3.71/1.80	YES
3.71/1.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.71/1.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.71/1.81	
3.71/1.81	
3.71/1.81	Termination w.r.t. Q of the given QTRS could be proven:
3.71/1.81	
3.71/1.81	(0) QTRS
3.71/1.81	(1) QTRSToCSRProof [SOUND, 0 ms]
3.71/1.81	(2) CSR
3.71/1.81	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
3.71/1.81	(4) CSR
3.71/1.81	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.71/1.81	(6) QCSDP
3.71/1.81	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.71/1.81	(8) AND
3.71/1.81	    (9) QCSDP
3.71/1.81	        (10) QCSDPSubtermProof [EQUIVALENT, 15 ms]
3.71/1.81	        (11) QCSDP
3.71/1.81	        (12) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.71/1.81	        (13) TRUE
3.71/1.81	    (14) QCSDP
3.71/1.81	        (15) QCSDPSubtermProof [EQUIVALENT, 4 ms]
3.71/1.81	        (16) QCSDP
3.71/1.81	        (17) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.71/1.81	        (18) TRUE
3.71/1.81	
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(0)
3.71/1.81	Obligation:
3.71/1.81	Q restricted rewrite system:
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
3.71/1.81	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
3.71/1.81	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
3.71/1.81	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
3.71/1.81	   active(plus(N, 0)) -> mark(N)
3.71/1.81	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
3.71/1.81	   active(x(N, 0)) -> mark(0)
3.71/1.81	   active(x(N, s(M))) -> mark(U21(tt, M, N))
3.71/1.81	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
3.71/1.81	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
3.71/1.81	   active(s(X)) -> s(active(X))
3.71/1.81	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.71/1.81	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.71/1.81	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
3.71/1.81	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
3.71/1.81	   active(x(X1, X2)) -> x(active(X1), X2)
3.71/1.81	   active(x(X1, X2)) -> x(X1, active(X2))
3.71/1.81	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
3.71/1.81	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
3.71/1.81	   s(mark(X)) -> mark(s(X))
3.71/1.81	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.71/1.81	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.71/1.81	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
3.71/1.81	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
3.71/1.81	   x(mark(X1), X2) -> mark(x(X1, X2))
3.71/1.81	   x(X1, mark(X2)) -> mark(x(X1, X2))
3.71/1.81	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(tt) -> ok(tt)
3.71/1.81	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(s(X)) -> s(proper(X))
3.71/1.81	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.71/1.81	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
3.71/1.81	   proper(0) -> ok(0)
3.71/1.81	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
3.71/1.81	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
3.71/1.81	   s(ok(X)) -> ok(s(X))
3.71/1.81	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.71/1.81	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
3.71/1.81	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
3.71/1.81	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
3.71/1.81	   top(mark(X)) -> top(proper(X))
3.71/1.81	   top(ok(X)) -> top(active(X))
3.71/1.81	
3.71/1.81	The set Q consists of the following terms:
3.71/1.81	
3.71/1.81	   active(U11(x0, x1, x2))
3.71/1.81	   active(U12(x0, x1, x2))
3.71/1.81	   active(s(x0))
3.71/1.81	   active(plus(x0, x1))
3.71/1.81	   active(U21(x0, x1, x2))
3.71/1.81	   active(U22(x0, x1, x2))
3.71/1.81	   active(x(x0, x1))
3.71/1.81	   U11(mark(x0), x1, x2)
3.71/1.81	   U12(mark(x0), x1, x2)
3.71/1.81	   s(mark(x0))
3.71/1.81	   plus(mark(x0), x1)
3.71/1.81	   plus(x0, mark(x1))
3.71/1.81	   U21(mark(x0), x1, x2)
3.71/1.81	   U22(mark(x0), x1, x2)
3.71/1.81	   x(mark(x0), x1)
3.71/1.81	   x(x0, mark(x1))
3.71/1.81	   proper(U11(x0, x1, x2))
3.71/1.81	   proper(tt)
3.71/1.81	   proper(U12(x0, x1, x2))
3.71/1.81	   proper(s(x0))
3.71/1.81	   proper(plus(x0, x1))
3.71/1.81	   proper(U21(x0, x1, x2))
3.71/1.81	   proper(U22(x0, x1, x2))
3.71/1.81	   proper(x(x0, x1))
3.71/1.81	   proper(0)
3.71/1.81	   U11(ok(x0), ok(x1), ok(x2))
3.71/1.81	   U12(ok(x0), ok(x1), ok(x2))
3.71/1.81	   s(ok(x0))
3.71/1.81	   plus(ok(x0), ok(x1))
3.71/1.81	   U21(ok(x0), ok(x1), ok(x2))
3.71/1.81	   U22(ok(x0), ok(x1), ok(x2))
3.71/1.81	   x(ok(x0), ok(x1))
3.71/1.81	   top(mark(x0))
3.71/1.81	   top(ok(x0))
3.71/1.81	
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(1) QTRSToCSRProof (SOUND)
3.71/1.81	The following Q TRS is given: Q restricted rewrite system:
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   active(U11(tt, M, N)) -> mark(U12(tt, M, N))
3.71/1.81	   active(U12(tt, M, N)) -> mark(s(plus(N, M)))
3.71/1.81	   active(U21(tt, M, N)) -> mark(U22(tt, M, N))
3.71/1.81	   active(U22(tt, M, N)) -> mark(plus(x(N, M), N))
3.71/1.81	   active(plus(N, 0)) -> mark(N)
3.71/1.81	   active(plus(N, s(M))) -> mark(U11(tt, M, N))
3.71/1.81	   active(x(N, 0)) -> mark(0)
3.71/1.81	   active(x(N, s(M))) -> mark(U21(tt, M, N))
3.71/1.81	   active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3)
3.71/1.81	   active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3)
3.71/1.81	   active(s(X)) -> s(active(X))
3.71/1.81	   active(plus(X1, X2)) -> plus(active(X1), X2)
3.71/1.81	   active(plus(X1, X2)) -> plus(X1, active(X2))
3.71/1.81	   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
3.71/1.81	   active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3)
3.71/1.81	   active(x(X1, X2)) -> x(active(X1), X2)
3.71/1.81	   active(x(X1, X2)) -> x(X1, active(X2))
3.71/1.81	   U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3))
3.71/1.81	   U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3))
3.71/1.81	   s(mark(X)) -> mark(s(X))
3.71/1.81	   plus(mark(X1), X2) -> mark(plus(X1, X2))
3.71/1.81	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
3.71/1.81	   U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3))
3.71/1.81	   U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3))
3.71/1.81	   x(mark(X1), X2) -> mark(x(X1, X2))
3.71/1.81	   x(X1, mark(X2)) -> mark(x(X1, X2))
3.71/1.81	   proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(tt) -> ok(tt)
3.71/1.81	   proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(s(X)) -> s(proper(X))
3.71/1.81	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
3.71/1.81	   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3))
3.71/1.81	   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
3.71/1.81	   proper(0) -> ok(0)
3.71/1.81	   U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3))
3.71/1.81	   U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3))
3.71/1.81	   s(ok(X)) -> ok(s(X))
3.71/1.81	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
3.71/1.81	   U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3))
3.71/1.81	   U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3))
3.71/1.81	   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
3.71/1.81	   top(mark(X)) -> top(proper(X))
3.71/1.81	   top(ok(X)) -> top(active(X))
3.71/1.81	
3.71/1.81	The set Q consists of the following terms:
3.71/1.81	
3.71/1.81	   active(U11(x0, x1, x2))
3.71/1.81	   active(U12(x0, x1, x2))
3.71/1.81	   active(s(x0))
3.71/1.81	   active(plus(x0, x1))
3.71/1.81	   active(U21(x0, x1, x2))
3.71/1.81	   active(U22(x0, x1, x2))
3.71/1.81	   active(x(x0, x1))
3.71/1.81	   U11(mark(x0), x1, x2)
3.71/1.81	   U12(mark(x0), x1, x2)
3.71/1.81	   s(mark(x0))
3.71/1.81	   plus(mark(x0), x1)
3.71/1.81	   plus(x0, mark(x1))
3.71/1.81	   U21(mark(x0), x1, x2)
3.71/1.81	   U22(mark(x0), x1, x2)
3.71/1.81	   x(mark(x0), x1)
3.71/1.81	   x(x0, mark(x1))
3.71/1.81	   proper(U11(x0, x1, x2))
3.71/1.81	   proper(tt)
3.71/1.81	   proper(U12(x0, x1, x2))
3.71/1.81	   proper(s(x0))
3.71/1.81	   proper(plus(x0, x1))
3.71/1.81	   proper(U21(x0, x1, x2))
3.71/1.81	   proper(U22(x0, x1, x2))
3.71/1.81	   proper(x(x0, x1))
3.71/1.81	   proper(0)
3.71/1.81	   U11(ok(x0), ok(x1), ok(x2))
3.71/1.81	   U12(ok(x0), ok(x1), ok(x2))
3.71/1.81	   s(ok(x0))
3.71/1.81	   plus(ok(x0), ok(x1))
3.71/1.81	   U21(ok(x0), ok(x1), ok(x2))
3.71/1.81	   U22(ok(x0), ok(x1), ok(x2))
3.71/1.81	   x(ok(x0), ok(x1))
3.71/1.81	   top(mark(x0))
3.71/1.81	   top(ok(x0))
3.71/1.81	
3.71/1.81	Special symbols used for the transformation (see [GM04]):
3.71/1.81	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.71/1.81	The replacement map contains the following entries:
3.71/1.81	
3.71/1.81	U11: {1}
3.71/1.81	tt: empty set
3.71/1.81	U12: {1}
3.71/1.81	s: {1}
3.71/1.81	plus: {1, 2}
3.71/1.81	U21: {1}
3.71/1.81	U22: {1}
3.71/1.81	x: {1, 2}
3.71/1.81	0: empty set
3.71/1.81	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(2)
3.71/1.81	Obligation:
3.71/1.81	Context-sensitive rewrite system:
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.81	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.81	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.81	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.81	   plus(N, 0) -> N
3.71/1.81	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.81	   x(N, 0) -> 0
3.71/1.81	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.81	
3.71/1.81	The replacement map contains the following entries:
3.71/1.81	
3.71/1.81	U11: {1}
3.71/1.81	tt: empty set
3.71/1.81	U12: {1}
3.71/1.81	s: {1}
3.71/1.81	plus: {1, 2}
3.71/1.81	U21: {1}
3.71/1.81	U22: {1}
3.71/1.81	x: {1, 2}
3.71/1.81	0: empty set
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(3) CSRInnermostProof (EQUIVALENT)
3.71/1.81	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(4)
3.71/1.81	Obligation:
3.71/1.81	Context-sensitive rewrite system:
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.81	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.81	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.81	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.81	   plus(N, 0) -> N
3.71/1.81	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.81	   x(N, 0) -> 0
3.71/1.81	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.81	
3.71/1.81	The replacement map contains the following entries:
3.71/1.81	
3.71/1.81	U11: {1}
3.71/1.81	tt: empty set
3.71/1.81	U12: {1}
3.71/1.81	s: {1}
3.71/1.81	plus: {1, 2}
3.71/1.81	U21: {1}
3.71/1.81	U22: {1}
3.71/1.81	x: {1, 2}
3.71/1.81	0: empty set
3.71/1.81	
3.71/1.81	
3.71/1.81	Innermost Strategy.
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(5) CSDependencyPairsProof (EQUIVALENT)
3.71/1.81	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(6)
3.71/1.81	Obligation:
3.71/1.81	Q-restricted context-sensitive dependency pair problem:
3.71/1.81	The symbols in {s_1, plus_2, x_2, PLUS_2, X_2} are replacing on all positions.
3.71/1.81	For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3, U22'_3, U21'_3} we have mu(f) = {1}.
3.71/1.81	The symbols in {U_1} are not replacing on any position.
3.71/1.81	
3.71/1.81	The ordinary context-sensitive dependency pairs DP_o are:
3.71/1.81	   U11'(tt, M, N) -> U12'(tt, M, N)
3.71/1.81	   U12'(tt, M, N) -> PLUS(N, M)
3.71/1.81	   U21'(tt, M, N) -> U22'(tt, M, N)
3.71/1.81	   U22'(tt, M, N) -> PLUS(x(N, M), N)
3.71/1.81	   U22'(tt, M, N) -> X(N, M)
3.71/1.81	   PLUS(N, s(M)) -> U11'(tt, M, N)
3.71/1.81	   X(N, s(M)) -> U21'(tt, M, N)
3.71/1.81	
3.71/1.81	The collapsing dependency pairs are DP_c:
3.71/1.81	   U12'(tt, M, N) -> N
3.71/1.81	   U12'(tt, M, N) -> M
3.71/1.81	   U22'(tt, M, N) -> N
3.71/1.81	   U22'(tt, M, N) -> M
3.71/1.81	
3.71/1.81	
3.71/1.81	The hidden terms of R are:
3.71/1.81	none
3.71/1.81	
3.71/1.81	Every hiding context is built from:none
3.71/1.81	
3.71/1.81	Hence, the new unhiding pairs DP_u are :
3.71/1.81	   U12'(tt, M, N) -> U(N)
3.71/1.81	   U12'(tt, M, N) -> U(M)
3.71/1.81	   U22'(tt, M, N) -> U(N)
3.71/1.81	   U22'(tt, M, N) -> U(M)
3.71/1.81	
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.81	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.81	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.81	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.81	   plus(N, 0) -> N
3.71/1.81	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.81	   x(N, 0) -> 0
3.71/1.81	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.81	
3.71/1.81	The set Q consists of the following terms:
3.71/1.81	
3.71/1.81	   U11(tt, x0, x1)
3.71/1.81	   U12(tt, x0, x1)
3.71/1.81	   U21(tt, x0, x1)
3.71/1.81	   U22(tt, x0, x1)
3.71/1.81	   plus(x0, 0)
3.71/1.81	   plus(x0, s(x1))
3.71/1.81	   x(x0, 0)
3.71/1.81	   x(x0, s(x1))
3.71/1.81	
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(7) QCSDependencyGraphProof (EQUIVALENT)
3.71/1.81	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 5 less nodes.
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(8)
3.71/1.81	Complex Obligation (AND)
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(9)
3.71/1.81	Obligation:
3.71/1.81	Q-restricted context-sensitive dependency pair problem:
3.71/1.81	The symbols in {s_1, plus_2, x_2, PLUS_2} are replacing on all positions.
3.71/1.81	For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3} we have mu(f) = {1}.
3.71/1.81	
3.71/1.81	The TRS P consists of the following rules:
3.71/1.81	
3.71/1.81	   U12'(tt, M, N) -> PLUS(N, M)
3.71/1.81	   PLUS(N, s(M)) -> U11'(tt, M, N)
3.71/1.81	   U11'(tt, M, N) -> U12'(tt, M, N)
3.71/1.81	
3.71/1.81	The TRS R consists of the following rules:
3.71/1.81	
3.71/1.81	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.81	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.81	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.81	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.81	   plus(N, 0) -> N
3.71/1.81	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.81	   x(N, 0) -> 0
3.71/1.81	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.81	
3.71/1.81	The set Q consists of the following terms:
3.71/1.81	
3.71/1.81	   U11(tt, x0, x1)
3.71/1.81	   U12(tt, x0, x1)
3.71/1.81	   U21(tt, x0, x1)
3.71/1.81	   U22(tt, x0, x1)
3.71/1.81	   plus(x0, 0)
3.71/1.81	   plus(x0, s(x1))
3.71/1.81	   x(x0, 0)
3.71/1.81	   x(x0, s(x1))
3.71/1.81	
3.71/1.81	
3.71/1.81	----------------------------------------
3.71/1.81	
3.71/1.81	(10) QCSDPSubtermProof (EQUIVALENT)
3.71/1.82	We use the subterm processor [DA_EMMES].
3.71/1.82	
3.71/1.82	
3.71/1.82	The following pairs can be oriented strictly and are deleted.
3.71/1.82	
3.71/1.82	   PLUS(N, s(M)) -> U11'(tt, M, N)
3.71/1.82	The remaining pairs can at least be oriented weakly.
3.71/1.82	
3.71/1.82	   U12'(tt, M, N) -> PLUS(N, M)
3.71/1.82	   U11'(tt, M, N) -> U12'(tt, M, N)
3.71/1.82	Used ordering:  Combined order from the following AFS and order.
3.71/1.82	PLUS(x1, x2)  =  x2
3.71/1.82	
3.71/1.82	U12'(x1, x2, x3)  =  x2
3.71/1.82	
3.71/1.82	U11'(x1, x2, x3)  =  x2
3.71/1.82	
3.71/1.82	
3.71/1.82	Subterm Order
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(11)
3.71/1.82	Obligation:
3.71/1.82	Q-restricted context-sensitive dependency pair problem:
3.71/1.82	The symbols in {s_1, plus_2, x_2, PLUS_2} are replacing on all positions.
3.71/1.82	For all symbols f in {U11_3, U12_3, U21_3, U22_3, U12'_3, U11'_3} we have mu(f) = {1}.
3.71/1.82	
3.71/1.82	The TRS P consists of the following rules:
3.71/1.82	
3.71/1.82	   U12'(tt, M, N) -> PLUS(N, M)
3.71/1.82	   U11'(tt, M, N) -> U12'(tt, M, N)
3.71/1.82	
3.71/1.82	The TRS R consists of the following rules:
3.71/1.82	
3.71/1.82	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.82	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.82	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.82	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.82	   plus(N, 0) -> N
3.71/1.82	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.82	   x(N, 0) -> 0
3.71/1.82	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.82	
3.71/1.82	The set Q consists of the following terms:
3.71/1.82	
3.71/1.82	   U11(tt, x0, x1)
3.71/1.82	   U12(tt, x0, x1)
3.71/1.82	   U21(tt, x0, x1)
3.71/1.82	   U22(tt, x0, x1)
3.71/1.82	   plus(x0, 0)
3.71/1.82	   plus(x0, s(x1))
3.71/1.82	   x(x0, 0)
3.71/1.82	   x(x0, s(x1))
3.71/1.82	
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(12) QCSDependencyGraphProof (EQUIVALENT)
3.71/1.82	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(13)
3.71/1.82	TRUE
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(14)
3.71/1.82	Obligation:
3.71/1.82	Q-restricted context-sensitive dependency pair problem:
3.71/1.82	The symbols in {s_1, plus_2, x_2, X_2} are replacing on all positions.
3.71/1.82	For all symbols f in {U11_3, U12_3, U21_3, U22_3, U22'_3, U21'_3} we have mu(f) = {1}.
3.71/1.82	
3.71/1.82	The TRS P consists of the following rules:
3.71/1.82	
3.71/1.82	   U22'(tt, M, N) -> X(N, M)
3.71/1.82	   X(N, s(M)) -> U21'(tt, M, N)
3.71/1.82	   U21'(tt, M, N) -> U22'(tt, M, N)
3.71/1.82	
3.71/1.82	The TRS R consists of the following rules:
3.71/1.82	
3.71/1.82	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.82	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.82	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.82	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.82	   plus(N, 0) -> N
3.71/1.82	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.82	   x(N, 0) -> 0
3.71/1.82	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.82	
3.71/1.82	The set Q consists of the following terms:
3.71/1.82	
3.71/1.82	   U11(tt, x0, x1)
3.71/1.82	   U12(tt, x0, x1)
3.71/1.82	   U21(tt, x0, x1)
3.71/1.82	   U22(tt, x0, x1)
3.71/1.82	   plus(x0, 0)
3.71/1.82	   plus(x0, s(x1))
3.71/1.82	   x(x0, 0)
3.71/1.82	   x(x0, s(x1))
3.71/1.82	
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(15) QCSDPSubtermProof (EQUIVALENT)
3.71/1.82	We use the subterm processor [DA_EMMES].
3.71/1.82	
3.71/1.82	
3.71/1.82	The following pairs can be oriented strictly and are deleted.
3.71/1.82	
3.71/1.82	   X(N, s(M)) -> U21'(tt, M, N)
3.71/1.82	The remaining pairs can at least be oriented weakly.
3.71/1.82	
3.71/1.82	   U22'(tt, M, N) -> X(N, M)
3.71/1.82	   U21'(tt, M, N) -> U22'(tt, M, N)
3.71/1.82	Used ordering:  Combined order from the following AFS and order.
3.71/1.82	X(x1, x2)  =  x2
3.71/1.82	
3.71/1.82	U22'(x1, x2, x3)  =  x2
3.71/1.82	
3.71/1.82	U21'(x1, x2, x3)  =  x2
3.71/1.82	
3.71/1.82	
3.71/1.82	Subterm Order
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(16)
3.71/1.82	Obligation:
3.71/1.82	Q-restricted context-sensitive dependency pair problem:
3.71/1.82	The symbols in {s_1, plus_2, x_2, X_2} are replacing on all positions.
3.71/1.82	For all symbols f in {U11_3, U12_3, U21_3, U22_3, U22'_3, U21'_3} we have mu(f) = {1}.
3.71/1.82	
3.71/1.82	The TRS P consists of the following rules:
3.71/1.82	
3.71/1.82	   U22'(tt, M, N) -> X(N, M)
3.71/1.82	   U21'(tt, M, N) -> U22'(tt, M, N)
3.71/1.82	
3.71/1.82	The TRS R consists of the following rules:
3.71/1.82	
3.71/1.82	   U11(tt, M, N) -> U12(tt, M, N)
3.71/1.82	   U12(tt, M, N) -> s(plus(N, M))
3.71/1.82	   U21(tt, M, N) -> U22(tt, M, N)
3.71/1.82	   U22(tt, M, N) -> plus(x(N, M), N)
3.71/1.82	   plus(N, 0) -> N
3.71/1.82	   plus(N, s(M)) -> U11(tt, M, N)
3.71/1.82	   x(N, 0) -> 0
3.71/1.82	   x(N, s(M)) -> U21(tt, M, N)
3.71/1.82	
3.71/1.82	The set Q consists of the following terms:
3.71/1.82	
3.71/1.82	   U11(tt, x0, x1)
3.71/1.82	   U12(tt, x0, x1)
3.71/1.82	   U21(tt, x0, x1)
3.71/1.82	   U22(tt, x0, x1)
3.71/1.82	   plus(x0, 0)
3.71/1.82	   plus(x0, s(x1))
3.71/1.82	   x(x0, 0)
3.71/1.82	   x(x0, s(x1))
3.71/1.82	
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(17) QCSDependencyGraphProof (EQUIVALENT)
3.71/1.82	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
3.71/1.82	
3.71/1.82	----------------------------------------
3.71/1.82	
3.71/1.82	(18)
3.71/1.82	TRUE
3.86/1.85	EOF