3.72/1.78	YES
3.72/1.80	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
3.72/1.80	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.72/1.80	
3.72/1.80	
3.72/1.80	Left Termination of the query pattern
3.72/1.80	
3.72/1.80	append3(g,g,g,a)
3.72/1.80	
3.72/1.80	w.r.t. the given Prolog program could successfully be proven:
3.72/1.80	
3.72/1.80	(0) Prolog
3.72/1.80	(1) PrologToPiTRSProof [SOUND, 0 ms]
3.72/1.80	(2) PiTRS
3.72/1.80	(3) DependencyPairsProof [EQUIVALENT, 0 ms]
3.72/1.80	(4) PiDP
3.72/1.80	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
3.72/1.80	(6) PiDP
3.72/1.80	(7) UsableRulesProof [EQUIVALENT, 0 ms]
3.72/1.80	(8) PiDP
3.72/1.80	(9) PiDPToQDPProof [SOUND, 0 ms]
3.72/1.80	(10) QDP
3.72/1.80	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
3.72/1.80	(12) YES
3.72/1.80	
3.72/1.80	
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(0)
3.72/1.80	Obligation:
3.72/1.80	Clauses:
3.72/1.80	
3.72/1.80	append([], L, L).
3.72/1.80	append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3).
3.72/1.80	append3(A, B, C, D) :- ','(append(A, B, E), append(E, C, D)).
3.72/1.80	
3.72/1.80	
3.72/1.80	Query: append3(g,g,g,a)
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(1) PrologToPiTRSProof (SOUND)
3.72/1.80	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
3.72/1.80	
3.72/1.80	append3_in_4: (b,b,b,f)
3.72/1.80	
3.72/1.80	append_in_3: (b,b,f)
3.72/1.80	
3.72/1.80	Transforming Prolog into the following Term Rewriting System:
3.72/1.80	
3.72/1.80	Pi-finite rewrite system:
3.72/1.80	The TRS R consists of the following rules:
3.72/1.80	
3.72/1.80	   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   append_in_gga([], L, L) -> append_out_gga([], L, L)
3.72/1.80	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
3.72/1.80	   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)
3.72/1.80	
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
3.72/1.80	
3.72/1.80	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.72/1.80	
3.72/1.80	[]  =  []
3.72/1.80	
3.72/1.80	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.72/1.80	
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
3.72/1.80	
3.72/1.80	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
3.72/1.80	
3.72/1.80	append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
3.72/1.80	
3.72/1.80	
3.72/1.80	
3.72/1.80	
3.72/1.80	
3.72/1.80	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
3.72/1.80	
3.72/1.80	
3.72/1.80	
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(2)
3.72/1.80	Obligation:
3.72/1.80	Pi-finite rewrite system:
3.72/1.80	The TRS R consists of the following rules:
3.72/1.80	
3.72/1.80	   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   append_in_gga([], L, L) -> append_out_gga([], L, L)
3.72/1.80	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
3.72/1.80	   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)
3.72/1.80	
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
3.72/1.80	
3.72/1.80	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.72/1.80	
3.72/1.80	[]  =  []
3.72/1.80	
3.72/1.80	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.72/1.80	
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
3.72/1.80	
3.72/1.80	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
3.72/1.80	
3.72/1.80	append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
3.72/1.80	
3.72/1.80	
3.72/1.80	
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(3) DependencyPairsProof (EQUIVALENT)
3.72/1.80	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
3.72/1.80	Pi DP problem:
3.72/1.80	The TRS P consists of the following rules:
3.72/1.80	
3.72/1.80	   APPEND3_IN_GGGA(A, B, C, D) -> U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   APPEND3_IN_GGGA(A, B, C, D) -> APPEND_IN_GGA(A, B, E)
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
3.72/1.80	   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> APPEND_IN_GGA(E, C, D)
3.72/1.80	
3.72/1.80	The TRS R consists of the following rules:
3.72/1.80	
3.72/1.80	   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   append_in_gga([], L, L) -> append_out_gga([], L, L)
3.72/1.80	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
3.72/1.80	   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)
3.72/1.80	
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
3.72/1.80	
3.72/1.80	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.72/1.80	
3.72/1.80	[]  =  []
3.72/1.80	
3.72/1.80	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.72/1.80	
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
3.72/1.80	
3.72/1.80	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
3.72/1.80	
3.72/1.80	append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
3.72/1.80	
3.72/1.80	APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x3, x5)
3.72/1.80	
3.72/1.80	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.72/1.80	
3.72/1.80	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
3.72/1.80	
3.72/1.80	U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
3.72/1.80	
3.72/1.80	
3.72/1.80	We have to consider all (P,R,Pi)-chains
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(4)
3.72/1.80	Obligation:
3.72/1.80	Pi DP problem:
3.72/1.80	The TRS P consists of the following rules:
3.72/1.80	
3.72/1.80	   APPEND3_IN_GGGA(A, B, C, D) -> U2_GGGA(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   APPEND3_IN_GGGA(A, B, C, D) -> APPEND_IN_GGA(A, B, E)
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U1_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
3.72/1.80	   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> U3_GGGA(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U2_GGGA(A, B, C, D, append_out_gga(A, B, E)) -> APPEND_IN_GGA(E, C, D)
3.72/1.80	
3.72/1.80	The TRS R consists of the following rules:
3.72/1.80	
3.72/1.80	   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   append_in_gga([], L, L) -> append_out_gga([], L, L)
3.72/1.80	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
3.72/1.80	   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)
3.72/1.80	
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
3.72/1.80	
3.72/1.80	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.72/1.80	
3.72/1.80	[]  =  []
3.72/1.80	
3.72/1.80	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.72/1.80	
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
3.72/1.80	
3.72/1.80	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
3.72/1.80	
3.72/1.80	append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
3.72/1.80	
3.72/1.80	APPEND3_IN_GGGA(x1, x2, x3, x4)  =  APPEND3_IN_GGGA(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_GGGA(x1, x2, x3, x4, x5)  =  U2_GGGA(x3, x5)
3.72/1.80	
3.72/1.80	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.72/1.80	
3.72/1.80	U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
3.72/1.80	
3.72/1.80	U3_GGGA(x1, x2, x3, x4, x5)  =  U3_GGGA(x5)
3.72/1.80	
3.72/1.80	
3.72/1.80	We have to consider all (P,R,Pi)-chains
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(5) DependencyGraphProof (EQUIVALENT)
3.72/1.80	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(6)
3.72/1.80	Obligation:
3.72/1.80	Pi DP problem:
3.72/1.80	The TRS P consists of the following rules:
3.72/1.80	
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
3.72/1.80	
3.72/1.80	The TRS R consists of the following rules:
3.72/1.80	
3.72/1.80	   append3_in_ggga(A, B, C, D) -> U2_ggga(A, B, C, D, append_in_gga(A, B, E))
3.72/1.80	   append_in_gga([], L, L) -> append_out_gga([], L, L)
3.72/1.80	   append_in_gga(.(H, L1), L2, .(H, L3)) -> U1_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3))
3.72/1.80	   U1_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3))
3.72/1.80	   U2_ggga(A, B, C, D, append_out_gga(A, B, E)) -> U3_ggga(A, B, C, D, append_in_gga(E, C, D))
3.72/1.80	   U3_ggga(A, B, C, D, append_out_gga(E, C, D)) -> append3_out_ggga(A, B, C, D)
3.72/1.80	
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	append3_in_ggga(x1, x2, x3, x4)  =  append3_in_ggga(x1, x2, x3)
3.72/1.80	
3.72/1.80	U2_ggga(x1, x2, x3, x4, x5)  =  U2_ggga(x3, x5)
3.72/1.80	
3.72/1.80	append_in_gga(x1, x2, x3)  =  append_in_gga(x1, x2)
3.72/1.80	
3.72/1.80	[]  =  []
3.72/1.80	
3.72/1.80	append_out_gga(x1, x2, x3)  =  append_out_gga(x3)
3.72/1.80	
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
3.72/1.80	
3.72/1.80	U3_ggga(x1, x2, x3, x4, x5)  =  U3_ggga(x5)
3.72/1.80	
3.72/1.80	append3_out_ggga(x1, x2, x3, x4)  =  append3_out_ggga(x4)
3.72/1.80	
3.72/1.80	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.72/1.80	
3.72/1.80	
3.72/1.80	We have to consider all (P,R,Pi)-chains
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(7) UsableRulesProof (EQUIVALENT)
3.72/1.80	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(8)
3.72/1.80	Obligation:
3.72/1.80	Pi DP problem:
3.72/1.80	The TRS P consists of the following rules:
3.72/1.80	
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3)
3.72/1.80	
3.72/1.80	R is empty.
3.72/1.80	The argument filtering Pi contains the following mapping:
3.72/1.80	.(x1, x2)  =  .(x1, x2)
3.72/1.80	
3.72/1.80	APPEND_IN_GGA(x1, x2, x3)  =  APPEND_IN_GGA(x1, x2)
3.72/1.80	
3.72/1.80	
3.72/1.80	We have to consider all (P,R,Pi)-chains
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(9) PiDPToQDPProof (SOUND)
3.72/1.80	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(10)
3.72/1.80	Obligation:
3.72/1.80	Q DP problem:
3.72/1.80	The TRS P consists of the following rules:
3.72/1.80	
3.72/1.80	   APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)
3.72/1.80	
3.72/1.80	R is empty.
3.72/1.80	Q is empty.
3.72/1.80	We have to consider all (P,Q,R)-chains.
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(11) QDPSizeChangeProof (EQUIVALENT)
3.72/1.80	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. 
3.72/1.80	
3.72/1.80	From the DPs we obtained the following set of size-change graphs:
3.72/1.80	*APPEND_IN_GGA(.(H, L1), L2) -> APPEND_IN_GGA(L1, L2)
3.72/1.80	The graph contains the following edges 1 > 1, 2 >= 2
3.72/1.80	
3.72/1.80	
3.72/1.80	----------------------------------------
3.72/1.80	
3.72/1.80	(12)
3.72/1.80	YES
3.93/1.84	EOF