6.03/2.37	MAYBE
6.03/2.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.03/2.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.03/2.40	
6.03/2.40	
6.03/2.40	Left Termination of the query pattern
6.03/2.40	
6.03/2.40	p(g)
6.03/2.40	
6.03/2.40	w.r.t. the given Prolog program could not be shown:
6.03/2.40	
6.03/2.40	(0) Prolog
6.03/2.40	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.03/2.40	(2) PiTRS
6.03/2.40	    (3) DependencyPairsProof [EQUIVALENT, 0 ms]
6.03/2.40	    (4) PiDP
6.03/2.40	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.03/2.40	    (6) PiDP
6.03/2.40	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
6.03/2.40	    (8) PiDP
6.03/2.40	    (9) PiDPToQDPProof [SOUND, 0 ms]
6.03/2.40	    (10) QDP
6.03/2.40	(11) PrologToPiTRSProof [SOUND, 0 ms]
6.03/2.40	(12) PiTRS
6.03/2.40	    (13) DependencyPairsProof [EQUIVALENT, 0 ms]
6.03/2.40	    (14) PiDP
6.03/2.40	    (15) DependencyGraphProof [EQUIVALENT, 0 ms]
6.03/2.40	    (16) PiDP
6.03/2.40	    (17) UsableRulesProof [EQUIVALENT, 0 ms]
6.03/2.40	    (18) PiDP
6.03/2.40	    (19) PiDPToQDPProof [SOUND, 0 ms]
6.03/2.40	    (20) QDP
6.03/2.40	(21) PrologToDTProblemTransformerProof [SOUND, 0 ms]
6.03/2.40	(22) TRIPLES
6.03/2.40	    (23) TriplesToPiDPProof [SOUND, 0 ms]
6.03/2.40	    (24) PiDP
6.03/2.40	    (25) DependencyGraphProof [EQUIVALENT, 0 ms]
6.03/2.40	    (26) PiDP
6.03/2.40	    (27) PiDPToQDPProof [SOUND, 0 ms]
6.03/2.40	    (28) QDP
6.03/2.40	(29) PrologToTRSTransformerProof [SOUND, 0 ms]
6.03/2.40	(30) QTRS
6.03/2.40	    (31) QTRSRRRProof [EQUIVALENT, 52 ms]
6.03/2.40	    (32) QTRS
6.03/2.40	    (33) QTRSRRRProof [EQUIVALENT, 9 ms]
6.03/2.40	    (34) QTRS
6.03/2.40	    (35) QTRSRRRProof [EQUIVALENT, 6 ms]
6.03/2.40	    (36) QTRS
6.03/2.40	    (37) QTRSRRRProof [EQUIVALENT, 3 ms]
6.03/2.40	    (38) QTRS
6.03/2.40	    (39) Overlay + Local Confluence [EQUIVALENT, 1 ms]
6.03/2.40	    (40) QTRS
6.03/2.40	    (41) DependencyPairsProof [EQUIVALENT, 0 ms]
6.03/2.40	    (42) QDP
6.03/2.40	    (43) UsableRulesProof [EQUIVALENT, 0 ms]
6.03/2.40	    (44) QDP
6.03/2.40	(45) PrologToIRSwTTransformerProof [SOUND, 0 ms]
6.03/2.40	(46) IRSwT
6.03/2.40	    (47) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.03/2.40	    (48) IRSwT
6.03/2.40	    (49) IntTRSCompressionProof [EQUIVALENT, 22 ms]
6.03/2.40	    (50) IRSwT
6.03/2.40	    (51) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.03/2.40	    (52) IRSwT
6.03/2.40	    (53) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.03/2.40	    (54) IRSwT
6.03/2.40	    (55) FilterProof [EQUIVALENT, 0 ms]
6.03/2.40	    (56) IntTRS
6.03/2.40	    (57) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
6.03/2.40	    (58) NO
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(0)
6.03/2.40	Obligation:
6.03/2.40	Clauses:
6.03/2.40	
6.03/2.40	p(b).
6.03/2.40	p(a) :- p1(X).
6.03/2.40	p1(b).
6.03/2.40	p1(a) :- p1(X).
6.03/2.40	
6.03/2.40	
6.03/2.40	Query: p(g)
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(1) PrologToPiTRSProof (SOUND)
6.03/2.40	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.03/2.40	
6.03/2.40	p_in_1: (b)
6.03/2.40	
6.03/2.40	p1_in_1: (f)
6.03/2.40	
6.03/2.40	Transforming Prolog into the following Term Rewriting System:
6.03/2.40	
6.03/2.40	Pi-finite rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(2)
6.03/2.40	Obligation:
6.03/2.40	Pi-finite rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(3) DependencyPairsProof (EQUIVALENT)
6.03/2.40	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P_IN_G(a) -> U1_G(p1_in_a(X))
6.03/2.40	   P_IN_G(a) -> P1_IN_A(X)
6.03/2.40	   P1_IN_A(a) -> U2_A(p1_in_a(X))
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P_IN_G(x1)  =  P_IN_G(x1)
6.03/2.40	
6.03/2.40	U1_G(x1)  =  U1_G(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	U2_A(x1)  =  U2_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(4)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P_IN_G(a) -> U1_G(p1_in_a(X))
6.03/2.40	   P_IN_G(a) -> P1_IN_A(X)
6.03/2.40	   P1_IN_A(a) -> U2_A(p1_in_a(X))
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P_IN_G(x1)  =  P_IN_G(x1)
6.03/2.40	
6.03/2.40	U1_G(x1)  =  U1_G(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	U2_A(x1)  =  U2_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(5) DependencyGraphProof (EQUIVALENT)
6.03/2.40	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(6)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(7) UsableRulesProof (EQUIVALENT)
6.03/2.40	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(8)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(9) PiDPToQDPProof (SOUND)
6.03/2.40	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(10)
6.03/2.40	Obligation:
6.03/2.40	Q DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A -> P1_IN_A
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	Q is empty.
6.03/2.40	We have to consider all (P,Q,R)-chains.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(11) PrologToPiTRSProof (SOUND)
6.03/2.40	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.03/2.40	
6.03/2.40	p_in_1: (b)
6.03/2.40	
6.03/2.40	p1_in_1: (f)
6.03/2.40	
6.03/2.40	Transforming Prolog into the following Term Rewriting System:
6.03/2.40	
6.03/2.40	Pi-finite rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g(x1)
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(12)
6.03/2.40	Obligation:
6.03/2.40	Pi-finite rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g(x1)
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(13) DependencyPairsProof (EQUIVALENT)
6.03/2.40	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P_IN_G(a) -> U1_G(p1_in_a(X))
6.03/2.40	   P_IN_G(a) -> P1_IN_A(X)
6.03/2.40	   P1_IN_A(a) -> U2_A(p1_in_a(X))
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g(x1)
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P_IN_G(x1)  =  P_IN_G(x1)
6.03/2.40	
6.03/2.40	U1_G(x1)  =  U1_G(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	U2_A(x1)  =  U2_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(14)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P_IN_G(a) -> U1_G(p1_in_a(X))
6.03/2.40	   P_IN_G(a) -> P1_IN_A(X)
6.03/2.40	   P1_IN_A(a) -> U2_A(p1_in_a(X))
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g(x1)
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P_IN_G(x1)  =  P_IN_G(x1)
6.03/2.40	
6.03/2.40	U1_G(x1)  =  U1_G(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	U2_A(x1)  =  U2_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(15) DependencyGraphProof (EQUIVALENT)
6.03/2.40	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(16)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   p_in_g(b) -> p_out_g(b)
6.03/2.40	   p_in_g(a) -> U1_g(p1_in_a(X))
6.03/2.40	   p1_in_a(b) -> p1_out_a(b)
6.03/2.40	   p1_in_a(a) -> U2_a(p1_in_a(X))
6.03/2.40	   U2_a(p1_out_a(X)) -> p1_out_a(a)
6.03/2.40	   U1_g(p1_out_a(X)) -> p_out_g(a)
6.03/2.40	
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	p_in_g(x1)  =  p_in_g(x1)
6.03/2.40	
6.03/2.40	b  =  b
6.03/2.40	
6.03/2.40	p_out_g(x1)  =  p_out_g(x1)
6.03/2.40	
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	U1_g(x1)  =  U1_g(x1)
6.03/2.40	
6.03/2.40	p1_in_a(x1)  =  p1_in_a
6.03/2.40	
6.03/2.40	p1_out_a(x1)  =  p1_out_a(x1)
6.03/2.40	
6.03/2.40	U2_a(x1)  =  U2_a(x1)
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(17) UsableRulesProof (EQUIVALENT)
6.03/2.40	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(18)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A(a) -> P1_IN_A(X)
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	P1_IN_A(x1)  =  P1_IN_A
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(19) PiDPToQDPProof (SOUND)
6.03/2.40	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(20)
6.03/2.40	Obligation:
6.03/2.40	Q DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1_IN_A -> P1_IN_A
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	Q is empty.
6.03/2.40	We have to consider all (P,Q,R)-chains.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(21) PrologToDTProblemTransformerProof (SOUND)
6.03/2.40	Built DT problem from termination graph DT10.
6.03/2.40	
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6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 65,
6.03/2.40	                "to": 71,
6.03/2.40	                "label": "EVAL with clause\np(a) :- p1(X1).\nand substitutionT1 -> a"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 65,
6.03/2.40	                "to": 72,
6.03/2.40	                "label": "EVAL-BACKTRACK"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 66,
6.03/2.40	                "to": 67,
6.03/2.40	                "label": "BACKTRACK\nfor clause: p(a) :- p1(X)because of non-unification"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 71,
6.03/2.40	                "to": 74,
6.03/2.40	                "label": "CASE"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 74,
6.03/2.40	                "to": 75,
6.03/2.40	                "label": "PARALLEL"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 74,
6.03/2.40	                "to": 76,
6.03/2.40	                "label": "PARALLEL"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 75,
6.03/2.40	                "to": 78,
6.03/2.40	                "label": "ONLY EVAL with clause\np1(b).\nand substitutionX1 -> b"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 76,
6.03/2.40	                "to": 82,
6.03/2.40	                "label": "ONLY EVAL with clause\np1(a) :- p1(X3).\nand substitutionX1 -> a"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 78,
6.03/2.40	                "to": 79,
6.03/2.40	                "label": "SUCCESS"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 82,
6.03/2.40	                "to": 71,
6.03/2.40	                "label": "INSTANCE with matching:\nX1 -> X3"
6.03/2.40	            }
6.03/2.40	        ],
6.03/2.40	        "type": "Graph"
6.03/2.40	    }
6.03/2.40	}
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(22)
6.03/2.40	Obligation:
6.03/2.40	Triples:
6.03/2.40	
6.03/2.40	p1A(a) :- p1A(X1).
6.03/2.40	pB(a) :- p1A(X1).
6.03/2.40	
6.03/2.40	Clauses:
6.03/2.40	
6.03/2.40	p1cA(b).
6.03/2.40	p1cA(a) :- p1cA(X1).
6.03/2.40	
6.03/2.40	Afs:
6.03/2.40	
6.03/2.40	pB(x1)  =  pB(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(23) TriplesToPiDPProof (SOUND)
6.03/2.40	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.03/2.40	
6.03/2.40	pB_in_1: (b)
6.03/2.40	
6.03/2.40	p1A_in_1: (f)
6.03/2.40	
6.03/2.40	Transforming TRIPLES into the following Term Rewriting System:
6.03/2.40	
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   PB_IN_G(a) -> U2_G(p1A_in_a(X1))
6.03/2.40	   PB_IN_G(a) -> P1A_IN_A(X1)
6.03/2.40	   P1A_IN_A(a) -> U1_A(p1A_in_a(X1))
6.03/2.40	   P1A_IN_A(a) -> P1A_IN_A(X1)
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	p1A_in_a(x1)  =  p1A_in_a
6.03/2.40	
6.03/2.40	PB_IN_G(x1)  =  PB_IN_G(x1)
6.03/2.40	
6.03/2.40	U2_G(x1)  =  U2_G(x1)
6.03/2.40	
6.03/2.40	P1A_IN_A(x1)  =  P1A_IN_A
6.03/2.40	
6.03/2.40	U1_A(x1)  =  U1_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	
6.03/2.40	
6.03/2.40	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(24)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   PB_IN_G(a) -> U2_G(p1A_in_a(X1))
6.03/2.40	   PB_IN_G(a) -> P1A_IN_A(X1)
6.03/2.40	   P1A_IN_A(a) -> U1_A(p1A_in_a(X1))
6.03/2.40	   P1A_IN_A(a) -> P1A_IN_A(X1)
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	p1A_in_a(x1)  =  p1A_in_a
6.03/2.40	
6.03/2.40	PB_IN_G(x1)  =  PB_IN_G(x1)
6.03/2.40	
6.03/2.40	U2_G(x1)  =  U2_G(x1)
6.03/2.40	
6.03/2.40	P1A_IN_A(x1)  =  P1A_IN_A
6.03/2.40	
6.03/2.40	U1_A(x1)  =  U1_A(x1)
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(25) DependencyGraphProof (EQUIVALENT)
6.03/2.40	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(26)
6.03/2.40	Obligation:
6.03/2.40	Pi DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1A_IN_A(a) -> P1A_IN_A(X1)
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	The argument filtering Pi contains the following mapping:
6.03/2.40	a  =  a
6.03/2.40	
6.03/2.40	P1A_IN_A(x1)  =  P1A_IN_A
6.03/2.40	
6.03/2.40	
6.03/2.40	We have to consider all (P,R,Pi)-chains
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(27) PiDPToQDPProof (SOUND)
6.03/2.40	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(28)
6.03/2.40	Obligation:
6.03/2.40	Q DP problem:
6.03/2.40	The TRS P consists of the following rules:
6.03/2.40	
6.03/2.40	   P1A_IN_A -> P1A_IN_A
6.03/2.40	
6.03/2.40	R is empty.
6.03/2.40	Q is empty.
6.03/2.40	We have to consider all (P,Q,R)-chains.
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(29) PrologToTRSTransformerProof (SOUND)
6.03/2.40	Transformed Prolog program to TRS.
6.03/2.40	
6.03/2.40	{
6.03/2.40	    "root": 10,
6.03/2.40	    "program": {
6.03/2.40	        "directives": [],
6.03/2.40	        "clauses": [
6.03/2.40	            [
6.03/2.40	                "(p (b))",
6.03/2.40	                null
6.03/2.40	            ],
6.03/2.40	            [
6.03/2.40	                "(p (a))",
6.03/2.40	                "(p1 X)"
6.03/2.40	            ],
6.03/2.40	            [
6.03/2.40	                "(p1 (b))",
6.03/2.40	                null
6.03/2.40	            ],
6.03/2.40	            [
6.03/2.40	                "(p1 (a))",
6.03/2.40	                "(p1 X)"
6.03/2.40	            ]
6.03/2.40	        ]
6.03/2.40	    },
6.03/2.40	    "graph": {
6.03/2.40	        "nodes": {
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6.03/2.40	                "goal": [
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6.03/2.40	                        "clause": 0,
6.03/2.40	                        "scope": 1,
6.03/2.40	                        "term": "(p T1)"
6.03/2.40	                    },
6.03/2.40	                    {
6.03/2.40	                        "clause": 1,
6.03/2.40	                        "scope": 1,
6.03/2.40	                        "term": "(p T1)"
6.03/2.40	                    }
6.03/2.40	                ],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
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6.03/2.40	                    },
6.03/2.40	                    "ground": ["T1"],
6.03/2.40	                    "free": [],
6.03/2.40	                    "exprvars": []
6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "34": {
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6.03/2.40	                    "clause": -1,
6.03/2.40	                    "scope": -1,
6.03/2.40	                    "term": "(true)"
6.03/2.40	                }],
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6.03/2.40	                    },
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "35": {
6.03/2.40	                "goal": [],
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6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "14": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": 0,
6.03/2.40	                    "scope": 1,
6.03/2.40	                    "term": "(p T1)"
6.03/2.40	                }],
6.03/2.40	                "kb": {
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6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
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6.03/2.40	                    "ground": ["T1"],
6.03/2.40	                    "free": [],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "15": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": 1,
6.03/2.40	                    "scope": 1,
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6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
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6.03/2.40	                    "ground": ["T1"],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "37": {
6.03/2.40	                "goal": [],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
6.03/2.40	                    "free": [],
6.03/2.40	                    "exprvars": []
6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "38": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": -1,
6.03/2.40	                    "scope": -1,
6.03/2.40	                    "term": "(p1 X2)"
6.03/2.40	                }],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
6.03/2.40	                    "free": ["X2"],
6.03/2.40	                    "exprvars": []
6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "39": {
6.03/2.40	                "goal": [],
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6.03/2.40	                    },
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6.03/2.40	                    "free": [],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "type": "Nodes",
6.03/2.40	            "93": {
6.03/2.40	                "goal": [
6.03/2.40	                    {
6.03/2.40	                        "clause": 2,
6.03/2.40	                        "scope": 2,
6.03/2.40	                        "term": "(p1 X2)"
6.03/2.40	                    },
6.03/2.40	                    {
6.03/2.40	                        "clause": 3,
6.03/2.40	                        "scope": 2,
6.03/2.40	                        "term": "(p1 X2)"
6.03/2.40	                    }
6.03/2.40	                ],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
6.03/2.40	                    "free": ["X2"],
6.03/2.40	                    "exprvars": []
6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "94": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": 2,
6.03/2.40	                    "scope": 2,
6.03/2.40	                    "term": "(p1 X2)"
6.03/2.40	                }],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "95": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": 3,
6.03/2.40	                    "scope": 2,
6.03/2.40	                    "term": "(p1 X2)"
6.03/2.40	                }],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
6.03/2.40	                    "intvars": {},
6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
6.03/2.40	                    "free": ["X2"],
6.03/2.40	                    "exprvars": []
6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "96": {
6.03/2.40	                "goal": [{
6.03/2.40	                    "clause": -1,
6.03/2.40	                    "scope": -1,
6.03/2.40	                    "term": "(true)"
6.03/2.40	                }],
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6.03/2.40	                        "type": "PlainIntegerRelationState",
6.03/2.40	                        "relations": []
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "97": {
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6.03/2.40	                    "arithmetic": {
6.03/2.40	                        "type": "PlainIntegerRelationState",
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6.03/2.40	            "10": {
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6.03/2.40	                        "relations": []
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6.03/2.40	                    "ground": ["T1"],
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6.03/2.40	                }
6.03/2.40	            },
6.03/2.40	            "98": {
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6.03/2.40	                }],
6.03/2.40	                "kb": {
6.03/2.40	                    "nonunifying": [],
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6.03/2.40	                        "relations": []
6.03/2.40	                    },
6.03/2.40	                    "ground": [],
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6.03/2.40	                }
6.03/2.40	            }
6.03/2.40	        },
6.03/2.40	        "edges": [
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6.03/2.40	                "to": 11,
6.03/2.40	                "label": "CASE"
6.03/2.40	            },
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6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 15,
6.03/2.40	                "to": 39,
6.03/2.40	                "label": "EVAL-BACKTRACK"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 34,
6.03/2.40	                "to": 37,
6.03/2.40	                "label": "SUCCESS"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 38,
6.03/2.40	                "to": 93,
6.03/2.40	                "label": "CASE"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 93,
6.03/2.40	                "to": 94,
6.03/2.40	                "label": "PARALLEL"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 93,
6.03/2.40	                "to": 95,
6.03/2.40	                "label": "PARALLEL"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 94,
6.03/2.40	                "to": 96,
6.03/2.40	                "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 95,
6.03/2.40	                "to": 98,
6.03/2.40	                "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 96,
6.03/2.40	                "to": 97,
6.03/2.40	                "label": "SUCCESS"
6.03/2.40	            },
6.03/2.40	            {
6.03/2.40	                "from": 98,
6.03/2.40	                "to": 38,
6.03/2.40	                "label": "INSTANCE with matching:\nX2 -> X4"
6.03/2.40	            }
6.03/2.40	        ],
6.03/2.40	        "type": "Graph"
6.03/2.40	    }
6.03/2.40	}
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(30)
6.03/2.40	Obligation:
6.03/2.40	Q restricted rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   f10_in(b) -> f10_out1
6.03/2.40	   f10_in(a) -> U1(f38_in, a)
6.03/2.40	   U1(f38_out1(X2), a) -> f10_out1
6.03/2.40	   f38_in -> f38_out1(b)
6.03/2.40	   f38_in -> U2(f38_in)
6.03/2.40	   U2(f38_out1(X4)) -> f38_out1(a)
6.03/2.40	
6.03/2.40	Q is empty.
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(31) QTRSRRRProof (EQUIVALENT)
6.03/2.40	Used ordering:
6.03/2.40	Polynomial interpretation [POLO]:
6.03/2.40	
6.03/2.40	   POL(U1(x_1, x_2)) = x_1 + 2*x_2
6.03/2.40	   POL(U2(x_1)) = 2*x_1
6.03/2.40	   POL(a) = 0
6.03/2.40	   POL(b) = 0
6.03/2.40	   POL(f10_in(x_1)) = 1 + 2*x_1
6.03/2.40	   POL(f10_out1) = 0
6.03/2.40	   POL(f38_in) = 0
6.03/2.40	   POL(f38_out1(x_1)) = x_1
6.03/2.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.03/2.40	
6.03/2.40	   f10_in(b) -> f10_out1
6.03/2.40	   f10_in(a) -> U1(f38_in, a)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(32)
6.03/2.40	Obligation:
6.03/2.40	Q restricted rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   U1(f38_out1(X2), a) -> f10_out1
6.03/2.40	   f38_in -> f38_out1(b)
6.03/2.40	   f38_in -> U2(f38_in)
6.03/2.40	   U2(f38_out1(X4)) -> f38_out1(a)
6.03/2.40	
6.03/2.40	Q is empty.
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(33) QTRSRRRProof (EQUIVALENT)
6.03/2.40	Used ordering:
6.03/2.40	Polynomial interpretation [POLO]:
6.03/2.40	
6.03/2.40	   POL(U1(x_1, x_2)) = 2 + x_1 + 2*x_2
6.03/2.40	   POL(U2(x_1)) = 2*x_1
6.03/2.40	   POL(a) = 0
6.03/2.40	   POL(b) = 0
6.03/2.40	   POL(f10_out1) = 0
6.03/2.40	   POL(f38_in) = 0
6.03/2.40	   POL(f38_out1(x_1)) = 2*x_1
6.03/2.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.03/2.40	
6.03/2.40	   U1(f38_out1(X2), a) -> f10_out1
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(34)
6.03/2.40	Obligation:
6.03/2.40	Q restricted rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   f38_in -> f38_out1(b)
6.03/2.40	   f38_in -> U2(f38_in)
6.03/2.40	   U2(f38_out1(X4)) -> f38_out1(a)
6.03/2.40	
6.03/2.40	Q is empty.
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(35) QTRSRRRProof (EQUIVALENT)
6.03/2.40	Used ordering:
6.03/2.40	Polynomial interpretation [POLO]:
6.03/2.40	
6.03/2.40	   POL(U2(x_1)) = x_1
6.03/2.40	   POL(a) = 0
6.03/2.40	   POL(b) = 0
6.03/2.40	   POL(f38_in) = 1
6.03/2.40	   POL(f38_out1(x_1)) = 2*x_1
6.03/2.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.03/2.40	
6.03/2.40	   f38_in -> f38_out1(b)
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(36)
6.03/2.40	Obligation:
6.03/2.40	Q restricted rewrite system:
6.03/2.40	The TRS R consists of the following rules:
6.03/2.40	
6.03/2.40	   f38_in -> U2(f38_in)
6.03/2.40	   U2(f38_out1(X4)) -> f38_out1(a)
6.03/2.40	
6.03/2.40	Q is empty.
6.03/2.40	
6.03/2.40	----------------------------------------
6.03/2.40	
6.03/2.40	(37) QTRSRRRProof (EQUIVALENT)
6.03/2.40	Used ordering:
6.03/2.40	Polynomial interpretation [POLO]:
6.03/2.40	
6.03/2.40	   POL(U2(x_1)) = 2*x_1
6.03/2.40	   POL(a) = 0
6.03/2.40	   POL(f38_in) = 0
6.03/2.41	   POL(f38_out1(x_1)) = 1 + 2*x_1
6.03/2.41	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
6.03/2.41	
6.03/2.41	   U2(f38_out1(X4)) -> f38_out1(a)
6.03/2.41	
6.03/2.41	
6.03/2.41	
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(38)
6.03/2.41	Obligation:
6.03/2.41	Q restricted rewrite system:
6.03/2.41	The TRS R consists of the following rules:
6.03/2.41	
6.03/2.41	   f38_in -> U2(f38_in)
6.03/2.41	
6.03/2.41	Q is empty.
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(39) Overlay + Local Confluence (EQUIVALENT)
6.03/2.41	The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(40)
6.03/2.41	Obligation:
6.03/2.41	Q restricted rewrite system:
6.03/2.41	The TRS R consists of the following rules:
6.03/2.41	
6.03/2.41	   f38_in -> U2(f38_in)
6.03/2.41	
6.03/2.41	The set Q consists of the following terms:
6.03/2.41	
6.03/2.41	   f38_in
6.03/2.41	
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(41) DependencyPairsProof (EQUIVALENT)
6.03/2.41	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(42)
6.03/2.41	Obligation:
6.03/2.41	Q DP problem:
6.03/2.41	The TRS P consists of the following rules:
6.03/2.41	
6.03/2.41	   F38_IN -> F38_IN
6.03/2.41	
6.03/2.41	The TRS R consists of the following rules:
6.03/2.41	
6.03/2.41	   f38_in -> U2(f38_in)
6.03/2.41	
6.03/2.41	The set Q consists of the following terms:
6.03/2.41	
6.03/2.41	   f38_in
6.03/2.41	
6.03/2.41	We have to consider all minimal (P,Q,R)-chains.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(43) UsableRulesProof (EQUIVALENT)
6.03/2.41	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.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(44)
6.03/2.41	Obligation:
6.03/2.41	Q DP problem:
6.03/2.41	The TRS P consists of the following rules:
6.03/2.41	
6.03/2.41	   F38_IN -> F38_IN
6.03/2.41	
6.03/2.41	R is empty.
6.03/2.41	The set Q consists of the following terms:
6.03/2.41	
6.03/2.41	   f38_in
6.03/2.41	
6.03/2.41	We have to consider all minimal (P,Q,R)-chains.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(45) PrologToIRSwTTransformerProof (SOUND)
6.03/2.41	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.03/2.41	
6.03/2.41	{
6.03/2.41	    "root": 3,
6.03/2.41	    "program": {
6.03/2.41	        "directives": [],
6.03/2.41	        "clauses": [
6.03/2.41	            [
6.03/2.41	                "(p (b))",
6.03/2.41	                null
6.03/2.41	            ],
6.03/2.41	            [
6.03/2.41	                "(p (a))",
6.03/2.41	                "(p1 X)"
6.03/2.41	            ],
6.03/2.41	            [
6.03/2.41	                "(p1 (b))",
6.03/2.41	                null
6.03/2.41	            ],
6.03/2.41	            [
6.03/2.41	                "(p1 (a))",
6.03/2.41	                "(p1 X)"
6.03/2.41	            ]
6.03/2.41	        ]
6.03/2.41	    },
6.03/2.41	    "graph": {
6.03/2.41	        "nodes": {
6.03/2.41	            "33": {
6.03/2.41	                "goal": [],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "77": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": -1,
6.03/2.41	                    "scope": -1,
6.03/2.41	                    "term": "(p1 X4)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": ["X4"],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "12": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": 0,
6.03/2.41	                    "scope": 1,
6.03/2.41	                    "term": "(p T1)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": ["T1"],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "13": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": 1,
6.03/2.41	                    "scope": 1,
6.03/2.41	                    "term": "(p T1)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": ["T1"],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "68": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": 2,
6.03/2.41	                    "scope": 2,
6.03/2.41	                    "term": "(p1 X2)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": ["X2"],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "36": {
6.03/2.41	                "goal": [],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "47": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": -1,
6.03/2.41	                    "scope": -1,
6.03/2.41	                    "term": "(p1 X2)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": ["X2"],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "69": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": 3,
6.03/2.41	                    "scope": 2,
6.03/2.41	                    "term": "(p1 X2)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": ["X2"],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "48": {
6.03/2.41	                "goal": [],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "type": "Nodes",
6.03/2.41	            "3": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": -1,
6.03/2.41	                    "scope": -1,
6.03/2.41	                    "term": "(p T1)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": ["T1"],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "9": {
6.03/2.41	                "goal": [
6.03/2.41	                    {
6.03/2.41	                        "clause": 0,
6.03/2.41	                        "scope": 1,
6.03/2.41	                        "term": "(p T1)"
6.03/2.41	                    },
6.03/2.41	                    {
6.03/2.41	                        "clause": 1,
6.03/2.41	                        "scope": 1,
6.03/2.41	                        "term": "(p T1)"
6.03/2.41	                    }
6.03/2.41	                ],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": ["T1"],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "70": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": -1,
6.03/2.41	                    "scope": -1,
6.03/2.41	                    "term": "(true)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "73": {
6.03/2.41	                "goal": [],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "64": {
6.03/2.41	                "goal": [
6.03/2.41	                    {
6.03/2.41	                        "clause": 2,
6.03/2.41	                        "scope": 2,
6.03/2.41	                        "term": "(p1 X2)"
6.03/2.41	                    },
6.03/2.41	                    {
6.03/2.41	                        "clause": 3,
6.03/2.41	                        "scope": 2,
6.03/2.41	                        "term": "(p1 X2)"
6.03/2.41	                    }
6.03/2.41	                ],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": ["X2"],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            },
6.03/2.41	            "32": {
6.03/2.41	                "goal": [{
6.03/2.41	                    "clause": -1,
6.03/2.41	                    "scope": -1,
6.03/2.41	                    "term": "(true)"
6.03/2.41	                }],
6.03/2.41	                "kb": {
6.03/2.41	                    "nonunifying": [],
6.03/2.41	                    "intvars": {},
6.03/2.41	                    "arithmetic": {
6.03/2.41	                        "type": "PlainIntegerRelationState",
6.03/2.41	                        "relations": []
6.03/2.41	                    },
6.03/2.41	                    "ground": [],
6.03/2.41	                    "free": [],
6.03/2.41	                    "exprvars": []
6.03/2.41	                }
6.03/2.41	            }
6.03/2.41	        },
6.03/2.41	        "edges": [
6.03/2.41	            {
6.03/2.41	                "from": 3,
6.03/2.41	                "to": 9,
6.03/2.41	                "label": "CASE"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 9,
6.03/2.41	                "to": 12,
6.03/2.41	                "label": "PARALLEL"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 9,
6.03/2.41	                "to": 13,
6.03/2.41	                "label": "PARALLEL"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 12,
6.03/2.41	                "to": 32,
6.03/2.41	                "label": "EVAL with clause\np(b).\nand substitutionT1 -> b"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 12,
6.03/2.41	                "to": 33,
6.03/2.41	                "label": "EVAL-BACKTRACK"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 13,
6.03/2.41	                "to": 47,
6.03/2.41	                "label": "EVAL with clause\np(a) :- p1(X2).\nand substitutionT1 -> a"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 13,
6.03/2.41	                "to": 48,
6.03/2.41	                "label": "EVAL-BACKTRACK"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 32,
6.03/2.41	                "to": 36,
6.03/2.41	                "label": "SUCCESS"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 47,
6.03/2.41	                "to": 64,
6.03/2.41	                "label": "CASE"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 64,
6.03/2.41	                "to": 68,
6.03/2.41	                "label": "PARALLEL"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 64,
6.03/2.41	                "to": 69,
6.03/2.41	                "label": "PARALLEL"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 68,
6.03/2.41	                "to": 70,
6.03/2.41	                "label": "ONLY EVAL with clause\np1(b).\nand substitutionX2 -> b"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 69,
6.03/2.41	                "to": 77,
6.03/2.41	                "label": "ONLY EVAL with clause\np1(a) :- p1(X4).\nand substitutionX2 -> a"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 70,
6.03/2.41	                "to": 73,
6.03/2.41	                "label": "SUCCESS"
6.03/2.41	            },
6.03/2.41	            {
6.03/2.41	                "from": 77,
6.03/2.41	                "to": 47,
6.03/2.41	                "label": "INSTANCE with matching:\nX2 -> X4"
6.03/2.41	            }
6.03/2.41	        ],
6.03/2.41	        "type": "Graph"
6.03/2.41	    }
6.03/2.41	}
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(46)
6.03/2.41	Obligation:
6.03/2.41	Rules:
6.03/2.41	f69_out -> f64_out :|: TRUE
6.03/2.41	f68_out -> f64_out :|: TRUE
6.03/2.41	f64_in -> f68_in :|: TRUE
6.03/2.41	f64_in -> f69_in :|: TRUE
6.03/2.41	f64_out -> f47_out :|: TRUE
6.03/2.41	f47_in -> f64_in :|: TRUE
6.03/2.41	f77_in -> f47_in :|: TRUE
6.03/2.41	f47_out -> f77_out :|: TRUE
6.03/2.41	f69_in -> f77_in :|: TRUE
6.03/2.41	f77_out -> f69_out :|: TRUE
6.03/2.41	f9_out(T1) -> f3_out(T1) :|: TRUE
6.03/2.41	f3_in(x) -> f9_in(x) :|: TRUE
6.03/2.41	f9_in(x1) -> f12_in(x1) :|: TRUE
6.03/2.41	f9_in(x2) -> f13_in(x2) :|: TRUE
6.03/2.41	f13_out(x3) -> f9_out(x3) :|: TRUE
6.03/2.41	f12_out(x4) -> f9_out(x4) :|: TRUE
6.03/2.41	f48_out -> f13_out(x5) :|: TRUE
6.03/2.41	f13_in(a) -> f47_in :|: TRUE
6.03/2.41	f13_in(x6) -> f48_in :|: TRUE
6.03/2.41	f47_out -> f13_out(a) :|: TRUE
6.03/2.41	Start term: f3_in(T1)
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(47) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.03/2.41	Constructed simple dependency graph.
6.03/2.41	
6.03/2.41	Simplified to the following IRSwTs:
6.03/2.41	
6.03/2.41	intTRSProblem:
6.03/2.41	f64_in -> f69_in :|: TRUE
6.03/2.41	f47_in -> f64_in :|: TRUE
6.03/2.41	f77_in -> f47_in :|: TRUE
6.03/2.41	f69_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(48)
6.03/2.41	Obligation:
6.03/2.41	Rules:
6.03/2.41	f64_in -> f69_in :|: TRUE
6.03/2.41	f47_in -> f64_in :|: TRUE
6.03/2.41	f77_in -> f47_in :|: TRUE
6.03/2.41	f69_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(49) IntTRSCompressionProof (EQUIVALENT)
6.03/2.41	Compressed rules.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(50)
6.03/2.41	Obligation:
6.03/2.41	Rules:
6.03/2.41	f77_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(51) IRSFormatTransformerProof (EQUIVALENT)
6.03/2.41	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(52)
6.03/2.41	Obligation:
6.03/2.41	Rules:
6.03/2.41	f77_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(53) IRSwTTerminationDigraphProof (EQUIVALENT)
6.03/2.41	Constructed termination digraph!
6.03/2.41	Nodes:
6.03/2.41	(1) f77_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	Arcs:
6.03/2.41	(1) -> (1)
6.03/2.41	
6.03/2.41	This digraph is fully evaluated!
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(54)
6.03/2.41	Obligation:
6.03/2.41	
6.03/2.41	Termination digraph:
6.03/2.41	Nodes:
6.03/2.41	(1) f77_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	Arcs:
6.03/2.41	(1) -> (1)
6.03/2.41	
6.03/2.41	This digraph is fully evaluated!
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(55) FilterProof (EQUIVALENT)
6.03/2.41	Used the following sort dictionary for filtering: 
6.03/2.41	f77_in()
6.03/2.41	Replaced non-predefined constructor symbols by 0.
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(56)
6.03/2.41	Obligation:
6.03/2.41	Rules:
6.03/2.41	f77_in -> f77_in :|: TRUE
6.03/2.41	
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(57) IntTRSPeriodicNontermProof (COMPLETE)
6.03/2.41	Normalized system to the following form:
6.03/2.41	f(pc) -> f(1) :|: pc = 1 && TRUE
6.03/2.41	Witness term starting non-terminating reduction: f(1)
6.03/2.41	----------------------------------------
6.03/2.41	
6.03/2.41	(58)
6.03/2.41	NO
6.33/2.47	EOF