36.12/10.89	WORST_CASE(Omega(n^1), O(n^1))
36.12/10.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
36.12/10.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
36.12/10.91	
36.12/10.91	
36.12/10.91	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
36.12/10.91	
36.12/10.91	(0) CpxTRS
36.12/10.91	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
36.12/10.91	(2) CpxTRS
36.12/10.91	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
36.12/10.91	    (4) CpxTRS
36.12/10.91	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 525 ms]
36.12/10.91	    (6) BOUNDS(1, n^1)
36.12/10.91	(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
36.12/10.91	(8) CpxTRS
36.12/10.91	    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
36.12/10.91	    (10) typed CpxTrs
36.12/10.91	    (11) OrderProof [LOWER BOUND(ID), 0 ms]
36.12/10.91	    (12) typed CpxTrs
36.12/10.91	    (13) RewriteLemmaProof [LOWER BOUND(ID), 463 ms]
36.12/10.91	    (14) BEST
36.12/10.91	        (15) proven lower bound
36.12/10.91	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
36.12/10.91	            (17) BOUNDS(n^1, INF)
36.12/10.91	        (18) typed CpxTrs
36.12/10.91	            (19) RewriteLemmaProof [LOWER BOUND(ID), 115 ms]
36.12/10.91	            (20) typed CpxTrs
36.12/10.91	            (21) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
36.12/10.91	            (22) typed CpxTrs
36.12/10.91	            (23) RewriteLemmaProof [LOWER BOUND(ID), 102 ms]
36.12/10.91	            (24) typed CpxTrs
36.12/10.91	            (25) RewriteLemmaProof [LOWER BOUND(ID), 104 ms]
36.12/10.91	            (26) typed CpxTrs
36.12/10.91	            (27) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
36.12/10.91	            (28) typed CpxTrs
36.12/10.91	            (29) RewriteLemmaProof [LOWER BOUND(ID), 123 ms]
36.12/10.91	            (30) typed CpxTrs
36.12/10.91	            (31) RewriteLemmaProof [LOWER BOUND(ID), 126 ms]
36.12/10.91	            (32) typed CpxTrs
36.12/10.91	            (33) RewriteLemmaProof [LOWER BOUND(ID), 179 ms]
36.12/10.91	            (34) typed CpxTrs
36.12/10.91	            (35) RewriteLemmaProof [LOWER BOUND(ID), 113 ms]
36.12/10.91	            (36) typed CpxTrs
36.12/10.91	            (37) RewriteLemmaProof [LOWER BOUND(ID), 174 ms]
36.12/10.91	            (38) typed CpxTrs
36.12/10.91	            (39) RewriteLemmaProof [LOWER BOUND(ID), 157 ms]
36.12/10.91	            (40) typed CpxTrs
36.12/10.91	            (41) RewriteLemmaProof [LOWER BOUND(ID), 141 ms]
36.12/10.91	            (42) typed CpxTrs
36.12/10.91	            (43) RewriteLemmaProof [LOWER BOUND(ID), 144 ms]
36.12/10.91	            (44) typed CpxTrs
36.12/10.91	            (45) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
36.12/10.91	            (46) typed CpxTrs
36.12/10.91	            (47) RewriteLemmaProof [LOWER BOUND(ID), 143 ms]
36.12/10.91	            (48) typed CpxTrs
36.12/10.91	
36.12/10.91	
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(0)
36.12/10.91	Obligation:
36.12/10.91	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
36.12/10.91	
36.12/10.91	
36.12/10.91	The TRS R consists of the following rules:
36.12/10.91	
36.12/10.91	   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.91	   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.91	   active(U13(tt)) -> mark(tt)
36.12/10.91	   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.91	   active(U22(tt)) -> mark(tt)
36.12/10.91	   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.91	   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.91	   active(U33(tt)) -> mark(tt)
36.12/10.91	   active(U41(tt, N)) -> mark(N)
36.12/10.91	   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.91	   active(U61(tt)) -> mark(0)
36.12/10.91	   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.91	   active(and(tt, X)) -> mark(X)
36.12/10.91	   active(isNat(0)) -> mark(tt)
36.12/10.91	   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.91	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.91	   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.91	   active(isNatKind(0)) -> mark(tt)
36.12/10.91	   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.91	   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.91	   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.91	   active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.91	   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.91	   active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.91	   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.91	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.91	   active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.91	   active(U13(X)) -> U13(active(X))
36.12/10.91	   active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.91	   active(U22(X)) -> U22(active(X))
36.12/10.91	   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.91	   active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.91	   active(U33(X)) -> U33(active(X))
36.12/10.91	   active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.91	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.91	   active(s(X)) -> s(active(X))
36.12/10.91	   active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.91	   active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.91	   active(U61(X)) -> U61(active(X))
36.12/10.91	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.91	   active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.91	   active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.91	   active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.91	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.91	   U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.91	   U13(mark(X)) -> mark(U13(X))
36.12/10.91	   U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.91	   U22(mark(X)) -> mark(U22(X))
36.12/10.91	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.91	   U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.91	   U33(mark(X)) -> mark(U33(X))
36.12/10.91	   U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.91	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.91	   s(mark(X)) -> mark(s(X))
36.12/10.91	   plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.91	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.91	   U61(mark(X)) -> mark(U61(X))
36.12/10.91	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.91	   x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.91	   x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.91	   and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.91	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.91	   proper(tt) -> ok(tt)
36.12/10.91	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.91	   proper(isNat(X)) -> isNat(proper(X))
36.12/10.91	   proper(U13(X)) -> U13(proper(X))
36.12/10.91	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.91	   proper(U22(X)) -> U22(proper(X))
36.12/10.91	   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.91	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.91	   proper(U33(X)) -> U33(proper(X))
36.12/10.91	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.91	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.91	   proper(s(X)) -> s(proper(X))
36.12/10.91	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.91	   proper(U61(X)) -> U61(proper(X))
36.12/10.91	   proper(0) -> ok(0)
36.12/10.91	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.91	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.91	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.91	   proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.91	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.91	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.91	   isNat(ok(X)) -> ok(isNat(X))
36.12/10.91	   U13(ok(X)) -> ok(U13(X))
36.12/10.91	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.91	   U22(ok(X)) -> ok(U22(X))
36.12/10.91	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.91	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.91	   U33(ok(X)) -> ok(U33(X))
36.12/10.91	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.91	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.91	   s(ok(X)) -> ok(s(X))
36.12/10.91	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.91	   U61(ok(X)) -> ok(U61(X))
36.12/10.91	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.91	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.91	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.91	   isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.91	   top(mark(X)) -> top(proper(X))
36.12/10.91	   top(ok(X)) -> top(active(X))
36.12/10.91	
36.12/10.91	S is empty.
36.12/10.91	Rewrite Strategy: FULL
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
36.12/10.91	The following defined symbols can occur below the 0th argument of top: proper, active
36.12/10.91	The following defined symbols can occur below the 0th argument of proper: proper, active
36.12/10.91	The following defined symbols can occur below the 0th argument of active: proper, active
36.12/10.91	
36.12/10.91	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
36.12/10.91	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.91	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.91	active(U13(tt)) -> mark(tt)
36.12/10.91	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.91	active(U22(tt)) -> mark(tt)
36.12/10.91	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.91	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.91	active(U33(tt)) -> mark(tt)
36.12/10.91	active(U41(tt, N)) -> mark(N)
36.12/10.91	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.91	active(U61(tt)) -> mark(0)
36.12/10.91	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.91	active(and(tt, X)) -> mark(X)
36.12/10.91	active(isNat(0)) -> mark(tt)
36.12/10.91	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.91	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.91	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.91	active(isNatKind(0)) -> mark(tt)
36.12/10.91	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.91	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.91	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.91	active(plus(N, 0)) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.91	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.91	active(x(N, 0)) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.91	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.91	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.91	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.91	active(U13(X)) -> U13(active(X))
36.12/10.91	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.91	active(U22(X)) -> U22(active(X))
36.12/10.91	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.91	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.91	active(U33(X)) -> U33(active(X))
36.12/10.91	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.91	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.91	active(s(X)) -> s(active(X))
36.12/10.91	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.91	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.91	active(U61(X)) -> U61(active(X))
36.12/10.91	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.91	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.91	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.91	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.91	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.91	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.91	proper(isNat(X)) -> isNat(proper(X))
36.12/10.91	proper(U13(X)) -> U13(proper(X))
36.12/10.91	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.91	proper(U22(X)) -> U22(proper(X))
36.12/10.91	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.91	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.91	proper(U33(X)) -> U33(proper(X))
36.12/10.91	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.91	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.91	proper(s(X)) -> s(proper(X))
36.12/10.91	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.91	proper(U61(X)) -> U61(proper(X))
36.12/10.91	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.91	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.91	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.91	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.91	
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(2)
36.12/10.91	Obligation:
36.12/10.91	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
36.12/10.91	
36.12/10.91	
36.12/10.91	The TRS R consists of the following rules:
36.12/10.91	
36.12/10.91	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.91	   U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.91	   U13(mark(X)) -> mark(U13(X))
36.12/10.91	   U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.91	   U22(mark(X)) -> mark(U22(X))
36.12/10.91	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.91	   U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.91	   U33(mark(X)) -> mark(U33(X))
36.12/10.91	   U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.91	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.91	   s(mark(X)) -> mark(s(X))
36.12/10.91	   plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.91	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.91	   U61(mark(X)) -> mark(U61(X))
36.12/10.91	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.91	   x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.91	   x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.91	   and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.91	   proper(tt) -> ok(tt)
36.12/10.91	   proper(0) -> ok(0)
36.12/10.91	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.91	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.91	   isNat(ok(X)) -> ok(isNat(X))
36.12/10.91	   U13(ok(X)) -> ok(U13(X))
36.12/10.91	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.91	   U22(ok(X)) -> ok(U22(X))
36.12/10.91	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.91	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.91	   U33(ok(X)) -> ok(U33(X))
36.12/10.91	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.91	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.91	   s(ok(X)) -> ok(s(X))
36.12/10.91	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.91	   U61(ok(X)) -> ok(U61(X))
36.12/10.91	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.91	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.91	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.91	   isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.91	   top(mark(X)) -> top(proper(X))
36.12/10.91	   top(ok(X)) -> top(active(X))
36.12/10.91	
36.12/10.91	S is empty.
36.12/10.91	Rewrite Strategy: FULL
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(3) RelTrsToTrsProof (UPPER BOUND(ID))
36.12/10.91	transformed relative TRS to TRS
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(4)
36.12/10.91	Obligation:
36.12/10.91	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
36.12/10.91	
36.12/10.91	
36.12/10.91	The TRS R consists of the following rules:
36.12/10.91	
36.12/10.91	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.91	   U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.91	   U13(mark(X)) -> mark(U13(X))
36.12/10.91	   U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.91	   U22(mark(X)) -> mark(U22(X))
36.12/10.91	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.91	   U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.91	   U33(mark(X)) -> mark(U33(X))
36.12/10.91	   U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.91	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.91	   s(mark(X)) -> mark(s(X))
36.12/10.91	   plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.91	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.91	   U61(mark(X)) -> mark(U61(X))
36.12/10.91	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.91	   x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.91	   x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.91	   and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.91	   proper(tt) -> ok(tt)
36.12/10.91	   proper(0) -> ok(0)
36.12/10.91	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.91	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.91	   isNat(ok(X)) -> ok(isNat(X))
36.12/10.91	   U13(ok(X)) -> ok(U13(X))
36.12/10.91	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.91	   U22(ok(X)) -> ok(U22(X))
36.12/10.91	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.91	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.91	   U33(ok(X)) -> ok(U33(X))
36.12/10.91	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.91	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.91	   s(ok(X)) -> ok(s(X))
36.12/10.91	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.91	   U61(ok(X)) -> ok(U61(X))
36.12/10.91	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.91	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.91	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.91	   isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.91	   top(mark(X)) -> top(proper(X))
36.12/10.91	   top(ok(X)) -> top(active(X))
36.12/10.91	
36.12/10.91	S is empty.
36.12/10.91	Rewrite Strategy: FULL
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(5) CpxTrsMatchBoundsTAProof (FINISHED)
36.12/10.91	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. 
36.12/10.91	
36.12/10.91	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
36.12/10.91	final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
36.12/10.91	transitions: 
36.12/10.91	mark0(0) -> 0
36.12/10.91	tt0() -> 0
36.12/10.91	ok0(0) -> 0
36.12/10.91	00() -> 0
36.12/10.91	active0(0) -> 0
36.12/10.91	U110(0, 0, 0) -> 1
36.12/10.91	U120(0, 0) -> 2
36.12/10.91	U130(0) -> 3
36.12/10.91	U210(0, 0) -> 4
36.12/10.91	U220(0) -> 5
36.12/10.91	U310(0, 0, 0) -> 6
36.12/10.91	U320(0, 0) -> 7
36.12/10.91	U330(0) -> 8
36.12/10.91	U410(0, 0) -> 9
36.12/10.91	U510(0, 0, 0) -> 10
36.12/10.91	s0(0) -> 11
36.12/10.91	plus0(0, 0) -> 12
36.12/10.91	U610(0) -> 13
36.12/10.91	U710(0, 0, 0) -> 14
36.12/10.91	x0(0, 0) -> 15
36.12/10.91	and0(0, 0) -> 16
36.12/10.91	proper0(0) -> 17
36.12/10.91	isNat0(0) -> 18
36.12/10.91	isNatKind0(0) -> 19
36.12/10.91	top0(0) -> 20
36.12/10.91	U111(0, 0, 0) -> 21
36.12/10.91	mark1(21) -> 1
36.12/10.91	U121(0, 0) -> 22
36.12/10.91	mark1(22) -> 2
36.12/10.91	U131(0) -> 23
36.12/10.91	mark1(23) -> 3
36.12/10.91	U211(0, 0) -> 24
36.12/10.91	mark1(24) -> 4
36.12/10.91	U221(0) -> 25
36.12/10.91	mark1(25) -> 5
36.12/10.91	U311(0, 0, 0) -> 26
36.12/10.91	mark1(26) -> 6
36.12/10.91	U321(0, 0) -> 27
36.12/10.91	mark1(27) -> 7
36.12/10.91	U331(0) -> 28
36.12/10.91	mark1(28) -> 8
36.12/10.91	U411(0, 0) -> 29
36.12/10.91	mark1(29) -> 9
36.12/10.91	U511(0, 0, 0) -> 30
36.12/10.91	mark1(30) -> 10
36.12/10.91	s1(0) -> 31
36.12/10.91	mark1(31) -> 11
36.12/10.91	plus1(0, 0) -> 32
36.12/10.91	mark1(32) -> 12
36.12/10.91	U611(0) -> 33
36.12/10.91	mark1(33) -> 13
36.12/10.91	U711(0, 0, 0) -> 34
36.12/10.91	mark1(34) -> 14
36.12/10.91	x1(0, 0) -> 35
36.12/10.91	mark1(35) -> 15
36.12/10.91	and1(0, 0) -> 36
36.12/10.91	mark1(36) -> 16
36.12/10.91	tt1() -> 37
36.12/10.91	ok1(37) -> 17
36.12/10.91	01() -> 38
36.12/10.91	ok1(38) -> 17
36.12/10.91	U111(0, 0, 0) -> 39
36.12/10.91	ok1(39) -> 1
36.12/10.91	U121(0, 0) -> 40
36.12/10.91	ok1(40) -> 2
36.12/10.91	isNat1(0) -> 41
36.12/10.91	ok1(41) -> 18
36.12/10.91	U131(0) -> 42
36.12/10.91	ok1(42) -> 3
36.12/10.91	U211(0, 0) -> 43
36.12/10.91	ok1(43) -> 4
36.12/10.91	U221(0) -> 44
36.12/10.91	ok1(44) -> 5
36.12/10.91	U311(0, 0, 0) -> 45
36.12/10.91	ok1(45) -> 6
36.12/10.91	U321(0, 0) -> 46
36.12/10.91	ok1(46) -> 7
36.12/10.91	U331(0) -> 47
36.12/10.91	ok1(47) -> 8
36.12/10.91	U411(0, 0) -> 48
36.12/10.91	ok1(48) -> 9
36.12/10.91	U511(0, 0, 0) -> 49
36.12/10.91	ok1(49) -> 10
36.12/10.91	s1(0) -> 50
36.12/10.91	ok1(50) -> 11
36.12/10.91	plus1(0, 0) -> 51
36.12/10.91	ok1(51) -> 12
36.12/10.91	U611(0) -> 52
36.12/10.91	ok1(52) -> 13
36.12/10.91	U711(0, 0, 0) -> 53
36.12/10.91	ok1(53) -> 14
36.12/10.91	x1(0, 0) -> 54
36.12/10.91	ok1(54) -> 15
36.12/10.91	and1(0, 0) -> 55
36.12/10.91	ok1(55) -> 16
36.12/10.91	isNatKind1(0) -> 56
36.12/10.91	ok1(56) -> 19
36.12/10.91	proper1(0) -> 57
36.12/10.91	top1(57) -> 20
36.12/10.91	active1(0) -> 58
36.12/10.91	top1(58) -> 20
36.12/10.91	mark1(21) -> 21
36.12/10.91	mark1(21) -> 39
36.12/10.91	mark1(22) -> 22
36.12/10.91	mark1(22) -> 40
36.12/10.91	mark1(23) -> 23
36.12/10.91	mark1(23) -> 42
36.12/10.91	mark1(24) -> 24
36.12/10.91	mark1(24) -> 43
36.12/10.91	mark1(25) -> 25
36.12/10.91	mark1(25) -> 44
36.12/10.91	mark1(26) -> 26
36.12/10.91	mark1(26) -> 45
36.12/10.91	mark1(27) -> 27
36.12/10.91	mark1(27) -> 46
36.12/10.91	mark1(28) -> 28
36.12/10.91	mark1(28) -> 47
36.12/10.91	mark1(29) -> 29
36.12/10.91	mark1(29) -> 48
36.12/10.91	mark1(30) -> 30
36.12/10.91	mark1(30) -> 49
36.12/10.91	mark1(31) -> 31
36.12/10.91	mark1(31) -> 50
36.12/10.91	mark1(32) -> 32
36.12/10.91	mark1(32) -> 51
36.12/10.91	mark1(33) -> 33
36.12/10.91	mark1(33) -> 52
36.12/10.91	mark1(34) -> 34
36.12/10.91	mark1(34) -> 53
36.12/10.91	mark1(35) -> 35
36.12/10.91	mark1(35) -> 54
36.12/10.91	mark1(36) -> 36
36.12/10.91	mark1(36) -> 55
36.12/10.91	ok1(37) -> 57
36.12/10.91	ok1(38) -> 57
36.12/10.91	ok1(39) -> 21
36.12/10.91	ok1(39) -> 39
36.12/10.91	ok1(40) -> 22
36.12/10.91	ok1(40) -> 40
36.12/10.91	ok1(41) -> 41
36.12/10.91	ok1(42) -> 23
36.12/10.91	ok1(42) -> 42
36.12/10.91	ok1(43) -> 24
36.12/10.91	ok1(43) -> 43
36.12/10.91	ok1(44) -> 25
36.12/10.91	ok1(44) -> 44
36.12/10.91	ok1(45) -> 26
36.12/10.91	ok1(45) -> 45
36.12/10.91	ok1(46) -> 27
36.12/10.91	ok1(46) -> 46
36.12/10.91	ok1(47) -> 28
36.12/10.91	ok1(47) -> 47
36.12/10.91	ok1(48) -> 29
36.12/10.91	ok1(48) -> 48
36.12/10.91	ok1(49) -> 30
36.12/10.91	ok1(49) -> 49
36.12/10.91	ok1(50) -> 31
36.12/10.91	ok1(50) -> 50
36.12/10.91	ok1(51) -> 32
36.12/10.91	ok1(51) -> 51
36.12/10.91	ok1(52) -> 33
36.12/10.91	ok1(52) -> 52
36.12/10.91	ok1(53) -> 34
36.12/10.91	ok1(53) -> 53
36.12/10.91	ok1(54) -> 35
36.12/10.91	ok1(54) -> 54
36.12/10.91	ok1(55) -> 36
36.12/10.91	ok1(55) -> 55
36.12/10.91	ok1(56) -> 56
36.12/10.91	active2(37) -> 59
36.12/10.91	top2(59) -> 20
36.12/10.91	active2(38) -> 59
36.12/10.91	
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(6)
36.12/10.91	BOUNDS(1, n^1)
36.12/10.91	
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(7) RenamingProof (BOTH BOUNDS(ID, ID))
36.12/10.91	Renamed function symbols to avoid clashes with predefined symbol.
36.12/10.91	----------------------------------------
36.12/10.91	
36.12/10.91	(8)
36.12/10.91	Obligation:
36.12/10.91	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
36.12/10.91	
36.12/10.91	
36.12/10.91	The TRS R consists of the following rules:
36.12/10.91	
36.12/10.91	   active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.91	   active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.91	   active(U13(tt)) -> mark(tt)
36.12/10.91	   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.91	   active(U22(tt)) -> mark(tt)
36.12/10.91	   active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.91	   active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.91	   active(U33(tt)) -> mark(tt)
36.12/10.91	   active(U41(tt, N)) -> mark(N)
36.12/10.91	   active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.91	   active(U61(tt)) -> mark(0')
36.12/10.91	   active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.91	   active(and(tt, X)) -> mark(X)
36.12/10.91	   active(isNat(0')) -> mark(tt)
36.12/10.91	   active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.91	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.91	   active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	   active(isNatKind(0')) -> mark(tt)
36.12/10.92	   active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	   active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	   active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	   active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	   active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	   active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	   active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	   active(U13(X)) -> U13(active(X))
36.12/10.92	   active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	   active(U22(X)) -> U22(active(X))
36.12/10.92	   active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	   active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	   active(U33(X)) -> U33(active(X))
36.12/10.92	   active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	   active(s(X)) -> s(active(X))
36.12/10.92	   active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	   active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	   active(U61(X)) -> U61(active(X))
36.12/10.92	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	   active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	   active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	   active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	   U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	   U13(mark(X)) -> mark(U13(X))
36.12/10.92	   U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	   U22(mark(X)) -> mark(U22(X))
36.12/10.92	   U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	   U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	   U33(mark(X)) -> mark(U33(X))
36.12/10.92	   U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	   s(mark(X)) -> mark(s(X))
36.12/10.92	   plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	   U61(mark(X)) -> mark(U61(X))
36.12/10.92	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	   x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	   x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	   and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	   proper(tt) -> ok(tt)
36.12/10.92	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	   proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	   proper(U13(X)) -> U13(proper(X))
36.12/10.92	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	   proper(U22(X)) -> U22(proper(X))
36.12/10.92	   proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	   proper(U33(X)) -> U33(proper(X))
36.12/10.92	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	   proper(s(X)) -> s(proper(X))
36.12/10.92	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	   proper(U61(X)) -> U61(proper(X))
36.12/10.92	   proper(0') -> ok(0')
36.12/10.92	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	   proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	   isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	   U13(ok(X)) -> ok(U13(X))
36.12/10.92	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	   U22(ok(X)) -> ok(U22(X))
36.12/10.92	   U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	   U33(ok(X)) -> ok(U33(X))
36.12/10.92	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	   s(ok(X)) -> ok(s(X))
36.12/10.92	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	   U61(ok(X)) -> ok(U61(X))
36.12/10.92	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	   isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	   top(mark(X)) -> top(proper(X))
36.12/10.92	   top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	S is empty.
36.12/10.92	Rewrite Strategy: FULL
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
36.12/10.92	Infered types.
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(10)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(11) OrderProof (LOWER BOUND(ID))
36.12/10.92	Heuristically decided to analyse the following defined symbols:
36.12/10.92	active, U12, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U12 < active
36.12/10.92	isNat < active
36.12/10.92	U13 < active
36.12/10.92	U22 < active
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U12 < proper
36.12/10.92	isNat < proper
36.12/10.92	U13 < proper
36.12/10.92	U22 < proper
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(12)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	U12, active, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U12 < active
36.12/10.92	isNat < active
36.12/10.92	U13 < active
36.12/10.92	U22 < active
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U12 < proper
36.12/10.92	isNat < proper
36.12/10.92	U13 < proper
36.12/10.92	U22 < proper
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(13) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.12/10.92	mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(14)
36.12/10.92	Complex Obligation (BEST)
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(15)
36.12/10.92	Obligation:
36.12/10.92	Proved the lower bound n^1 for the following obligation:
36.12/10.92	
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	U12, active, isNat, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U12 < active
36.12/10.92	isNat < active
36.12/10.92	U13 < active
36.12/10.92	U22 < active
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U12 < proper
36.12/10.92	isNat < proper
36.12/10.92	U13 < proper
36.12/10.92	U22 < proper
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(16) LowerBoundPropagationProof (FINISHED)
36.12/10.92	Propagated lower bound.
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(17)
36.12/10.92	BOUNDS(n^1, INF)
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(18)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Lemmas:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	isNat, active, U13, U22, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	isNat < active
36.12/10.92	U13 < active
36.12/10.92	U22 < active
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	isNat < proper
36.12/10.92	U13 < proper
36.12/10.92	U22 < proper
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(19) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, 0)))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, +(n1863_0, 1)))) ->_R^Omega(1)
36.12/10.92	mark(U13(gen_tt:mark:0':ok3_0(+(1, n1863_0)))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(20)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Lemmas:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	U22, active, U32, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U22 < active
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U22 < proper
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(21) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, 0)))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, +(n2583_0, 1)))) ->_R^Omega(1)
36.12/10.92	mark(U22(gen_tt:mark:0':ok3_0(+(1, n2583_0)))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(22)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Lemmas:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	U32, active, U33, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U32 < active
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U32 < proper
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(23) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	U32(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	U32(gen_tt:mark:0':ok3_0(+(1, +(n3404_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.12/10.92	mark(U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(24)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Lemmas:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.12/10.92	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	U33, active, s, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	U33 < active
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	U33 < proper
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(25) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	U33(gen_tt:mark:0':ok3_0(+(1, 0)))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	U33(gen_tt:mark:0':ok3_0(+(1, +(n5975_0, 1)))) ->_R^Omega(1)
36.12/10.92	mark(U33(gen_tt:mark:0':ok3_0(+(1, n5975_0)))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(26)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.12/10.92	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.12/10.92	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.12/10.92	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.12/10.92	active(U12(X1, X2)) -> U12(active(X1), X2)
36.12/10.92	active(U13(X)) -> U13(active(X))
36.12/10.92	active(U21(X1, X2)) -> U21(active(X1), X2)
36.12/10.92	active(U22(X)) -> U22(active(X))
36.12/10.92	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.12/10.92	active(U32(X1, X2)) -> U32(active(X1), X2)
36.12/10.92	active(U33(X)) -> U33(active(X))
36.12/10.92	active(U41(X1, X2)) -> U41(active(X1), X2)
36.12/10.92	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.12/10.92	active(s(X)) -> s(active(X))
36.12/10.92	active(plus(X1, X2)) -> plus(active(X1), X2)
36.12/10.92	active(plus(X1, X2)) -> plus(X1, active(X2))
36.12/10.92	active(U61(X)) -> U61(active(X))
36.12/10.92	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.12/10.92	active(x(X1, X2)) -> x(active(X1), X2)
36.12/10.92	active(x(X1, X2)) -> x(X1, active(X2))
36.12/10.92	active(and(X1, X2)) -> and(active(X1), X2)
36.12/10.92	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.12/10.92	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.12/10.92	U13(mark(X)) -> mark(U13(X))
36.12/10.92	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.12/10.92	U22(mark(X)) -> mark(U22(X))
36.12/10.92	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.12/10.92	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.12/10.92	U33(mark(X)) -> mark(U33(X))
36.12/10.92	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.12/10.92	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.12/10.92	s(mark(X)) -> mark(s(X))
36.12/10.92	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.12/10.92	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.12/10.92	U61(mark(X)) -> mark(U61(X))
36.12/10.92	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.12/10.92	x(mark(X1), X2) -> mark(x(X1, X2))
36.12/10.92	x(X1, mark(X2)) -> mark(x(X1, X2))
36.12/10.92	and(mark(X1), X2) -> mark(and(X1, X2))
36.12/10.92	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(tt) -> ok(tt)
36.12/10.92	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.12/10.92	proper(isNat(X)) -> isNat(proper(X))
36.12/10.92	proper(U13(X)) -> U13(proper(X))
36.12/10.92	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.12/10.92	proper(U22(X)) -> U22(proper(X))
36.12/10.92	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.12/10.92	proper(U33(X)) -> U33(proper(X))
36.12/10.92	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.12/10.92	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(s(X)) -> s(proper(X))
36.12/10.92	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.12/10.92	proper(U61(X)) -> U61(proper(X))
36.12/10.92	proper(0') -> ok(0')
36.12/10.92	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.12/10.92	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.12/10.92	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.12/10.92	proper(isNatKind(X)) -> isNatKind(proper(X))
36.12/10.92	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.12/10.92	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.12/10.92	isNat(ok(X)) -> ok(isNat(X))
36.12/10.92	U13(ok(X)) -> ok(U13(X))
36.12/10.92	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.12/10.92	U22(ok(X)) -> ok(U22(X))
36.12/10.92	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.12/10.92	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.12/10.92	U33(ok(X)) -> ok(U33(X))
36.12/10.92	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.12/10.92	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.12/10.92	s(ok(X)) -> ok(s(X))
36.12/10.92	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.12/10.92	U61(ok(X)) -> ok(U61(X))
36.12/10.92	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.12/10.92	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.12/10.92	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.12/10.92	isNatKind(ok(X)) -> ok(isNatKind(X))
36.12/10.92	top(mark(X)) -> top(proper(X))
36.12/10.92	top(ok(X)) -> top(active(X))
36.12/10.92	
36.12/10.92	Types:
36.12/10.92	active :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	tt :: tt:mark:0':ok
36.12/10.92	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	s :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	0' :: tt:mark:0':ok
36.12/10.92	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.12/10.92	top :: tt:mark:0':ok -> top
36.12/10.92	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.12/10.92	hole_top2_0 :: top
36.12/10.92	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.12/10.92	
36.12/10.92	
36.12/10.92	Lemmas:
36.12/10.92	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.12/10.92	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.12/10.92	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.12/10.92	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.12/10.92	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.12/10.92	
36.12/10.92	
36.12/10.92	Generator Equations:
36.12/10.92	gen_tt:mark:0':ok3_0(0) <=> tt
36.12/10.92	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.12/10.92	
36.12/10.92	
36.12/10.92	The following defined symbols remain to be analysed:
36.12/10.92	s, active, plus, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.12/10.92	
36.12/10.92	They will be analysed ascendingly in the following order:
36.12/10.92	s < active
36.12/10.92	plus < active
36.12/10.92	x < active
36.12/10.92	U11 < active
36.12/10.92	and < active
36.12/10.92	isNatKind < active
36.12/10.92	U21 < active
36.12/10.92	U31 < active
36.12/10.92	U41 < active
36.12/10.92	U51 < active
36.12/10.92	U61 < active
36.12/10.92	U71 < active
36.12/10.92	active < top
36.12/10.92	s < proper
36.12/10.92	plus < proper
36.12/10.92	x < proper
36.12/10.92	U11 < proper
36.12/10.92	and < proper
36.12/10.92	isNatKind < proper
36.12/10.92	U21 < proper
36.12/10.92	U31 < proper
36.12/10.92	U41 < proper
36.12/10.92	U51 < proper
36.12/10.92	U61 < proper
36.12/10.92	U71 < proper
36.12/10.92	proper < top
36.12/10.92	
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(27) RewriteLemmaProof (LOWER BOUND(ID))
36.12/10.92	Proved the following rewrite lemma:
36.12/10.92	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.12/10.92	
36.12/10.92	Induction Base:
36.12/10.92	s(gen_tt:mark:0':ok3_0(+(1, 0)))
36.12/10.92	
36.12/10.92	Induction Step:
36.12/10.92	s(gen_tt:mark:0':ok3_0(+(1, +(n7047_0, 1)))) ->_R^Omega(1)
36.12/10.92	mark(s(gen_tt:mark:0':ok3_0(+(1, n7047_0)))) ->_IH
36.12/10.92	mark(*4_0)
36.12/10.92	
36.12/10.92	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.12/10.92	----------------------------------------
36.12/10.92	
36.12/10.92	(28)
36.12/10.92	Obligation:
36.12/10.92	TRS:
36.12/10.92	Rules:
36.12/10.92	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.12/10.92	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.12/10.92	active(U13(tt)) -> mark(tt)
36.12/10.92	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.12/10.92	active(U22(tt)) -> mark(tt)
36.12/10.92	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.12/10.92	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.12/10.92	active(U33(tt)) -> mark(tt)
36.12/10.92	active(U41(tt, N)) -> mark(N)
36.12/10.92	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.12/10.92	active(U61(tt)) -> mark(0')
36.12/10.92	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.12/10.92	active(and(tt, X)) -> mark(X)
36.12/10.92	active(isNat(0')) -> mark(tt)
36.12/10.92	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.12/10.92	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.12/10.92	active(isNatKind(0')) -> mark(tt)
36.12/10.92	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.12/10.92	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.12/10.92	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	plus, active, x, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	plus < active
36.34/10.93	x < active
36.34/10.93	U11 < active
36.34/10.93	and < active
36.34/10.93	isNatKind < active
36.34/10.93	U21 < active
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	plus < proper
36.34/10.93	x < proper
36.34/10.93	U11 < proper
36.34/10.93	and < proper
36.34/10.93	isNatKind < proper
36.34/10.93	U21 < proper
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(29) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, +(n8220_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.34/10.93	mark(plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(30)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.93	active(and(tt, X)) -> mark(X)
36.34/10.93	active(isNat(0')) -> mark(tt)
36.34/10.93	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.93	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNatKind(0')) -> mark(tt)
36.34/10.93	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.93	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	x, active, U11, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	x < active
36.34/10.93	U11 < active
36.34/10.93	and < active
36.34/10.93	isNatKind < active
36.34/10.93	U21 < active
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	x < proper
36.34/10.93	U11 < proper
36.34/10.93	and < proper
36.34/10.93	isNatKind < proper
36.34/10.93	U21 < proper
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(31) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, +(n11712_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.34/10.93	mark(x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(32)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.93	active(and(tt, X)) -> mark(X)
36.34/10.93	active(isNat(0')) -> mark(tt)
36.34/10.93	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.93	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNatKind(0')) -> mark(tt)
36.34/10.93	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.93	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	U11, active, and, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	U11 < active
36.34/10.93	and < active
36.34/10.93	isNatKind < active
36.34/10.93	U21 < active
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	U11 < proper
36.34/10.93	and < proper
36.34/10.93	isNatKind < proper
36.34/10.93	U21 < proper
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(33) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, +(n15510_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
36.34/10.93	mark(U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(34)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.93	active(and(tt, X)) -> mark(X)
36.34/10.93	active(isNat(0')) -> mark(tt)
36.34/10.93	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.93	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNatKind(0')) -> mark(tt)
36.34/10.93	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.93	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	and, active, isNatKind, U21, U31, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	and < active
36.34/10.93	isNatKind < active
36.34/10.93	U21 < active
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	and < proper
36.34/10.93	isNatKind < proper
36.34/10.93	U21 < proper
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(35) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	and(gen_tt:mark:0':ok3_0(+(1, +(n21746_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.34/10.93	mark(and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(36)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.93	active(and(tt, X)) -> mark(X)
36.34/10.93	active(isNat(0')) -> mark(tt)
36.34/10.93	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.93	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNatKind(0')) -> mark(tt)
36.34/10.93	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.93	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.93	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	isNatKind, active, U21, U31, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	isNatKind < active
36.34/10.93	U21 < active
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	isNatKind < proper
36.34/10.93	U21 < proper
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(37) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	U21(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	U21(gen_tt:mark:0':ok3_0(+(1, +(n26103_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.34/10.93	mark(U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(38)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.93	active(and(tt, X)) -> mark(X)
36.34/10.93	active(isNat(0')) -> mark(tt)
36.34/10.93	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.93	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.93	active(isNatKind(0')) -> mark(tt)
36.34/10.93	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.93	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.93	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.93	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.93	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.93	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.93	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.93	active(U13(X)) -> U13(active(X))
36.34/10.93	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.93	active(U22(X)) -> U22(active(X))
36.34/10.93	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.93	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.93	active(U33(X)) -> U33(active(X))
36.34/10.93	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.93	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.93	active(s(X)) -> s(active(X))
36.34/10.93	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.93	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.93	active(U61(X)) -> U61(active(X))
36.34/10.93	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.93	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.93	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.93	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.93	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.93	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.93	U13(mark(X)) -> mark(U13(X))
36.34/10.93	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.93	U22(mark(X)) -> mark(U22(X))
36.34/10.93	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.93	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.93	U33(mark(X)) -> mark(U33(X))
36.34/10.93	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.93	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.93	s(mark(X)) -> mark(s(X))
36.34/10.93	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.93	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.93	U61(mark(X)) -> mark(U61(X))
36.34/10.93	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.93	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.93	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.93	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.93	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(tt) -> ok(tt)
36.34/10.93	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.93	proper(isNat(X)) -> isNat(proper(X))
36.34/10.93	proper(U13(X)) -> U13(proper(X))
36.34/10.93	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.93	proper(U22(X)) -> U22(proper(X))
36.34/10.93	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.93	proper(U33(X)) -> U33(proper(X))
36.34/10.93	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.93	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(s(X)) -> s(proper(X))
36.34/10.93	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.93	proper(U61(X)) -> U61(proper(X))
36.34/10.93	proper(0') -> ok(0')
36.34/10.93	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.93	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.93	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.93	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.93	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.93	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.93	isNat(ok(X)) -> ok(isNat(X))
36.34/10.93	U13(ok(X)) -> ok(U13(X))
36.34/10.93	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.93	U22(ok(X)) -> ok(U22(X))
36.34/10.93	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.93	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.93	U33(ok(X)) -> ok(U33(X))
36.34/10.93	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.93	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.93	s(ok(X)) -> ok(s(X))
36.34/10.93	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.93	U61(ok(X)) -> ok(U61(X))
36.34/10.93	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.93	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.93	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.93	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.93	top(mark(X)) -> top(proper(X))
36.34/10.93	top(ok(X)) -> top(active(X))
36.34/10.93	
36.34/10.93	Types:
36.34/10.93	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	tt :: tt:mark:0':ok
36.34/10.93	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	0' :: tt:mark:0':ok
36.34/10.93	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.93	top :: tt:mark:0':ok -> top
36.34/10.93	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.93	hole_top2_0 :: top
36.34/10.93	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.93	
36.34/10.93	
36.34/10.93	Lemmas:
36.34/10.93	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.93	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.93	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.93	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.93	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.93	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.93	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.93	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.93	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.93	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.93	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.93	
36.34/10.93	
36.34/10.93	Generator Equations:
36.34/10.93	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.93	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.93	
36.34/10.93	
36.34/10.93	The following defined symbols remain to be analysed:
36.34/10.93	U31, active, U41, U51, U61, U71, proper, top
36.34/10.93	
36.34/10.93	They will be analysed ascendingly in the following order:
36.34/10.93	U31 < active
36.34/10.93	U41 < active
36.34/10.93	U51 < active
36.34/10.93	U61 < active
36.34/10.93	U71 < active
36.34/10.93	active < top
36.34/10.93	U31 < proper
36.34/10.93	U41 < proper
36.34/10.93	U51 < proper
36.34/10.93	U61 < proper
36.34/10.93	U71 < proper
36.34/10.93	proper < top
36.34/10.93	
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(39) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.93	Proved the following rewrite lemma:
36.34/10.93	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.93	
36.34/10.93	Induction Base:
36.34/10.93	U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
36.34/10.93	
36.34/10.93	Induction Step:
36.34/10.93	U31(gen_tt:mark:0':ok3_0(+(1, +(n30718_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
36.34/10.93	mark(U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
36.34/10.93	mark(*4_0)
36.34/10.93	
36.34/10.93	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.93	----------------------------------------
36.34/10.93	
36.34/10.93	(40)
36.34/10.93	Obligation:
36.34/10.93	TRS:
36.34/10.93	Rules:
36.34/10.93	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.93	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.93	active(U13(tt)) -> mark(tt)
36.34/10.93	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.93	active(U22(tt)) -> mark(tt)
36.34/10.93	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.93	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.93	active(U33(tt)) -> mark(tt)
36.34/10.93	active(U41(tt, N)) -> mark(N)
36.34/10.93	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.93	active(U61(tt)) -> mark(0')
36.34/10.93	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.94	active(and(tt, X)) -> mark(X)
36.34/10.94	active(isNat(0')) -> mark(tt)
36.34/10.94	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.94	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNatKind(0')) -> mark(tt)
36.34/10.94	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.94	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.94	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.94	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.94	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.94	active(U13(X)) -> U13(active(X))
36.34/10.94	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.94	active(U22(X)) -> U22(active(X))
36.34/10.94	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.94	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.94	active(U33(X)) -> U33(active(X))
36.34/10.94	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.94	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.94	active(s(X)) -> s(active(X))
36.34/10.94	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.94	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.94	active(U61(X)) -> U61(active(X))
36.34/10.94	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.94	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.94	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.94	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.94	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.94	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.94	U13(mark(X)) -> mark(U13(X))
36.34/10.94	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.94	U22(mark(X)) -> mark(U22(X))
36.34/10.94	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.94	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.94	U33(mark(X)) -> mark(U33(X))
36.34/10.94	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.94	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.94	s(mark(X)) -> mark(s(X))
36.34/10.94	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.94	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.94	U61(mark(X)) -> mark(U61(X))
36.34/10.94	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.94	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.94	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.94	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.94	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(tt) -> ok(tt)
36.34/10.94	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.94	proper(isNat(X)) -> isNat(proper(X))
36.34/10.94	proper(U13(X)) -> U13(proper(X))
36.34/10.94	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.94	proper(U22(X)) -> U22(proper(X))
36.34/10.94	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.94	proper(U33(X)) -> U33(proper(X))
36.34/10.94	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.94	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(s(X)) -> s(proper(X))
36.34/10.94	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.94	proper(U61(X)) -> U61(proper(X))
36.34/10.94	proper(0') -> ok(0')
36.34/10.94	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.94	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.94	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.94	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.94	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.94	isNat(ok(X)) -> ok(isNat(X))
36.34/10.94	U13(ok(X)) -> ok(U13(X))
36.34/10.94	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.94	U22(ok(X)) -> ok(U22(X))
36.34/10.94	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.94	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.94	U33(ok(X)) -> ok(U33(X))
36.34/10.94	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.94	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.94	s(ok(X)) -> ok(s(X))
36.34/10.94	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.94	U61(ok(X)) -> ok(U61(X))
36.34/10.94	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.94	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.94	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.94	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.94	top(mark(X)) -> top(proper(X))
36.34/10.94	top(ok(X)) -> top(active(X))
36.34/10.94	
36.34/10.94	Types:
36.34/10.94	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	tt :: tt:mark:0':ok
36.34/10.94	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	0' :: tt:mark:0':ok
36.34/10.94	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	top :: tt:mark:0':ok -> top
36.34/10.94	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.94	hole_top2_0 :: top
36.34/10.94	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.94	
36.34/10.94	
36.34/10.94	Lemmas:
36.34/10.94	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.94	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.94	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.94	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.94	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.94	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.94	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.94	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.94	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.94	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.94	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.94	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.94	
36.34/10.94	
36.34/10.94	Generator Equations:
36.34/10.94	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.94	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.94	
36.34/10.94	
36.34/10.94	The following defined symbols remain to be analysed:
36.34/10.94	U41, active, U51, U61, U71, proper, top
36.34/10.94	
36.34/10.94	They will be analysed ascendingly in the following order:
36.34/10.94	U41 < active
36.34/10.94	U51 < active
36.34/10.94	U61 < active
36.34/10.94	U71 < active
36.34/10.94	active < top
36.34/10.94	U41 < proper
36.34/10.94	U51 < proper
36.34/10.94	U61 < proper
36.34/10.94	U71 < proper
36.34/10.94	proper < top
36.34/10.94	
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(41) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.94	Proved the following rewrite lemma:
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0)
36.34/10.94	
36.34/10.94	Induction Base:
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
36.34/10.94	
36.34/10.94	Induction Step:
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, +(n38487_0, 1))), gen_tt:mark:0':ok3_0(b)) ->_R^Omega(1)
36.34/10.94	mark(U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b))) ->_IH
36.34/10.94	mark(*4_0)
36.34/10.94	
36.34/10.94	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(42)
36.34/10.94	Obligation:
36.34/10.94	TRS:
36.34/10.94	Rules:
36.34/10.94	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.94	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.94	active(U13(tt)) -> mark(tt)
36.34/10.94	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.94	active(U22(tt)) -> mark(tt)
36.34/10.94	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.94	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.94	active(U33(tt)) -> mark(tt)
36.34/10.94	active(U41(tt, N)) -> mark(N)
36.34/10.94	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.94	active(U61(tt)) -> mark(0')
36.34/10.94	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.94	active(and(tt, X)) -> mark(X)
36.34/10.94	active(isNat(0')) -> mark(tt)
36.34/10.94	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.94	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNatKind(0')) -> mark(tt)
36.34/10.94	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.94	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.94	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.94	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.94	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.94	active(U13(X)) -> U13(active(X))
36.34/10.94	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.94	active(U22(X)) -> U22(active(X))
36.34/10.94	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.94	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.94	active(U33(X)) -> U33(active(X))
36.34/10.94	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.94	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.94	active(s(X)) -> s(active(X))
36.34/10.94	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.94	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.94	active(U61(X)) -> U61(active(X))
36.34/10.94	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.94	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.94	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.94	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.94	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.94	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.94	U13(mark(X)) -> mark(U13(X))
36.34/10.94	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.94	U22(mark(X)) -> mark(U22(X))
36.34/10.94	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.94	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.94	U33(mark(X)) -> mark(U33(X))
36.34/10.94	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.94	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.94	s(mark(X)) -> mark(s(X))
36.34/10.94	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.94	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.94	U61(mark(X)) -> mark(U61(X))
36.34/10.94	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.94	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.94	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.94	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.94	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(tt) -> ok(tt)
36.34/10.94	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.94	proper(isNat(X)) -> isNat(proper(X))
36.34/10.94	proper(U13(X)) -> U13(proper(X))
36.34/10.94	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.94	proper(U22(X)) -> U22(proper(X))
36.34/10.94	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.94	proper(U33(X)) -> U33(proper(X))
36.34/10.94	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.94	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(s(X)) -> s(proper(X))
36.34/10.94	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.94	proper(U61(X)) -> U61(proper(X))
36.34/10.94	proper(0') -> ok(0')
36.34/10.94	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.94	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.94	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.94	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.94	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.94	isNat(ok(X)) -> ok(isNat(X))
36.34/10.94	U13(ok(X)) -> ok(U13(X))
36.34/10.94	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.94	U22(ok(X)) -> ok(U22(X))
36.34/10.94	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.94	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.94	U33(ok(X)) -> ok(U33(X))
36.34/10.94	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.94	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.94	s(ok(X)) -> ok(s(X))
36.34/10.94	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.94	U61(ok(X)) -> ok(U61(X))
36.34/10.94	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.94	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.94	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.94	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.94	top(mark(X)) -> top(proper(X))
36.34/10.94	top(ok(X)) -> top(active(X))
36.34/10.94	
36.34/10.94	Types:
36.34/10.94	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	tt :: tt:mark:0':ok
36.34/10.94	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	0' :: tt:mark:0':ok
36.34/10.94	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	top :: tt:mark:0':ok -> top
36.34/10.94	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.94	hole_top2_0 :: top
36.34/10.94	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.94	
36.34/10.94	
36.34/10.94	Lemmas:
36.34/10.94	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.94	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.94	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.94	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.94	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.94	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.94	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.94	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.94	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.94	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.94	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.94	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0)
36.34/10.94	
36.34/10.94	
36.34/10.94	Generator Equations:
36.34/10.94	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.94	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.94	
36.34/10.94	
36.34/10.94	The following defined symbols remain to be analysed:
36.34/10.94	U51, active, U61, U71, proper, top
36.34/10.94	
36.34/10.94	They will be analysed ascendingly in the following order:
36.34/10.94	U51 < active
36.34/10.94	U61 < active
36.34/10.94	U71 < active
36.34/10.94	active < top
36.34/10.94	U51 < proper
36.34/10.94	U61 < proper
36.34/10.94	U71 < proper
36.34/10.94	proper < top
36.34/10.94	
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(43) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.94	Proved the following rewrite lemma:
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0)
36.34/10.94	
36.34/10.94	Induction Base:
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
36.34/10.94	
36.34/10.94	Induction Step:
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, +(n43812_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
36.34/10.94	mark(U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
36.34/10.94	mark(*4_0)
36.34/10.94	
36.34/10.94	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(44)
36.34/10.94	Obligation:
36.34/10.94	TRS:
36.34/10.94	Rules:
36.34/10.94	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.94	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.94	active(U13(tt)) -> mark(tt)
36.34/10.94	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.94	active(U22(tt)) -> mark(tt)
36.34/10.94	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.94	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.94	active(U33(tt)) -> mark(tt)
36.34/10.94	active(U41(tt, N)) -> mark(N)
36.34/10.94	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.94	active(U61(tt)) -> mark(0')
36.34/10.94	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.94	active(and(tt, X)) -> mark(X)
36.34/10.94	active(isNat(0')) -> mark(tt)
36.34/10.94	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.94	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNatKind(0')) -> mark(tt)
36.34/10.94	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.94	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.94	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.94	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.94	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.94	active(U13(X)) -> U13(active(X))
36.34/10.94	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.94	active(U22(X)) -> U22(active(X))
36.34/10.94	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.94	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.94	active(U33(X)) -> U33(active(X))
36.34/10.94	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.94	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.94	active(s(X)) -> s(active(X))
36.34/10.94	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.94	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.94	active(U61(X)) -> U61(active(X))
36.34/10.94	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.94	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.94	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.94	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.94	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.94	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.94	U13(mark(X)) -> mark(U13(X))
36.34/10.94	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.94	U22(mark(X)) -> mark(U22(X))
36.34/10.94	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.94	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.94	U33(mark(X)) -> mark(U33(X))
36.34/10.94	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.94	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.94	s(mark(X)) -> mark(s(X))
36.34/10.94	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.94	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.94	U61(mark(X)) -> mark(U61(X))
36.34/10.94	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.94	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.94	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.94	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.94	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(tt) -> ok(tt)
36.34/10.94	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.94	proper(isNat(X)) -> isNat(proper(X))
36.34/10.94	proper(U13(X)) -> U13(proper(X))
36.34/10.94	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.94	proper(U22(X)) -> U22(proper(X))
36.34/10.94	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.94	proper(U33(X)) -> U33(proper(X))
36.34/10.94	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.94	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(s(X)) -> s(proper(X))
36.34/10.94	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.94	proper(U61(X)) -> U61(proper(X))
36.34/10.94	proper(0') -> ok(0')
36.34/10.94	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.94	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.94	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.94	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.94	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.94	isNat(ok(X)) -> ok(isNat(X))
36.34/10.94	U13(ok(X)) -> ok(U13(X))
36.34/10.94	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.94	U22(ok(X)) -> ok(U22(X))
36.34/10.94	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.94	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.94	U33(ok(X)) -> ok(U33(X))
36.34/10.94	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.94	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.94	s(ok(X)) -> ok(s(X))
36.34/10.94	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.94	U61(ok(X)) -> ok(U61(X))
36.34/10.94	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.94	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.94	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.94	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.94	top(mark(X)) -> top(proper(X))
36.34/10.94	top(ok(X)) -> top(active(X))
36.34/10.94	
36.34/10.94	Types:
36.34/10.94	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	tt :: tt:mark:0':ok
36.34/10.94	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	0' :: tt:mark:0':ok
36.34/10.94	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	top :: tt:mark:0':ok -> top
36.34/10.94	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.94	hole_top2_0 :: top
36.34/10.94	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.94	
36.34/10.94	
36.34/10.94	Lemmas:
36.34/10.94	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.94	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.94	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.94	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.94	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.94	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.94	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.94	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.94	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.94	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.94	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.94	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0)
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0)
36.34/10.94	
36.34/10.94	
36.34/10.94	Generator Equations:
36.34/10.94	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.94	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.94	
36.34/10.94	
36.34/10.94	The following defined symbols remain to be analysed:
36.34/10.94	U61, active, U71, proper, top
36.34/10.94	
36.34/10.94	They will be analysed ascendingly in the following order:
36.34/10.94	U61 < active
36.34/10.94	U71 < active
36.34/10.94	active < top
36.34/10.94	U61 < proper
36.34/10.94	U71 < proper
36.34/10.94	proper < top
36.34/10.94	
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(45) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.94	Proved the following rewrite lemma:
36.34/10.94	U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0)
36.34/10.94	
36.34/10.94	Induction Base:
36.34/10.94	U61(gen_tt:mark:0':ok3_0(+(1, 0)))
36.34/10.94	
36.34/10.94	Induction Step:
36.34/10.94	U61(gen_tt:mark:0':ok3_0(+(1, +(n52652_0, 1)))) ->_R^Omega(1)
36.34/10.94	mark(U61(gen_tt:mark:0':ok3_0(+(1, n52652_0)))) ->_IH
36.34/10.94	mark(*4_0)
36.34/10.94	
36.34/10.94	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(46)
36.34/10.94	Obligation:
36.34/10.94	TRS:
36.34/10.94	Rules:
36.34/10.94	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.94	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.94	active(U13(tt)) -> mark(tt)
36.34/10.94	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.94	active(U22(tt)) -> mark(tt)
36.34/10.94	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.94	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.94	active(U33(tt)) -> mark(tt)
36.34/10.94	active(U41(tt, N)) -> mark(N)
36.34/10.94	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.94	active(U61(tt)) -> mark(0')
36.34/10.94	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.94	active(and(tt, X)) -> mark(X)
36.34/10.94	active(isNat(0')) -> mark(tt)
36.34/10.94	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.94	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNatKind(0')) -> mark(tt)
36.34/10.94	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.94	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.94	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.94	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.94	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.94	active(U13(X)) -> U13(active(X))
36.34/10.94	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.94	active(U22(X)) -> U22(active(X))
36.34/10.94	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.94	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.94	active(U33(X)) -> U33(active(X))
36.34/10.94	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.94	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.94	active(s(X)) -> s(active(X))
36.34/10.94	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.94	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.94	active(U61(X)) -> U61(active(X))
36.34/10.94	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.94	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.94	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.94	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.94	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.94	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.94	U13(mark(X)) -> mark(U13(X))
36.34/10.94	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.94	U22(mark(X)) -> mark(U22(X))
36.34/10.94	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.94	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.94	U33(mark(X)) -> mark(U33(X))
36.34/10.94	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.94	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.94	s(mark(X)) -> mark(s(X))
36.34/10.94	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.94	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.94	U61(mark(X)) -> mark(U61(X))
36.34/10.94	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.94	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.94	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.94	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.94	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(tt) -> ok(tt)
36.34/10.94	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.94	proper(isNat(X)) -> isNat(proper(X))
36.34/10.94	proper(U13(X)) -> U13(proper(X))
36.34/10.94	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.94	proper(U22(X)) -> U22(proper(X))
36.34/10.94	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.94	proper(U33(X)) -> U33(proper(X))
36.34/10.94	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.94	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(s(X)) -> s(proper(X))
36.34/10.94	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.94	proper(U61(X)) -> U61(proper(X))
36.34/10.94	proper(0') -> ok(0')
36.34/10.94	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.94	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.94	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.94	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.94	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.94	isNat(ok(X)) -> ok(isNat(X))
36.34/10.94	U13(ok(X)) -> ok(U13(X))
36.34/10.94	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.94	U22(ok(X)) -> ok(U22(X))
36.34/10.94	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.94	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.94	U33(ok(X)) -> ok(U33(X))
36.34/10.94	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.94	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.94	s(ok(X)) -> ok(s(X))
36.34/10.94	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.94	U61(ok(X)) -> ok(U61(X))
36.34/10.94	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.94	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.94	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.94	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.94	top(mark(X)) -> top(proper(X))
36.34/10.94	top(ok(X)) -> top(active(X))
36.34/10.94	
36.34/10.94	Types:
36.34/10.94	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	tt :: tt:mark:0':ok
36.34/10.94	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	0' :: tt:mark:0':ok
36.34/10.94	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	top :: tt:mark:0':ok -> top
36.34/10.94	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.94	hole_top2_0 :: top
36.34/10.94	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.94	
36.34/10.94	
36.34/10.94	Lemmas:
36.34/10.94	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.94	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.94	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.94	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.94	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.94	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.94	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.94	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.94	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.94	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.94	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.94	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0)
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0)
36.34/10.94	U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0)
36.34/10.94	
36.34/10.94	
36.34/10.94	Generator Equations:
36.34/10.94	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.94	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.94	
36.34/10.94	
36.34/10.94	The following defined symbols remain to be analysed:
36.34/10.94	U71, active, proper, top
36.34/10.94	
36.34/10.94	They will be analysed ascendingly in the following order:
36.34/10.94	U71 < active
36.34/10.94	active < top
36.34/10.94	U71 < proper
36.34/10.94	proper < top
36.34/10.94	
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(47) RewriteLemmaProof (LOWER BOUND(ID))
36.34/10.94	Proved the following rewrite lemma:
36.34/10.94	U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n55273_0)
36.34/10.94	
36.34/10.94	Induction Base:
36.34/10.94	U71(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))
36.34/10.94	
36.34/10.94	Induction Step:
36.34/10.94	U71(gen_tt:mark:0':ok3_0(+(1, +(n55273_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) ->_R^Omega(1)
36.34/10.94	mark(U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) ->_IH
36.34/10.94	mark(*4_0)
36.34/10.94	
36.34/10.94	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
36.34/10.94	----------------------------------------
36.34/10.94	
36.34/10.94	(48)
36.34/10.94	Obligation:
36.34/10.94	TRS:
36.34/10.94	Rules:
36.34/10.94	active(U11(tt, V1, V2)) -> mark(U12(isNat(V1), V2))
36.34/10.94	active(U12(tt, V2)) -> mark(U13(isNat(V2)))
36.34/10.94	active(U13(tt)) -> mark(tt)
36.34/10.94	active(U21(tt, V1)) -> mark(U22(isNat(V1)))
36.34/10.94	active(U22(tt)) -> mark(tt)
36.34/10.94	active(U31(tt, V1, V2)) -> mark(U32(isNat(V1), V2))
36.34/10.94	active(U32(tt, V2)) -> mark(U33(isNat(V2)))
36.34/10.94	active(U33(tt)) -> mark(tt)
36.34/10.94	active(U41(tt, N)) -> mark(N)
36.34/10.94	active(U51(tt, M, N)) -> mark(s(plus(N, M)))
36.34/10.94	active(U61(tt)) -> mark(0')
36.34/10.94	active(U71(tt, M, N)) -> mark(plus(x(N, M), N))
36.34/10.94	active(and(tt, X)) -> mark(X)
36.34/10.94	active(isNat(0')) -> mark(tt)
36.34/10.94	active(isNat(plus(V1, V2))) -> mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
36.34/10.94	active(isNat(x(V1, V2))) -> mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2))
36.34/10.94	active(isNatKind(0')) -> mark(tt)
36.34/10.94	active(isNatKind(plus(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(isNatKind(s(V1))) -> mark(isNatKind(V1))
36.34/10.94	active(isNatKind(x(V1, V2))) -> mark(and(isNatKind(V1), isNatKind(V2)))
36.34/10.94	active(plus(N, 0')) -> mark(U41(and(isNat(N), isNatKind(N)), N))
36.34/10.94	active(plus(N, s(M))) -> mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(x(N, 0')) -> mark(U61(and(isNat(N), isNatKind(N))))
36.34/10.94	active(x(N, s(M))) -> mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
36.34/10.94	active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
36.34/10.94	active(U12(X1, X2)) -> U12(active(X1), X2)
36.34/10.94	active(U13(X)) -> U13(active(X))
36.34/10.94	active(U21(X1, X2)) -> U21(active(X1), X2)
36.34/10.94	active(U22(X)) -> U22(active(X))
36.34/10.94	active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3)
36.34/10.94	active(U32(X1, X2)) -> U32(active(X1), X2)
36.34/10.94	active(U33(X)) -> U33(active(X))
36.34/10.94	active(U41(X1, X2)) -> U41(active(X1), X2)
36.34/10.94	active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
36.34/10.94	active(s(X)) -> s(active(X))
36.34/10.94	active(plus(X1, X2)) -> plus(active(X1), X2)
36.34/10.94	active(plus(X1, X2)) -> plus(X1, active(X2))
36.34/10.94	active(U61(X)) -> U61(active(X))
36.34/10.94	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
36.34/10.94	active(x(X1, X2)) -> x(active(X1), X2)
36.34/10.94	active(x(X1, X2)) -> x(X1, active(X2))
36.34/10.94	active(and(X1, X2)) -> and(active(X1), X2)
36.34/10.94	U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
36.34/10.94	U12(mark(X1), X2) -> mark(U12(X1, X2))
36.34/10.94	U13(mark(X)) -> mark(U13(X))
36.34/10.94	U21(mark(X1), X2) -> mark(U21(X1, X2))
36.34/10.94	U22(mark(X)) -> mark(U22(X))
36.34/10.94	U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3))
36.34/10.94	U32(mark(X1), X2) -> mark(U32(X1, X2))
36.34/10.94	U33(mark(X)) -> mark(U33(X))
36.34/10.94	U41(mark(X1), X2) -> mark(U41(X1, X2))
36.34/10.94	U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
36.34/10.94	s(mark(X)) -> mark(s(X))
36.34/10.94	plus(mark(X1), X2) -> mark(plus(X1, X2))
36.34/10.94	plus(X1, mark(X2)) -> mark(plus(X1, X2))
36.34/10.94	U61(mark(X)) -> mark(U61(X))
36.34/10.94	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
36.34/10.94	x(mark(X1), X2) -> mark(x(X1, X2))
36.34/10.94	x(X1, mark(X2)) -> mark(x(X1, X2))
36.34/10.94	and(mark(X1), X2) -> mark(and(X1, X2))
36.34/10.94	proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(tt) -> ok(tt)
36.34/10.94	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
36.34/10.94	proper(isNat(X)) -> isNat(proper(X))
36.34/10.94	proper(U13(X)) -> U13(proper(X))
36.34/10.94	proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
36.34/10.94	proper(U22(X)) -> U22(proper(X))
36.34/10.94	proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
36.34/10.94	proper(U33(X)) -> U33(proper(X))
36.34/10.94	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
36.34/10.94	proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(s(X)) -> s(proper(X))
36.34/10.94	proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
36.34/10.94	proper(U61(X)) -> U61(proper(X))
36.34/10.94	proper(0') -> ok(0')
36.34/10.94	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
36.34/10.94	proper(x(X1, X2)) -> x(proper(X1), proper(X2))
36.34/10.94	proper(and(X1, X2)) -> and(proper(X1), proper(X2))
36.34/10.94	proper(isNatKind(X)) -> isNatKind(proper(X))
36.34/10.94	U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
36.34/10.94	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
36.34/10.94	isNat(ok(X)) -> ok(isNat(X))
36.34/10.94	U13(ok(X)) -> ok(U13(X))
36.34/10.94	U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
36.34/10.94	U22(ok(X)) -> ok(U22(X))
36.34/10.94	U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3))
36.34/10.94	U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
36.34/10.94	U33(ok(X)) -> ok(U33(X))
36.34/10.94	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
36.34/10.94	U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
36.34/10.94	s(ok(X)) -> ok(s(X))
36.34/10.94	plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
36.34/10.94	U61(ok(X)) -> ok(U61(X))
36.34/10.94	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
36.34/10.94	x(ok(X1), ok(X2)) -> ok(x(X1, X2))
36.34/10.94	and(ok(X1), ok(X2)) -> ok(and(X1, X2))
36.34/10.94	isNatKind(ok(X)) -> ok(isNatKind(X))
36.34/10.94	top(mark(X)) -> top(proper(X))
36.34/10.94	top(ok(X)) -> top(active(X))
36.34/10.94	
36.34/10.94	Types:
36.34/10.94	active :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U11 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	tt :: tt:mark:0':ok
36.34/10.94	mark :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U12 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNat :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U13 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U21 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U22 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U31 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U32 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U33 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U41 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U51 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	s :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	plus :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	U61 :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	0' :: tt:mark:0':ok
36.34/10.94	U71 :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	x :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	and :: tt:mark:0':ok -> tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	isNatKind :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	proper :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	ok :: tt:mark:0':ok -> tt:mark:0':ok
36.34/10.94	top :: tt:mark:0':ok -> top
36.34/10.94	hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
36.34/10.94	hole_top2_0 :: top
36.34/10.94	gen_tt:mark:0':ok3_0 :: Nat -> tt:mark:0':ok
36.34/10.94	
36.34/10.94	
36.34/10.94	Lemmas:
36.34/10.94	U12(gen_tt:mark:0':ok3_0(+(1, n5_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
36.34/10.94	U13(gen_tt:mark:0':ok3_0(+(1, n1863_0))) -> *4_0, rt in Omega(n1863_0)
36.34/10.94	U22(gen_tt:mark:0':ok3_0(+(1, n2583_0))) -> *4_0, rt in Omega(n2583_0)
36.34/10.94	U32(gen_tt:mark:0':ok3_0(+(1, n3404_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n3404_0)
36.34/10.94	U33(gen_tt:mark:0':ok3_0(+(1, n5975_0))) -> *4_0, rt in Omega(n5975_0)
36.34/10.94	s(gen_tt:mark:0':ok3_0(+(1, n7047_0))) -> *4_0, rt in Omega(n7047_0)
36.34/10.94	plus(gen_tt:mark:0':ok3_0(+(1, n8220_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n8220_0)
36.34/10.94	x(gen_tt:mark:0':ok3_0(+(1, n11712_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n11712_0)
36.34/10.94	U11(gen_tt:mark:0':ok3_0(+(1, n15510_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n15510_0)
36.34/10.94	and(gen_tt:mark:0':ok3_0(+(1, n21746_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n21746_0)
36.34/10.94	U21(gen_tt:mark:0':ok3_0(+(1, n26103_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n26103_0)
36.34/10.94	U31(gen_tt:mark:0':ok3_0(+(1, n30718_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n30718_0)
36.34/10.94	U41(gen_tt:mark:0':ok3_0(+(1, n38487_0)), gen_tt:mark:0':ok3_0(b)) -> *4_0, rt in Omega(n38487_0)
36.34/10.94	U51(gen_tt:mark:0':ok3_0(+(1, n43812_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n43812_0)
36.34/10.94	U61(gen_tt:mark:0':ok3_0(+(1, n52652_0))) -> *4_0, rt in Omega(n52652_0)
36.34/10.94	U71(gen_tt:mark:0':ok3_0(+(1, n55273_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) -> *4_0, rt in Omega(n55273_0)
36.34/10.94	
36.34/10.94	
36.34/10.94	Generator Equations:
36.34/10.94	gen_tt:mark:0':ok3_0(0) <=> tt
36.34/10.94	gen_tt:mark:0':ok3_0(+(x, 1)) <=> mark(gen_tt:mark:0':ok3_0(x))
36.34/10.94	
36.34/10.94	
36.34/10.94	The following defined symbols remain to be analysed:
36.34/10.94	active, proper, top
36.34/10.94	
36.34/10.94	They will be analysed ascendingly in the following order:
36.34/10.94	active < top
36.34/10.94	proper < top
36.41/11.00	EOF