AND(isNat(M), isNatKind(M))
12.12/3.78	   PLUS(N, s(M)) -> ISNAT(M)
12.12/3.78	   X(N, 0) -> U61'(and(isNat(N), isNatKind(N)))
12.12/3.78	   X(N, 0) -> AND(isNat(N), isNatKind(N))
12.12/3.78	   X(N, 0) -> ISNAT(N)
12.12/3.78	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   X(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
12.12/3.78	   X(N, s(M)) -> AND(isNat(M), isNatKind(M))
12.12/3.78	   X(N, s(M)) -> ISNAT(M)
12.12/3.78	
12.12/3.78	The collapsing dependency pairs are DP_c:
12.12/3.78	   U41'(tt, N) -> N
12.12/3.78	   U51'(tt, M, N) -> N
12.12/3.78	   U51'(tt, M, N) -> M
12.12/3.78	   U71'(tt, M, N) -> N
12.12/3.78	   U71'(tt, M, N) -> M
12.12/3.78	   AND(tt, X) -> X
12.12/3.78	
12.12/3.78	
12.12/3.78	The hidden terms of R are:
12.12/3.78	
12.12/3.78	   isNatKind(x0)
12.12/3.78	   and(isNat(x0), isNatKind(x0))
12.12/3.78	   isNat(x0)
12.12/3.78	
12.12/3.78	Every hiding context is built from:
12.12/3.78	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@16f58148
12.12/3.78	
12.12/3.78	Hence, the new unhiding pairs DP_u are :
12.12/3.78	   U41'(tt, N) -> U(N)
12.12/3.78	   U51'(tt, M, N) -> U(N)
12.12/3.78	   U51'(tt, M, N) -> U(M)
12.12/3.78	   U71'(tt, M, N) -> U(N)
12.12/3.78	   U71'(tt, M, N) -> U(M)
12.12/3.78	   AND(tt, X) -> U(X)
12.12/3.78	   U(and(x_0, x_1)) -> U(x_0)
12.12/3.78	   U(isNatKind(x0)) -> ISNATKIND(x0)
12.12/3.78	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
12.12/3.78	   U(isNat(x0)) -> ISNAT(x0)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(3) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 21 less nodes.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(4)
12.12/3.78	Complex Obligation (AND)
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(5)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNAT_1, U_1, ISNATKIND_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U12'(tt, V2) -> ISNAT(V2)
12.12/3.78	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
12.12/3.78	   U11'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   AND(tt, X) -> U(X)
12.12/3.78	   U(and(x_0, x_1)) -> U(x_0)
12.12/3.78	   U(isNatKind(x0)) -> ISNATKIND(x0)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
12.12/3.78	   U(isNat(x0)) -> ISNAT(x0)
12.12/3.78	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
12.12/3.78	   U21'(tt, V1) -> ISNAT(V1)
12.12/3.78	   ISNAT(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
12.12/3.78	   U32'(tt, V2) -> ISNAT(V2)
12.12/3.78	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   U31'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(6) QCSDPReductionPairProof (EQUIVALENT)
12.12/3.78	Using the order
12.12/3.78	
12.12/3.78	Polynomial interpretation with max and min functions [POLO,MAXPOLO]:
12.12/3.78	
12.12/3.78	   POL(0) = 0
12.12/3.78	   POL(AND(x_1, x_2)) = x_2
12.12/3.78	   POL(ISNAT(x_1)) = 0
12.12/3.78	   POL(ISNATKIND(x_1)) = 0
12.12/3.78	   POL(U(x_1)) = x_1
12.12/3.78	   POL(U11(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(U11'(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(U12(x_1, x_2)) = 0
12.12/3.78	   POL(U12'(x_1, x_2)) = 0
12.12/3.78	   POL(U13(x_1)) = 0
12.12/3.78	   POL(U21(x_1, x_2)) = 1 + x_1
12.12/3.78	   POL(U21'(x_1, x_2)) = 0
12.12/3.78	   POL(U22(x_1)) = x_1
12.12/3.78	   POL(U31(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(U31'(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(U32(x_1, x_2)) = 0
12.12/3.78	   POL(U32'(x_1, x_2)) = 0
12.12/3.78	   POL(U33(x_1)) = 0
12.12/3.78	   POL(U41(x_1, x_2)) = x_2
12.12/3.78	   POL(U51(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(U61(x_1)) = 0
12.12/3.78	   POL(U71(x_1, x_2, x_3)) = 0
12.12/3.78	   POL(and(x_1, x_2)) = x_1 + x_2
12.12/3.78	   POL(isNat(x_1)) = 1
12.12/3.78	   POL(isNatKind(x_1)) = 0
12.12/3.78	   POL(plus(x_1, x_2)) = x_1
12.12/3.78	   POL(s(x_1)) = 0
12.12/3.78	   POL(tt) = 0
12.12/3.78	   POL(x(x_1, x_2)) = 0
12.12/3.78	
12.12/3.78	
12.12/3.78	the following usable rules
12.12/3.78	
12.12/3.78	
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	
12.12/3.78	
12.12/3.78	could all be oriented weakly.
12.12/3.78	
12.12/3.78	Furthermore, the pairs
12.12/3.78	
12.12/3.78	
12.12/3.78	   U(and(isNat(x0), isNatKind(x0))) -> AND(isNat(x0), isNatKind(x0))
12.12/3.78	   U(isNat(x0)) -> ISNAT(x0)
12.12/3.78	
12.12/3.78	
12.12/3.78	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
12.12/3.78	
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(7)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNAT_1, U_1, ISNATKIND_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U12'(tt, V2) -> ISNAT(V2)
12.12/3.78	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
12.12/3.78	   U11'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   AND(tt, X) -> U(X)
12.12/3.78	   U(and(x_0, x_1)) -> U(x_0)
12.12/3.78	   U(isNatKind(x0)) -> ISNATKIND(x0)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
12.12/3.78	   U21'(tt, V1) -> ISNAT(V1)
12.12/3.78	   ISNAT(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
12.12/3.78	   U32'(tt, V2) -> ISNAT(V2)
12.12/3.78	   ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   U31'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(8) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 5 less nodes.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(9)
12.12/3.78	Complex Obligation (AND)
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(10)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, U_1, ISNATKIND_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U(and(x_0, x_1)) -> U(x_0)
12.12/3.78	   U(isNatKind(x0)) -> ISNATKIND(x0)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   AND(tt, X) -> U(X)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(11) QCSDPReductionPairProof (EQUIVALENT)
12.12/3.78	Using the order
12.12/3.78	
12.12/3.78	U/1(YES)
12.12/3.78	and/2(YES,YES)
12.12/3.78	isNatKind/1)YES(
12.12/3.78	ISNATKIND/1(YES)
12.12/3.78	plus/2(YES,YES)
12.12/3.78	AND/2(NO,YES)
12.12/3.78	tt/0)
12.12/3.78	s/1(YES)
12.12/3.78	x/2(YES,YES)
12.12/3.78	0/0)
12.12/3.78	U41/2)NO,YES(
12.12/3.78	isNat/1)YES(
12.12/3.78	U51/3(YES,YES,YES)
12.12/3.78	U11/3(NO,NO,NO)
12.12/3.78	U21/2(NO,YES)
12.12/3.78	U31/3(NO,NO,YES)
12.12/3.78	U12/2(NO,NO)
12.12/3.78	U13/1(NO)
12.12/3.78	U22/1)YES(
12.12/3.78	U61/1(YES)
12.12/3.78	U71/3(YES,YES,YES)
12.12/3.78	U32/2)NO,YES(
12.12/3.78	U33/1)YES(
12.12/3.78	
12.12/3.78	Quasi precedence:
12.12/3.78	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > [plus_2, U51_3, U11] > [and_2, tt, s_1, U21_1, U12, U13]
12.12/3.78	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > [0, U61_1] > [and_2, tt, s_1, U21_1, U12, U13]
12.12/3.78	[U_1, ISNATKIND_1, AND_1, x_2, U71_3] > U31_1
12.12/3.78	
12.12/3.78	
12.12/3.78	Status:
12.12/3.78	U_1: multiset status
12.12/3.78	and_2: [1,2]
12.12/3.78	ISNATKIND_1: multiset status
12.12/3.78	plus_2: [1,2]
12.12/3.78	AND_1: multiset status
12.12/3.78	tt: multiset status
12.12/3.78	s_1: [1]
12.12/3.78	x_2: [2,1]
12.12/3.78	0: multiset status
12.12/3.78	U51_3: [3,2,1]
12.12/3.78	U11: []
12.12/3.78	U21_1: [1]
12.12/3.78	U31_1: multiset status
12.12/3.78	U12: []
12.12/3.78	U13: []
12.12/3.78	U61_1: multiset status
12.12/3.78	U71_3: [2,3,1]
12.12/3.78	
12.12/3.78	
12.12/3.78	
12.12/3.78	the following usable rules
12.12/3.78	
12.12/3.78	
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	
12.12/3.78	
12.12/3.78	could all be oriented weakly.
12.12/3.78	
12.12/3.78	Furthermore, the pairs
12.12/3.78	
12.12/3.78	
12.12/3.78	   U(and(x_0, x_1)) -> U(x_0)
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
12.12/3.78	   ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
12.12/3.78	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
12.12/3.78	
12.12/3.78	
12.12/3.78	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
12.12/3.78	
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(12)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNATKIND_1, U_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U(isNatKind(x0)) -> ISNATKIND(x0)
12.12/3.78	   AND(tt, X) -> U(X)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(13) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(14)
12.12/3.78	TRUE
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(15)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNAT_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
12.12/3.78	   U12'(tt, V2) -> ISNAT(V2)
12.12/3.78	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
12.12/3.78	   U21'(tt, V1) -> ISNAT(V1)
12.12/3.78	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
12.12/3.78	   U32'(tt, V2) -> ISNAT(V2)
12.12/3.78	   U31'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   U11'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(16) QCSDPReductionPairProof (EQUIVALENT)
12.12/3.78	Using the order
12.12/3.78	
12.12/3.78	ISNAT/1(YES)
12.12/3.78	plus/2(YES,YES)
12.12/3.78	U11'/3(NO,YES,YES)
12.12/3.78	and/2)NO,YES(
12.12/3.78	isNatKind/1(YES)
12.12/3.78	tt/0)
12.12/3.78	U12'/2(NO,YES)
12.12/3.78	isNat/1(NO)
12.12/3.78	s/1(YES)
12.12/3.78	U21'/2(NO,YES)
12.12/3.78	x/2(YES,YES)
12.12/3.78	U31'/3(YES,YES,YES)
12.12/3.78	U32'/2(NO,YES)
12.12/3.78	0/0)
12.12/3.78	U41/2(YES,YES)
12.12/3.78	U51/3(YES,YES,YES)
12.12/3.78	U11/3(NO,NO,NO)
12.12/3.78	U21/2(NO,NO)
12.12/3.78	U31/3(NO,NO,NO)
12.12/3.78	U12/2(NO,NO)
12.12/3.78	U13/1(NO)
12.12/3.78	U61/1(NO)
12.12/3.78	U71/3(YES,YES,YES)
12.12/3.78	U22/1)YES(
12.12/3.78	U32/2(NO,NO)
12.12/3.78	U33/1(NO)
12.12/3.78	
12.12/3.78	Quasi precedence:
12.12/3.78	[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
12.12/3.78	[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
12.12/3.78	[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
12.12/3.78	
12.12/3.78	
12.12/3.78	Status:
12.12/3.78	ISNAT_1: multiset status
12.12/3.78	plus_2: [1,2]
12.12/3.78	U11'_2: multiset status
12.12/3.78	isNatKind_1: [1]
12.12/3.78	tt: multiset status
12.12/3.78	U12'_1: multiset status
12.12/3.78	isNat: multiset status
12.12/3.78	s_1: [1]
12.12/3.78	U21'_1: multiset status
12.12/3.78	x_2: [2,1]
12.12/3.78	U31'_3: multiset status
12.12/3.78	U32'_1: multiset status
12.12/3.78	0: multiset status
12.12/3.78	U41_2: [1,2]
12.12/3.78	U51_3: [3,2,1]
12.12/3.78	U11: multiset status
12.12/3.78	U21: multiset status
12.12/3.78	U31: []
12.12/3.78	U12: []
12.12/3.78	U13: multiset status
12.12/3.78	U61: multiset status
12.12/3.78	U71_3: [2,3,1]
12.12/3.78	U32: multiset status
12.12/3.78	U33: multiset status
12.12/3.78	
12.12/3.78	
12.12/3.78	
12.12/3.78	the following usable rules
12.12/3.78	
12.12/3.78	
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	
12.12/3.78	
12.12/3.78	could all be oriented weakly.
12.12/3.78	
12.12/3.78	Furthermore, the pairs
12.12/3.78	
12.12/3.78	
12.12/3.78	   ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
12.12/3.78	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
12.12/3.78	   ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
12.12/3.78	   U31'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	   U11'(tt, V1, V2) -> ISNAT(V1)
12.12/3.78	
12.12/3.78	
12.12/3.78	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
12.12/3.78	
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(17)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1, ISNAT_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U12'(tt, V2) -> ISNAT(V2)
12.12/3.78	   U21'(tt, V1) -> ISNAT(V1)
12.12/3.78	   U32'(tt, V2) -> ISNAT(V2)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(18) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(19)
12.12/3.78	TRUE
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(20)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   U51'(tt, M, N) -> PLUS(N, M)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(21) QCSDPSubtermProof (EQUIVALENT)
12.12/3.78	We use the subterm processor [DA_EMMES].
12.12/3.78	
12.12/3.78	
12.12/3.78	The following pairs can be oriented strictly and are deleted.
12.12/3.78	
12.12/3.78	   PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	The remaining pairs can at least be oriented weakly.
12.12/3.78	
12.12/3.78	   U51'(tt, M, N) -> PLUS(N, M)
12.12/3.78	Used ordering:  Combined order from the following AFS and order.
12.12/3.78	U51'(x1, x2, x3)  =  x2
12.12/3.78	
12.12/3.78	PLUS(x1, x2)  =  x2
12.12/3.78	
12.12/3.78	
12.12/3.78	Subterm Order
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(22)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, PLUS_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U51'(tt, M, N) -> PLUS(N, M)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(23) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(24)
12.12/3.78	TRUE
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(25)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U71'(tt, M, N) -> X(N, M)
12.12/3.78	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(26) QCSDPSubtermProof (EQUIVALENT)
12.12/3.78	We use the subterm processor [DA_EMMES].
12.12/3.78	
12.12/3.78	
12.12/3.78	The following pairs can be oriented strictly and are deleted.
12.12/3.78	
12.12/3.78	   X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	The remaining pairs can at least be oriented weakly.
12.12/3.78	
12.12/3.78	   U71'(tt, M, N) -> X(N, M)
12.12/3.78	Used ordering:  Combined order from the following AFS and order.
12.12/3.78	X(x1, x2)  =  x2
12.12/3.78	
12.12/3.78	U71'(x1, x2, x3)  =  x2
12.12/3.78	
12.12/3.78	
12.12/3.78	Subterm Order
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(27)
12.12/3.78	Obligation:
12.12/3.78	Q-restricted context-sensitive dependency pair problem:
12.12/3.78	The symbols in {U13_1, U22_1, U33_1, s_1, plus_2, U61_1, x_2, X_2} are replacing on all positions.
12.12/3.78	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}.
12.12/3.78	The symbols in {isNat_1, isNatKind_1} are not replacing on any position.
12.12/3.78	
12.12/3.78	The TRS P consists of the following rules:
12.12/3.78	
12.12/3.78	   U71'(tt, M, N) -> X(N, M)
12.12/3.78	
12.12/3.78	The TRS R consists of the following rules:
12.12/3.78	
12.12/3.78	   U11(tt, V1, V2) -> U12(isNat(V1), V2)
12.12/3.78	   U12(tt, V2) -> U13(isNat(V2))
12.12/3.78	   U13(tt) -> tt
12.12/3.78	   U21(tt, V1) -> U22(isNat(V1))
12.12/3.78	   U22(tt) -> tt
12.12/3.78	   U31(tt, V1, V2) -> U32(isNat(V1), V2)
12.12/3.78	   U32(tt, V2) -> U33(isNat(V2))
12.12/3.78	   U33(tt) -> tt
12.12/3.78	   U41(tt, N) -> N
12.12/3.78	   U51(tt, M, N) -> s(plus(N, M))
12.12/3.78	   U61(tt) -> 0
12.12/3.78	   U71(tt, M, N) -> plus(x(N, M), N)
12.12/3.78	   and(tt, X) -> X
12.12/3.78	   isNat(0) -> tt
12.12/3.78	   isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
12.12/3.78	   isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
12.12/3.78	   isNatKind(0) -> tt
12.12/3.78	   isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   isNatKind(s(V1)) -> isNatKind(V1)
12.12/3.78	   isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
12.12/3.78	   plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
12.12/3.78	   plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	   x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
12.12/3.78	   x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
12.12/3.78	
12.12/3.78	Q is empty.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(28) QCSDependencyGraphProof (EQUIVALENT)
12.12/3.78	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
12.12/3.78	
12.12/3.78	----------------------------------------
12.12/3.78	
12.12/3.78	(29)
12.12/3.78	TRUE
12.60/3.83	EOF