5.02/2.13	YES
5.35/2.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
5.35/2.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.35/2.15	
5.35/2.15	
5.35/2.15	Left Termination of the query pattern
5.35/2.15	
5.35/2.15	perm(g,a)
5.35/2.15	
5.35/2.15	w.r.t. the given Prolog program could successfully be proven:
5.35/2.15	
5.35/2.15	(0) Prolog
5.35/2.15	(1) PrologToPiTRSProof [SOUND, 21 ms]
5.35/2.15	(2) PiTRS
5.35/2.15	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.35/2.15	(4) PiDP
5.35/2.15	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
5.35/2.15	(6) AND
5.35/2.15	    (7) PiDP
5.35/2.15	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
5.35/2.15	        (9) PiDP
5.35/2.15	        (10) PiDPToQDPProof [SOUND, 0 ms]
5.35/2.15	        (11) QDP
5.35/2.15	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
5.35/2.15	        (13) YES
5.35/2.15	    (14) PiDP
5.35/2.15	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
5.35/2.15	        (16) PiDP
5.35/2.15	        (17) PiDPToQDPProof [SOUND, 3 ms]
5.35/2.15	        (18) QDP
5.35/2.15	        (19) MRRProof [EQUIVALENT, 2 ms]
5.35/2.15	        (20) QDP
5.35/2.15	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
5.35/2.15	        (22) YES
5.35/2.15	
5.35/2.15	
5.35/2.15	----------------------------------------
5.35/2.15	
5.35/2.15	(0)
5.35/2.15	Obligation:
5.35/2.15	Clauses:
5.35/2.15	
5.35/2.15	perm([], []).
5.35/2.15	perm(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), perm(W, V)).
5.35/2.15	delete(X, .(X, Y), Y).
5.35/2.15	delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).
5.35/2.15	
5.35/2.15	
5.35/2.15	Query: perm(g,a)
5.35/2.15	----------------------------------------
5.35/2.15	
5.35/2.15	(1) PrologToPiTRSProof (SOUND)
5.35/2.15	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.35/2.15	
5.35/2.15	perm_in_2: (b,f)
5.35/2.15	
5.35/2.15	delete_in_3: (f,b,f)
5.35/2.15	
5.35/2.15	Transforming Prolog into the following Term Rewriting System:
5.35/2.15	
5.35/2.15	Pi-finite rewrite system:
5.35/2.15	The TRS R consists of the following rules:
5.35/2.15	
5.35/2.15	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.15	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.15	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.15	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.15	
5.35/2.15	The argument filtering Pi contains the following mapping:
5.35/2.15	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.15	
5.35/2.15	[]  =  []
5.35/2.15	
5.35/2.15	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.15	
5.35/2.15	.(x1, x2)  =  .(x1, x2)
5.35/2.15	
5.35/2.15	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.15	
5.35/2.15	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.15	
5.35/2.15	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.15	
5.35/2.15	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.15	
5.35/2.15	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.15	
5.35/2.15	
5.35/2.15	
5.35/2.15	
5.35/2.15	
5.35/2.15	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.35/2.15	
5.35/2.15	
5.35/2.15	
5.35/2.15	----------------------------------------
5.35/2.15	
5.35/2.15	(2)
5.35/2.15	Obligation:
5.35/2.15	Pi-finite rewrite system:
5.35/2.15	The TRS R consists of the following rules:
5.35/2.15	
5.35/2.15	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.15	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.15	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.15	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.15	
5.35/2.15	The argument filtering Pi contains the following mapping:
5.35/2.15	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.15	
5.35/2.15	[]  =  []
5.35/2.15	
5.35/2.15	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.15	
5.35/2.15	.(x1, x2)  =  .(x1, x2)
5.35/2.15	
5.35/2.15	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.15	
5.35/2.15	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.15	
5.35/2.15	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.15	
5.35/2.15	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.15	
5.35/2.15	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.15	
5.35/2.15	
5.35/2.15	
5.35/2.15	----------------------------------------
5.35/2.15	
5.35/2.15	(3) DependencyPairsProof (EQUIVALENT)
5.35/2.15	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.35/2.15	Pi DP problem:
5.35/2.15	The TRS P consists of the following rules:
5.35/2.15	
5.35/2.15	   PERM_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   PERM_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W)
5.35/2.15	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z)
5.35/2.15	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERM_IN_GA(W, V)
5.35/2.15	
5.35/2.15	The TRS R consists of the following rules:
5.35/2.15	
5.35/2.15	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.15	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.15	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.15	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.15	
5.35/2.15	The argument filtering Pi contains the following mapping:
5.35/2.15	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.15	
5.35/2.15	[]  =  []
5.35/2.15	
5.35/2.15	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.15	
5.35/2.15	.(x1, x2)  =  .(x1, x2)
5.35/2.15	
5.35/2.15	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.15	
5.35/2.15	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.15	
5.35/2.15	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.15	
5.35/2.15	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.15	
5.35/2.15	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.15	
5.35/2.15	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
5.35/2.15	
5.35/2.15	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
5.35/2.15	
5.35/2.15	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
5.35/2.15	
5.35/2.15	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
5.35/2.15	
5.35/2.15	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
5.35/2.15	
5.35/2.15	
5.35/2.15	We have to consider all (P,R,Pi)-chains
5.35/2.15	----------------------------------------
5.35/2.15	
5.35/2.15	(4)
5.35/2.15	Obligation:
5.35/2.15	Pi DP problem:
5.35/2.15	The TRS P consists of the following rules:
5.35/2.15	
5.35/2.15	   PERM_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   PERM_IN_GA(.(X, Y), .(U, V)) -> DELETE_IN_AGA(U, .(X, Y), W)
5.35/2.15	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z)
5.35/2.15	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_GA(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERM_IN_GA(W, V)
5.35/2.15	
5.35/2.15	The TRS R consists of the following rules:
5.35/2.15	
5.35/2.15	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.15	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.15	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.15	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.15	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.15	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.15	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.15	
5.35/2.15	The argument filtering Pi contains the following mapping:
5.35/2.15	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.15	
5.35/2.15	[]  =  []
5.35/2.15	
5.35/2.15	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.15	
5.35/2.15	.(x1, x2)  =  .(x1, x2)
5.35/2.15	
5.35/2.15	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.15	
5.35/2.15	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.15	
5.35/2.15	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.15	
5.35/2.15	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.15	
5.35/2.15	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.15	
5.35/2.15	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
5.35/2.15	
5.35/2.15	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
5.35/2.16	
5.35/2.16	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
5.35/2.16	
5.35/2.16	U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
5.35/2.16	
5.35/2.16	U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
5.35/2.16	
5.35/2.16	
5.35/2.16	We have to consider all (P,R,Pi)-chains
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(5) DependencyGraphProof (EQUIVALENT)
5.35/2.16	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(6)
5.35/2.16	Complex Obligation (AND)
5.35/2.16	
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(7)
5.35/2.16	Obligation:
5.35/2.16	Pi DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z)
5.35/2.16	
5.35/2.16	The TRS R consists of the following rules:
5.35/2.16	
5.35/2.16	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.16	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.16	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.16	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.16	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.16	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.16	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.16	
5.35/2.16	The argument filtering Pi contains the following mapping:
5.35/2.16	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.16	
5.35/2.16	[]  =  []
5.35/2.16	
5.35/2.16	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.16	
5.35/2.16	.(x1, x2)  =  .(x1, x2)
5.35/2.16	
5.35/2.16	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.16	
5.35/2.16	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.16	
5.35/2.16	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.16	
5.35/2.16	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.16	
5.35/2.16	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.16	
5.35/2.16	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
5.35/2.16	
5.35/2.16	
5.35/2.16	We have to consider all (P,R,Pi)-chains
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(8) UsableRulesProof (EQUIVALENT)
5.35/2.16	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(9)
5.35/2.16	Obligation:
5.35/2.16	Pi DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   DELETE_IN_AGA(U, .(X, Y), .(X, Z)) -> DELETE_IN_AGA(U, Y, Z)
5.35/2.16	
5.35/2.16	R is empty.
5.35/2.16	The argument filtering Pi contains the following mapping:
5.35/2.16	.(x1, x2)  =  .(x1, x2)
5.35/2.16	
5.35/2.16	DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
5.35/2.16	
5.35/2.16	
5.35/2.16	We have to consider all (P,R,Pi)-chains
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(10) PiDPToQDPProof (SOUND)
5.35/2.16	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(11)
5.35/2.16	Obligation:
5.35/2.16	Q DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y)
5.35/2.16	
5.35/2.16	R is empty.
5.35/2.16	Q is empty.
5.35/2.16	We have to consider all (P,Q,R)-chains.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(12) QDPSizeChangeProof (EQUIVALENT)
5.35/2.16	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
5.35/2.16	
5.35/2.16	From the DPs we obtained the following set of size-change graphs:
5.35/2.16	*DELETE_IN_AGA(.(X, Y)) -> DELETE_IN_AGA(Y)
5.35/2.16	The graph contains the following edges 1 > 1
5.35/2.16	
5.35/2.16	
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(13)
5.35/2.16	YES
5.35/2.16	
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(14)
5.35/2.16	Obligation:
5.35/2.16	Pi DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERM_IN_GA(W, V)
5.35/2.16	   PERM_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.16	
5.35/2.16	The TRS R consists of the following rules:
5.35/2.16	
5.35/2.16	   perm_in_ga([], []) -> perm_out_ga([], [])
5.35/2.16	   perm_in_ga(.(X, Y), .(U, V)) -> U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.16	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.16	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.16	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.16	   U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> U2_ga(X, Y, U, V, perm_in_ga(W, V))
5.35/2.16	   U2_ga(X, Y, U, V, perm_out_ga(W, V)) -> perm_out_ga(.(X, Y), .(U, V))
5.35/2.16	
5.35/2.16	The argument filtering Pi contains the following mapping:
5.35/2.16	perm_in_ga(x1, x2)  =  perm_in_ga(x1)
5.35/2.16	
5.35/2.16	[]  =  []
5.35/2.16	
5.35/2.16	perm_out_ga(x1, x2)  =  perm_out_ga(x2)
5.35/2.16	
5.35/2.16	.(x1, x2)  =  .(x1, x2)
5.35/2.16	
5.35/2.16	U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
5.35/2.16	
5.35/2.16	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.16	
5.35/2.16	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.16	
5.35/2.16	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.16	
5.35/2.16	U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
5.35/2.16	
5.35/2.16	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
5.35/2.16	
5.35/2.16	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
5.35/2.16	
5.35/2.16	
5.35/2.16	We have to consider all (P,R,Pi)-chains
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(15) UsableRulesProof (EQUIVALENT)
5.35/2.16	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(16)
5.35/2.16	Obligation:
5.35/2.16	Pi DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) -> PERM_IN_GA(W, V)
5.35/2.16	   PERM_IN_GA(.(X, Y), .(U, V)) -> U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
5.35/2.16	
5.35/2.16	The TRS R consists of the following rules:
5.35/2.16	
5.35/2.16	   delete_in_aga(X, .(X, Y), Y) -> delete_out_aga(X, .(X, Y), Y)
5.35/2.16	   delete_in_aga(U, .(X, Y), .(X, Z)) -> U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
5.35/2.16	   U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) -> delete_out_aga(U, .(X, Y), .(X, Z))
5.35/2.16	
5.35/2.16	The argument filtering Pi contains the following mapping:
5.35/2.16	.(x1, x2)  =  .(x1, x2)
5.35/2.16	
5.35/2.16	delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
5.35/2.16	
5.35/2.16	delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
5.35/2.16	
5.35/2.16	U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
5.35/2.16	
5.35/2.16	PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
5.35/2.16	
5.35/2.16	U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
5.35/2.16	
5.35/2.16	
5.35/2.16	We have to consider all (P,R,Pi)-chains
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(17) PiDPToQDPProof (SOUND)
5.35/2.16	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(18)
5.35/2.16	Obligation:
5.35/2.16	Q DP problem:
5.35/2.16	The TRS P consists of the following rules:
5.35/2.16	
5.35/2.16	   U1_GA(delete_out_aga(U, W)) -> PERM_IN_GA(W)
5.35/2.16	   PERM_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y)))
5.35/2.16	
5.35/2.16	The TRS R consists of the following rules:
5.35/2.16	
5.35/2.16	   delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y)
5.35/2.16	   delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y))
5.35/2.16	   U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z))
5.35/2.16	
5.35/2.16	The set Q consists of the following terms:
5.35/2.16	
5.35/2.16	   delete_in_aga(x0)
5.35/2.16	   U3_aga(x0, x1)
5.35/2.16	
5.35/2.16	We have to consider all (P,Q,R)-chains.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(19) MRRProof (EQUIVALENT)
5.35/2.16	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
5.35/2.16	
5.35/2.16	Strictly oriented dependency pairs:
5.35/2.16	
5.35/2.16	   U1_GA(delete_out_aga(U, W)) -> PERM_IN_GA(W)
5.35/2.16	   PERM_IN_GA(.(X, Y)) -> U1_GA(delete_in_aga(.(X, Y)))
5.35/2.16	
5.35/2.16	Strictly oriented rules of the TRS R:
5.35/2.16	
5.35/2.16	   delete_in_aga(.(X, Y)) -> delete_out_aga(X, Y)
5.35/2.16	   delete_in_aga(.(X, Y)) -> U3_aga(X, delete_in_aga(Y))
5.35/2.16	   U3_aga(X, delete_out_aga(U, Z)) -> delete_out_aga(U, .(X, Z))
5.35/2.16	
5.35/2.16	Used ordering: Knuth-Bendix order [KBO] with precedence:._2 > delete_in_aga_1 > U3_aga_2 > U1_GA_1 > PERM_IN_GA_1 > delete_out_aga_2
5.35/2.16	
5.35/2.16	and weight map:
5.35/2.16	
5.35/2.16	   delete_in_aga_1=1
5.35/2.16	   U1_GA_1=1
5.35/2.16	   PERM_IN_GA_1=3
5.35/2.16	   ._2=0
5.35/2.16	   delete_out_aga_2=1
5.35/2.16	   U3_aga_2=0
5.35/2.16	
5.35/2.16	The variable weight is 1
5.35/2.16	
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(20)
5.35/2.16	Obligation:
5.35/2.16	Q DP problem:
5.35/2.16	P is empty.
5.35/2.16	R is empty.
5.35/2.16	The set Q consists of the following terms:
5.35/2.16	
5.35/2.16	   delete_in_aga(x0)
5.35/2.16	   U3_aga(x0, x1)
5.35/2.16	
5.35/2.16	We have to consider all (P,Q,R)-chains.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(21) PisEmptyProof (EQUIVALENT)
5.35/2.16	The TRS P is empty. Hence, there is no (P,Q,R) chain.
5.35/2.16	----------------------------------------
5.35/2.16	
5.35/2.16	(22)
5.35/2.16	YES
5.35/2.19	EOF