5.89/2.32	MAYBE
5.89/2.34	proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
5.89/2.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.89/2.34	
5.89/2.34	
5.89/2.34	Left Termination of the query pattern
5.89/2.34	
5.89/2.34	p(a,a,a)
5.89/2.34	
5.89/2.34	w.r.t. the given Prolog program could not be shown:
5.89/2.34	
5.89/2.34	(0) Prolog
5.89/2.34	(1) PrologToPiTRSProof [SOUND, 0 ms]
5.89/2.34	(2) PiTRS
5.89/2.34	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
5.89/2.34	    (4) PiDP
5.89/2.34	    (5) DependencyGraphProof [EQUIVALENT, 1 ms]
5.89/2.34	    (6) PiDP
5.89/2.34	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
5.89/2.34	    (8) PiDP
5.89/2.34	    (9) PiDPToQDPProof [SOUND, 0 ms]
5.89/2.34	    (10) QDP
5.89/2.34	(11) PrologToTRSTransformerProof [SOUND, 0 ms]
5.89/2.34	(12) QTRS
5.89/2.34	    (13) QTRSRRRProof [EQUIVALENT, 0 ms]
5.89/2.34	    (14) QTRS
5.89/2.34	    (15) Overlay + Local Confluence [EQUIVALENT, 0 ms]
5.89/2.34	    (16) QTRS
5.89/2.34	    (17) DependencyPairsProof [EQUIVALENT, 0 ms]
5.89/2.34	    (18) QDP
5.89/2.34	    (19) UsableRulesProof [EQUIVALENT, 0 ms]
5.89/2.34	    (20) QDP
5.89/2.34	    (21) QReductionProof [EQUIVALENT, 0 ms]
5.89/2.34	    (22) QDP
5.89/2.34	(23) PrologToPiTRSProof [SOUND, 0 ms]
5.89/2.34	(24) PiTRS
5.89/2.34	    (25) DependencyPairsProof [EQUIVALENT, 0 ms]
5.89/2.34	    (26) PiDP
5.89/2.34	    (27) DependencyGraphProof [EQUIVALENT, 0 ms]
5.89/2.34	    (28) PiDP
5.89/2.34	    (29) UsableRulesProof [EQUIVALENT, 0 ms]
5.89/2.34	    (30) PiDP
5.89/2.34	    (31) PiDPToQDPProof [SOUND, 0 ms]
5.89/2.34	    (32) QDP
5.89/2.34	(33) PrologToDTProblemTransformerProof [SOUND, 0 ms]
5.89/2.34	(34) TRIPLES
5.89/2.34	    (35) TriplesToPiDPProof [SOUND, 1 ms]
5.89/2.34	    (36) PiDP
5.89/2.34	    (37) DependencyGraphProof [EQUIVALENT, 0 ms]
5.89/2.34	    (38) PiDP
5.89/2.34	    (39) PiDPToQDPProof [SOUND, 0 ms]
5.89/2.34	    (40) QDP
5.89/2.34	(41) PrologToIRSwTTransformerProof [SOUND, 0 ms]
5.89/2.34	(42) IRSwT
5.89/2.34	    (43) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
5.89/2.34	    (44) IRSwT
5.89/2.34	    (45) IntTRSCompressionProof [EQUIVALENT, 18 ms]
5.89/2.34	    (46) IRSwT
5.89/2.34	    (47) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
5.89/2.34	    (48) IRSwT
5.89/2.34	    (49) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
5.89/2.34	    (50) IRSwT
5.89/2.34	    (51) FilterProof [EQUIVALENT, 0 ms]
5.89/2.34	    (52) IntTRS
5.89/2.34	    (53) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
5.89/2.34	    (54) NO
5.89/2.34	
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(0)
5.89/2.34	Obligation:
5.89/2.34	Clauses:
5.89/2.34	
5.89/2.34	p(X, Y, Z) :- append(.(X, Y), Z, Y).
5.89/2.34	append([], Y, Y).
5.89/2.34	append(.(H, Xs), Ys, .(H, Zs)) :- append(Xs, Ys, Zs).
5.89/2.34	
5.89/2.34	
5.89/2.34	Query: p(a,a,a)
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(1) PrologToPiTRSProof (SOUND)
5.89/2.34	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.89/2.34	
5.89/2.34	p_in_3: (f,f,f)
5.89/2.34	
5.89/2.34	append_in_3: (f,f,f)
5.89/2.34	
5.89/2.34	Transforming Prolog into the following Term Rewriting System:
5.89/2.34	
5.89/2.34	Pi-finite rewrite system:
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.34	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.34	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.34	
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.34	
5.89/2.34	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.34	
5.89/2.34	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.34	
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.34	
5.89/2.34	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.34	
5.89/2.34	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(2)
5.89/2.34	Obligation:
5.89/2.34	Pi-finite rewrite system:
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.34	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.34	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.34	
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.34	
5.89/2.34	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.34	
5.89/2.34	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.34	
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.34	
5.89/2.34	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.34	
5.89/2.34	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(3) DependencyPairsProof (EQUIVALENT)
5.89/2.34	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.89/2.34	Pi DP problem:
5.89/2.34	The TRS P consists of the following rules:
5.89/2.34	
5.89/2.34	   P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(.(X, Y), Z, Y)
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.34	
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.34	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.34	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.34	
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.34	
5.89/2.34	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.34	
5.89/2.34	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.34	
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.34	
5.89/2.34	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.34	
5.89/2.34	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.34	
5.89/2.34	P_IN_AAA(x1, x2, x3)  =  P_IN_AAA
5.89/2.34	
5.89/2.34	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.34	
5.89/2.34	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.34	
5.89/2.34	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
5.89/2.34	
5.89/2.34	
5.89/2.34	We have to consider all (P,R,Pi)-chains
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(4)
5.89/2.34	Obligation:
5.89/2.34	Pi DP problem:
5.89/2.34	The TRS P consists of the following rules:
5.89/2.34	
5.89/2.34	   P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(.(X, Y), Z, Y)
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.34	
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.34	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.34	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.34	
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.34	
5.89/2.34	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.34	
5.89/2.34	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.34	
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.34	
5.89/2.34	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.34	
5.89/2.34	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.34	
5.89/2.34	P_IN_AAA(x1, x2, x3)  =  P_IN_AAA
5.89/2.34	
5.89/2.34	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.34	
5.89/2.34	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.34	
5.89/2.34	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
5.89/2.34	
5.89/2.34	
5.89/2.34	We have to consider all (P,R,Pi)-chains
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(5) DependencyGraphProof (EQUIVALENT)
5.89/2.34	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(6)
5.89/2.34	Obligation:
5.89/2.34	Pi DP problem:
5.89/2.34	The TRS P consists of the following rules:
5.89/2.34	
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.34	
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.34	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.34	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.34	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.34	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.34	
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.34	
5.89/2.34	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.34	
5.89/2.34	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.34	
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.34	
5.89/2.34	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.34	
5.89/2.34	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.34	
5.89/2.34	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.34	
5.89/2.34	
5.89/2.34	We have to consider all (P,R,Pi)-chains
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(7) UsableRulesProof (EQUIVALENT)
5.89/2.34	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(8)
5.89/2.34	Obligation:
5.89/2.34	Pi DP problem:
5.89/2.34	The TRS P consists of the following rules:
5.89/2.34	
5.89/2.34	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.34	
5.89/2.34	R is empty.
5.89/2.34	The argument filtering Pi contains the following mapping:
5.89/2.34	.(x1, x2)  =  .(x2)
5.89/2.34	
5.89/2.34	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.34	
5.89/2.34	
5.89/2.34	We have to consider all (P,R,Pi)-chains
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(9) PiDPToQDPProof (SOUND)
5.89/2.34	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(10)
5.89/2.34	Obligation:
5.89/2.34	Q DP problem:
5.89/2.34	The TRS P consists of the following rules:
5.89/2.34	
5.89/2.34	   APPEND_IN_AAA -> APPEND_IN_AAA
5.89/2.34	
5.89/2.34	R is empty.
5.89/2.34	Q is empty.
5.89/2.34	We have to consider all (P,Q,R)-chains.
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(11) PrologToTRSTransformerProof (SOUND)
5.89/2.34	Transformed Prolog program to TRS.
5.89/2.34	
5.89/2.34	{
5.89/2.34	    "root": 1,
5.89/2.34	    "program": {
5.89/2.34	        "directives": [],
5.89/2.34	        "clauses": [
5.89/2.34	            [
5.89/2.34	                "(p X Y Z)",
5.89/2.34	                "(append (. X Y) Z Y)"
5.89/2.34	            ],
5.89/2.34	            [
5.89/2.34	                "(append ([]) Y Y)",
5.89/2.34	                null
5.89/2.34	            ],
5.89/2.34	            [
5.89/2.34	                "(append (. H Xs) Ys (. H Zs))",
5.89/2.34	                "(append Xs Ys Zs)"
5.89/2.34	            ]
5.89/2.34	        ]
5.89/2.34	    },
5.89/2.34	    "graph": {
5.89/2.34	        "nodes": {
5.89/2.34	            "110": {
5.89/2.34	                "goal": [{
5.89/2.34	                    "clause": -1,
5.89/2.34	                    "scope": -1,
5.89/2.34	                    "term": "(append (. T33 T32) T31 T32)"
5.89/2.34	                }],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "1": {
5.89/2.34	                "goal": [{
5.89/2.34	                    "clause": -1,
5.89/2.34	                    "scope": -1,
5.89/2.34	                    "term": "(p T1 T2 T3)"
5.89/2.34	                }],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "111": {
5.89/2.34	                "goal": [],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "102": {
5.89/2.34	                "goal": [{
5.89/2.34	                    "clause": 0,
5.89/2.34	                    "scope": 1,
5.89/2.34	                    "term": "(p T1 T2 T3)"
5.89/2.34	                }],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "105": {
5.89/2.34	                "goal": [{
5.89/2.34	                    "clause": -1,
5.89/2.34	                    "scope": -1,
5.89/2.34	                    "term": "(append (. T18 T17) T16 T17)"
5.89/2.34	                }],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "106": {
5.89/2.34	                "goal": [
5.89/2.34	                    {
5.89/2.34	                        "clause": 1,
5.89/2.34	                        "scope": 2,
5.89/2.34	                        "term": "(append (. T18 T17) T16 T17)"
5.89/2.34	                    },
5.89/2.34	                    {
5.89/2.34	                        "clause": 2,
5.89/2.34	                        "scope": 2,
5.89/2.34	                        "term": "(append (. T18 T17) T16 T17)"
5.89/2.34	                    }
5.89/2.34	                ],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "107": {
5.89/2.34	                "goal": [{
5.89/2.34	                    "clause": 2,
5.89/2.34	                    "scope": 2,
5.89/2.34	                    "term": "(append (. T18 T17) T16 T17)"
5.89/2.34	                }],
5.89/2.34	                "kb": {
5.89/2.34	                    "nonunifying": [],
5.89/2.34	                    "intvars": {},
5.89/2.34	                    "arithmetic": {
5.89/2.34	                        "type": "PlainIntegerRelationState",
5.89/2.34	                        "relations": []
5.89/2.34	                    },
5.89/2.34	                    "ground": [],
5.89/2.34	                    "free": [],
5.89/2.34	                    "exprvars": []
5.89/2.34	                }
5.89/2.34	            },
5.89/2.34	            "type": "Nodes"
5.89/2.34	        },
5.89/2.34	        "edges": [
5.89/2.34	            {
5.89/2.34	                "from": 1,
5.89/2.34	                "to": 102,
5.89/2.34	                "label": "CASE"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 102,
5.89/2.34	                "to": 105,
5.89/2.34	                "label": "ONLY EVAL with clause\np(X9, X10, X11) :- append(.(X9, X10), X11, X10).\nand substitutionT1 -> T18,\nX9 -> T18,\nT2 -> T17,\nX10 -> T17,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16,\nT14 -> T17,\nT13 -> T18"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 105,
5.89/2.34	                "to": 106,
5.89/2.34	                "label": "CASE"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 106,
5.89/2.34	                "to": 107,
5.89/2.34	                "label": "BACKTRACK\nfor clause: append([], Y, Y)because of non-unification"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 107,
5.89/2.34	                "to": 110,
5.89/2.34	                "label": "EVAL with clause\nappend(.(X23, X24), X25, .(X23, X26)) :- append(X24, X25, X26).\nand substitutionT18 -> T33,\nX23 -> T33,\nT17 -> .(T33, T32),\nX24 -> .(T33, T32),\nT16 -> T31,\nX25 -> T31,\nX26 -> T32,\nT28 -> .(T33, T32),\nT29 -> T31,\nT30 -> T32,\nT27 -> T33"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 107,
5.89/2.34	                "to": 111,
5.89/2.34	                "label": "EVAL-BACKTRACK"
5.89/2.34	            },
5.89/2.34	            {
5.89/2.34	                "from": 110,
5.89/2.34	                "to": 105,
5.89/2.34	                "label": "INSTANCE with matching:\nT18 -> T33\nT17 -> T32\nT16 -> T31"
5.89/2.34	            }
5.89/2.34	        ],
5.89/2.34	        "type": "Graph"
5.89/2.34	    }
5.89/2.34	}
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(12)
5.89/2.34	Obligation:
5.89/2.34	Q restricted rewrite system:
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   f1_in -> U1(f105_in)
5.89/2.34	   U1(f105_out1) -> f1_out1
5.89/2.34	   f105_in -> U2(f105_in)
5.89/2.34	   U2(f105_out1) -> f105_out1
5.89/2.34	
5.89/2.34	Q is empty.
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(13) QTRSRRRProof (EQUIVALENT)
5.89/2.34	Used ordering:
5.89/2.34	Polynomial interpretation [POLO]:
5.89/2.34	
5.89/2.34	   POL(U1(x_1)) = 2*x_1
5.89/2.34	   POL(U2(x_1)) = 2*x_1
5.89/2.34	   POL(f105_in) = 0
5.89/2.34	   POL(f105_out1) = 2
5.89/2.34	   POL(f1_in) = 1
5.89/2.34	   POL(f1_out1) = 0
5.89/2.34	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.89/2.34	
5.89/2.34	   f1_in -> U1(f105_in)
5.89/2.34	   U1(f105_out1) -> f1_out1
5.89/2.34	   U2(f105_out1) -> f105_out1
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(14)
5.89/2.34	Obligation:
5.89/2.34	Q restricted rewrite system:
5.89/2.34	The TRS R consists of the following rules:
5.89/2.34	
5.89/2.34	   f105_in -> U2(f105_in)
5.89/2.34	
5.89/2.34	Q is empty.
5.89/2.34	
5.89/2.34	----------------------------------------
5.89/2.34	
5.89/2.34	(15) Overlay + Local Confluence (EQUIVALENT)
5.89/2.34	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
5.89/2.34	----------------------------------------
5.89/2.35	
5.89/2.35	(16)
5.89/2.35	Obligation:
5.89/2.35	Q restricted rewrite system:
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   f105_in -> U2(f105_in)
5.89/2.35	
5.89/2.35	The set Q consists of the following terms:
5.89/2.35	
5.89/2.35	   f105_in
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(17) DependencyPairsProof (EQUIVALENT)
5.89/2.35	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(18)
5.89/2.35	Obligation:
5.89/2.35	Q DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   F105_IN -> F105_IN
5.89/2.35	
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   f105_in -> U2(f105_in)
5.89/2.35	
5.89/2.35	The set Q consists of the following terms:
5.89/2.35	
5.89/2.35	   f105_in
5.89/2.35	
5.89/2.35	We have to consider all minimal (P,Q,R)-chains.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(19) UsableRulesProof (EQUIVALENT)
5.89/2.35	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(20)
5.89/2.35	Obligation:
5.89/2.35	Q DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   F105_IN -> F105_IN
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	The set Q consists of the following terms:
5.89/2.35	
5.89/2.35	   f105_in
5.89/2.35	
5.89/2.35	We have to consider all minimal (P,Q,R)-chains.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(21) QReductionProof (EQUIVALENT)
5.89/2.35	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
5.89/2.35	
5.89/2.35	   f105_in
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(22)
5.89/2.35	Obligation:
5.89/2.35	Q DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   F105_IN -> F105_IN
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	Q is empty.
5.89/2.35	We have to consider all minimal (P,Q,R)-chains.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(23) PrologToPiTRSProof (SOUND)
5.89/2.35	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.89/2.35	
5.89/2.35	p_in_3: (f,f,f)
5.89/2.35	
5.89/2.35	append_in_3: (f,f,f)
5.89/2.35	
5.89/2.35	Transforming Prolog into the following Term Rewriting System:
5.89/2.35	
5.89/2.35	Pi-finite rewrite system:
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.35	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.35	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.35	
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.35	
5.89/2.35	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.35	
5.89/2.35	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.35	
5.89/2.35	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.35	
5.89/2.35	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(24)
5.89/2.35	Obligation:
5.89/2.35	Pi-finite rewrite system:
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.35	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.35	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.35	
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.35	
5.89/2.35	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.35	
5.89/2.35	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.35	
5.89/2.35	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.35	
5.89/2.35	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(25) DependencyPairsProof (EQUIVALENT)
5.89/2.35	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(.(X, Y), Z, Y)
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.35	
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.35	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.35	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.35	
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.35	
5.89/2.35	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.35	
5.89/2.35	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.35	
5.89/2.35	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.35	
5.89/2.35	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.35	
5.89/2.35	P_IN_AAA(x1, x2, x3)  =  P_IN_AAA
5.89/2.35	
5.89/2.35	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.35	
5.89/2.35	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.35	
5.89/2.35	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(26)
5.89/2.35	Obligation:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   P_IN_AAA(X, Y, Z) -> U1_AAA(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   P_IN_AAA(X, Y, Z) -> APPEND_IN_AAA(.(X, Y), Z, Y)
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> U2_AAA(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.35	
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.35	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.35	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.35	
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.35	
5.89/2.35	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.35	
5.89/2.35	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.35	
5.89/2.35	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.35	
5.89/2.35	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.35	
5.89/2.35	P_IN_AAA(x1, x2, x3)  =  P_IN_AAA
5.89/2.35	
5.89/2.35	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.35	
5.89/2.35	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.35	
5.89/2.35	U2_AAA(x1, x2, x3, x4, x5)  =  U2_AAA(x5)
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(27) DependencyGraphProof (EQUIVALENT)
5.89/2.35	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(28)
5.89/2.35	Obligation:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.35	
5.89/2.35	The TRS R consists of the following rules:
5.89/2.35	
5.89/2.35	   p_in_aaa(X, Y, Z) -> U1_aaa(X, Y, Z, append_in_aaa(.(X, Y), Z, Y))
5.89/2.35	   append_in_aaa([], Y, Y) -> append_out_aaa([], Y, Y)
5.89/2.35	   append_in_aaa(.(H, Xs), Ys, .(H, Zs)) -> U2_aaa(H, Xs, Ys, Zs, append_in_aaa(Xs, Ys, Zs))
5.89/2.35	   U2_aaa(H, Xs, Ys, Zs, append_out_aaa(Xs, Ys, Zs)) -> append_out_aaa(.(H, Xs), Ys, .(H, Zs))
5.89/2.35	   U1_aaa(X, Y, Z, append_out_aaa(.(X, Y), Z, Y)) -> p_out_aaa(X, Y, Z)
5.89/2.35	
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	p_in_aaa(x1, x2, x3)  =  p_in_aaa
5.89/2.35	
5.89/2.35	U1_aaa(x1, x2, x3, x4)  =  U1_aaa(x4)
5.89/2.35	
5.89/2.35	append_in_aaa(x1, x2, x3)  =  append_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	append_out_aaa(x1, x2, x3)  =  append_out_aaa(x1)
5.89/2.35	
5.89/2.35	U2_aaa(x1, x2, x3, x4, x5)  =  U2_aaa(x5)
5.89/2.35	
5.89/2.35	p_out_aaa(x1, x2, x3)  =  p_out_aaa(x2)
5.89/2.35	
5.89/2.35	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(29) UsableRulesProof (EQUIVALENT)
5.89/2.35	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(30)
5.89/2.35	Obligation:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   APPEND_IN_AAA(.(H, Xs), Ys, .(H, Zs)) -> APPEND_IN_AAA(Xs, Ys, Zs)
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	.(x1, x2)  =  .(x2)
5.89/2.35	
5.89/2.35	APPEND_IN_AAA(x1, x2, x3)  =  APPEND_IN_AAA
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(31) PiDPToQDPProof (SOUND)
5.89/2.35	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(32)
5.89/2.35	Obligation:
5.89/2.35	Q DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   APPEND_IN_AAA -> APPEND_IN_AAA
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	Q is empty.
5.89/2.35	We have to consider all (P,Q,R)-chains.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(33) PrologToDTProblemTransformerProof (SOUND)
5.89/2.35	Built DT problem from termination graph DT10.
5.89/2.35	
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5.89/2.35	    "program": {
5.89/2.35	        "directives": [],
5.89/2.35	        "clauses": [
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5.89/2.35	                "(p X Y Z)",
5.89/2.35	                "(append (. X Y) Z Y)"
5.89/2.35	            ],
5.89/2.35	            [
5.89/2.35	                "(append ([]) Y Y)",
5.89/2.35	                null
5.89/2.35	            ],
5.89/2.35	            [
5.89/2.35	                "(append (. H Xs) Ys (. H Zs))",
5.89/2.35	                "(append Xs Ys Zs)"
5.89/2.35	            ]
5.89/2.35	        ]
5.89/2.35	    },
5.89/2.35	    "graph": {
5.89/2.35	        "nodes": {
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5.89/2.35	                }],
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5.89/2.35	                    },
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "112": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": 0,
5.89/2.35	                    "scope": 1,
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5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
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5.89/2.35	                    },
5.89/2.35	                    "ground": [],
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "113": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": -1,
5.89/2.35	                    "scope": -1,
5.89/2.35	                    "term": "(append (. T12 T11) T10 T11)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
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5.89/2.35	                        "relations": []
5.89/2.35	                    },
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "114": {
5.89/2.35	                "goal": [
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5.89/2.35	                        "scope": 2,
5.89/2.35	                        "term": "(append (. T12 T11) T10 T11)"
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5.89/2.35	                        "term": "(append (. T12 T11) T10 T11)"
5.89/2.35	                    }
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5.89/2.35	                        "relations": []
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5.89/2.35	                    "ground": [],
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "115": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": 2,
5.89/2.35	                    "scope": 2,
5.89/2.35	                    "term": "(append (. T12 T11) T10 T11)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
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5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "116": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": -1,
5.89/2.35	                    "scope": -1,
5.89/2.35	                    "term": "(append (. T27 T26) T25 T26)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "117": {
5.89/2.35	                "goal": [],
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5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
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5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
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5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "type": "Nodes"
5.89/2.35	        },
5.89/2.35	        "edges": [
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5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 112,
5.89/2.35	                "to": 113,
5.89/2.35	                "label": "ONLY EVAL with clause\np(X4, X5, X6) :- append(.(X4, X5), X6, X5).\nand substitutionT1 -> T12,\nX4 -> T12,\nT2 -> T11,\nX5 -> T11,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10,\nT8 -> T11,\nT7 -> T12"
5.89/2.35	            },
5.89/2.35	            {
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5.89/2.35	                "to": 114,
5.89/2.35	                "label": "CASE"
5.89/2.35	            },
5.89/2.35	            {
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5.89/2.35	                "to": 115,
5.89/2.35	                "label": "BACKTRACK\nfor clause: append([], Y, Y)because of non-unification"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 115,
5.89/2.35	                "to": 116,
5.89/2.35	                "label": "EVAL with clause\nappend(.(X16, X17), X18, .(X16, X19)) :- append(X17, X18, X19).\nand substitutionT12 -> T27,\nX16 -> T27,\nT11 -> .(T27, T26),\nX17 -> .(T27, T26),\nT10 -> T25,\nX18 -> T25,\nX19 -> T26,\nT22 -> .(T27, T26),\nT23 -> T25,\nT24 -> T26,\nT21 -> T27"
5.89/2.35	            },
5.89/2.35	            {
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5.89/2.35	                "to": 117,
5.89/2.35	                "label": "EVAL-BACKTRACK"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 116,
5.89/2.35	                "to": 113,
5.89/2.35	                "label": "INSTANCE with matching:\nT12 -> T27\nT11 -> T26\nT10 -> T25"
5.89/2.35	            }
5.89/2.35	        ],
5.89/2.35	        "type": "Graph"
5.89/2.35	    }
5.89/2.35	}
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(34)
5.89/2.35	Obligation:
5.89/2.35	Triples:
5.89/2.35	
5.89/2.35	appendA(X1, .(X1, X2), X3) :- appendA(X1, X2, X3).
5.89/2.35	pB(X1, X2, X3) :- appendA(X1, X2, X3).
5.89/2.35	
5.89/2.35	Clauses:
5.89/2.35	
5.89/2.35	appendcA(X1, .(X1, X2), X3) :- appendcA(X1, X2, X3).
5.89/2.35	
5.89/2.35	Afs:
5.89/2.35	
5.89/2.35	pB(x1, x2, x3)  =  pB
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(35) TriplesToPiDPProof (SOUND)
5.89/2.35	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
5.89/2.35	
5.89/2.35	pB_in_3: (f,f,f)
5.89/2.35	
5.89/2.35	appendA_in_3: (f,f,f)
5.89/2.35	
5.89/2.35	Transforming TRIPLES into the following Term Rewriting System:
5.89/2.35	
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   PB_IN_AAA(X1, X2, X3) -> U2_AAA(X1, X2, X3, appendA_in_aaa(X1, X2, X3))
5.89/2.35	   PB_IN_AAA(X1, X2, X3) -> APPENDA_IN_AAA(X1, X2, X3)
5.89/2.35	   APPENDA_IN_AAA(X1, .(X1, X2), X3) -> U1_AAA(X1, X2, X3, appendA_in_aaa(X1, X2, X3))
5.89/2.35	   APPENDA_IN_AAA(X1, .(X1, X2), X3) -> APPENDA_IN_AAA(X1, X2, X3)
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	appendA_in_aaa(x1, x2, x3)  =  appendA_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x1, x2)
5.89/2.35	
5.89/2.35	PB_IN_AAA(x1, x2, x3)  =  PB_IN_AAA
5.89/2.35	
5.89/2.35	U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
5.89/2.35	
5.89/2.35	APPENDA_IN_AAA(x1, x2, x3)  =  APPENDA_IN_AAA
5.89/2.35	
5.89/2.35	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	
5.89/2.35	
5.89/2.35	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
5.89/2.35	
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(36)
5.89/2.35	Obligation:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   PB_IN_AAA(X1, X2, X3) -> U2_AAA(X1, X2, X3, appendA_in_aaa(X1, X2, X3))
5.89/2.35	   PB_IN_AAA(X1, X2, X3) -> APPENDA_IN_AAA(X1, X2, X3)
5.89/2.35	   APPENDA_IN_AAA(X1, .(X1, X2), X3) -> U1_AAA(X1, X2, X3, appendA_in_aaa(X1, X2, X3))
5.89/2.35	   APPENDA_IN_AAA(X1, .(X1, X2), X3) -> APPENDA_IN_AAA(X1, X2, X3)
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	appendA_in_aaa(x1, x2, x3)  =  appendA_in_aaa
5.89/2.35	
5.89/2.35	.(x1, x2)  =  .(x1, x2)
5.89/2.35	
5.89/2.35	PB_IN_AAA(x1, x2, x3)  =  PB_IN_AAA
5.89/2.35	
5.89/2.35	U2_AAA(x1, x2, x3, x4)  =  U2_AAA(x4)
5.89/2.35	
5.89/2.35	APPENDA_IN_AAA(x1, x2, x3)  =  APPENDA_IN_AAA
5.89/2.35	
5.89/2.35	U1_AAA(x1, x2, x3, x4)  =  U1_AAA(x4)
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(37) DependencyGraphProof (EQUIVALENT)
5.89/2.35	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(38)
5.89/2.35	Obligation:
5.89/2.35	Pi DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   APPENDA_IN_AAA(X1, .(X1, X2), X3) -> APPENDA_IN_AAA(X1, X2, X3)
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	The argument filtering Pi contains the following mapping:
5.89/2.35	.(x1, x2)  =  .(x1, x2)
5.89/2.35	
5.89/2.35	APPENDA_IN_AAA(x1, x2, x3)  =  APPENDA_IN_AAA
5.89/2.35	
5.89/2.35	
5.89/2.35	We have to consider all (P,R,Pi)-chains
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(39) PiDPToQDPProof (SOUND)
5.89/2.35	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(40)
5.89/2.35	Obligation:
5.89/2.35	Q DP problem:
5.89/2.35	The TRS P consists of the following rules:
5.89/2.35	
5.89/2.35	   APPENDA_IN_AAA -> APPENDA_IN_AAA
5.89/2.35	
5.89/2.35	R is empty.
5.89/2.35	Q is empty.
5.89/2.35	We have to consider all (P,Q,R)-chains.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(41) PrologToIRSwTTransformerProof (SOUND)
5.89/2.35	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
5.89/2.35	
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5.89/2.35	    "root": 3,
5.89/2.35	    "program": {
5.89/2.35	        "directives": [],
5.89/2.35	        "clauses": [
5.89/2.35	            [
5.89/2.35	                "(p X Y Z)",
5.89/2.35	                "(append (. X Y) Z Y)"
5.89/2.35	            ],
5.89/2.35	            [
5.89/2.35	                "(append ([]) Y Y)",
5.89/2.35	                null
5.89/2.35	            ],
5.89/2.35	            [
5.89/2.35	                "(append (. H Xs) Ys (. H Zs))",
5.89/2.35	                "(append Xs Ys Zs)"
5.89/2.35	            ]
5.89/2.35	        ]
5.89/2.35	    },
5.89/2.35	    "graph": {
5.89/2.35	        "nodes": {
5.89/2.35	            "101": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": -1,
5.89/2.35	                    "scope": -1,
5.89/2.35	                    "term": "(append (. T18 T17) T16 T17)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "3": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": -1,
5.89/2.35	                    "scope": -1,
5.89/2.35	                    "term": "(p T1 T2 T3)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "103": {
5.89/2.35	                "goal": [
5.89/2.35	                    {
5.89/2.35	                        "clause": 1,
5.89/2.35	                        "scope": 2,
5.89/2.35	                        "term": "(append (. T18 T17) T16 T17)"
5.89/2.35	                    },
5.89/2.35	                    {
5.89/2.35	                        "clause": 2,
5.89/2.35	                        "scope": 2,
5.89/2.35	                        "term": "(append (. T18 T17) T16 T17)"
5.89/2.35	                    }
5.89/2.35	                ],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "104": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": 2,
5.89/2.35	                    "scope": 2,
5.89/2.35	                    "term": "(append (. T18 T17) T16 T17)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "8": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": 0,
5.89/2.35	                    "scope": 1,
5.89/2.35	                    "term": "(p T1 T2 T3)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "108": {
5.89/2.35	                "goal": [{
5.89/2.35	                    "clause": -1,
5.89/2.35	                    "scope": -1,
5.89/2.35	                    "term": "(append (. T33 T32) T31 T32)"
5.89/2.35	                }],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "109": {
5.89/2.35	                "goal": [],
5.89/2.35	                "kb": {
5.89/2.35	                    "nonunifying": [],
5.89/2.35	                    "intvars": {},
5.89/2.35	                    "arithmetic": {
5.89/2.35	                        "type": "PlainIntegerRelationState",
5.89/2.35	                        "relations": []
5.89/2.35	                    },
5.89/2.35	                    "ground": [],
5.89/2.35	                    "free": [],
5.89/2.35	                    "exprvars": []
5.89/2.35	                }
5.89/2.35	            },
5.89/2.35	            "type": "Nodes"
5.89/2.35	        },
5.89/2.35	        "edges": [
5.89/2.35	            {
5.89/2.35	                "from": 3,
5.89/2.35	                "to": 8,
5.89/2.35	                "label": "CASE"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 8,
5.89/2.35	                "to": 101,
5.89/2.35	                "label": "ONLY EVAL with clause\np(X9, X10, X11) :- append(.(X9, X10), X11, X10).\nand substitutionT1 -> T18,\nX9 -> T18,\nT2 -> T17,\nX10 -> T17,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16,\nT14 -> T17,\nT13 -> T18"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 101,
5.89/2.35	                "to": 103,
5.89/2.35	                "label": "CASE"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 103,
5.89/2.35	                "to": 104,
5.89/2.35	                "label": "BACKTRACK\nfor clause: append([], Y, Y)because of non-unification"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 104,
5.89/2.35	                "to": 108,
5.89/2.35	                "label": "EVAL with clause\nappend(.(X23, X24), X25, .(X23, X26)) :- append(X24, X25, X26).\nand substitutionT18 -> T33,\nX23 -> T33,\nT17 -> .(T33, T32),\nX24 -> .(T33, T32),\nT16 -> T31,\nX25 -> T31,\nX26 -> T32,\nT28 -> .(T33, T32),\nT29 -> T31,\nT30 -> T32,\nT27 -> T33"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 104,
5.89/2.35	                "to": 109,
5.89/2.35	                "label": "EVAL-BACKTRACK"
5.89/2.35	            },
5.89/2.35	            {
5.89/2.35	                "from": 108,
5.89/2.35	                "to": 101,
5.89/2.35	                "label": "INSTANCE with matching:\nT18 -> T33\nT17 -> T32\nT16 -> T31"
5.89/2.35	            }
5.89/2.35	        ],
5.89/2.35	        "type": "Graph"
5.89/2.35	    }
5.89/2.35	}
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(42)
5.89/2.35	Obligation:
5.89/2.35	Rules:
5.89/2.35	f109_out -> f104_out :|: TRUE
5.89/2.35	f104_in -> f109_in :|: TRUE
5.89/2.35	f108_out -> f104_out :|: TRUE
5.89/2.35	f104_in -> f108_in :|: TRUE
5.89/2.35	f103_in -> f104_in :|: TRUE
5.89/2.35	f104_out -> f103_out :|: TRUE
5.89/2.35	f103_out -> f101_out :|: TRUE
5.89/2.35	f101_in -> f103_in :|: TRUE
5.89/2.35	f108_in -> f101_in :|: TRUE
5.89/2.35	f101_out -> f108_out :|: TRUE
5.89/2.35	f8_out -> f3_out :|: TRUE
5.89/2.35	f3_in -> f8_in :|: TRUE
5.89/2.35	f8_in -> f101_in :|: TRUE
5.89/2.35	f101_out -> f8_out :|: TRUE
5.89/2.35	Start term: f3_in
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(43) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
5.89/2.35	Constructed simple dependency graph.
5.89/2.35	
5.89/2.35	Simplified to the following IRSwTs:
5.89/2.35	
5.89/2.35	intTRSProblem:
5.89/2.35	f104_in -> f108_in :|: TRUE
5.89/2.35	f103_in -> f104_in :|: TRUE
5.89/2.35	f101_in -> f103_in :|: TRUE
5.89/2.35	f108_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(44)
5.89/2.35	Obligation:
5.89/2.35	Rules:
5.89/2.35	f104_in -> f108_in :|: TRUE
5.89/2.35	f103_in -> f104_in :|: TRUE
5.89/2.35	f101_in -> f103_in :|: TRUE
5.89/2.35	f108_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(45) IntTRSCompressionProof (EQUIVALENT)
5.89/2.35	Compressed rules.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(46)
5.89/2.35	Obligation:
5.89/2.35	Rules:
5.89/2.35	f101_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(47) IRSFormatTransformerProof (EQUIVALENT)
5.89/2.35	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(48)
5.89/2.35	Obligation:
5.89/2.35	Rules:
5.89/2.35	f101_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(49) IRSwTTerminationDigraphProof (EQUIVALENT)
5.89/2.35	Constructed termination digraph!
5.89/2.35	Nodes:
5.89/2.35	(1) f101_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	Arcs:
5.89/2.35	(1) -> (1)
5.89/2.35	
5.89/2.35	This digraph is fully evaluated!
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(50)
5.89/2.35	Obligation:
5.89/2.35	
5.89/2.35	Termination digraph:
5.89/2.35	Nodes:
5.89/2.35	(1) f101_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	Arcs:
5.89/2.35	(1) -> (1)
5.89/2.35	
5.89/2.35	This digraph is fully evaluated!
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(51) FilterProof (EQUIVALENT)
5.89/2.35	Used the following sort dictionary for filtering: 
5.89/2.35	f101_in()
5.89/2.35	Replaced non-predefined constructor symbols by 0.
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(52)
5.89/2.35	Obligation:
5.89/2.35	Rules:
5.89/2.35	f101_in -> f101_in :|: TRUE
5.89/2.35	
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(53) IntTRSPeriodicNontermProof (COMPLETE)
5.89/2.35	Normalized system to the following form:
5.89/2.35	f(pc) -> f(1) :|: pc = 1 && TRUE
5.89/2.35	Witness term starting non-terminating reduction: f(1)
5.89/2.35	----------------------------------------
5.89/2.35	
5.89/2.35	(54)
5.89/2.35	NO
6.11/2.38	EOF