34.79/10.06	WORST_CASE(Omega(n^1), O(n^1))
34.79/10.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.79/10.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.79/10.08	
34.79/10.08	
34.79/10.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
34.79/10.08	
34.79/10.08	(0) CpxTRS
34.79/10.08	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
34.79/10.08	(2) CpxTRS
34.79/10.08	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
34.79/10.08	    (4) CpxTRS
34.79/10.08	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 568 ms]
34.79/10.08	    (6) BOUNDS(1, n^1)
34.79/10.08	(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
34.79/10.08	(8) CpxTRS
34.79/10.08	    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
34.79/10.08	    (10) typed CpxTrs
34.79/10.08	    (11) OrderProof [LOWER BOUND(ID), 9 ms]
34.79/10.08	    (12) typed CpxTrs
34.79/10.08	    (13) RewriteLemmaProof [LOWER BOUND(ID), 496 ms]
34.79/10.08	    (14) BEST
34.79/10.08	        (15) proven lower bound
34.79/10.08	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
34.79/10.08	            (17) BOUNDS(n^1, INF)
34.79/10.08	        (18) typed CpxTrs
34.79/10.08	            (19) RewriteLemmaProof [LOWER BOUND(ID), 142 ms]
34.79/10.08	            (20) typed CpxTrs
34.79/10.08	            (21) RewriteLemmaProof [LOWER BOUND(ID), 161 ms]
34.79/10.08	            (22) typed CpxTrs
34.79/10.08	            (23) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
34.79/10.08	            (24) typed CpxTrs
34.79/10.08	            (25) RewriteLemmaProof [LOWER BOUND(ID), 95 ms]
34.79/10.08	            (26) typed CpxTrs
34.79/10.08	            (27) RewriteLemmaProof [LOWER BOUND(ID), 146 ms]
34.79/10.08	            (28) typed CpxTrs
34.79/10.08	            (29) RewriteLemmaProof [LOWER BOUND(ID), 65 ms]
34.79/10.08	            (30) typed CpxTrs
34.79/10.08	            (31) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
34.79/10.08	            (32) typed CpxTrs
34.79/10.08	            (33) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
34.79/10.08	            (34) typed CpxTrs
34.79/10.08	            (35) RewriteLemmaProof [LOWER BOUND(ID), 196 ms]
34.79/10.08	            (36) typed CpxTrs
34.79/10.08	            (37) RewriteLemmaProof [LOWER BOUND(ID), 195 ms]
34.79/10.08	            (38) typed CpxTrs
34.79/10.08	            (39) RewriteLemmaProof [LOWER BOUND(ID), 111 ms]
34.79/10.08	            (40) typed CpxTrs
34.79/10.08	            (41) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
34.79/10.08	            (42) typed CpxTrs
34.79/10.08	            (43) RewriteLemmaProof [LOWER BOUND(ID), 119 ms]
34.79/10.08	            (44) typed CpxTrs
34.79/10.08	            (45) RewriteLemmaProof [LOWER BOUND(ID), 165 ms]
34.79/10.08	            (46) typed CpxTrs
34.79/10.08	            (47) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
34.79/10.08	            (48) typed CpxTrs
34.79/10.08	            (49) RewriteLemmaProof [LOWER BOUND(ID), 87 ms]
34.79/10.08	            (50) typed CpxTrs
34.79/10.08	            (51) RewriteLemmaProof [LOWER BOUND(ID), 15 ms]
34.79/10.08	            (52) typed CpxTrs
34.79/10.08	            (53) RewriteLemmaProof [LOWER BOUND(ID), 130 ms]
34.79/10.08	            (54) typed CpxTrs
34.79/10.08	            (55) RewriteLemmaProof [LOWER BOUND(ID), 153 ms]
34.79/10.08	            (56) typed CpxTrs
34.79/10.08	
34.79/10.08	
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(0)
34.79/10.08	Obligation:
34.79/10.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
34.79/10.08	
34.79/10.08	
34.79/10.08	The TRS R consists of the following rules:
34.79/10.08	
34.79/10.08	   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
34.79/10.08	   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
34.79/10.08	   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
34.79/10.08	   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
34.79/10.08	   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
34.79/10.08	   active(U16(tt)) -> mark(tt)
34.79/10.08	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
34.79/10.08	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
34.79/10.08	   active(U23(tt)) -> mark(tt)
34.79/10.08	   active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
34.79/10.08	   active(U32(tt)) -> mark(tt)
34.79/10.08	   active(U41(tt)) -> mark(tt)
34.79/10.08	   active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
34.79/10.08	   active(U52(tt, N)) -> mark(N)
34.79/10.08	   active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
34.79/10.08	   active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
34.79/10.08	   active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
34.79/10.08	   active(U64(tt, M, N)) -> mark(s(plus(N, M)))
34.79/10.08	   active(isNat(0)) -> mark(tt)
34.79/10.08	   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
34.79/10.08	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
34.79/10.08	   active(isNatKind(0)) -> mark(tt)
34.79/10.08	   active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
34.79/10.08	   active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
34.79/10.08	   active(plus(N, 0)) -> mark(U51(isNat(N), N))
34.79/10.08	   active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
34.79/10.08	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
34.79/10.08	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
34.79/10.08	   active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
34.79/10.08	   active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
34.79/10.08	   active(U15(X1, X2)) -> U15(active(X1), X2)
34.79/10.08	   active(U16(X)) -> U16(active(X))
34.79/10.08	   active(U21(X1, X2)) -> U21(active(X1), X2)
34.79/10.08	   active(U22(X1, X2)) -> U22(active(X1), X2)
34.79/10.08	   active(U23(X)) -> U23(active(X))
34.79/10.08	   active(U31(X1, X2)) -> U31(active(X1), X2)
34.79/10.08	   active(U32(X)) -> U32(active(X))
34.79/10.08	   active(U41(X)) -> U41(active(X))
34.79/10.08	   active(U51(X1, X2)) -> U51(active(X1), X2)
34.79/10.08	   active(U52(X1, X2)) -> U52(active(X1), X2)
34.79/10.08	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
34.79/10.08	   active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
34.79/10.08	   active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
34.79/10.08	   active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
34.79/10.08	   active(s(X)) -> s(active(X))
34.79/10.08	   active(plus(X1, X2)) -> plus(active(X1), X2)
34.79/10.08	   active(plus(X1, X2)) -> plus(X1, active(X2))
34.79/10.08	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
34.79/10.08	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
34.79/10.08	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
34.79/10.08	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
34.79/10.08	   U15(mark(X1), X2) -> mark(U15(X1, X2))
34.79/10.08	   U16(mark(X)) -> mark(U16(X))
34.79/10.08	   U21(mark(X1), X2) -> mark(U21(X1, X2))
34.79/10.08	   U22(mark(X1), X2) -> mark(U22(X1, X2))
34.79/10.08	   U23(mark(X)) -> mark(U23(X))
34.79/10.08	   U31(mark(X1), X2) -> mark(U31(X1, X2))
34.79/10.08	   U32(mark(X)) -> mark(U32(X))
34.79/10.08	   U41(mark(X)) -> mark(U41(X))
34.79/10.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
34.79/10.08	   U52(mark(X1), X2) -> mark(U52(X1, X2))
34.79/10.08	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
34.79/10.08	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
34.79/10.08	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
34.79/10.08	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
34.79/10.08	   s(mark(X)) -> mark(s(X))
34.79/10.08	   plus(mark(X1), X2) -> mark(plus(X1, X2))
34.79/10.08	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
34.79/10.08	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(tt) -> ok(tt)
34.79/10.08	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(isNatKind(X)) -> isNatKind(proper(X))
34.79/10.08	   proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
34.79/10.08	   proper(isNat(X)) -> isNat(proper(X))
34.79/10.08	   proper(U16(X)) -> U16(proper(X))
34.79/10.08	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
34.79/10.08	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
34.79/10.08	   proper(U23(X)) -> U23(proper(X))
34.79/10.08	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
34.79/10.08	   proper(U32(X)) -> U32(proper(X))
34.79/10.08	   proper(U41(X)) -> U41(proper(X))
34.79/10.08	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
34.79/10.08	   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
34.79/10.08	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(s(X)) -> s(proper(X))
34.79/10.08	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
34.79/10.08	   proper(0) -> ok(0)
34.79/10.08	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
34.79/10.08	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
34.79/10.08	   isNatKind(ok(X)) -> ok(isNatKind(X))
34.79/10.08	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
34.79/10.08	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
34.79/10.08	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
34.79/10.08	   isNat(ok(X)) -> ok(isNat(X))
34.79/10.08	   U16(ok(X)) -> ok(U16(X))
34.79/10.08	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
34.79/10.08	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
34.79/10.08	   U23(ok(X)) -> ok(U23(X))
34.79/10.08	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
34.79/10.08	   U32(ok(X)) -> ok(U32(X))
34.79/10.08	   U41(ok(X)) -> ok(U41(X))
34.79/10.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
34.79/10.08	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
34.79/10.08	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
34.79/10.08	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
34.79/10.08	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
34.79/10.08	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
34.79/10.08	   s(ok(X)) -> ok(s(X))
34.79/10.08	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
34.79/10.08	   top(mark(X)) -> top(proper(X))
34.79/10.08	   top(ok(X)) -> top(active(X))
34.79/10.08	
34.79/10.08	S is empty.
34.79/10.08	Rewrite Strategy: FULL
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
34.79/10.08	The following defined symbols can occur below the 0th argument of top: proper, active
34.79/10.08	The following defined symbols can occur below the 0th argument of proper: proper, active
34.79/10.08	The following defined symbols can occur below the 0th argument of active: proper, active
34.79/10.08	
34.79/10.08	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
34.79/10.08	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
34.79/10.08	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
34.79/10.08	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
34.79/10.08	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
34.79/10.08	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
34.79/10.08	active(U16(tt)) -> mark(tt)
34.79/10.08	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
34.79/10.08	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
34.79/10.08	active(U23(tt)) -> mark(tt)
34.79/10.08	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
34.79/10.08	active(U32(tt)) -> mark(tt)
34.79/10.08	active(U41(tt)) -> mark(tt)
34.79/10.08	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
34.79/10.08	active(U52(tt, N)) -> mark(N)
34.79/10.08	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
34.79/10.08	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
34.79/10.08	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
34.79/10.08	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
34.79/10.08	active(isNat(0)) -> mark(tt)
34.79/10.08	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
34.79/10.08	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
34.79/10.08	active(isNatKind(0)) -> mark(tt)
34.79/10.08	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
34.79/10.08	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
34.79/10.08	active(plus(N, 0)) -> mark(U51(isNat(N), N))
34.79/10.08	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
34.79/10.08	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
34.79/10.08	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
34.79/10.08	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
34.79/10.08	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
34.79/10.08	active(U15(X1, X2)) -> U15(active(X1), X2)
34.79/10.08	active(U16(X)) -> U16(active(X))
34.79/10.08	active(U21(X1, X2)) -> U21(active(X1), X2)
34.79/10.08	active(U22(X1, X2)) -> U22(active(X1), X2)
34.79/10.08	active(U23(X)) -> U23(active(X))
34.79/10.08	active(U31(X1, X2)) -> U31(active(X1), X2)
34.79/10.08	active(U32(X)) -> U32(active(X))
34.79/10.08	active(U41(X)) -> U41(active(X))
34.79/10.08	active(U51(X1, X2)) -> U51(active(X1), X2)
34.79/10.08	active(U52(X1, X2)) -> U52(active(X1), X2)
34.79/10.08	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
34.79/10.08	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
34.79/10.08	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
34.79/10.08	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
34.79/10.08	active(s(X)) -> s(active(X))
34.79/10.08	active(plus(X1, X2)) -> plus(active(X1), X2)
34.79/10.08	active(plus(X1, X2)) -> plus(X1, active(X2))
34.79/10.08	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(isNatKind(X)) -> isNatKind(proper(X))
34.79/10.08	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
34.79/10.08	proper(isNat(X)) -> isNat(proper(X))
34.79/10.08	proper(U16(X)) -> U16(proper(X))
34.79/10.08	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
34.79/10.08	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
34.79/10.08	proper(U23(X)) -> U23(proper(X))
34.79/10.08	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
34.79/10.08	proper(U32(X)) -> U32(proper(X))
34.79/10.08	proper(U41(X)) -> U41(proper(X))
34.79/10.08	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
34.79/10.08	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
34.79/10.08	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(s(X)) -> s(proper(X))
34.79/10.08	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
34.79/10.08	
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(2)
34.79/10.08	Obligation:
34.79/10.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
34.79/10.08	
34.79/10.08	
34.79/10.08	The TRS R consists of the following rules:
34.79/10.08	
34.79/10.08	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
34.79/10.08	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
34.79/10.08	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
34.79/10.08	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
34.79/10.08	   U15(mark(X1), X2) -> mark(U15(X1, X2))
34.79/10.08	   U16(mark(X)) -> mark(U16(X))
34.79/10.08	   U21(mark(X1), X2) -> mark(U21(X1, X2))
34.79/10.08	   U22(mark(X1), X2) -> mark(U22(X1, X2))
34.79/10.08	   U23(mark(X)) -> mark(U23(X))
34.79/10.08	   U31(mark(X1), X2) -> mark(U31(X1, X2))
34.79/10.08	   U32(mark(X)) -> mark(U32(X))
34.79/10.08	   U41(mark(X)) -> mark(U41(X))
34.79/10.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
34.79/10.08	   U52(mark(X1), X2) -> mark(U52(X1, X2))
34.79/10.08	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
34.79/10.08	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
34.79/10.08	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
34.79/10.08	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
34.79/10.08	   s(mark(X)) -> mark(s(X))
34.79/10.08	   plus(mark(X1), X2) -> mark(plus(X1, X2))
34.79/10.08	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
34.79/10.08	   proper(tt) -> ok(tt)
34.79/10.08	   proper(0) -> ok(0)
34.79/10.08	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
34.79/10.08	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
34.79/10.08	   isNatKind(ok(X)) -> ok(isNatKind(X))
34.79/10.08	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
34.79/10.08	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
34.79/10.08	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
34.79/10.08	   isNat(ok(X)) -> ok(isNat(X))
34.79/10.08	   U16(ok(X)) -> ok(U16(X))
34.79/10.08	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
34.79/10.08	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
34.79/10.08	   U23(ok(X)) -> ok(U23(X))
34.79/10.08	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
34.79/10.08	   U32(ok(X)) -> ok(U32(X))
34.79/10.08	   U41(ok(X)) -> ok(U41(X))
34.79/10.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
34.79/10.08	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
34.79/10.08	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
34.79/10.08	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
34.79/10.08	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
34.79/10.08	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
34.79/10.08	   s(ok(X)) -> ok(s(X))
34.79/10.08	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
34.79/10.08	   top(mark(X)) -> top(proper(X))
34.79/10.08	   top(ok(X)) -> top(active(X))
34.79/10.08	
34.79/10.08	S is empty.
34.79/10.08	Rewrite Strategy: FULL
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(3) RelTrsToTrsProof (UPPER BOUND(ID))
34.79/10.08	transformed relative TRS to TRS
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(4)
34.79/10.08	Obligation:
34.79/10.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
34.79/10.08	
34.79/10.08	
34.79/10.08	The TRS R consists of the following rules:
34.79/10.08	
34.79/10.08	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
34.79/10.08	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
34.79/10.08	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
34.79/10.08	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
34.79/10.08	   U15(mark(X1), X2) -> mark(U15(X1, X2))
34.79/10.08	   U16(mark(X)) -> mark(U16(X))
34.79/10.08	   U21(mark(X1), X2) -> mark(U21(X1, X2))
34.79/10.08	   U22(mark(X1), X2) -> mark(U22(X1, X2))
34.79/10.08	   U23(mark(X)) -> mark(U23(X))
34.79/10.08	   U31(mark(X1), X2) -> mark(U31(X1, X2))
34.79/10.08	   U32(mark(X)) -> mark(U32(X))
34.79/10.08	   U41(mark(X)) -> mark(U41(X))
34.79/10.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
34.79/10.08	   U52(mark(X1), X2) -> mark(U52(X1, X2))
34.79/10.08	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
34.79/10.08	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
34.79/10.08	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
34.79/10.08	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
34.79/10.08	   s(mark(X)) -> mark(s(X))
34.79/10.08	   plus(mark(X1), X2) -> mark(plus(X1, X2))
34.79/10.08	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
34.79/10.08	   proper(tt) -> ok(tt)
34.79/10.08	   proper(0) -> ok(0)
34.79/10.08	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
34.79/10.08	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
34.79/10.08	   isNatKind(ok(X)) -> ok(isNatKind(X))
34.79/10.08	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
34.79/10.08	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
34.79/10.08	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
34.79/10.08	   isNat(ok(X)) -> ok(isNat(X))
34.79/10.08	   U16(ok(X)) -> ok(U16(X))
34.79/10.08	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
34.79/10.08	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
34.79/10.08	   U23(ok(X)) -> ok(U23(X))
34.79/10.08	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
34.79/10.08	   U32(ok(X)) -> ok(U32(X))
34.79/10.08	   U41(ok(X)) -> ok(U41(X))
34.79/10.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
34.79/10.08	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
34.79/10.08	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
34.79/10.08	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
34.79/10.08	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
34.79/10.08	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
34.79/10.08	   s(ok(X)) -> ok(s(X))
34.79/10.08	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
34.79/10.08	   top(mark(X)) -> top(proper(X))
34.79/10.08	   top(ok(X)) -> top(active(X))
34.79/10.08	
34.79/10.08	S is empty.
34.79/10.08	Rewrite Strategy: FULL
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(5) CpxTrsMatchBoundsTAProof (FINISHED)
34.79/10.08	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 
34.79/10.08	
34.79/10.08	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
34.79/10.08	final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
34.79/10.08	transitions: 
34.79/10.08	mark0(0) -> 0
34.79/10.08	tt0() -> 0
34.79/10.08	ok0(0) -> 0
34.79/10.08	00() -> 0
34.79/10.08	active0(0) -> 0
34.79/10.08	U110(0, 0, 0) -> 1
34.79/10.08	U120(0, 0, 0) -> 2
34.79/10.08	U130(0, 0, 0) -> 3
34.79/10.08	U140(0, 0, 0) -> 4
34.79/10.08	U150(0, 0) -> 5
34.79/10.08	U160(0) -> 6
34.79/10.08	U210(0, 0) -> 7
34.79/10.08	U220(0, 0) -> 8
34.79/10.08	U230(0) -> 9
34.79/10.08	U310(0, 0) -> 10
34.79/10.08	U320(0) -> 11
34.79/10.08	U410(0) -> 12
34.79/10.08	U510(0, 0) -> 13
34.79/10.08	U520(0, 0) -> 14
34.79/10.08	U610(0, 0, 0) -> 15
34.79/10.08	U620(0, 0, 0) -> 16
34.79/10.08	U630(0, 0, 0) -> 17
34.79/10.08	U640(0, 0, 0) -> 18
34.79/10.08	s0(0) -> 19
34.79/10.08	plus0(0, 0) -> 20
34.79/10.08	proper0(0) -> 21
34.79/10.08	isNatKind0(0) -> 22
34.79/10.08	isNat0(0) -> 23
34.79/10.08	top0(0) -> 24
34.79/10.08	U111(0, 0, 0) -> 25
34.79/10.08	mark1(25) -> 1
34.79/10.08	U121(0, 0, 0) -> 26
34.79/10.08	mark1(26) -> 2
34.79/10.08	U131(0, 0, 0) -> 27
34.79/10.08	mark1(27) -> 3
34.79/10.08	U141(0, 0, 0) -> 28
34.79/10.08	mark1(28) -> 4
34.79/10.08	U151(0, 0) -> 29
34.79/10.08	mark1(29) -> 5
34.79/10.08	U161(0) -> 30
34.79/10.08	mark1(30) -> 6
34.79/10.08	U211(0, 0) -> 31
34.79/10.08	mark1(31) -> 7
34.79/10.08	U221(0, 0) -> 32
34.79/10.08	mark1(32) -> 8
34.79/10.08	U231(0) -> 33
34.79/10.08	mark1(33) -> 9
34.79/10.08	U311(0, 0) -> 34
34.79/10.08	mark1(34) -> 10
34.79/10.08	U321(0) -> 35
34.79/10.08	mark1(35) -> 11
34.79/10.08	U411(0) -> 36
34.79/10.08	mark1(36) -> 12
34.79/10.08	U511(0, 0) -> 37
34.79/10.08	mark1(37) -> 13
34.79/10.08	U521(0, 0) -> 38
34.79/10.08	mark1(38) -> 14
34.79/10.08	U611(0, 0, 0) -> 39
34.79/10.08	mark1(39) -> 15
34.79/10.08	U621(0, 0, 0) -> 40
34.79/10.08	mark1(40) -> 16
34.79/10.08	U631(0, 0, 0) -> 41
34.79/10.08	mark1(41) -> 17
34.79/10.08	U641(0, 0, 0) -> 42
34.79/10.08	mark1(42) -> 18
34.79/10.08	s1(0) -> 43
34.79/10.08	mark1(43) -> 19
34.79/10.08	plus1(0, 0) -> 44
34.79/10.08	mark1(44) -> 20
34.79/10.08	tt1() -> 45
34.79/10.08	ok1(45) -> 21
34.79/10.08	01() -> 46
34.79/10.08	ok1(46) -> 21
34.79/10.08	U111(0, 0, 0) -> 47
34.79/10.08	ok1(47) -> 1
34.79/10.08	U121(0, 0, 0) -> 48
34.79/10.08	ok1(48) -> 2
34.79/10.08	isNatKind1(0) -> 49
34.79/10.08	ok1(49) -> 22
34.79/10.08	U131(0, 0, 0) -> 50
34.79/10.08	ok1(50) -> 3
34.79/10.08	U141(0, 0, 0) -> 51
34.79/10.08	ok1(51) -> 4
34.79/10.08	U151(0, 0) -> 52
34.79/10.08	ok1(52) -> 5
34.79/10.08	isNat1(0) -> 53
34.79/10.08	ok1(53) -> 23
34.79/10.08	U161(0) -> 54
34.79/10.08	ok1(54) -> 6
34.79/10.08	U211(0, 0) -> 55
34.79/10.08	ok1(55) -> 7
34.79/10.08	U221(0, 0) -> 56
34.79/10.08	ok1(56) -> 8
34.79/10.08	U231(0) -> 57
34.79/10.08	ok1(57) -> 9
34.79/10.08	U311(0, 0) -> 58
34.79/10.08	ok1(58) -> 10
34.79/10.08	U321(0) -> 59
34.79/10.08	ok1(59) -> 11
34.79/10.08	U411(0) -> 60
34.79/10.08	ok1(60) -> 12
34.79/10.08	U511(0, 0) -> 61
34.79/10.08	ok1(61) -> 13
34.79/10.08	U521(0, 0) -> 62
34.79/10.08	ok1(62) -> 14
34.79/10.08	U611(0, 0, 0) -> 63
34.79/10.08	ok1(63) -> 15
34.79/10.08	U621(0, 0, 0) -> 64
34.79/10.08	ok1(64) -> 16
34.79/10.08	U631(0, 0, 0) -> 65
34.79/10.08	ok1(65) -> 17
34.79/10.08	U641(0, 0, 0) -> 66
34.79/10.08	ok1(66) -> 18
34.79/10.08	s1(0) -> 67
34.79/10.08	ok1(67) -> 19
34.79/10.08	plus1(0, 0) -> 68
34.79/10.08	ok1(68) -> 20
34.79/10.08	proper1(0) -> 69
34.79/10.08	top1(69) -> 24
34.79/10.08	active1(0) -> 70
34.79/10.08	top1(70) -> 24
34.79/10.08	mark1(25) -> 25
34.79/10.08	mark1(25) -> 47
34.79/10.08	mark1(26) -> 26
34.79/10.08	mark1(26) -> 48
34.79/10.08	mark1(27) -> 27
34.79/10.08	mark1(27) -> 50
34.79/10.08	mark1(28) -> 28
34.79/10.08	mark1(28) -> 51
34.79/10.08	mark1(29) -> 29
34.79/10.08	mark1(29) -> 52
34.79/10.08	mark1(30) -> 30
34.79/10.08	mark1(30) -> 54
34.79/10.08	mark1(31) -> 31
34.79/10.08	mark1(31) -> 55
34.79/10.08	mark1(32) -> 32
34.79/10.08	mark1(32) -> 56
34.79/10.08	mark1(33) -> 33
34.79/10.08	mark1(33) -> 57
34.79/10.08	mark1(34) -> 34
34.79/10.08	mark1(34) -> 58
34.79/10.08	mark1(35) -> 35
34.79/10.08	mark1(35) -> 59
34.79/10.08	mark1(36) -> 36
34.79/10.08	mark1(36) -> 60
34.79/10.08	mark1(37) -> 37
34.79/10.08	mark1(37) -> 61
34.79/10.08	mark1(38) -> 38
34.79/10.08	mark1(38) -> 62
34.79/10.08	mark1(39) -> 39
34.79/10.08	mark1(39) -> 63
34.79/10.08	mark1(40) -> 40
34.79/10.08	mark1(40) -> 64
34.79/10.08	mark1(41) -> 41
34.79/10.08	mark1(41) -> 65
34.79/10.08	mark1(42) -> 42
34.79/10.08	mark1(42) -> 66
34.79/10.08	mark1(43) -> 43
34.79/10.08	mark1(43) -> 67
34.79/10.08	mark1(44) -> 44
34.79/10.08	mark1(44) -> 68
34.79/10.08	ok1(45) -> 69
34.79/10.08	ok1(46) -> 69
34.79/10.08	ok1(47) -> 25
34.79/10.08	ok1(47) -> 47
34.79/10.08	ok1(48) -> 26
34.79/10.08	ok1(48) -> 48
34.79/10.08	ok1(49) -> 49
34.79/10.08	ok1(50) -> 27
34.79/10.08	ok1(50) -> 50
34.79/10.08	ok1(51) -> 28
34.79/10.08	ok1(51) -> 51
34.79/10.08	ok1(52) -> 29
34.79/10.08	ok1(52) -> 52
34.79/10.08	ok1(53) -> 53
34.79/10.08	ok1(54) -> 30
34.79/10.08	ok1(54) -> 54
34.79/10.08	ok1(55) -> 31
34.79/10.08	ok1(55) -> 55
34.79/10.08	ok1(56) -> 32
34.79/10.08	ok1(56) -> 56
34.79/10.08	ok1(57) -> 33
34.79/10.08	ok1(57) -> 57
34.79/10.08	ok1(58) -> 34
34.79/10.08	ok1(58) -> 58
34.79/10.08	ok1(59) -> 35
34.79/10.08	ok1(59) -> 59
34.79/10.08	ok1(60) -> 36
34.79/10.08	ok1(60) -> 60
34.79/10.08	ok1(61) -> 37
34.79/10.08	ok1(61) -> 61
34.79/10.08	ok1(62) -> 38
34.79/10.08	ok1(62) -> 62
34.79/10.08	ok1(63) -> 39
34.79/10.08	ok1(63) -> 63
34.79/10.08	ok1(64) -> 40
34.79/10.08	ok1(64) -> 64
34.79/10.08	ok1(65) -> 41
34.79/10.08	ok1(65) -> 65
34.79/10.08	ok1(66) -> 42
34.79/10.08	ok1(66) -> 66
34.79/10.08	ok1(67) -> 43
34.79/10.08	ok1(67) -> 67
34.79/10.08	ok1(68) -> 44
34.79/10.08	ok1(68) -> 68
34.79/10.08	active2(45) -> 71
34.79/10.08	top2(71) -> 24
34.79/10.08	active2(46) -> 71
34.79/10.08	
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(6)
34.79/10.08	BOUNDS(1, n^1)
34.79/10.08	
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(7) RenamingProof (BOTH BOUNDS(ID, ID))
34.79/10.08	Renamed function symbols to avoid clashes with predefined symbol.
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(8)
34.79/10.08	Obligation:
34.79/10.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
34.79/10.08	
34.79/10.08	
34.79/10.08	The TRS R consists of the following rules:
34.79/10.08	
34.79/10.08	   active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
34.79/10.08	   active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
34.79/10.08	   active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
34.79/10.08	   active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
34.79/10.08	   active(U15(tt, V2)) -> mark(U16(isNat(V2)))
34.79/10.08	   active(U16(tt)) -> mark(tt)
34.79/10.08	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
34.79/10.08	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
34.79/10.08	   active(U23(tt)) -> mark(tt)
34.79/10.08	   active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
34.79/10.08	   active(U32(tt)) -> mark(tt)
34.79/10.08	   active(U41(tt)) -> mark(tt)
34.79/10.08	   active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
34.79/10.08	   active(U52(tt, N)) -> mark(N)
34.79/10.08	   active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
34.79/10.08	   active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
34.79/10.08	   active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
34.79/10.08	   active(U64(tt, M, N)) -> mark(s(plus(N, M)))
34.79/10.08	   active(isNat(0')) -> mark(tt)
34.79/10.08	   active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
34.79/10.08	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
34.79/10.08	   active(isNatKind(0')) -> mark(tt)
34.79/10.08	   active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
34.79/10.08	   active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
34.79/10.08	   active(plus(N, 0')) -> mark(U51(isNat(N), N))
34.79/10.08	   active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
34.79/10.08	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
34.79/10.08	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
34.79/10.08	   active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
34.79/10.08	   active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
34.79/10.08	   active(U15(X1, X2)) -> U15(active(X1), X2)
34.79/10.08	   active(U16(X)) -> U16(active(X))
34.79/10.08	   active(U21(X1, X2)) -> U21(active(X1), X2)
34.79/10.08	   active(U22(X1, X2)) -> U22(active(X1), X2)
34.79/10.08	   active(U23(X)) -> U23(active(X))
34.79/10.08	   active(U31(X1, X2)) -> U31(active(X1), X2)
34.79/10.08	   active(U32(X)) -> U32(active(X))
34.79/10.08	   active(U41(X)) -> U41(active(X))
34.79/10.08	   active(U51(X1, X2)) -> U51(active(X1), X2)
34.79/10.08	   active(U52(X1, X2)) -> U52(active(X1), X2)
34.79/10.08	   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
34.79/10.08	   active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
34.79/10.08	   active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
34.79/10.08	   active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
34.79/10.08	   active(s(X)) -> s(active(X))
34.79/10.08	   active(plus(X1, X2)) -> plus(active(X1), X2)
34.79/10.08	   active(plus(X1, X2)) -> plus(X1, active(X2))
34.79/10.08	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
34.79/10.08	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
34.79/10.08	   U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
34.79/10.08	   U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
34.79/10.08	   U15(mark(X1), X2) -> mark(U15(X1, X2))
34.79/10.08	   U16(mark(X)) -> mark(U16(X))
34.79/10.08	   U21(mark(X1), X2) -> mark(U21(X1, X2))
34.79/10.08	   U22(mark(X1), X2) -> mark(U22(X1, X2))
34.79/10.08	   U23(mark(X)) -> mark(U23(X))
34.79/10.08	   U31(mark(X1), X2) -> mark(U31(X1, X2))
34.79/10.08	   U32(mark(X)) -> mark(U32(X))
34.79/10.08	   U41(mark(X)) -> mark(U41(X))
34.79/10.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
34.79/10.08	   U52(mark(X1), X2) -> mark(U52(X1, X2))
34.79/10.08	   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
34.79/10.08	   U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
34.79/10.08	   U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
34.79/10.08	   U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
34.79/10.08	   s(mark(X)) -> mark(s(X))
34.79/10.08	   plus(mark(X1), X2) -> mark(plus(X1, X2))
34.79/10.08	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
34.79/10.08	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(tt) -> ok(tt)
34.79/10.08	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(isNatKind(X)) -> isNatKind(proper(X))
34.79/10.08	   proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
34.79/10.08	   proper(isNat(X)) -> isNat(proper(X))
34.79/10.08	   proper(U16(X)) -> U16(proper(X))
34.79/10.08	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
34.79/10.08	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
34.79/10.08	   proper(U23(X)) -> U23(proper(X))
34.79/10.08	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
34.79/10.08	   proper(U32(X)) -> U32(proper(X))
34.79/10.08	   proper(U41(X)) -> U41(proper(X))
34.79/10.08	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
34.79/10.08	   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
34.79/10.08	   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
34.79/10.08	   proper(s(X)) -> s(proper(X))
34.79/10.08	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
34.79/10.08	   proper(0') -> ok(0')
34.79/10.08	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
34.79/10.08	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
34.79/10.08	   isNatKind(ok(X)) -> ok(isNatKind(X))
34.79/10.08	   U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
34.79/10.08	   U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
34.79/10.08	   U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
34.79/10.08	   isNat(ok(X)) -> ok(isNat(X))
34.79/10.08	   U16(ok(X)) -> ok(U16(X))
34.79/10.08	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
34.79/10.08	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
34.79/10.08	   U23(ok(X)) -> ok(U23(X))
34.79/10.08	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
34.79/10.08	   U32(ok(X)) -> ok(U32(X))
34.79/10.08	   U41(ok(X)) -> ok(U41(X))
34.79/10.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
34.79/10.08	   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
34.79/10.08	   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
34.79/10.08	   U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
34.79/10.08	   U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
34.79/10.08	   U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
34.79/10.08	   s(ok(X)) -> ok(s(X))
34.79/10.08	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
34.79/10.08	   top(mark(X)) -> top(proper(X))
34.79/10.08	   top(ok(X)) -> top(active(X))
34.79/10.08	
34.79/10.08	S is empty.
34.79/10.08	Rewrite Strategy: FULL
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
34.79/10.08	Infered types.
34.79/10.08	----------------------------------------
34.79/10.08	
34.79/10.08	(10)
34.79/10.08	Obligation:
34.79/10.08	TRS:
34.79/10.08	Rules:
34.79/10.08	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
34.79/10.08	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
34.79/10.08	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
34.79/10.08	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
34.79/10.08	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
34.79/10.08	active(U16(tt)) -> mark(tt)
34.79/10.08	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
34.79/10.08	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
34.79/10.08	active(U23(tt)) -> mark(tt)
34.79/10.08	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
34.79/10.08	active(U32(tt)) -> mark(tt)
34.79/10.08	active(U41(tt)) -> mark(tt)
34.79/10.08	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
34.79/10.08	active(U52(tt, N)) -> mark(N)
34.79/10.08	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
34.79/10.08	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
34.79/10.08	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
34.79/10.08	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
34.79/10.08	active(isNat(0')) -> mark(tt)
34.79/10.08	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
34.79/10.08	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
34.79/10.08	active(isNatKind(0')) -> mark(tt)
34.79/10.08	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
34.79/10.08	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
34.79/10.08	active(plus(N, 0')) -> mark(U51(isNat(N), N))
34.79/10.08	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
34.79/10.08	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
34.79/10.08	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
34.79/10.08	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
34.79/10.08	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
34.79/10.08	active(U15(X1, X2)) -> U15(active(X1), X2)
34.79/10.08	active(U16(X)) -> U16(active(X))
34.79/10.08	active(U21(X1, X2)) -> U21(active(X1), X2)
34.79/10.08	active(U22(X1, X2)) -> U22(active(X1), X2)
34.79/10.08	active(U23(X)) -> U23(active(X))
34.79/10.08	active(U31(X1, X2)) -> U31(active(X1), X2)
34.79/10.08	active(U32(X)) -> U32(active(X))
34.79/10.08	active(U41(X)) -> U41(active(X))
34.79/10.08	active(U51(X1, X2)) -> U51(active(X1), X2)
34.79/10.08	active(U52(X1, X2)) -> U52(active(X1), X2)
34.79/10.08	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
34.79/10.08	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
34.79/10.08	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
34.79/10.08	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
34.79/10.08	active(s(X)) -> s(active(X))
34.79/10.08	active(plus(X1, X2)) -> plus(active(X1), X2)
34.79/10.08	active(plus(X1, X2)) -> plus(X1, active(X2))
34.79/10.08	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
34.79/10.08	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
34.79/10.08	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
34.79/10.08	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
34.79/10.08	U15(mark(X1), X2) -> mark(U15(X1, X2))
34.79/10.08	U16(mark(X)) -> mark(U16(X))
34.79/10.08	U21(mark(X1), X2) -> mark(U21(X1, X2))
34.79/10.08	U22(mark(X1), X2) -> mark(U22(X1, X2))
34.79/10.08	U23(mark(X)) -> mark(U23(X))
34.79/10.08	U31(mark(X1), X2) -> mark(U31(X1, X2))
34.79/10.08	U32(mark(X)) -> mark(U32(X))
34.79/10.08	U41(mark(X)) -> mark(U41(X))
34.79/10.08	U51(mark(X1), X2) -> mark(U51(X1, X2))
34.79/10.08	U52(mark(X1), X2) -> mark(U52(X1, X2))
34.79/10.08	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
34.79/10.08	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
34.79/10.08	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
34.79/10.08	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
34.79/10.08	s(mark(X)) -> mark(s(X))
34.79/10.08	plus(mark(X1), X2) -> mark(plus(X1, X2))
34.79/10.08	plus(X1, mark(X2)) -> mark(plus(X1, X2))
34.79/10.08	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(tt) -> ok(tt)
34.79/10.08	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(isNatKind(X)) -> isNatKind(proper(X))
34.79/10.08	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
34.79/10.08	proper(isNat(X)) -> isNat(proper(X))
34.79/10.08	proper(U16(X)) -> U16(proper(X))
34.79/10.08	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
34.79/10.08	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
34.79/10.08	proper(U23(X)) -> U23(proper(X))
34.79/10.08	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
34.79/10.08	proper(U32(X)) -> U32(proper(X))
34.79/10.08	proper(U41(X)) -> U41(proper(X))
34.79/10.08	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
34.79/10.08	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
34.79/10.08	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
34.79/10.08	proper(s(X)) -> s(proper(X))
34.79/10.08	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
34.79/10.08	proper(0') -> ok(0')
34.79/10.08	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
34.79/10.08	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
34.79/10.08	isNatKind(ok(X)) -> ok(isNatKind(X))
34.79/10.08	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
34.79/10.08	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
34.79/10.08	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
34.79/10.08	isNat(ok(X)) -> ok(isNat(X))
34.79/10.08	U16(ok(X)) -> ok(U16(X))
34.79/10.08	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
34.79/10.08	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
34.79/10.08	U23(ok(X)) -> ok(U23(X))
34.79/10.08	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
34.79/10.08	U32(ok(X)) -> ok(U32(X))
34.79/10.08	U41(ok(X)) -> ok(U41(X))
34.79/10.08	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
34.79/10.08	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
34.79/10.08	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
34.79/10.08	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
34.79/10.08	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
34.79/10.08	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
34.79/10.08	s(ok(X)) -> ok(s(X))
34.79/10.08	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
34.79/10.08	top(mark(X)) -> top(proper(X))
34.79/10.08	top(ok(X)) -> top(active(X))
34.79/10.08	
34.79/10.08	Types:
34.79/10.08	active :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	tt :: tt:mark:0':ok
34.79/10.08	mark :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	isNat :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U16 :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U23 :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U32 :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U41 :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	s :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	0' :: tt:mark:0':ok
34.79/10.08	proper :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	ok :: tt:mark:0':ok -> tt:mark:0':ok
34.79/10.08	top :: tt:mark:0':ok -> top
34.79/10.08	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
34.79/10.08	hole_top2_0 :: top
35.09/10.09	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(11) OrderProof (LOWER BOUND(ID))
35.09/10.09	Heuristically decided to analyse the following defined symbols:
35.09/10.09	active, U12, isNatKind, U13, U14, U15, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.09	
35.09/10.09	They will be analysed ascendingly in the following order:
35.09/10.09	U12 < active
35.09/10.09	isNatKind < active
35.09/10.09	U13 < active
35.09/10.09	U14 < active
35.09/10.09	U15 < active
35.09/10.09	isNat < active
35.09/10.09	U16 < active
35.09/10.09	U22 < active
35.09/10.09	U23 < active
35.09/10.09	U32 < active
35.09/10.09	U52 < active
35.09/10.09	U62 < active
35.09/10.09	U63 < active
35.09/10.09	U64 < active
35.09/10.09	s < active
35.09/10.09	plus < active
35.09/10.09	U11 < active
35.09/10.09	U21 < active
35.09/10.09	U31 < active
35.09/10.09	U41 < active
35.09/10.09	U51 < active
35.09/10.09	U61 < active
35.09/10.09	active < top
35.09/10.09	U12 < proper
35.09/10.09	isNatKind < proper
35.09/10.09	U13 < proper
35.09/10.09	U14 < proper
35.09/10.09	U15 < proper
35.09/10.09	isNat < proper
35.09/10.09	U16 < proper
35.09/10.09	U22 < proper
35.09/10.09	U23 < proper
35.09/10.09	U32 < proper
35.09/10.09	U52 < proper
35.09/10.09	U62 < proper
35.09/10.09	U63 < proper
35.09/10.09	U64 < proper
35.09/10.09	s < proper
35.09/10.09	plus < proper
35.09/10.09	U11 < proper
35.09/10.09	U21 < proper
35.09/10.09	U31 < proper
35.09/10.09	U41 < proper
35.09/10.09	U51 < proper
35.09/10.09	U61 < proper
35.09/10.09	proper < top
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(12)
35.09/10.09	Obligation:
35.09/10.09	TRS:
35.09/10.09	Rules:
35.09/10.09	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.09	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.09	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.09	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.09	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.09	active(U16(tt)) -> mark(tt)
35.09/10.09	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.09	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.09	active(U23(tt)) -> mark(tt)
35.09/10.09	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.09	active(U32(tt)) -> mark(tt)
35.09/10.09	active(U41(tt)) -> mark(tt)
35.09/10.09	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.09	active(U52(tt, N)) -> mark(N)
35.09/10.09	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.09	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.09	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.09	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.09	active(isNat(0')) -> mark(tt)
35.09/10.09	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.09	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.09	active(isNatKind(0')) -> mark(tt)
35.09/10.09	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.09	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.09	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.09	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.09	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.09	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.09	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.09	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.09	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.09	active(U16(X)) -> U16(active(X))
35.09/10.09	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.09	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.09	active(U23(X)) -> U23(active(X))
35.09/10.09	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.09	active(U32(X)) -> U32(active(X))
35.09/10.09	active(U41(X)) -> U41(active(X))
35.09/10.09	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.09	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.09	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.09	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.09	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.09	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.09	active(s(X)) -> s(active(X))
35.09/10.09	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.09	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.09	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.09	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.09	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.09	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.09	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.09	U16(mark(X)) -> mark(U16(X))
35.09/10.09	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.09	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.09	U23(mark(X)) -> mark(U23(X))
35.09/10.09	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.09	U32(mark(X)) -> mark(U32(X))
35.09/10.09	U41(mark(X)) -> mark(U41(X))
35.09/10.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.09	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.09	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.09	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.09	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.09	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.09	s(mark(X)) -> mark(s(X))
35.09/10.09	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.09	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.09	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(tt) -> ok(tt)
35.09/10.09	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.09	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.09	proper(isNat(X)) -> isNat(proper(X))
35.09/10.09	proper(U16(X)) -> U16(proper(X))
35.09/10.09	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.09	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.09	proper(U23(X)) -> U23(proper(X))
35.09/10.09	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.09	proper(U32(X)) -> U32(proper(X))
35.09/10.09	proper(U41(X)) -> U41(proper(X))
35.09/10.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.09	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.09	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(s(X)) -> s(proper(X))
35.09/10.09	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.09	proper(0') -> ok(0')
35.09/10.09	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.09	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.09	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.09	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.09	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.09	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.09	isNat(ok(X)) -> ok(isNat(X))
35.09/10.09	U16(ok(X)) -> ok(U16(X))
35.09/10.09	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.09	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.09	U23(ok(X)) -> ok(U23(X))
35.09/10.09	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.09	U32(ok(X)) -> ok(U32(X))
35.09/10.09	U41(ok(X)) -> ok(U41(X))
35.09/10.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.09	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.09	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.09	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.09	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.09	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.09	s(ok(X)) -> ok(s(X))
35.09/10.09	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.09	top(mark(X)) -> top(proper(X))
35.09/10.09	top(ok(X)) -> top(active(X))
35.09/10.09	
35.09/10.09	Types:
35.09/10.09	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	tt :: tt:mark:0':ok
35.09/10.09	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	0' :: tt:mark:0':ok
35.09/10.09	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	top :: tt:mark:0':ok -> top
35.09/10.09	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.09	hole_top2_0 :: top
35.09/10.09	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.09	
35.09/10.09	
35.09/10.09	Generator Equations:
35.09/10.09	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.09	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.09	
35.09/10.09	
35.09/10.09	The following defined symbols remain to be analysed:
35.09/10.09	U12, active, isNatKind, U13, U14, U15, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.09	
35.09/10.09	They will be analysed ascendingly in the following order:
35.09/10.09	U12 < active
35.09/10.09	isNatKind < active
35.09/10.09	U13 < active
35.09/10.09	U14 < active
35.09/10.09	U15 < active
35.09/10.09	isNat < active
35.09/10.09	U16 < active
35.09/10.09	U22 < active
35.09/10.09	U23 < active
35.09/10.09	U32 < active
35.09/10.09	U52 < active
35.09/10.09	U62 < active
35.09/10.09	U63 < active
35.09/10.09	U64 < active
35.09/10.09	s < active
35.09/10.09	plus < active
35.09/10.09	U11 < active
35.09/10.09	U21 < active
35.09/10.09	U31 < active
35.09/10.09	U41 < active
35.09/10.09	U51 < active
35.09/10.09	U61 < active
35.09/10.09	active < top
35.09/10.09	U12 < proper
35.09/10.09	isNatKind < proper
35.09/10.09	U13 < proper
35.09/10.09	U14 < proper
35.09/10.09	U15 < proper
35.09/10.09	isNat < proper
35.09/10.09	U16 < proper
35.09/10.09	U22 < proper
35.09/10.09	U23 < proper
35.09/10.09	U32 < proper
35.09/10.09	U52 < proper
35.09/10.09	U62 < proper
35.09/10.09	U63 < proper
35.09/10.09	U64 < proper
35.09/10.09	s < proper
35.09/10.09	plus < proper
35.09/10.09	U11 < proper
35.09/10.09	U21 < proper
35.09/10.09	U31 < proper
35.09/10.09	U41 < proper
35.09/10.09	U51 < proper
35.09/10.09	U61 < proper
35.09/10.09	proper < top
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(13) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.09	Proved the following rewrite lemma:
35.09/10.09	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.09	
35.09/10.09	Induction Base:
35.09/10.09	U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.09	
35.09/10.09	Induction Step:
35.09/10.09	U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.09	mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.09	mark(*4_0)
35.09/10.09	
35.09/10.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(14)
35.09/10.09	Complex Obligation (BEST)
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(15)
35.09/10.09	Obligation:
35.09/10.09	Proved the lower bound n^1 for the following obligation:
35.09/10.09	
35.09/10.09	TRS:
35.09/10.09	Rules:
35.09/10.09	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.09	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.09	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.09	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.09	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.09	active(U16(tt)) -> mark(tt)
35.09/10.09	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.09	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.09	active(U23(tt)) -> mark(tt)
35.09/10.09	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.09	active(U32(tt)) -> mark(tt)
35.09/10.09	active(U41(tt)) -> mark(tt)
35.09/10.09	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.09	active(U52(tt, N)) -> mark(N)
35.09/10.09	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.09	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.09	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.09	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.09	active(isNat(0')) -> mark(tt)
35.09/10.09	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.09	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.09	active(isNatKind(0')) -> mark(tt)
35.09/10.09	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.09	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.09	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.09	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.09	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.09	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.09	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.09	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.09	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.09	active(U16(X)) -> U16(active(X))
35.09/10.09	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.09	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.09	active(U23(X)) -> U23(active(X))
35.09/10.09	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.09	active(U32(X)) -> U32(active(X))
35.09/10.09	active(U41(X)) -> U41(active(X))
35.09/10.09	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.09	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.09	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.09	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.09	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.09	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.09	active(s(X)) -> s(active(X))
35.09/10.09	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.09	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.09	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.09	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.09	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.09	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.09	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.09	U16(mark(X)) -> mark(U16(X))
35.09/10.09	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.09	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.09	U23(mark(X)) -> mark(U23(X))
35.09/10.09	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.09	U32(mark(X)) -> mark(U32(X))
35.09/10.09	U41(mark(X)) -> mark(U41(X))
35.09/10.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.09	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.09	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.09	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.09	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.09	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.09	s(mark(X)) -> mark(s(X))
35.09/10.09	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.09	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.09	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(tt) -> ok(tt)
35.09/10.09	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.09	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.09	proper(isNat(X)) -> isNat(proper(X))
35.09/10.09	proper(U16(X)) -> U16(proper(X))
35.09/10.09	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.09	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.09	proper(U23(X)) -> U23(proper(X))
35.09/10.09	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.09	proper(U32(X)) -> U32(proper(X))
35.09/10.09	proper(U41(X)) -> U41(proper(X))
35.09/10.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.09	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.09	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(s(X)) -> s(proper(X))
35.09/10.09	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.09	proper(0') -> ok(0')
35.09/10.09	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.09	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.09	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.09	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.09	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.09	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.09	isNat(ok(X)) -> ok(isNat(X))
35.09/10.09	U16(ok(X)) -> ok(U16(X))
35.09/10.09	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.09	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.09	U23(ok(X)) -> ok(U23(X))
35.09/10.09	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.09	U32(ok(X)) -> ok(U32(X))
35.09/10.09	U41(ok(X)) -> ok(U41(X))
35.09/10.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.09	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.09	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.09	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.09	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.09	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.09	s(ok(X)) -> ok(s(X))
35.09/10.09	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.09	top(mark(X)) -> top(proper(X))
35.09/10.09	top(ok(X)) -> top(active(X))
35.09/10.09	
35.09/10.09	Types:
35.09/10.09	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	tt :: tt:mark:0':ok
35.09/10.09	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	0' :: tt:mark:0':ok
35.09/10.09	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	top :: tt:mark:0':ok -> top
35.09/10.09	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.09	hole_top2_0 :: top
35.09/10.09	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.09	
35.09/10.09	
35.09/10.09	Generator Equations:
35.09/10.09	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.09	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.09	
35.09/10.09	
35.09/10.09	The following defined symbols remain to be analysed:
35.09/10.09	U12, active, isNatKind, U13, U14, U15, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.09	
35.09/10.09	They will be analysed ascendingly in the following order:
35.09/10.09	U12 < active
35.09/10.09	isNatKind < active
35.09/10.09	U13 < active
35.09/10.09	U14 < active
35.09/10.09	U15 < active
35.09/10.09	isNat < active
35.09/10.09	U16 < active
35.09/10.09	U22 < active
35.09/10.09	U23 < active
35.09/10.09	U32 < active
35.09/10.09	U52 < active
35.09/10.09	U62 < active
35.09/10.09	U63 < active
35.09/10.09	U64 < active
35.09/10.09	s < active
35.09/10.09	plus < active
35.09/10.09	U11 < active
35.09/10.09	U21 < active
35.09/10.09	U31 < active
35.09/10.09	U41 < active
35.09/10.09	U51 < active
35.09/10.09	U61 < active
35.09/10.09	active < top
35.09/10.09	U12 < proper
35.09/10.09	isNatKind < proper
35.09/10.09	U13 < proper
35.09/10.09	U14 < proper
35.09/10.09	U15 < proper
35.09/10.09	isNat < proper
35.09/10.09	U16 < proper
35.09/10.09	U22 < proper
35.09/10.09	U23 < proper
35.09/10.09	U32 < proper
35.09/10.09	U52 < proper
35.09/10.09	U62 < proper
35.09/10.09	U63 < proper
35.09/10.09	U64 < proper
35.09/10.09	s < proper
35.09/10.09	plus < proper
35.09/10.09	U11 < proper
35.09/10.09	U21 < proper
35.09/10.09	U31 < proper
35.09/10.09	U41 < proper
35.09/10.09	U51 < proper
35.09/10.09	U61 < proper
35.09/10.09	proper < top
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(16) LowerBoundPropagationProof (FINISHED)
35.09/10.09	Propagated lower bound.
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(17)
35.09/10.09	BOUNDS(n^1, INF)
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(18)
35.09/10.09	Obligation:
35.09/10.09	TRS:
35.09/10.09	Rules:
35.09/10.09	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.09	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.09	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.09	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.09	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.09	active(U16(tt)) -> mark(tt)
35.09/10.09	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.09	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.09	active(U23(tt)) -> mark(tt)
35.09/10.09	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.09	active(U32(tt)) -> mark(tt)
35.09/10.09	active(U41(tt)) -> mark(tt)
35.09/10.09	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.09	active(U52(tt, N)) -> mark(N)
35.09/10.09	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.09	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.09	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.09	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.09	active(isNat(0')) -> mark(tt)
35.09/10.09	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.09	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.09	active(isNatKind(0')) -> mark(tt)
35.09/10.09	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.09	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.09	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.09	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.09	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.09	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.09	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.09	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.09	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.09	active(U16(X)) -> U16(active(X))
35.09/10.09	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.09	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.09	active(U23(X)) -> U23(active(X))
35.09/10.09	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.09	active(U32(X)) -> U32(active(X))
35.09/10.09	active(U41(X)) -> U41(active(X))
35.09/10.09	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.09	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.09	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.09	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.09	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.09	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.09	active(s(X)) -> s(active(X))
35.09/10.09	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.09	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.09	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.09	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.09	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.09	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.09	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.09	U16(mark(X)) -> mark(U16(X))
35.09/10.09	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.09	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.09	U23(mark(X)) -> mark(U23(X))
35.09/10.09	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.09	U32(mark(X)) -> mark(U32(X))
35.09/10.09	U41(mark(X)) -> mark(U41(X))
35.09/10.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.09	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.09	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.09	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.09	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.09	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.09	s(mark(X)) -> mark(s(X))
35.09/10.09	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.09	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.09	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(tt) -> ok(tt)
35.09/10.09	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.09	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.09	proper(isNat(X)) -> isNat(proper(X))
35.09/10.09	proper(U16(X)) -> U16(proper(X))
35.09/10.09	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.09	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.09	proper(U23(X)) -> U23(proper(X))
35.09/10.09	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.09	proper(U32(X)) -> U32(proper(X))
35.09/10.09	proper(U41(X)) -> U41(proper(X))
35.09/10.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.09	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.09	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(s(X)) -> s(proper(X))
35.09/10.09	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.09	proper(0') -> ok(0')
35.09/10.09	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.09	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.09	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.09	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.09	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.09	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.09	isNat(ok(X)) -> ok(isNat(X))
35.09/10.09	U16(ok(X)) -> ok(U16(X))
35.09/10.09	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.09	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.09	U23(ok(X)) -> ok(U23(X))
35.09/10.09	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.09	U32(ok(X)) -> ok(U32(X))
35.09/10.09	U41(ok(X)) -> ok(U41(X))
35.09/10.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.09	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.09	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.09	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.09	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.09	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.09	s(ok(X)) -> ok(s(X))
35.09/10.09	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.09	top(mark(X)) -> top(proper(X))
35.09/10.09	top(ok(X)) -> top(active(X))
35.09/10.09	
35.09/10.09	Types:
35.09/10.09	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	tt :: tt:mark:0':ok
35.09/10.09	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	0' :: tt:mark:0':ok
35.09/10.09	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	top :: tt:mark:0':ok -> top
35.09/10.09	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.09	hole_top2_0 :: top
35.09/10.09	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.09	
35.09/10.09	
35.09/10.09	Lemmas:
35.09/10.09	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.09	
35.09/10.09	
35.09/10.09	Generator Equations:
35.09/10.09	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.09	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.09	
35.09/10.09	
35.09/10.09	The following defined symbols remain to be analysed:
35.09/10.09	isNatKind, active, U13, U14, U15, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.09	
35.09/10.09	They will be analysed ascendingly in the following order:
35.09/10.09	isNatKind < active
35.09/10.09	U13 < active
35.09/10.09	U14 < active
35.09/10.09	U15 < active
35.09/10.09	isNat < active
35.09/10.09	U16 < active
35.09/10.09	U22 < active
35.09/10.09	U23 < active
35.09/10.09	U32 < active
35.09/10.09	U52 < active
35.09/10.09	U62 < active
35.09/10.09	U63 < active
35.09/10.09	U64 < active
35.09/10.09	s < active
35.09/10.09	plus < active
35.09/10.09	U11 < active
35.09/10.09	U21 < active
35.09/10.09	U31 < active
35.09/10.09	U41 < active
35.09/10.09	U51 < active
35.09/10.09	U61 < active
35.09/10.09	active < top
35.09/10.09	isNatKind < proper
35.09/10.09	U13 < proper
35.09/10.09	U14 < proper
35.09/10.09	U15 < proper
35.09/10.09	isNat < proper
35.09/10.09	U16 < proper
35.09/10.09	U22 < proper
35.09/10.09	U23 < proper
35.09/10.09	U32 < proper
35.09/10.09	U52 < proper
35.09/10.09	U62 < proper
35.09/10.09	U63 < proper
35.09/10.09	U64 < proper
35.09/10.09	s < proper
35.09/10.09	plus < proper
35.09/10.09	U11 < proper
35.09/10.09	U21 < proper
35.09/10.09	U31 < proper
35.09/10.09	U41 < proper
35.09/10.09	U51 < proper
35.09/10.09	U61 < proper
35.09/10.09	proper < top
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(19) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.09	Proved the following rewrite lemma:
35.09/10.09	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.09	
35.09/10.09	Induction Base:
35.09/10.09	U13(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.09	
35.09/10.09	Induction Step:
35.09/10.09	U13(gen_tt:mark:0':ok3_0(+(1, +(n3519_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.09	mark(U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.09	mark(*4_0)
35.09/10.09	
35.09/10.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(20)
35.09/10.09	Obligation:
35.09/10.09	TRS:
35.09/10.09	Rules:
35.09/10.09	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.09	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.09	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.09	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.09	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.09	active(U16(tt)) -> mark(tt)
35.09/10.09	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.09	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.09	active(U23(tt)) -> mark(tt)
35.09/10.09	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.09	active(U32(tt)) -> mark(tt)
35.09/10.09	active(U41(tt)) -> mark(tt)
35.09/10.09	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.09	active(U52(tt, N)) -> mark(N)
35.09/10.09	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.09	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.09	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.09	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.09	active(isNat(0')) -> mark(tt)
35.09/10.09	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.09	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.09	active(isNatKind(0')) -> mark(tt)
35.09/10.09	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.09	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.09	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.09	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.09	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.09	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.09	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.09	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.09	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.09	active(U16(X)) -> U16(active(X))
35.09/10.09	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.09	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.09	active(U23(X)) -> U23(active(X))
35.09/10.09	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.09	active(U32(X)) -> U32(active(X))
35.09/10.09	active(U41(X)) -> U41(active(X))
35.09/10.09	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.09	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.09	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.09	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.09	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.09	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.09	active(s(X)) -> s(active(X))
35.09/10.09	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.09	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.09	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.09	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.09	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.09	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.09	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.09	U16(mark(X)) -> mark(U16(X))
35.09/10.09	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.09	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.09	U23(mark(X)) -> mark(U23(X))
35.09/10.09	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.09	U32(mark(X)) -> mark(U32(X))
35.09/10.09	U41(mark(X)) -> mark(U41(X))
35.09/10.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.09	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.09	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.09	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.09	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.09	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.09	s(mark(X)) -> mark(s(X))
35.09/10.09	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.09	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.09	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(tt) -> ok(tt)
35.09/10.09	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.09	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.09	proper(isNat(X)) -> isNat(proper(X))
35.09/10.09	proper(U16(X)) -> U16(proper(X))
35.09/10.09	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.09	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.09	proper(U23(X)) -> U23(proper(X))
35.09/10.09	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.09	proper(U32(X)) -> U32(proper(X))
35.09/10.09	proper(U41(X)) -> U41(proper(X))
35.09/10.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.09	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.09	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.09	proper(s(X)) -> s(proper(X))
35.09/10.09	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.09	proper(0') -> ok(0')
35.09/10.09	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.09	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.09	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.09	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.09	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.09	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.09	isNat(ok(X)) -> ok(isNat(X))
35.09/10.09	U16(ok(X)) -> ok(U16(X))
35.09/10.09	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.09	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.09	U23(ok(X)) -> ok(U23(X))
35.09/10.09	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.09	U32(ok(X)) -> ok(U32(X))
35.09/10.09	U41(ok(X)) -> ok(U41(X))
35.09/10.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.09	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.09	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.09	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.09	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.09	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.09	s(ok(X)) -> ok(s(X))
35.09/10.09	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.09	top(mark(X)) -> top(proper(X))
35.09/10.09	top(ok(X)) -> top(active(X))
35.09/10.09	
35.09/10.09	Types:
35.09/10.09	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	tt :: tt:mark:0':ok
35.09/10.09	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	0' :: tt:mark:0':ok
35.09/10.09	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.09	top :: tt:mark:0':ok -> top
35.09/10.09	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.09	hole_top2_0 :: top
35.09/10.09	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.09	
35.09/10.09	
35.09/10.09	Lemmas:
35.09/10.09	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.09	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.09	
35.09/10.09	
35.09/10.09	Generator Equations:
35.09/10.09	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.09	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.09	
35.09/10.09	
35.09/10.09	The following defined symbols remain to be analysed:
35.09/10.09	U14, active, U15, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.09	
35.09/10.09	They will be analysed ascendingly in the following order:
35.09/10.09	U14 < active
35.09/10.09	U15 < active
35.09/10.09	isNat < active
35.09/10.09	U16 < active
35.09/10.09	U22 < active
35.09/10.09	U23 < active
35.09/10.09	U32 < active
35.09/10.09	U52 < active
35.09/10.09	U62 < active
35.09/10.09	U63 < active
35.09/10.09	U64 < active
35.09/10.09	s < active
35.09/10.09	plus < active
35.09/10.09	U11 < active
35.09/10.09	U21 < active
35.09/10.09	U31 < active
35.09/10.09	U41 < active
35.09/10.09	U51 < active
35.09/10.09	U61 < active
35.09/10.09	active < top
35.09/10.09	U14 < proper
35.09/10.09	U15 < proper
35.09/10.09	isNat < proper
35.09/10.09	U16 < proper
35.09/10.09	U22 < proper
35.09/10.09	U23 < proper
35.09/10.09	U32 < proper
35.09/10.09	U52 < proper
35.09/10.09	U62 < proper
35.09/10.09	U63 < proper
35.09/10.09	U64 < proper
35.09/10.09	s < proper
35.09/10.09	plus < proper
35.09/10.09	U11 < proper
35.09/10.09	U21 < proper
35.09/10.09	U31 < proper
35.09/10.09	U41 < proper
35.09/10.09	U51 < proper
35.09/10.09	U61 < proper
35.09/10.09	proper < top
35.09/10.09	
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(21) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.09	Proved the following rewrite lemma:
35.09/10.09	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.09	
35.09/10.09	Induction Base:
35.09/10.09	U14(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.09	
35.09/10.09	Induction Step:
35.09/10.09	U14(gen_tt:mark:0':ok3_0(+(1, +(n7631_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.09	mark(U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.09	mark(*4_0)
35.09/10.09	
35.09/10.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.09	----------------------------------------
35.09/10.09	
35.09/10.09	(22)
35.09/10.09	Obligation:
35.09/10.09	TRS:
35.09/10.09	Rules:
35.09/10.09	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.09	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.09	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.09	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.09	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.09	active(U16(tt)) -> mark(tt)
35.09/10.09	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.09	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.09	active(U23(tt)) -> mark(tt)
35.09/10.09	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.09	active(U32(tt)) -> mark(tt)
35.09/10.09	active(U41(tt)) -> mark(tt)
35.09/10.09	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.09	active(U52(tt, N)) -> mark(N)
35.09/10.09	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.09	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.09	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.09	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.09	active(isNat(0')) -> mark(tt)
35.09/10.09	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.09	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.09	active(isNatKind(0')) -> mark(tt)
35.09/10.09	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.09	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.09	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.09	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.09	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.09	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.09	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.09	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.09	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.09	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U15, active, isNat, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U15 < active
35.09/10.10	isNat < active
35.09/10.10	U16 < active
35.09/10.10	U22 < active
35.09/10.10	U23 < active
35.09/10.10	U32 < active
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U15 < proper
35.09/10.10	isNat < proper
35.09/10.10	U16 < proper
35.09/10.10	U22 < proper
35.09/10.10	U23 < proper
35.09/10.10	U32 < proper
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(23) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, +(n12352_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.10	mark(U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(24)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	isNat, active, U16, U22, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	isNat < active
35.09/10.10	U16 < active
35.09/10.10	U22 < active
35.09/10.10	U23 < active
35.09/10.10	U32 < active
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	isNat < proper
35.09/10.10	U16 < proper
35.09/10.10	U22 < proper
35.09/10.10	U23 < proper
35.09/10.10	U32 < proper
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(25) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, 0)))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, +(n15653_0, 1)))) ->_R^Omega(1)
35.09/10.10	mark(U16(gen_tt:mark:0':ok3_0(+(1, n15653_0)))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(26)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U22, active, U23, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U22 < active
35.09/10.10	U23 < active
35.09/10.10	U32 < active
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U22 < proper
35.09/10.10	U23 < proper
35.09/10.10	U32 < proper
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(27) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, +(n17015_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.10	mark(U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(28)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U23, active, U32, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U23 < active
35.09/10.10	U32 < active
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U23 < proper
35.09/10.10	U32 < proper
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(29) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, 0)))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, +(n20800_0, 1)))) ->_R^Omega(1)
35.09/10.10	mark(U23(gen_tt:mark:0':ok3_0(+(1, n20800_0)))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(30)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U32, active, U52, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U32 < active
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U32 < proper
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(31) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U32(gen_tt:mark:0':ok3_0(+(1, 0)))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U32(gen_tt:mark:0':ok3_0(+(1, +(n22413_0, 1)))) ->_R^Omega(1)
35.09/10.10	mark(U32(gen_tt:mark:0':ok3_0(+(1, n22413_0)))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(32)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.10	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U52, active, U62, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U52 < active
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U52 < proper
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(33) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U52(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U52(gen_tt:mark:0':ok3_0(+(1, +(n24127_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.10	mark(U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(34)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.10	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.10	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.10	s(mark(X)) -> mark(s(X))
35.09/10.10	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.10	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.10	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(tt) -> ok(tt)
35.09/10.10	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.10	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.10	proper(isNat(X)) -> isNat(proper(X))
35.09/10.10	proper(U16(X)) -> U16(proper(X))
35.09/10.10	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.10	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.10	proper(U23(X)) -> U23(proper(X))
35.09/10.10	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.10	proper(U32(X)) -> U32(proper(X))
35.09/10.10	proper(U41(X)) -> U41(proper(X))
35.09/10.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.10	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.10	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.10	proper(s(X)) -> s(proper(X))
35.09/10.10	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.10	proper(0') -> ok(0')
35.09/10.10	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.10	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.10	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.10	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.10	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.10	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.10	isNat(ok(X)) -> ok(isNat(X))
35.09/10.10	U16(ok(X)) -> ok(U16(X))
35.09/10.10	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.10	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.10	U23(ok(X)) -> ok(U23(X))
35.09/10.10	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.10	U32(ok(X)) -> ok(U32(X))
35.09/10.10	U41(ok(X)) -> ok(U41(X))
35.09/10.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.10	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.10	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.10	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.10	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.10	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.10	s(ok(X)) -> ok(s(X))
35.09/10.10	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.10	top(mark(X)) -> top(proper(X))
35.09/10.10	top(ok(X)) -> top(active(X))
35.09/10.10	
35.09/10.10	Types:
35.09/10.10	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	tt :: tt:mark:0':ok
35.09/10.10	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	0' :: tt:mark:0':ok
35.09/10.10	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.10	top :: tt:mark:0':ok -> top
35.09/10.10	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.10	hole_top2_0 :: top
35.09/10.10	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.10	
35.09/10.10	
35.09/10.10	Lemmas:
35.09/10.10	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.10	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.10	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.10	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.10	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.10	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.10	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.10	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.10	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.10	
35.09/10.10	
35.09/10.10	Generator Equations:
35.09/10.10	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.10	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.10	
35.09/10.10	
35.09/10.10	The following defined symbols remain to be analysed:
35.09/10.10	U62, active, U63, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.10	
35.09/10.10	They will be analysed ascendingly in the following order:
35.09/10.10	U62 < active
35.09/10.10	U63 < active
35.09/10.10	U64 < active
35.09/10.10	s < active
35.09/10.10	plus < active
35.09/10.10	U11 < active
35.09/10.10	U21 < active
35.09/10.10	U31 < active
35.09/10.10	U41 < active
35.09/10.10	U51 < active
35.09/10.10	U61 < active
35.09/10.10	active < top
35.09/10.10	U62 < proper
35.09/10.10	U63 < proper
35.09/10.10	U64 < proper
35.09/10.10	s < proper
35.09/10.10	plus < proper
35.09/10.10	U11 < proper
35.09/10.10	U21 < proper
35.09/10.10	U31 < proper
35.09/10.10	U41 < proper
35.09/10.10	U51 < proper
35.09/10.10	U61 < proper
35.09/10.10	proper < top
35.09/10.10	
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(35) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.10	Proved the following rewrite lemma:
35.09/10.10	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.10	
35.09/10.10	Induction Base:
35.09/10.10	U62(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.10	
35.09/10.10	Induction Step:
35.09/10.10	U62(gen_tt:mark:0':ok3_0(+(1, +(n28634_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.10	mark(U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.10	mark(*4_0)
35.09/10.10	
35.09/10.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.10	----------------------------------------
35.09/10.10	
35.09/10.10	(36)
35.09/10.10	Obligation:
35.09/10.10	TRS:
35.09/10.10	Rules:
35.09/10.10	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.10	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.10	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.10	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.10	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.10	active(U16(tt)) -> mark(tt)
35.09/10.10	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.10	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.10	active(U23(tt)) -> mark(tt)
35.09/10.10	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.10	active(U32(tt)) -> mark(tt)
35.09/10.10	active(U41(tt)) -> mark(tt)
35.09/10.10	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.10	active(U52(tt, N)) -> mark(N)
35.09/10.10	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.10	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.10	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.10	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.10	active(isNat(0')) -> mark(tt)
35.09/10.10	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.10	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.10	active(isNatKind(0')) -> mark(tt)
35.09/10.10	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.10	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.10	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.10	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.10	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.10	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.10	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.10	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.10	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.10	active(U16(X)) -> U16(active(X))
35.09/10.10	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.10	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.10	active(U23(X)) -> U23(active(X))
35.09/10.10	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.10	active(U32(X)) -> U32(active(X))
35.09/10.10	active(U41(X)) -> U41(active(X))
35.09/10.10	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.10	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.10	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.10	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.10	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.10	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.10	active(s(X)) -> s(active(X))
35.09/10.10	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.10	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.10	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.10	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.10	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.10	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.10	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.10	U16(mark(X)) -> mark(U16(X))
35.09/10.10	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.10	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.10	U23(mark(X)) -> mark(U23(X))
35.09/10.10	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.10	U32(mark(X)) -> mark(U32(X))
35.09/10.10	U41(mark(X)) -> mark(U41(X))
35.09/10.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.10	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.10	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.10	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U63, active, U64, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U63 < active
35.09/10.11	U64 < active
35.09/10.11	s < active
35.09/10.11	plus < active
35.09/10.11	U11 < active
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U63 < proper
35.09/10.11	U64 < proper
35.09/10.11	s < proper
35.09/10.11	plus < proper
35.09/10.11	U11 < proper
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(37) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, +(n36295_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.11	mark(U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(38)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U64, active, s, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U64 < active
35.09/10.11	s < active
35.09/10.11	plus < active
35.09/10.11	U11 < active
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U64 < proper
35.09/10.11	s < proper
35.09/10.11	plus < proper
35.09/10.11	U11 < proper
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(39) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, +(n44565_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.11	mark(U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(40)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	s, active, plus, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	s < active
35.09/10.11	plus < active
35.09/10.11	U11 < active
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	s < proper
35.09/10.11	plus < proper
35.09/10.11	U11 < proper
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(41) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, 0)))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, +(n53444_0, 1)))) ->_R^Omega(1)
35.09/10.11	mark(s(gen_tt:mark:0':ok3_0(+(1, n53444_0)))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(42)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	plus, active, U11, U21, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	plus < active
35.09/10.11	U11 < active
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	plus < proper
35.09/10.11	U11 < proper
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(43) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, +(n56006_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.11	mark(plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(44)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U11, active, U21, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U11 < active
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U11 < proper
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(45) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, +(n62438_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.11	mark(U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(46)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U21, active, U31, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U21 < active
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U21 < proper
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(47) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, +(n72703_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.11	mark(U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(48)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U31, active, U41, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U31 < active
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U31 < proper
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(49) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U31(gen_tt:mark:0':ok3_0(+(1, +(n79646_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.11	mark(U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(50)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.11	U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U41, active, U51, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U41 < active
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U41 < proper
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(51) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U41(gen_tt:mark:0':ok3_0(+(1, 0)))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U41(gen_tt:mark:0':ok3_0(+(1, +(n86895_0, 1)))) ->_R^Omega(1)
35.09/10.11	mark(U41(gen_tt:mark:0':ok3_0(+(1, n86895_0)))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(52)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.11	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.11	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.11	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.11	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.11	active(U16(tt)) -> mark(tt)
35.09/10.11	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.11	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.11	active(U23(tt)) -> mark(tt)
35.09/10.11	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.11	active(U32(tt)) -> mark(tt)
35.09/10.11	active(U41(tt)) -> mark(tt)
35.09/10.11	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.11	active(U52(tt, N)) -> mark(N)
35.09/10.11	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.11	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.11	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.11	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.11	active(isNat(0')) -> mark(tt)
35.09/10.11	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.11	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.11	active(isNatKind(0')) -> mark(tt)
35.09/10.11	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.11	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.11	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.11	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.11	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.11	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.11	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.11	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.11	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.11	active(U16(X)) -> U16(active(X))
35.09/10.11	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.11	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.11	active(U23(X)) -> U23(active(X))
35.09/10.11	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.11	active(U32(X)) -> U32(active(X))
35.09/10.11	active(U41(X)) -> U41(active(X))
35.09/10.11	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.11	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.11	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.11	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.11	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.11	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.11	active(s(X)) -> s(active(X))
35.09/10.11	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.11	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.11	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.11	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.11	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.11	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.11	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.11	U16(mark(X)) -> mark(U16(X))
35.09/10.11	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.11	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.11	U23(mark(X)) -> mark(U23(X))
35.09/10.11	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.11	U32(mark(X)) -> mark(U32(X))
35.09/10.11	U41(mark(X)) -> mark(U41(X))
35.09/10.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.11	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.11	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.11	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.11	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.11	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.11	s(mark(X)) -> mark(s(X))
35.09/10.11	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.11	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.11	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(tt) -> ok(tt)
35.09/10.11	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.11	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.11	proper(isNat(X)) -> isNat(proper(X))
35.09/10.11	proper(U16(X)) -> U16(proper(X))
35.09/10.11	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.11	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.11	proper(U23(X)) -> U23(proper(X))
35.09/10.11	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.11	proper(U32(X)) -> U32(proper(X))
35.09/10.11	proper(U41(X)) -> U41(proper(X))
35.09/10.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.11	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.11	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.11	proper(s(X)) -> s(proper(X))
35.09/10.11	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.11	proper(0') -> ok(0')
35.09/10.11	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.11	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.11	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.11	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.11	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.11	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.11	isNat(ok(X)) -> ok(isNat(X))
35.09/10.11	U16(ok(X)) -> ok(U16(X))
35.09/10.11	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.11	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.11	U23(ok(X)) -> ok(U23(X))
35.09/10.11	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.11	U32(ok(X)) -> ok(U32(X))
35.09/10.11	U41(ok(X)) -> ok(U41(X))
35.09/10.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.11	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.11	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.11	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.11	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.11	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.11	s(ok(X)) -> ok(s(X))
35.09/10.11	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.11	top(mark(X)) -> top(proper(X))
35.09/10.11	top(ok(X)) -> top(active(X))
35.09/10.11	
35.09/10.11	Types:
35.09/10.11	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	tt :: tt:mark:0':ok
35.09/10.11	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	0' :: tt:mark:0':ok
35.09/10.11	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.11	top :: tt:mark:0':ok -> top
35.09/10.11	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.11	hole_top2_0 :: top
35.09/10.11	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.11	
35.09/10.11	
35.09/10.11	Lemmas:
35.09/10.11	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.11	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.11	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.11	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.11	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.11	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.11	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.11	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.11	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.11	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.11	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.11	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.11	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.11	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.11	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.11	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.11	U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
35.09/10.11	U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
35.09/10.11	
35.09/10.11	
35.09/10.11	Generator Equations:
35.09/10.11	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.11	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.11	
35.09/10.11	
35.09/10.11	The following defined symbols remain to be analysed:
35.09/10.11	U51, active, U61, proper, top
35.09/10.11	
35.09/10.11	They will be analysed ascendingly in the following order:
35.09/10.11	U51 < active
35.09/10.11	U61 < active
35.09/10.11	active < top
35.09/10.11	U51 < proper
35.09/10.11	U61 < proper
35.09/10.11	proper < top
35.09/10.11	
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(53) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.11	Proved the following rewrite lemma:
35.09/10.11	U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)
35.09/10.11	
35.09/10.11	Induction Base:
35.09/10.11	U51(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
35.09/10.11	
35.09/10.11	Induction Step:
35.09/10.11	U51(gen_tt:mark:0':ok3_0(+(1, +(n90207_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
35.09/10.11	mark(U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
35.09/10.11	mark(*4_0)
35.09/10.11	
35.09/10.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.11	----------------------------------------
35.09/10.11	
35.09/10.11	(54)
35.09/10.11	Obligation:
35.09/10.11	TRS:
35.09/10.11	Rules:
35.09/10.11	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.12	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.12	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.12	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.12	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.12	active(U16(tt)) -> mark(tt)
35.09/10.12	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.12	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.12	active(U23(tt)) -> mark(tt)
35.09/10.12	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.12	active(U32(tt)) -> mark(tt)
35.09/10.12	active(U41(tt)) -> mark(tt)
35.09/10.12	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.12	active(U52(tt, N)) -> mark(N)
35.09/10.12	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.12	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.12	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.12	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.12	active(isNat(0')) -> mark(tt)
35.09/10.12	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.12	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.12	active(isNatKind(0')) -> mark(tt)
35.09/10.12	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.12	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.12	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.12	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.12	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.12	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.12	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.12	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.12	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.12	active(U16(X)) -> U16(active(X))
35.09/10.12	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.12	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.12	active(U23(X)) -> U23(active(X))
35.09/10.12	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.12	active(U32(X)) -> U32(active(X))
35.09/10.12	active(U41(X)) -> U41(active(X))
35.09/10.12	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.12	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.12	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.12	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.12	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.12	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.12	active(s(X)) -> s(active(X))
35.09/10.12	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.12	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.12	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.12	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.12	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.12	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.12	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.12	U16(mark(X)) -> mark(U16(X))
35.09/10.12	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.12	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.12	U23(mark(X)) -> mark(U23(X))
35.09/10.12	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.12	U32(mark(X)) -> mark(U32(X))
35.09/10.12	U41(mark(X)) -> mark(U41(X))
35.09/10.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.12	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.12	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.12	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.12	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.12	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.12	s(mark(X)) -> mark(s(X))
35.09/10.12	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.12	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.12	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(tt) -> ok(tt)
35.09/10.12	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.12	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.12	proper(isNat(X)) -> isNat(proper(X))
35.09/10.12	proper(U16(X)) -> U16(proper(X))
35.09/10.12	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.12	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.12	proper(U23(X)) -> U23(proper(X))
35.09/10.12	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.12	proper(U32(X)) -> U32(proper(X))
35.09/10.12	proper(U41(X)) -> U41(proper(X))
35.09/10.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.12	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.12	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(s(X)) -> s(proper(X))
35.09/10.12	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.12	proper(0') -> ok(0')
35.09/10.12	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.12	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.12	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.12	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.12	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.12	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.12	isNat(ok(X)) -> ok(isNat(X))
35.09/10.12	U16(ok(X)) -> ok(U16(X))
35.09/10.12	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.12	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.12	U23(ok(X)) -> ok(U23(X))
35.09/10.12	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.12	U32(ok(X)) -> ok(U32(X))
35.09/10.12	U41(ok(X)) -> ok(U41(X))
35.09/10.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.12	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.12	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.12	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.12	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.12	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.12	s(ok(X)) -> ok(s(X))
35.09/10.12	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.12	top(mark(X)) -> top(proper(X))
35.09/10.12	top(ok(X)) -> top(active(X))
35.09/10.12	
35.09/10.12	Types:
35.09/10.12	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	tt :: tt:mark:0':ok
35.09/10.12	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	0' :: tt:mark:0':ok
35.09/10.12	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	top :: tt:mark:0':ok -> top
35.09/10.12	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.12	hole_top2_0 :: top
35.09/10.12	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.12	
35.09/10.12	
35.09/10.12	Lemmas:
35.09/10.12	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.12	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.12	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.12	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.12	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.12	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.12	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.12	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.12	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.12	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.12	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.12	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.12	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.12	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.12	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.12	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.12	U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
35.09/10.12	U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
35.09/10.12	U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)
35.09/10.12	
35.09/10.12	
35.09/10.12	Generator Equations:
35.09/10.12	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.12	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.12	
35.09/10.12	
35.09/10.12	The following defined symbols remain to be analysed:
35.09/10.12	U61, active, proper, top
35.09/10.12	
35.09/10.12	They will be analysed ascendingly in the following order:
35.09/10.12	U61 < active
35.09/10.12	active < top
35.09/10.12	U61 < proper
35.09/10.12	proper < top
35.09/10.12	
35.09/10.12	----------------------------------------
35.09/10.12	
35.09/10.12	(55) RewriteLemmaProof (LOWER BOUND(ID))
35.09/10.12	Proved the following rewrite lemma:
35.09/10.12	U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n97970_0)
35.09/10.12	
35.09/10.12	Induction Base:
35.09/10.12	U61(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
35.09/10.12	
35.09/10.12	Induction Step:
35.09/10.12	U61(gen_tt:mark:0':ok3_0(+(1, +(n97970_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
35.09/10.12	mark(U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
35.09/10.12	mark(*4_0)
35.09/10.12	
35.09/10.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
35.09/10.12	----------------------------------------
35.09/10.12	
35.09/10.12	(56)
35.09/10.12	Obligation:
35.09/10.12	TRS:
35.09/10.12	Rules:
35.09/10.12	active(U11(tt, V1, V2)) -> mark(U12(isNatKind(V1), V1, V2))
35.09/10.12	active(U12(tt, V1, V2)) -> mark(U13(isNatKind(V2), V1, V2))
35.09/10.12	active(U13(tt, V1, V2)) -> mark(U14(isNatKind(V2), V1, V2))
35.09/10.12	active(U14(tt, V1, V2)) -> mark(U15(isNat(V1), V2))
35.09/10.12	active(U15(tt, V2)) -> mark(U16(isNat(V2)))
35.09/10.12	active(U16(tt)) -> mark(tt)
35.09/10.12	active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
35.09/10.12	active(U22(tt, V1)) -> mark(U23(isNat(V1)))
35.09/10.12	active(U23(tt)) -> mark(tt)
35.09/10.12	active(U31(tt, V2)) -> mark(U32(isNatKind(V2)))
35.09/10.12	active(U32(tt)) -> mark(tt)
35.09/10.12	active(U41(tt)) -> mark(tt)
35.09/10.12	active(U51(tt, N)) -> mark(U52(isNatKind(N), N))
35.09/10.12	active(U52(tt, N)) -> mark(N)
35.09/10.12	active(U61(tt, M, N)) -> mark(U62(isNatKind(M), M, N))
35.09/10.12	active(U62(tt, M, N)) -> mark(U63(isNat(N), M, N))
35.09/10.12	active(U63(tt, M, N)) -> mark(U64(isNatKind(N), M, N))
35.09/10.12	active(U64(tt, M, N)) -> mark(s(plus(N, M)))
35.09/10.12	active(isNat(0')) -> mark(tt)
35.09/10.12	active(isNat(plus(V1, V2))) -> mark(U11(isNatKind(V1), V1, V2))
35.09/10.12	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
35.09/10.12	active(isNatKind(0')) -> mark(tt)
35.09/10.12	active(isNatKind(plus(V1, V2))) -> mark(U31(isNatKind(V1), V2))
35.09/10.12	active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1)))
35.09/10.12	active(plus(N, 0')) -> mark(U51(isNat(N), N))
35.09/10.12	active(plus(N, s(M))) -> mark(U61(isNat(M), M, N))
35.09/10.12	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
35.09/10.12	active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
35.09/10.12	active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3)
35.09/10.12	active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3)
35.09/10.12	active(U15(X1, X2)) -> U15(active(X1), X2)
35.09/10.12	active(U16(X)) -> U16(active(X))
35.09/10.12	active(U21(X1, X2)) -> U21(active(X1), X2)
35.09/10.12	active(U22(X1, X2)) -> U22(active(X1), X2)
35.09/10.12	active(U23(X)) -> U23(active(X))
35.09/10.12	active(U31(X1, X2)) -> U31(active(X1), X2)
35.09/10.12	active(U32(X)) -> U32(active(X))
35.09/10.12	active(U41(X)) -> U41(active(X))
35.09/10.12	active(U51(X1, X2)) -> U51(active(X1), X2)
35.09/10.12	active(U52(X1, X2)) -> U52(active(X1), X2)
35.09/10.12	active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
35.09/10.12	active(U62(X1, X2, X3)) -> U62(active(X1), X2, X3)
35.09/10.12	active(U63(X1, X2, X3)) -> U63(active(X1), X2, X3)
35.09/10.12	active(U64(X1, X2, X3)) -> U64(active(X1), X2, X3)
35.09/10.12	active(s(X)) -> s(active(X))
35.09/10.12	active(plus(X1, X2)) -> plus(active(X1), X2)
35.09/10.12	active(plus(X1, X2)) -> plus(X1, active(X2))
35.09/10.12	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
35.09/10.12	U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
35.09/10.12	U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3))
35.09/10.12	U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3))
35.09/10.12	U15(mark(X1), X2) -> mark(U15(X1, X2))
35.09/10.12	U16(mark(X)) -> mark(U16(X))
35.09/10.12	U21(mark(X1), X2) -> mark(U21(X1, X2))
35.09/10.12	U22(mark(X1), X2) -> mark(U22(X1, X2))
35.09/10.12	U23(mark(X)) -> mark(U23(X))
35.09/10.12	U31(mark(X1), X2) -> mark(U31(X1, X2))
35.09/10.12	U32(mark(X)) -> mark(U32(X))
35.09/10.12	U41(mark(X)) -> mark(U41(X))
35.09/10.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
35.09/10.12	U52(mark(X1), X2) -> mark(U52(X1, X2))
35.09/10.12	U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
35.09/10.12	U62(mark(X1), X2, X3) -> mark(U62(X1, X2, X3))
35.09/10.12	U63(mark(X1), X2, X3) -> mark(U63(X1, X2, X3))
35.09/10.12	U64(mark(X1), X2, X3) -> mark(U64(X1, X2, X3))
35.09/10.12	s(mark(X)) -> mark(s(X))
35.09/10.12	plus(mark(X1), X2) -> mark(plus(X1, X2))
35.09/10.12	plus(X1, mark(X2)) -> mark(plus(X1, X2))
35.09/10.12	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(tt) -> ok(tt)
35.09/10.12	proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(isNatKind(X)) -> isNatKind(proper(X))
35.09/10.12	proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U15(X1, X2)) -> U15(proper(X1), proper(X2))
35.09/10.12	proper(isNat(X)) -> isNat(proper(X))
35.09/10.12	proper(U16(X)) -> U16(proper(X))
35.09/10.12	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
35.09/10.12	proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
35.09/10.12	proper(U23(X)) -> U23(proper(X))
35.09/10.12	proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
35.09/10.12	proper(U32(X)) -> U32(proper(X))
35.09/10.12	proper(U41(X)) -> U41(proper(X))
35.09/10.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
35.09/10.12	proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
35.09/10.12	proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U62(X1, X2, X3)) -> U62(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U63(X1, X2, X3)) -> U63(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(U64(X1, X2, X3)) -> U64(proper(X1), proper(X2), proper(X3))
35.09/10.12	proper(s(X)) -> s(proper(X))
35.09/10.12	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
35.09/10.12	proper(0') -> ok(0')
35.09/10.12	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
35.09/10.12	U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
35.09/10.12	isNatKind(ok(X)) -> ok(isNatKind(X))
35.09/10.12	U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3))
35.09/10.12	U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3))
35.09/10.12	U15(ok(X1), ok(X2)) -> ok(U15(X1, X2))
35.09/10.12	isNat(ok(X)) -> ok(isNat(X))
35.09/10.12	U16(ok(X)) -> ok(U16(X))
35.09/10.12	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
35.09/10.12	U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
35.09/10.12	U23(ok(X)) -> ok(U23(X))
35.09/10.12	U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
35.09/10.12	U32(ok(X)) -> ok(U32(X))
35.09/10.12	U41(ok(X)) -> ok(U41(X))
35.09/10.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
35.09/10.12	U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
35.09/10.12	U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
35.09/10.12	U62(ok(X1), ok(X2), ok(X3)) -> ok(U62(X1, X2, X3))
35.09/10.12	U63(ok(X1), ok(X2), ok(X3)) -> ok(U63(X1, X2, X3))
35.09/10.12	U64(ok(X1), ok(X2), ok(X3)) -> ok(U64(X1, X2, X3))
35.09/10.12	s(ok(X)) -> ok(s(X))
35.09/10.12	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
35.09/10.12	top(mark(X)) -> top(proper(X))
35.09/10.12	top(ok(X)) -> top(active(X))
35.09/10.12	
35.09/10.12	Types:
35.09/10.12	active :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	tt :: tt:mark:0':ok
35.09/10.12	mark :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U13 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U14 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U15 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	isNat :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U16 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U22 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U23 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U32 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U41 :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U52 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U61 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U62 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U63 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	U64 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	s :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	0' :: tt:mark:0':ok
35.09/10.12	proper :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	ok :: tt:mark:0':ok -> tt:mark:0':ok
35.09/10.12	top :: tt:mark:0':ok -> top
35.09/10.12	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
35.09/10.12	hole_top2_0 :: top
35.09/10.12	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
35.09/10.12	
35.09/10.12	
35.09/10.12	Lemmas:
35.09/10.12	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n5_0)
35.09/10.12	U13(gen_tt:mark:0':ok3_0(+(1, n3519_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n3519_0)
35.09/10.12	U14(gen_tt:mark:0':ok3_0(+(1, n7631_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n7631_0)
35.09/10.12	U15(gen_tt:mark:0':ok3_0(+(1, n12352_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n12352_0)
35.09/10.12	U16(gen_tt:mark:0':ok3_0(+(1, n15653_0))) -> *4_0, rt in Omega(n15653_0)
35.09/10.12	U22(gen_tt:mark:0':ok3_0(+(1, n17015_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n17015_0)
35.09/10.12	U23(gen_tt:mark:0':ok3_0(+(1, n20800_0))) -> *4_0, rt in Omega(n20800_0)
35.09/10.12	U32(gen_tt:mark:0':ok3_0(+(1, n22413_0))) -> *4_0, rt in Omega(n22413_0)
35.09/10.12	U52(gen_tt:mark:0':ok3_0(+(1, n24127_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n24127_0)
35.09/10.12	U62(gen_tt:mark:0':ok3_0(+(1, n28634_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n28634_0)
35.09/10.12	U63(gen_tt:mark:0':ok3_0(+(1, n36295_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n36295_0)
35.09/10.12	U64(gen_tt:mark:0':ok3_0(+(1, n44565_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n44565_0)
35.09/10.12	s(gen_tt:mark:0':ok3_0(+(1, n53444_0))) -> *4_0, rt in Omega(n53444_0)
35.09/10.12	plus(gen_tt:mark:0':ok3_0(+(1, n56006_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n56006_0)
35.09/10.12	U11(gen_tt:mark:0':ok3_0(+(1, n62438_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n62438_0)
35.09/10.12	U21(gen_tt:mark:0':ok3_0(+(1, n72703_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n72703_0)
35.09/10.12	U31(gen_tt:mark:0':ok3_0(+(1, n79646_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n79646_0)
35.09/10.12	U41(gen_tt:mark:0':ok3_0(+(1, n86895_0))) -> *4_0, rt in Omega(n86895_0)
35.09/10.12	U51(gen_tt:mark:0':ok3_0(+(1, n90207_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n90207_0)
35.09/10.12	U61(gen_tt:mark:0':ok3_0(+(1, n97970_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n97970_0)
35.09/10.12	
35.09/10.12	
35.09/10.12	Generator Equations:
35.09/10.12	gen_tt:mark:0':ok3_0(0) <=> tt
35.09/10.12	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
35.09/10.12	
35.09/10.12	
35.09/10.12	The following defined symbols remain to be analysed:
35.09/10.12	active, proper, top
35.09/10.12	
35.09/10.12	They will be analysed ascendingly in the following order:
35.09/10.12	active < top
35.09/10.12	proper < top
35.09/10.17	EOF