3.43/1.73	YES
3.43/1.73	proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
3.43/1.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.43/1.73	
3.43/1.73	
3.43/1.73	Left Termination of the query pattern
3.43/1.73	
3.43/1.73	append(g,g,a)
3.43/1.73	
3.43/1.73	w.r.t. the given Prolog program could successfully be proven:
3.43/1.73	
3.43/1.73	(0) Prolog
3.43/1.73	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.43/1.73	(2) PiTRS
3.43/1.73	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.43/1.73	(4) PiDP
3.43/1.73	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.43/1.73	(6) PiDP
3.43/1.73	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.43/1.73	(8) PiDP
3.43/1.73	(9) PiDPToQDPProof [SOUND, 0 ms]
3.43/1.73	(10) QDP
3.43/1.73	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.43/1.73	(12) YES
3.43/1.73	
3.43/1.73	
3.43/1.73	----------------------------------------
3.43/1.73	
3.43/1.73	(0)
3.43/1.73	Obligation:
3.43/1.73	Clauses:
3.43/1.73	
3.43/1.73	append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
3.43/1.73	append([], Y, Y).
3.43/1.73	
3.43/1.73	
3.43/1.73	Query: append(g,g,a)
3.43/1.73	----------------------------------------
3.43/1.73	
3.43/1.73	(1) PrologToPiTRSProof (SOUND)
3.43/1.73	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.43/1.73	
3.43/1.73	append_in_3: (b,b,f)
3.43/1.73	
3.43/1.73	Transforming Prolog into the following Term Rewriting System:
3.43/1.73	
3.43/1.73	Pi-finite rewrite system:
3.43/1.73	The TRS R consists of the following rules:
3.43/1.73	
3.43/1.73	   append_in_gga(.(H, X), Y, .(X, Z)) -> U1_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.73	   append_in_gga([], Y, Y) -> append_out_gga([], Y, Y)
3.43/1.73	   U1_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) -> append_out_gga(.(H, X), Y, .(X, Z))
3.43/1.73	
3.43/1.73	The argument filtering Pi contains the following mapping:
3.43/1.73	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.43/1.73	
3.43/1.73	.(x1, x2)  =  .(x1, x2)
3.43/1.73	
3.43/1.73	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
3.43/1.73	
3.43/1.73	[]  =  []
3.43/1.73	
3.43/1.73	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.43/1.73	
3.43/1.73	
3.43/1.73	
3.43/1.73	
3.43/1.73	
3.43/1.73	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.43/1.73	
3.43/1.73	
3.43/1.73	
3.43/1.73	----------------------------------------
3.43/1.73	
3.43/1.73	(2)
3.43/1.73	Obligation:
3.43/1.73	Pi-finite rewrite system:
3.43/1.73	The TRS R consists of the following rules:
3.43/1.73	
3.43/1.73	   append_in_gga(.(H, X), Y, .(X, Z)) -> U1_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.73	   append_in_gga([], Y, Y) -> append_out_gga([], Y, Y)
3.43/1.73	   U1_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) -> append_out_gga(.(H, X), Y, .(X, Z))
3.43/1.73	
3.43/1.73	The argument filtering Pi contains the following mapping:
3.43/1.74	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.43/1.74	
3.43/1.74	.(x1, x2)  =  .(x1, x2)
3.43/1.74	
3.43/1.74	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
3.43/1.74	
3.43/1.74	[]  =  []
3.43/1.74	
3.43/1.74	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.43/1.74	
3.43/1.74	
3.43/1.74	
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(3) DependencyPairsProof (EQUIVALENT)
3.43/1.74	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.43/1.74	Pi DP problem:
3.43/1.74	The TRS P consists of the following rules:
3.43/1.74	
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> U1_GGA(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> APPEND_IN_GGA(X, Y, Z)
3.43/1.74	
3.43/1.74	The TRS R consists of the following rules:
3.43/1.74	
3.43/1.74	   append_in_gga(.(H, X), Y, .(X, Z)) -> U1_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.74	   append_in_gga([], Y, Y) -> append_out_gga([], Y, Y)
3.43/1.74	   U1_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) -> append_out_gga(.(H, X), Y, .(X, Z))
3.43/1.74	
3.43/1.74	The argument filtering Pi contains the following mapping:
3.43/1.74	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.43/1.74	
3.43/1.74	.(x1, x2)  =  .(x1, x2)
3.43/1.74	
3.43/1.74	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
3.43/1.74	
3.43/1.74	[]  =  []
3.43/1.74	
3.43/1.74	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.43/1.74	
3.43/1.74	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.43/1.74	
3.43/1.74	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x5)
3.43/1.74	
3.43/1.74	
3.43/1.74	We have to consider all (P,R,Pi)-chains
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(4)
3.43/1.74	Obligation:
3.43/1.74	Pi DP problem:
3.43/1.74	The TRS P consists of the following rules:
3.43/1.74	
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> U1_GGA(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> APPEND_IN_GGA(X, Y, Z)
3.43/1.74	
3.43/1.74	The TRS R consists of the following rules:
3.43/1.74	
3.43/1.74	   append_in_gga(.(H, X), Y, .(X, Z)) -> U1_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.74	   append_in_gga([], Y, Y) -> append_out_gga([], Y, Y)
3.43/1.74	   U1_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) -> append_out_gga(.(H, X), Y, .(X, Z))
3.43/1.74	
3.43/1.74	The argument filtering Pi contains the following mapping:
3.43/1.74	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.43/1.74	
3.43/1.74	.(x1, x2)  =  .(x1, x2)
3.43/1.74	
3.43/1.74	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
3.43/1.74	
3.43/1.74	[]  =  []
3.43/1.74	
3.43/1.74	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.43/1.74	
3.43/1.74	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.43/1.74	
3.43/1.74	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x2, x5)
3.43/1.74	
3.43/1.74	
3.43/1.74	We have to consider all (P,R,Pi)-chains
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(5) DependencyGraphProof (EQUIVALENT)
3.43/1.74	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(6)
3.43/1.74	Obligation:
3.43/1.74	Pi DP problem:
3.43/1.74	The TRS P consists of the following rules:
3.43/1.74	
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> APPEND_IN_GGA(X, Y, Z)
3.43/1.74	
3.43/1.74	The TRS R consists of the following rules:
3.43/1.74	
3.43/1.74	   append_in_gga(.(H, X), Y, .(X, Z)) -> U1_gga(H, X, Y, Z, append_in_gga(X, Y, Z))
3.43/1.74	   append_in_gga([], Y, Y) -> append_out_gga([], Y, Y)
3.43/1.74	   U1_gga(H, X, Y, Z, append_out_gga(X, Y, Z)) -> append_out_gga(.(H, X), Y, .(X, Z))
3.43/1.74	
3.43/1.74	The argument filtering Pi contains the following mapping:
3.43/1.74	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.43/1.74	
3.43/1.74	.(x1, x2)  =  .(x1, x2)
3.43/1.74	
3.43/1.74	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x2, x5)
3.43/1.74	
3.43/1.74	[]  =  []
3.43/1.74	
3.43/1.74	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.43/1.74	
3.43/1.74	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.43/1.74	
3.43/1.74	
3.43/1.74	We have to consider all (P,R,Pi)-chains
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(7) UsableRulesProof (EQUIVALENT)
3.43/1.74	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(8)
3.43/1.74	Obligation:
3.43/1.74	Pi DP problem:
3.43/1.74	The TRS P consists of the following rules:
3.43/1.74	
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y, .(X, Z)) -> APPEND_IN_GGA(X, Y, Z)
3.43/1.74	
3.43/1.74	R is empty.
3.43/1.74	The argument filtering Pi contains the following mapping:
3.43/1.74	.(x1, x2)  =  .(x1, x2)
3.43/1.74	
3.43/1.74	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.43/1.74	
3.43/1.74	
3.43/1.74	We have to consider all (P,R,Pi)-chains
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(9) PiDPToQDPProof (SOUND)
3.43/1.74	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(10)
3.43/1.74	Obligation:
3.43/1.74	Q DP problem:
3.43/1.74	The TRS P consists of the following rules:
3.43/1.74	
3.43/1.74	   APPEND_IN_GGA(.(H, X), Y) -> APPEND_IN_GGA(X, Y)
3.43/1.74	
3.43/1.74	R is empty.
3.43/1.74	Q is empty.
3.43/1.74	We have to consider all (P,Q,R)-chains.
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(11) QDPSizeChangeProof (EQUIVALENT)
3.43/1.74	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. 
3.43/1.74	
3.43/1.74	From the DPs we obtained the following set of size-change graphs:
3.43/1.74	*APPEND_IN_GGA(.(H, X), Y) -> APPEND_IN_GGA(X, Y)
3.43/1.74	The graph contains the following edges 1 > 1, 2 >= 2
3.43/1.74	
3.43/1.74	
3.43/1.74	----------------------------------------
3.43/1.74	
3.43/1.74	(12)
3.43/1.74	YES
3.43/1.76	EOF