5.31/2.02	YES
5.31/2.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.31/2.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.31/2.03	
5.31/2.03	
5.31/2.03	Termination of the given CSR could be proven:
5.31/2.03	
5.31/2.03	(0) CSR
5.31/2.03	(1) CSDependencyPairsProof [EQUIVALENT, 77 ms]
5.31/2.03	(2) QCSDP
5.31/2.03	(3) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.31/2.03	(4) AND
5.31/2.03	    (5) QCSDP
5.31/2.03	        (6) QCSDPSubtermProof [EQUIVALENT, 26 ms]
5.31/2.03	        (7) QCSDP
5.31/2.03	        (8) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.31/2.03	        (9) TRUE
5.31/2.03	    (10) QCSDP
5.31/2.03	        (11) QCSUsableRulesProof [EQUIVALENT, 0 ms]
5.31/2.03	        (12) QCSDP
5.31/2.03	        (13) QCSDPMuMonotonicPoloProof [EQUIVALENT, 62 ms]
5.31/2.03	        (14) QCSDP
5.31/2.03	        (15) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.31/2.03	        (16) TRUE
5.31/2.03	    (17) QCSDP
5.31/2.03	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
5.31/2.03	        (19) QCSDP
5.31/2.03	        (20) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.31/2.03	        (21) TRUE
5.31/2.03	    (22) QCSDP
5.31/2.03	        (23) QCSDPSubtermProof [EQUIVALENT, 0 ms]
5.31/2.03	        (24) QCSDP
5.31/2.03	        (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.31/2.03	        (26) TRUE
5.31/2.03	
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(0)
5.31/2.03	Obligation:
5.31/2.03	Context-sensitive rewrite system:
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	The replacement map contains the following entries:
5.31/2.03	
5.31/2.03	U101: {1}
5.31/2.03	tt: empty set
5.31/2.03	U102: {1}
5.31/2.03	isNatKind: empty set
5.31/2.03	U103: {1}
5.31/2.03	isNat: empty set
5.31/2.03	U104: {1}
5.31/2.03	plus: {1, 2}
5.31/2.03	x: {1, 2}
5.31/2.03	U11: {1}
5.31/2.03	U12: {1}
5.31/2.03	U13: {1}
5.31/2.03	U14: {1}
5.31/2.03	U15: {1}
5.31/2.03	U16: {1}
5.31/2.03	U21: {1}
5.31/2.03	U22: {1}
5.31/2.03	U23: {1}
5.31/2.03	U31: {1}
5.31/2.03	U32: {1}
5.31/2.03	U33: {1}
5.31/2.03	U34: {1}
5.31/2.03	U35: {1}
5.31/2.03	U36: {1}
5.31/2.03	U41: {1}
5.31/2.03	U42: {1}
5.31/2.03	U51: {1}
5.31/2.03	U61: {1}
5.31/2.03	U62: {1}
5.31/2.03	U71: {1}
5.31/2.03	U72: {1}
5.31/2.03	U81: {1}
5.31/2.03	U82: {1}
5.31/2.03	U83: {1}
5.31/2.03	U84: {1}
5.31/2.03	s: {1}
5.31/2.03	U91: {1}
5.31/2.03	U92: {1}
5.31/2.03	0: empty set
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(1) CSDependencyPairsProof (EQUIVALENT)
5.31/2.03	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(2)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1, PLUS_2, X_2, U16'_1, U23'_1, U36'_1, U42'_1, U62'_1, U92'_1, U51'_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U102'_3, U101'_3, U103'_3, U104'_3, U12'_3, U11'_3, U13'_3, U14'_3, U15'_2, U22'_2, U21'_2, U32'_3, U31'_3, U33'_3, U34'_3, U35'_2, U41'_2, U61'_2, U72'_2, U71'_2, U82'_3, U81'_3, U83'_3, U84'_3, U91'_2} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1, ISNATKIND_1, ISNAT_1, U_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The ordinary context-sensitive dependency pairs DP_o are:
5.31/2.03	   U101'(tt, M, N) -> U102'(isNatKind(M), M, N)
5.31/2.03	   U101'(tt, M, N) -> ISNATKIND(M)
5.31/2.03	   U102'(tt, M, N) -> U103'(isNat(N), M, N)
5.31/2.03	   U102'(tt, M, N) -> ISNAT(N)
5.31/2.03	   U103'(tt, M, N) -> U104'(isNatKind(N), M, N)
5.31/2.03	   U103'(tt, M, N) -> ISNATKIND(N)
5.31/2.03	   U104'(tt, M, N) -> PLUS(x(N, M), N)
5.31/2.03	   U104'(tt, M, N) -> X(N, M)
5.31/2.03	   U11'(tt, V1, V2) -> U12'(isNatKind(V1), V1, V2)
5.31/2.03	   U11'(tt, V1, V2) -> ISNATKIND(V1)
5.31/2.03	   U12'(tt, V1, V2) -> U13'(isNatKind(V2), V1, V2)
5.31/2.03	   U12'(tt, V1, V2) -> ISNATKIND(V2)
5.31/2.03	   U13'(tt, V1, V2) -> U14'(isNatKind(V2), V1, V2)
5.31/2.03	   U13'(tt, V1, V2) -> ISNATKIND(V2)
5.31/2.03	   U14'(tt, V1, V2) -> U15'(isNat(V1), V2)
5.31/2.03	   U14'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	   U15'(tt, V2) -> U16'(isNat(V2))
5.31/2.03	   U15'(tt, V2) -> ISNAT(V2)
5.31/2.03	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
5.31/2.03	   U21'(tt, V1) -> ISNATKIND(V1)
5.31/2.03	   U22'(tt, V1) -> U23'(isNat(V1))
5.31/2.03	   U22'(tt, V1) -> ISNAT(V1)
5.31/2.03	   U31'(tt, V1, V2) -> U32'(isNatKind(V1), V1, V2)
5.31/2.03	   U31'(tt, V1, V2) -> ISNATKIND(V1)
5.31/2.03	   U32'(tt, V1, V2) -> U33'(isNatKind(V2), V1, V2)
5.31/2.03	   U32'(tt, V1, V2) -> ISNATKIND(V2)
5.31/2.03	   U33'(tt, V1, V2) -> U34'(isNatKind(V2), V1, V2)
5.31/2.03	   U33'(tt, V1, V2) -> ISNATKIND(V2)
5.31/2.03	   U34'(tt, V1, V2) -> U35'(isNat(V1), V2)
5.31/2.03	   U34'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	   U35'(tt, V2) -> U36'(isNat(V2))
5.31/2.03	   U35'(tt, V2) -> ISNAT(V2)
5.31/2.03	   U41'(tt, V2) -> U42'(isNatKind(V2))
5.31/2.03	   U41'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   U61'(tt, V2) -> U62'(isNatKind(V2))
5.31/2.03	   U61'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   U71'(tt, N) -> U72'(isNatKind(N), N)
5.31/2.03	   U71'(tt, N) -> ISNATKIND(N)
5.31/2.03	   U81'(tt, M, N) -> U82'(isNatKind(M), M, N)
5.31/2.03	   U81'(tt, M, N) -> ISNATKIND(M)
5.31/2.03	   U82'(tt, M, N) -> U83'(isNat(N), M, N)
5.31/2.03	   U82'(tt, M, N) -> ISNAT(N)
5.31/2.03	   U83'(tt, M, N) -> U84'(isNatKind(N), M, N)
5.31/2.03	   U83'(tt, M, N) -> ISNATKIND(N)
5.31/2.03	   U84'(tt, M, N) -> PLUS(N, M)
5.31/2.03	   U91'(tt, N) -> U92'(isNatKind(N))
5.31/2.03	   U91'(tt, N) -> ISNATKIND(N)
5.31/2.03	   ISNAT(plus(V1, V2)) -> U11'(isNatKind(V1), V1, V2)
5.31/2.03	   ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
5.31/2.03	   ISNAT(s(V1)) -> ISNATKIND(V1)
5.31/2.03	   ISNAT(x(V1, V2)) -> U31'(isNatKind(V1), V1, V2)
5.31/2.03	   ISNAT(x(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> U41'(isNatKind(V1), V2)
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(s(V1)) -> U51'(isNatKind(V1))
5.31/2.03	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> U61'(isNatKind(V1), V2)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   PLUS(N, 0) -> U71'(isNat(N), N)
5.31/2.03	   PLUS(N, 0) -> ISNAT(N)
5.31/2.03	   PLUS(N, s(M)) -> U81'(isNat(M), M, N)
5.31/2.03	   PLUS(N, s(M)) -> ISNAT(M)
5.31/2.03	   X(N, 0) -> U91'(isNat(N), N)
5.31/2.03	   X(N, 0) -> ISNAT(N)
5.31/2.03	   X(N, s(M)) -> U101'(isNat(M), M, N)
5.31/2.03	   X(N, s(M)) -> ISNAT(M)
5.31/2.03	
5.31/2.03	The collapsing dependency pairs are DP_c:
5.31/2.03	   U104'(tt, M, N) -> N
5.31/2.03	   U104'(tt, M, N) -> M
5.31/2.03	   U72'(tt, N) -> N
5.31/2.03	   U84'(tt, M, N) -> N
5.31/2.03	   U84'(tt, M, N) -> M
5.31/2.03	
5.31/2.03	
5.31/2.03	The hidden terms of R are:
5.31/2.03	none
5.31/2.03	
5.31/2.03	Every hiding context is built from:none
5.31/2.03	
5.31/2.03	Hence, the new unhiding pairs DP_u are :
5.31/2.03	   U104'(tt, M, N) -> U(N)
5.31/2.03	   U104'(tt, M, N) -> U(M)
5.31/2.03	   U72'(tt, N) -> U(N)
5.31/2.03	   U84'(tt, M, N) -> U(N)
5.31/2.03	   U84'(tt, M, N) -> U(M)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(3) QCSDependencyGraphProof (EQUIVALENT)
5.31/2.03	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 38 less nodes.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(4)
5.31/2.03	Complex Obligation (AND)
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(5)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U41'_2, U61'_2} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1, ISNATKIND_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U41'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> U41'(isNatKind(V1), V2)
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> U61'(isNatKind(V1), V2)
5.31/2.03	   U61'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(6) QCSDPSubtermProof (EQUIVALENT)
5.31/2.03	We use the subterm processor [DA_EMMES].
5.31/2.03	
5.31/2.03	
5.31/2.03	The following pairs can be oriented strictly and are deleted.
5.31/2.03	
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> U41'(isNatKind(V1), V2)
5.31/2.03	   ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> U61'(isNatKind(V1), V2)
5.31/2.03	   ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
5.31/2.03	The remaining pairs can at least be oriented weakly.
5.31/2.03	
5.31/2.03	   U41'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   U61'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	Used ordering:  Combined order from the following AFS and order.
5.31/2.03	ISNATKIND(x1)  =  x1
5.31/2.03	
5.31/2.03	U41'(x1, x2)  =  x2
5.31/2.03	
5.31/2.03	U61'(x1, x2)  =  x2
5.31/2.03	
5.31/2.03	
5.31/2.03	Subterm Order
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(7)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U41'_2, U61'_2} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1, ISNATKIND_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U41'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	   U61'(tt, V2) -> ISNATKIND(V2)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(8) QCSDependencyGraphProof (EQUIVALENT)
5.31/2.03	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(9)
5.31/2.03	TRUE
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(10)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U12'_3, U11'_3, U13'_3, U14'_3, U15'_2, U21'_2, U22'_2, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1, ISNAT_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U11'(tt, V1, V2) -> U12'(isNatKind(V1), V1, V2)
5.31/2.03	   U12'(tt, V1, V2) -> U13'(isNatKind(V2), V1, V2)
5.31/2.03	   U13'(tt, V1, V2) -> U14'(isNatKind(V2), V1, V2)
5.31/2.03	   U14'(tt, V1, V2) -> U15'(isNat(V1), V2)
5.31/2.03	   U15'(tt, V2) -> ISNAT(V2)
5.31/2.03	   ISNAT(plus(V1, V2)) -> U11'(isNatKind(V1), V1, V2)
5.31/2.03	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
5.31/2.03	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
5.31/2.03	   U22'(tt, V1) -> ISNAT(V1)
5.31/2.03	   ISNAT(x(V1, V2)) -> U31'(isNatKind(V1), V1, V2)
5.31/2.03	   U31'(tt, V1, V2) -> U32'(isNatKind(V1), V1, V2)
5.31/2.03	   U32'(tt, V1, V2) -> U33'(isNatKind(V2), V1, V2)
5.31/2.03	   U33'(tt, V1, V2) -> U34'(isNatKind(V2), V1, V2)
5.31/2.03	   U34'(tt, V1, V2) -> U35'(isNat(V1), V2)
5.31/2.03	   U35'(tt, V2) -> ISNAT(V2)
5.31/2.03	   U34'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	   U14'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(11) QCSUsableRulesProof (EQUIVALENT)
5.31/2.03	The following rules are not useable [DA_EMMES] and can be deleted:
5.31/2.03	
5.31/2.03	   U101(tt, x0, x1) -> U102(isNatKind(x0), x0, x1)
5.31/2.03	   U102(tt, x0, x1) -> U103(isNat(x1), x0, x1)
5.31/2.03	   U103(tt, x0, x1) -> U104(isNatKind(x1), x0, x1)
5.31/2.03	   U104(tt, x0, x1) -> plus(x(x1, x0), x1)
5.31/2.03	   U71(tt, x0) -> U72(isNatKind(x0), x0)
5.31/2.03	   U72(tt, x0) -> x0
5.31/2.03	   U81(tt, x0, x1) -> U82(isNatKind(x0), x0, x1)
5.31/2.03	   U82(tt, x0, x1) -> U83(isNat(x1), x0, x1)
5.31/2.03	   U83(tt, x0, x1) -> U84(isNatKind(x1), x0, x1)
5.31/2.03	   U84(tt, x0, x1) -> s(plus(x1, x0))
5.31/2.03	   U91(tt, x0) -> U92(isNatKind(x0))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   plus(x0, 0) -> U71(isNat(x0), x0)
5.31/2.03	   plus(x0, s(x1)) -> U81(isNat(x1), x1, x0)
5.31/2.03	   x(x0, 0) -> U91(isNat(x0), x0)
5.31/2.03	   x(x0, s(x1)) -> U101(isNat(x1), x1, x0)
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(12)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, s_1, U51_1, x_2, U62_1, U42_1, U23_1, U36_1, U16_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U41_2, U61_2, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U12'_3, U11'_3, U13'_3, U14'_3, U15'_2, U21'_2, U22'_2, U31'_3, U32'_3, U33'_3, U34'_3, U35'_2} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1, ISNAT_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U11'(tt, V1, V2) -> U12'(isNatKind(V1), V1, V2)
5.31/2.03	   U12'(tt, V1, V2) -> U13'(isNatKind(V2), V1, V2)
5.31/2.03	   U13'(tt, V1, V2) -> U14'(isNatKind(V2), V1, V2)
5.31/2.03	   U14'(tt, V1, V2) -> U15'(isNat(V1), V2)
5.31/2.03	   U15'(tt, V2) -> ISNAT(V2)
5.31/2.03	   ISNAT(plus(V1, V2)) -> U11'(isNatKind(V1), V1, V2)
5.31/2.03	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
5.31/2.03	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
5.31/2.03	   U22'(tt, V1) -> ISNAT(V1)
5.31/2.03	   ISNAT(x(V1, V2)) -> U31'(isNatKind(V1), V1, V2)
5.31/2.03	   U31'(tt, V1, V2) -> U32'(isNatKind(V1), V1, V2)
5.31/2.03	   U32'(tt, V1, V2) -> U33'(isNatKind(V2), V1, V2)
5.31/2.03	   U33'(tt, V1, V2) -> U34'(isNatKind(V2), V1, V2)
5.31/2.03	   U34'(tt, V1, V2) -> U35'(isNat(V1), V2)
5.31/2.03	   U35'(tt, V2) -> ISNAT(V2)
5.31/2.03	   U34'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	   U14'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(13) QCSDPMuMonotonicPoloProof (EQUIVALENT)
5.31/2.03	By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
5.31/2.03	
5.31/2.03	Strictly oriented dependency pairs:
5.31/2.03	
5.31/2.03	   U14'(tt, V1, V2) -> U15'(isNat(V1), V2)
5.31/2.03	   U15'(tt, V2) -> ISNAT(V2)
5.31/2.03	   ISNAT(plus(V1, V2)) -> U11'(isNatKind(V1), V1, V2)
5.31/2.03	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
5.31/2.03	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
5.31/2.03	   U22'(tt, V1) -> ISNAT(V1)
5.31/2.03	   ISNAT(x(V1, V2)) -> U31'(isNatKind(V1), V1, V2)
5.31/2.03	   U34'(tt, V1, V2) -> U35'(isNat(V1), V2)
5.31/2.03	   U35'(tt, V2) -> ISNAT(V2)
5.31/2.03	   U34'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	   U14'(tt, V1, V2) -> ISNAT(V1)
5.31/2.03	
5.31/2.03	Strictly oriented rules of the TRS R:
5.31/2.03	
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	
5.31/2.03	Used ordering: POLO with Polynomial interpretation [POLO]:
5.31/2.03	
5.31/2.03	   POL(0) = 2
5.31/2.03	   POL(ISNAT(x_1)) = 2 + 2*x_1
5.31/2.03	   POL(U11(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U11'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U12(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U12'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U13(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U13'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U14(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U14'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U15(x_1, x_2)) = x_1 + 2*x_2
5.31/2.03	   POL(U15'(x_1, x_2)) = 1 + x_1 + 2*x_2
5.31/2.03	   POL(U16(x_1)) = 2*x_1
5.31/2.03	   POL(U21(x_1, x_2)) = x_1 + 2*x_2
5.31/2.03	   POL(U21'(x_1, x_2)) = 2*x_1 + 2*x_2
5.31/2.03	   POL(U22(x_1, x_2)) = x_1 + 2*x_2
5.31/2.03	   POL(U22'(x_1, x_2)) = 1 + x_1 + 2*x_2
5.31/2.03	   POL(U23(x_1)) = 1 + 2*x_1
5.31/2.03	   POL(U31(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U31'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U32(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
5.31/2.03	   POL(U32'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U33(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
5.31/2.03	   POL(U33'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U34(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3
5.31/2.03	   POL(U34'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
5.31/2.03	   POL(U35(x_1, x_2)) = 1 + 2*x_1 + x_2
5.31/2.03	   POL(U35'(x_1, x_2)) = 2*x_1 + 2*x_2
5.31/2.03	   POL(U36(x_1)) = x_1
5.31/2.03	   POL(U41(x_1, x_2)) = x_1
5.31/2.03	   POL(U42(x_1)) = x_1
5.31/2.03	   POL(U51(x_1)) = x_1
5.31/2.03	   POL(U61(x_1, x_2)) = x_1
5.31/2.03	   POL(U62(x_1)) = x_1
5.31/2.03	   POL(isNat(x_1)) = x_1
5.31/2.03	   POL(isNatKind(x_1)) = 2
5.31/2.03	   POL(plus(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
5.31/2.03	   POL(s(x_1)) = 2 + 2*x_1
5.31/2.03	   POL(tt) = 2
5.31/2.03	   POL(x(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
5.31/2.03	
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(14)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, s_1, U51_1, x_2, U62_1, U42_1, U36_1} are replacing on all positions.
5.31/2.03	For all symbols f in {U41_2, U61_2, U11_3, U12_3, U13_3, U14_3, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U12'_3, U11'_3, U13'_3, U14'_3, U32'_3, U31'_3, U33'_3, U34'_3} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U11'(tt, V1, V2) -> U12'(isNatKind(V1), V1, V2)
5.31/2.03	   U12'(tt, V1, V2) -> U13'(isNatKind(V2), V1, V2)
5.31/2.03	   U13'(tt, V1, V2) -> U14'(isNatKind(V2), V1, V2)
5.31/2.03	   U31'(tt, V1, V2) -> U32'(isNatKind(V1), V1, V2)
5.31/2.03	   U32'(tt, V1, V2) -> U33'(isNatKind(V2), V1, V2)
5.31/2.03	   U33'(tt, V1, V2) -> U34'(isNatKind(V2), V1, V2)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U36(tt) -> tt
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(15) QCSDependencyGraphProof (EQUIVALENT)
5.31/2.03	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(16)
5.31/2.03	TRUE
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(17)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1, PLUS_2} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U83'_3, U82'_3, U84'_3, U81'_3} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U82'(tt, M, N) -> U83'(isNat(N), M, N)
5.31/2.03	   U83'(tt, M, N) -> U84'(isNatKind(N), M, N)
5.31/2.03	   U84'(tt, M, N) -> PLUS(N, M)
5.31/2.03	   PLUS(N, s(M)) -> U81'(isNat(M), M, N)
5.31/2.03	   U81'(tt, M, N) -> U82'(isNatKind(M), M, N)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(18) QCSDPSubtermProof (EQUIVALENT)
5.31/2.03	We use the subterm processor [DA_EMMES].
5.31/2.03	
5.31/2.03	
5.31/2.03	The following pairs can be oriented strictly and are deleted.
5.31/2.03	
5.31/2.03	   PLUS(N, s(M)) -> U81'(isNat(M), M, N)
5.31/2.03	The remaining pairs can at least be oriented weakly.
5.31/2.03	
5.31/2.03	   U82'(tt, M, N) -> U83'(isNat(N), M, N)
5.31/2.03	   U83'(tt, M, N) -> U84'(isNatKind(N), M, N)
5.31/2.03	   U84'(tt, M, N) -> PLUS(N, M)
5.31/2.03	   U81'(tt, M, N) -> U82'(isNatKind(M), M, N)
5.31/2.03	Used ordering:  Combined order from the following AFS and order.
5.31/2.03	U83'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	U82'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	U84'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	PLUS(x1, x2)  =  x2
5.31/2.03	
5.31/2.03	U81'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	
5.31/2.03	Subterm Order
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(19)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1, PLUS_2} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U83'_3, U82'_3, U84'_3, U81'_3} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U82'(tt, M, N) -> U83'(isNat(N), M, N)
5.31/2.03	   U83'(tt, M, N) -> U84'(isNatKind(N), M, N)
5.31/2.03	   U84'(tt, M, N) -> PLUS(N, M)
5.31/2.03	   U81'(tt, M, N) -> U82'(isNatKind(M), M, N)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(20) QCSDependencyGraphProof (EQUIVALENT)
5.31/2.03	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 4 less nodes.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(21)
5.31/2.03	TRUE
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(22)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1, X_2} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U103'_3, U102'_3, U104'_3, U101'_3} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U102'(tt, M, N) -> U103'(isNat(N), M, N)
5.31/2.03	   U103'(tt, M, N) -> U104'(isNatKind(N), M, N)
5.31/2.03	   U104'(tt, M, N) -> X(N, M)
5.31/2.03	   X(N, s(M)) -> U101'(isNat(M), M, N)
5.31/2.03	   U101'(tt, M, N) -> U102'(isNatKind(M), M, N)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(23) QCSDPSubtermProof (EQUIVALENT)
5.31/2.03	We use the subterm processor [DA_EMMES].
5.31/2.03	
5.31/2.03	
5.31/2.03	The following pairs can be oriented strictly and are deleted.
5.31/2.03	
5.31/2.03	   X(N, s(M)) -> U101'(isNat(M), M, N)
5.31/2.03	The remaining pairs can at least be oriented weakly.
5.31/2.03	
5.31/2.03	   U102'(tt, M, N) -> U103'(isNat(N), M, N)
5.31/2.03	   U103'(tt, M, N) -> U104'(isNatKind(N), M, N)
5.31/2.03	   U104'(tt, M, N) -> X(N, M)
5.31/2.03	   U101'(tt, M, N) -> U102'(isNatKind(M), M, N)
5.31/2.03	Used ordering:  Combined order from the following AFS and order.
5.31/2.03	U103'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	U102'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	U104'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	X(x1, x2)  =  x2
5.31/2.03	
5.31/2.03	U101'(x1, x2, x3)  =  x2
5.31/2.03	
5.31/2.03	
5.31/2.03	Subterm Order
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(24)
5.31/2.03	Obligation:
5.31/2.03	Q-restricted context-sensitive dependency pair problem:
5.31/2.03	The symbols in {plus_2, x_2, U16_1, U23_1, U36_1, U42_1, U51_1, U62_1, s_1, U92_1, X_2} are replacing on all positions.
5.31/2.03	For all symbols f in {U101_3, U102_3, U103_3, U104_3, U11_3, U12_3, U13_3, U14_3, U15_2, U21_2, U22_2, U31_3, U32_3, U33_3, U34_3, U35_2, U41_2, U61_2, U71_2, U72_2, U81_3, U82_3, U83_3, U84_3, U91_2, U103'_3, U102'_3, U104'_3, U101'_3} we have mu(f) = {1}.
5.31/2.03	The symbols in {isNatKind_1, isNat_1} are not replacing on any position.
5.31/2.03	
5.31/2.03	The TRS P consists of the following rules:
5.31/2.03	
5.31/2.03	   U102'(tt, M, N) -> U103'(isNat(N), M, N)
5.31/2.03	   U103'(tt, M, N) -> U104'(isNatKind(N), M, N)
5.31/2.03	   U104'(tt, M, N) -> X(N, M)
5.31/2.03	   U101'(tt, M, N) -> U102'(isNatKind(M), M, N)
5.31/2.03	
5.31/2.03	The TRS R consists of the following rules:
5.31/2.03	
5.31/2.03	   U101(tt, M, N) -> U102(isNatKind(M), M, N)
5.31/2.03	   U102(tt, M, N) -> U103(isNat(N), M, N)
5.31/2.03	   U103(tt, M, N) -> U104(isNatKind(N), M, N)
5.31/2.03	   U104(tt, M, N) -> plus(x(N, M), N)
5.31/2.03	   U11(tt, V1, V2) -> U12(isNatKind(V1), V1, V2)
5.31/2.03	   U12(tt, V1, V2) -> U13(isNatKind(V2), V1, V2)
5.31/2.03	   U13(tt, V1, V2) -> U14(isNatKind(V2), V1, V2)
5.31/2.03	   U14(tt, V1, V2) -> U15(isNat(V1), V2)
5.31/2.03	   U15(tt, V2) -> U16(isNat(V2))
5.31/2.03	   U16(tt) -> tt
5.31/2.03	   U21(tt, V1) -> U22(isNatKind(V1), V1)
5.31/2.03	   U22(tt, V1) -> U23(isNat(V1))
5.31/2.03	   U23(tt) -> tt
5.31/2.03	   U31(tt, V1, V2) -> U32(isNatKind(V1), V1, V2)
5.31/2.03	   U32(tt, V1, V2) -> U33(isNatKind(V2), V1, V2)
5.31/2.03	   U33(tt, V1, V2) -> U34(isNatKind(V2), V1, V2)
5.31/2.03	   U34(tt, V1, V2) -> U35(isNat(V1), V2)
5.31/2.03	   U35(tt, V2) -> U36(isNat(V2))
5.31/2.03	   U36(tt) -> tt
5.31/2.03	   U41(tt, V2) -> U42(isNatKind(V2))
5.31/2.03	   U42(tt) -> tt
5.31/2.03	   U51(tt) -> tt
5.31/2.03	   U61(tt, V2) -> U62(isNatKind(V2))
5.31/2.03	   U62(tt) -> tt
5.31/2.03	   U71(tt, N) -> U72(isNatKind(N), N)
5.31/2.03	   U72(tt, N) -> N
5.31/2.03	   U81(tt, M, N) -> U82(isNatKind(M), M, N)
5.31/2.03	   U82(tt, M, N) -> U83(isNat(N), M, N)
5.31/2.03	   U83(tt, M, N) -> U84(isNatKind(N), M, N)
5.31/2.03	   U84(tt, M, N) -> s(plus(N, M))
5.31/2.03	   U91(tt, N) -> U92(isNatKind(N))
5.31/2.03	   U92(tt) -> 0
5.31/2.03	   isNat(0) -> tt
5.31/2.03	   isNat(plus(V1, V2)) -> U11(isNatKind(V1), V1, V2)
5.31/2.03	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
5.31/2.03	   isNat(x(V1, V2)) -> U31(isNatKind(V1), V1, V2)
5.31/2.03	   isNatKind(0) -> tt
5.31/2.03	   isNatKind(plus(V1, V2)) -> U41(isNatKind(V1), V2)
5.31/2.03	   isNatKind(s(V1)) -> U51(isNatKind(V1))
5.31/2.03	   isNatKind(x(V1, V2)) -> U61(isNatKind(V1), V2)
5.31/2.03	   plus(N, 0) -> U71(isNat(N), N)
5.31/2.03	   plus(N, s(M)) -> U81(isNat(M), M, N)
5.31/2.03	   x(N, 0) -> U91(isNat(N), N)
5.31/2.03	   x(N, s(M)) -> U101(isNat(M), M, N)
5.31/2.03	
5.31/2.03	Q is empty.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(25) QCSDependencyGraphProof (EQUIVALENT)
5.31/2.03	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 4 less nodes.
5.31/2.03	
5.31/2.03	----------------------------------------
5.31/2.03	
5.31/2.03	(26)
5.31/2.03	TRUE
5.53/2.11	EOF