6.82/2.67	MAYBE
6.82/2.68	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.82/2.68	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.82/2.68	
6.82/2.68	
6.82/2.68	Left Termination of the query pattern
6.82/2.68	
6.82/2.68	transpose_aux(a,g,a)
6.82/2.68	
6.82/2.68	w.r.t. the given Prolog program could not be shown:
6.82/2.68	
6.82/2.68	(0) Prolog
6.82/2.68	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.82/2.68	(2) PiTRS
6.82/2.68	    (3) DependencyPairsProof [EQUIVALENT, 9 ms]
6.82/2.68	    (4) PiDP
6.82/2.68	    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (6) PiDP
6.82/2.68	    (7) UsableRulesProof [EQUIVALENT, 0 ms]
6.82/2.68	    (8) PiDP
6.82/2.68	    (9) PiDPToQDPProof [SOUND, 0 ms]
6.82/2.68	    (10) QDP
6.82/2.68	    (11) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (12) QDP
6.82/2.68	    (13) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (14) QDP
6.82/2.68	    (15) NonTerminationLoopProof [COMPLETE, 0 ms]
6.82/2.68	    (16) NO
6.82/2.68	(17) PrologToPiTRSProof [SOUND, 0 ms]
6.82/2.68	(18) PiTRS
6.82/2.68	    (19) DependencyPairsProof [EQUIVALENT, 0 ms]
6.82/2.68	    (20) PiDP
6.82/2.68	    (21) DependencyGraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (22) PiDP
6.82/2.68	    (23) UsableRulesProof [EQUIVALENT, 0 ms]
6.82/2.68	    (24) PiDP
6.82/2.68	    (25) PiDPToQDPProof [SOUND, 0 ms]
6.82/2.68	    (26) QDP
6.82/2.68	    (27) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (28) QDP
6.82/2.68	    (29) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (30) QDP
6.82/2.68	    (31) NonTerminationLoopProof [COMPLETE, 0 ms]
6.82/2.68	    (32) NO
6.82/2.68	(33) PrologToDTProblemTransformerProof [SOUND, 57 ms]
6.82/2.68	(34) TRIPLES
6.82/2.68	    (35) TriplesToPiDPProof [SOUND, 0 ms]
6.82/2.68	    (36) PiDP
6.82/2.68	    (37) DependencyGraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (38) PiDP
6.82/2.68	    (39) PiDPToQDPProof [SOUND, 0 ms]
6.82/2.68	    (40) QDP
6.82/2.68	    (41) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (42) QDP
6.82/2.68	    (43) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (44) QDP
6.82/2.68	    (45) NonTerminationLoopProof [COMPLETE, 0 ms]
6.82/2.68	    (46) NO
6.82/2.68	(47) PrologToIRSwTTransformerProof [SOUND, 62 ms]
6.82/2.68	(48) IRSwT
6.82/2.68	    (49) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (50) IRSwT
6.82/2.68	    (51) IntTRSCompressionProof [EQUIVALENT, 51 ms]
6.82/2.68	    (52) IRSwT
6.82/2.68	    (53) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
6.82/2.68	    (54) IRSwT
6.82/2.68	    (55) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (56) IRSwT
6.82/2.68	    (57) IRSwTToIntTRSProof [SOUND, 0 ms]
6.82/2.68	    (58) IRSwT
6.82/2.68	(59) PrologToTRSTransformerProof [SOUND, 23 ms]
6.82/2.68	(60) QTRS
6.82/2.68	    (61) AAECC Innermost [EQUIVALENT, 0 ms]
6.82/2.68	    (62) QTRS
6.82/2.68	    (63) DependencyPairsProof [EQUIVALENT, 0 ms]
6.82/2.68	    (64) QDP
6.82/2.68	    (65) DependencyGraphProof [EQUIVALENT, 0 ms]
6.82/2.68	    (66) QDP
6.82/2.68	    (67) UsableRulesProof [EQUIVALENT, 0 ms]
6.82/2.68	    (68) QDP
6.82/2.68	    (69) QReductionProof [EQUIVALENT, 0 ms]
6.82/2.68	    (70) QDP
6.82/2.68	    (71) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (72) QDP
6.82/2.68	    (73) TransformationProof [EQUIVALENT, 0 ms]
6.82/2.68	    (74) QDP
6.82/2.68	    (75) NonTerminationLoopProof [COMPLETE, 0 ms]
6.82/2.68	    (76) NO
6.82/2.68	
6.82/2.68	
6.82/2.68	----------------------------------------
6.82/2.68	
6.82/2.68	(0)
6.82/2.68	Obligation:
6.82/2.68	Clauses:
6.82/2.68	
6.82/2.68	transpose_aux(.(R, Rs), X1, .(C, Cs)) :- row2col(R, .(C, Cs), Cols1, [], Accm).
6.82/2.68	row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) :- row2col(Xs, Cols, Cols1, .([], A), B).
6.82/2.68	
6.82/2.68	
6.82/2.68	Query: transpose_aux(a,g,a)
6.82/2.68	----------------------------------------
6.82/2.68	
6.82/2.68	(1) PrologToPiTRSProof (SOUND)
6.82/2.68	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.82/2.68	
6.82/2.68	transpose_aux_in_3: (f,b,f)
6.82/2.68	
6.82/2.68	row2col_in_5: (f,f,f,b,f)
6.82/2.68	
6.82/2.68	Transforming Prolog into the following Term Rewriting System:
6.82/2.68	
6.82/2.68	Pi-finite rewrite system:
6.82/2.68	The TRS R consists of the following rules:
6.82/2.68	
6.82/2.68	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.68	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.68	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.68	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.68	
6.82/2.68	The argument filtering Pi contains the following mapping:
6.82/2.68	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.68	
6.82/2.68	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
6.82/2.68	
6.82/2.68	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.68	
6.82/2.68	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
6.82/2.68	
6.82/2.68	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
6.82/2.68	
6.82/2.68	.(x1, x2)  =  .(x1, x2)
6.82/2.68	
6.82/2.68	[]  =  []
6.82/2.68	
6.82/2.68	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
6.82/2.68	
6.82/2.68	
6.82/2.68	
6.82/2.68	
6.82/2.68	
6.82/2.68	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.82/2.68	
6.82/2.68	
6.82/2.68	
6.82/2.68	----------------------------------------
6.82/2.68	
6.82/2.68	(2)
6.82/2.68	Obligation:
6.82/2.68	Pi-finite rewrite system:
6.82/2.68	The TRS R consists of the following rules:
6.82/2.68	
6.82/2.68	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.68	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.68	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.68	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.68	
6.82/2.68	The argument filtering Pi contains the following mapping:
6.82/2.68	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.68	
6.82/2.68	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
6.82/2.68	
6.82/2.68	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.68	
6.82/2.68	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
6.82/2.68	
6.82/2.68	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
6.82/2.68	
6.82/2.68	.(x1, x2)  =  .(x1, x2)
6.82/2.68	
6.82/2.68	[]  =  []
6.82/2.68	
6.82/2.68	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
6.82/2.68	
6.82/2.68	
6.82/2.68	
6.82/2.68	----------------------------------------
6.82/2.68	
6.82/2.68	(3) DependencyPairsProof (EQUIVALENT)
6.82/2.68	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.82/2.68	Pi DP problem:
6.82/2.68	The TRS P consists of the following rules:
6.82/2.68	
6.82/2.68	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.68	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
6.82/2.68	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.68	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.68	
6.82/2.68	The TRS R consists of the following rules:
6.82/2.68	
6.82/2.68	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.68	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.68	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.68	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.68	
6.82/2.68	The argument filtering Pi contains the following mapping:
6.82/2.68	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.68	
6.82/2.68	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
6.82/2.68	
6.82/2.68	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.68	
6.82/2.68	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
6.82/2.68	
6.82/2.68	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
6.82/2.69	
6.82/2.69	.(x1, x2)  =  .(x1, x2)
6.82/2.69	
6.82/2.69	[]  =  []
6.82/2.69	
6.82/2.69	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
6.82/2.69	
6.82/2.69	TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
6.82/2.69	
6.82/2.69	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x6)
6.82/2.69	
6.82/2.69	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.69	
6.82/2.69	U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x8)
6.82/2.69	
6.82/2.69	
6.82/2.69	We have to consider all (P,R,Pi)-chains
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(4)
6.82/2.69	Obligation:
6.82/2.69	Pi DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.69	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
6.82/2.69	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.69	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.69	
6.82/2.69	The TRS R consists of the following rules:
6.82/2.69	
6.82/2.69	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.69	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.69	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.69	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.69	
6.82/2.69	The argument filtering Pi contains the following mapping:
6.82/2.69	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.69	
6.82/2.69	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
6.82/2.69	
6.82/2.69	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.69	
6.82/2.69	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
6.82/2.69	
6.82/2.69	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
6.82/2.69	
6.82/2.69	.(x1, x2)  =  .(x1, x2)
6.82/2.69	
6.82/2.69	[]  =  []
6.82/2.69	
6.82/2.69	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
6.82/2.69	
6.82/2.69	TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
6.82/2.69	
6.82/2.69	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x6)
6.82/2.69	
6.82/2.69	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.69	
6.82/2.69	U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x8)
6.82/2.69	
6.82/2.69	
6.82/2.69	We have to consider all (P,R,Pi)-chains
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(5) DependencyGraphProof (EQUIVALENT)
6.82/2.69	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(6)
6.82/2.69	Obligation:
6.82/2.69	Pi DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.69	
6.82/2.69	The TRS R consists of the following rules:
6.82/2.69	
6.82/2.69	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.69	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.69	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.69	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.69	
6.82/2.69	The argument filtering Pi contains the following mapping:
6.82/2.69	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.69	
6.82/2.69	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x6)
6.82/2.69	
6.82/2.69	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.69	
6.82/2.69	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x8)
6.82/2.69	
6.82/2.69	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x5)
6.82/2.69	
6.82/2.69	.(x1, x2)  =  .(x1, x2)
6.82/2.69	
6.82/2.69	[]  =  []
6.82/2.69	
6.82/2.69	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga
6.82/2.69	
6.82/2.69	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.69	
6.82/2.69	
6.82/2.69	We have to consider all (P,R,Pi)-chains
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(7) UsableRulesProof (EQUIVALENT)
6.82/2.69	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(8)
6.82/2.69	Obligation:
6.82/2.69	Pi DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.69	
6.82/2.69	R is empty.
6.82/2.69	The argument filtering Pi contains the following mapping:
6.82/2.69	.(x1, x2)  =  .(x1, x2)
6.82/2.69	
6.82/2.69	[]  =  []
6.82/2.69	
6.82/2.69	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.69	
6.82/2.69	
6.82/2.69	We have to consider all (P,R,Pi)-chains
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(9) PiDPToQDPProof (SOUND)
6.82/2.69	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(10)
6.82/2.69	Obligation:
6.82/2.69	Q DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   ROW2COL_IN_AAAGA(A) -> ROW2COL_IN_AAAGA(.([], A))
6.82/2.69	
6.82/2.69	R is empty.
6.82/2.69	Q is empty.
6.82/2.69	We have to consider all (P,Q,R)-chains.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(11) TransformationProof (EQUIVALENT)
6.82/2.69	By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) -> ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:
6.82/2.69	
6.82/2.69	   (ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))),ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))))
6.82/2.69	
6.82/2.69	
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(12)
6.82/2.69	Obligation:
6.82/2.69	Q DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0)))
6.82/2.69	
6.82/2.69	R is empty.
6.82/2.69	Q is empty.
6.82/2.69	We have to consider all (P,Q,R)-chains.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(13) TransformationProof (EQUIVALENT)
6.82/2.69	By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:
6.82/2.69	
6.82/2.69	   (ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0)))),ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0)))))
6.82/2.69	
6.82/2.69	
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(14)
6.82/2.69	Obligation:
6.82/2.69	Q DP problem:
6.82/2.69	The TRS P consists of the following rules:
6.82/2.69	
6.82/2.69	   ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.69	
6.82/2.69	R is empty.
6.82/2.69	Q is empty.
6.82/2.69	We have to consider all (P,Q,R)-chains.
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(15) NonTerminationLoopProof (COMPLETE)
6.82/2.69	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.82/2.69	Found a loop by semiunifying a rule from P directly.
6.82/2.69	
6.82/2.69	s = ROW2COL_IN_AAAGA(.([], .([], z0))) evaluates to  t =ROW2COL_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.69	
6.82/2.69	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.82/2.69	* Matcher: [z0 / .([], z0)]
6.82/2.69	* Semiunifier: [ ]
6.82/2.69	
6.82/2.69	--------------------------------------------------------------------------------
6.82/2.69	Rewriting sequence
6.82/2.69	
6.82/2.69	The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(16)
6.82/2.69	NO
6.82/2.69	
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(17) PrologToPiTRSProof (SOUND)
6.82/2.69	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.82/2.69	
6.82/2.69	transpose_aux_in_3: (f,b,f)
6.82/2.69	
6.82/2.69	row2col_in_5: (f,f,f,b,f)
6.82/2.69	
6.82/2.69	Transforming Prolog into the following Term Rewriting System:
6.82/2.69	
6.82/2.69	Pi-finite rewrite system:
6.82/2.69	The TRS R consists of the following rules:
6.82/2.69	
6.82/2.69	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.69	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.69	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.69	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.69	
6.82/2.69	The argument filtering Pi contains the following mapping:
6.82/2.69	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.69	
6.82/2.69	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
6.82/2.69	
6.82/2.69	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.69	
6.82/2.69	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
6.82/2.69	
6.82/2.69	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
6.82/2.69	
6.82/2.69	.(x1, x2)  =  .(x1, x2)
6.82/2.69	
6.82/2.69	[]  =  []
6.82/2.69	
6.82/2.69	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.82/2.69	
6.82/2.69	
6.82/2.69	
6.82/2.69	----------------------------------------
6.82/2.69	
6.82/2.69	(18)
6.82/2.69	Obligation:
6.82/2.69	Pi-finite rewrite system:
6.82/2.69	The TRS R consists of the following rules:
6.82/2.69	
6.82/2.69	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.69	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.69	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.69	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.70	
6.82/2.70	The argument filtering Pi contains the following mapping:
6.82/2.70	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.70	
6.82/2.70	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
6.82/2.70	
6.82/2.70	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.70	
6.82/2.70	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
6.82/2.70	
6.82/2.70	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
6.82/2.70	
6.82/2.70	.(x1, x2)  =  .(x1, x2)
6.82/2.70	
6.82/2.70	[]  =  []
6.82/2.70	
6.82/2.70	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
6.82/2.70	
6.82/2.70	
6.82/2.70	
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(19) DependencyPairsProof (EQUIVALENT)
6.82/2.70	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.82/2.70	Pi DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.70	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.70	
6.82/2.70	The TRS R consists of the following rules:
6.82/2.70	
6.82/2.70	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.70	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.70	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.70	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.70	
6.82/2.70	The argument filtering Pi contains the following mapping:
6.82/2.70	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.70	
6.82/2.70	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
6.82/2.70	
6.82/2.70	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.70	
6.82/2.70	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
6.82/2.70	
6.82/2.70	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
6.82/2.70	
6.82/2.70	.(x1, x2)  =  .(x1, x2)
6.82/2.70	
6.82/2.70	[]  =  []
6.82/2.70	
6.82/2.70	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
6.82/2.70	
6.82/2.70	TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
6.82/2.70	
6.82/2.70	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x3, x6)
6.82/2.70	
6.82/2.70	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.70	
6.82/2.70	U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x6, x8)
6.82/2.70	
6.82/2.70	
6.82/2.70	We have to consider all (P,R,Pi)-chains
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(20)
6.82/2.70	Obligation:
6.82/2.70	Pi DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> U1_AGA(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.70	   TRANSPOSE_AUX_IN_AGA(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.70	
6.82/2.70	The TRS R consists of the following rules:
6.82/2.70	
6.82/2.70	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.70	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.70	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.70	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.70	
6.82/2.70	The argument filtering Pi contains the following mapping:
6.82/2.70	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.70	
6.82/2.70	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
6.82/2.70	
6.82/2.70	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.70	
6.82/2.70	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
6.82/2.70	
6.82/2.70	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
6.82/2.70	
6.82/2.70	.(x1, x2)  =  .(x1, x2)
6.82/2.70	
6.82/2.70	[]  =  []
6.82/2.70	
6.82/2.70	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
6.82/2.70	
6.82/2.70	TRANSPOSE_AUX_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGA(x2)
6.82/2.70	
6.82/2.70	U1_AGA(x1, x2, x3, x4, x5, x6)  =  U1_AGA(x3, x6)
6.82/2.70	
6.82/2.70	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.70	
6.82/2.70	U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AAAGA(x6, x8)
6.82/2.70	
6.82/2.70	
6.82/2.70	We have to consider all (P,R,Pi)-chains
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(21) DependencyGraphProof (EQUIVALENT)
6.82/2.70	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(22)
6.82/2.70	Obligation:
6.82/2.70	Pi DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.70	
6.82/2.70	The TRS R consists of the following rules:
6.82/2.70	
6.82/2.70	   transpose_aux_in_aga(.(R, Rs), X1, .(C, Cs)) -> U1_aga(R, Rs, X1, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
6.82/2.70	   row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
6.82/2.70	   U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) -> row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
6.82/2.70	   U1_aga(R, Rs, X1, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) -> transpose_aux_out_aga(.(R, Rs), X1, .(C, Cs))
6.82/2.70	
6.82/2.70	The argument filtering Pi contains the following mapping:
6.82/2.70	transpose_aux_in_aga(x1, x2, x3)  =  transpose_aux_in_aga(x2)
6.82/2.70	
6.82/2.70	U1_aga(x1, x2, x3, x4, x5, x6)  =  U1_aga(x3, x6)
6.82/2.70	
6.82/2.70	row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
6.82/2.70	
6.82/2.70	U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_aaaga(x6, x8)
6.82/2.70	
6.82/2.70	row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x4, x5)
6.82/2.70	
6.82/2.70	.(x1, x2)  =  .(x1, x2)
6.82/2.70	
6.82/2.70	[]  =  []
6.82/2.70	
6.82/2.70	transpose_aux_out_aga(x1, x2, x3)  =  transpose_aux_out_aga(x2)
6.82/2.70	
6.82/2.70	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.70	
6.82/2.70	
6.82/2.70	We have to consider all (P,R,Pi)-chains
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(23) UsableRulesProof (EQUIVALENT)
6.82/2.70	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(24)
6.82/2.70	Obligation:
6.82/2.70	Pi DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) -> ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
6.82/2.70	
6.82/2.70	R is empty.
6.82/2.70	The argument filtering Pi contains the following mapping:
6.82/2.70	.(x1, x2)  =  .(x1, x2)
6.82/2.70	
6.82/2.70	[]  =  []
6.82/2.70	
6.82/2.70	ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
6.82/2.70	
6.82/2.70	
6.82/2.70	We have to consider all (P,R,Pi)-chains
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(25) PiDPToQDPProof (SOUND)
6.82/2.70	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(26)
6.82/2.70	Obligation:
6.82/2.70	Q DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   ROW2COL_IN_AAAGA(A) -> ROW2COL_IN_AAAGA(.([], A))
6.82/2.70	
6.82/2.70	R is empty.
6.82/2.70	Q is empty.
6.82/2.70	We have to consider all (P,Q,R)-chains.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(27) TransformationProof (EQUIVALENT)
6.82/2.70	By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(A) -> ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules [LPAR04]:
6.82/2.70	
6.82/2.70	   (ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))),ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))))
6.82/2.70	
6.82/2.70	
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(28)
6.82/2.70	Obligation:
6.82/2.70	Q DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0)))
6.82/2.70	
6.82/2.70	R is empty.
6.82/2.70	Q is empty.
6.82/2.70	We have to consider all (P,Q,R)-chains.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(29) TransformationProof (EQUIVALENT)
6.82/2.70	By instantiating [LPAR04] the rule ROW2COL_IN_AAAGA(.([], z0)) -> ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:
6.82/2.70	
6.82/2.70	   (ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0)))),ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0)))))
6.82/2.70	
6.82/2.70	
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(30)
6.82/2.70	Obligation:
6.82/2.70	Q DP problem:
6.82/2.70	The TRS P consists of the following rules:
6.82/2.70	
6.82/2.70	   ROW2COL_IN_AAAGA(.([], .([], z0))) -> ROW2COL_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.70	
6.82/2.70	R is empty.
6.82/2.70	Q is empty.
6.82/2.70	We have to consider all (P,Q,R)-chains.
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(31) NonTerminationLoopProof (COMPLETE)
6.82/2.70	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.82/2.70	Found a loop by semiunifying a rule from P directly.
6.82/2.70	
6.82/2.70	s = ROW2COL_IN_AAAGA(.([], .([], z0))) evaluates to  t =ROW2COL_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.70	
6.82/2.70	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.82/2.70	* Matcher: [z0 / .([], z0)]
6.82/2.70	* Semiunifier: [ ]
6.82/2.70	
6.82/2.70	--------------------------------------------------------------------------------
6.82/2.70	Rewriting sequence
6.82/2.70	
6.82/2.70	The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).
6.82/2.70	
6.82/2.70	
6.82/2.70	
6.82/2.70	
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(32)
6.82/2.70	NO
6.82/2.70	
6.82/2.70	----------------------------------------
6.82/2.70	
6.82/2.70	(33) PrologToDTProblemTransformerProof (SOUND)
6.82/2.70	Built DT problem from termination graph DT10.
6.82/2.70	
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6.82/2.70	                    "term": "(transpose_aux T1 T2 T3)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": ["T2"],
6.82/2.70	                    "free": [],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "166": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": 0,
6.82/2.70	                    "scope": 1,
6.82/2.70	                    "term": "(transpose_aux T1 T2 T3)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": ["T2"],
6.82/2.70	                    "free": [],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "188": {
6.82/2.70	                "goal": [],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "243": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": -1,
6.82/2.70	                    "scope": -1,
6.82/2.70	                    "term": "(row2col T155 T156 X291 (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) ([]))))))))) X292)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [
6.82/2.70	                        "X291",
6.82/2.70	                        "X292"
6.82/2.70	                    ],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "244": {
6.82/2.70	                "goal": [],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "168": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": -1,
6.82/2.70	                    "scope": -1,
6.82/2.70	                    "term": "(row2col T14 (. T15 T16) X12 ([]) X13)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [
6.82/2.70	                        "X12",
6.82/2.70	                        "X13"
6.82/2.70	                    ],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "245": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": -1,
6.82/2.70	                    "scope": -1,
6.82/2.70	                    "term": "(row2col T155 T156 X291 (. ([]) T161) X292)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": ["T161"],
6.82/2.70	                    "free": [
6.82/2.70	                        "X291",
6.82/2.70	                        "X292"
6.82/2.70	                    ],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "169": {
6.82/2.70	                "goal": [],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "202": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": -1,
6.82/2.70	                    "scope": -1,
6.82/2.70	                    "term": "(row2col T83 T84 X147 (. ([]) (. ([]) (. ([]) (. ([]) ([]))))) X148)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [
6.82/2.70	                        "X147",
6.82/2.70	                        "X148"
6.82/2.70	                    ],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "224": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": 1,
6.82/2.70	                    "scope": 8,
6.82/2.70	                    "term": "(row2col T119 T120 X219 (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) (. ([]) ([]))))))) X220)"
6.82/2.70	                }],
6.82/2.70	                "kb": {
6.82/2.70	                    "nonunifying": [],
6.82/2.70	                    "intvars": {},
6.82/2.70	                    "arithmetic": {
6.82/2.70	                        "type": "PlainIntegerRelationState",
6.82/2.70	                        "relations": []
6.82/2.70	                    },
6.82/2.70	                    "ground": [],
6.82/2.70	                    "free": [
6.82/2.70	                        "X219",
6.82/2.70	                        "X220"
6.82/2.70	                    ],
6.82/2.70	                    "exprvars": []
6.82/2.70	                }
6.82/2.70	            },
6.82/2.70	            "246": {
6.82/2.70	                "goal": [{
6.82/2.70	                    "clause": 1,
6.82/2.70	                    "scope": 10,
6.82/2.71	                    "term": "(row2col T155 T156 X291 (. ([]) T161) X292)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": ["T161"],
6.82/2.71	                    "free": [
6.82/2.71	                        "X291",
6.82/2.71	                        "X292"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "203": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "247": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": -1,
6.82/2.71	                    "scope": -1,
6.82/2.71	                    "term": "(row2col T182 T183 X336 (. ([]) (. ([]) T181)) X337)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": ["T181"],
6.82/2.71	                    "free": [
6.82/2.71	                        "X336",
6.82/2.71	                        "X337"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "248": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "206": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": 1,
6.82/2.71	                    "scope": 6,
6.82/2.71	                    "term": "(row2col T83 T84 X147 (. ([]) (. ([]) (. ([]) (. ([]) ([]))))) X148)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X147",
6.82/2.71	                        "X148"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            }
6.82/2.71	        },
6.82/2.71	        "edges": [
6.82/2.71	            {
6.82/2.71	                "from": 1,
6.82/2.71	                "to": 166,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 166,
6.82/2.71	                "to": 168,
6.82/2.71	                "label": "EVAL with clause\ntranspose_aux(.(X7, X8), X9, .(X10, X11)) :- row2col(X7, .(X10, X11), X12, [], X13).\nand substitutionX7 -> T14,\nX8 -> T10,\nT1 -> .(T14, T10),\nT2 -> T11,\nX9 -> T11,\nX10 -> T15,\nX11 -> T16,\nT3 -> .(T15, T16),\nT9 -> T14,\nT12 -> T15,\nT13 -> T16"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 166,
6.82/2.71	                "to": 169,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 168,
6.82/2.71	                "to": 170,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 170,
6.82/2.71	                "to": 173,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X32, X33), .(.(X32, X34), X35), .(X34, X36), X37, X38) :- row2col(X33, X35, X36, .([], X37), X38).\nand substitutionX32 -> T25,\nX33 -> T29,\nT14 -> .(T25, T29),\nX34 -> T27,\nT15 -> .(T25, T27),\nT16 -> T30,\nX35 -> T30,\nX36 -> X39,\nX12 -> .(T27, X39),\nX37 -> [],\nX13 -> X40,\nX38 -> X40,\nT26 -> T29,\nT28 -> T30"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 170,
6.82/2.71	                "to": 174,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 173,
6.82/2.71	                "to": 176,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 176,
6.82/2.71	                "to": 178,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X68, X69), .(.(X68, X70), X71), .(X70, X72), X73, X74) :- row2col(X69, X71, X72, .([], X73), X74).\nand substitutionX68 -> T43,\nX69 -> T47,\nT29 -> .(T43, T47),\nX70 -> T45,\nX71 -> T48,\nT30 -> .(.(T43, T45), T48),\nX72 -> X75,\nX39 -> .(T45, X75),\nX73 -> .([], []),\nX40 -> X76,\nX74 -> X76,\nT44 -> T47,\nT46 -> T48"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 176,
6.82/2.71	                "to": 179,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 178,
6.82/2.71	                "to": 184,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 184,
6.82/2.71	                "to": 187,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X104, X105), .(.(X104, X106), X107), .(X106, X108), X109, X110) :- row2col(X105, X107, X108, .([], X109), X110).\nand substitutionX104 -> T61,\nX105 -> T65,\nT47 -> .(T61, T65),\nX106 -> T63,\nX107 -> T66,\nT48 -> .(.(T61, T63), T66),\nX108 -> X111,\nX75 -> .(T63, X111),\nX109 -> .([], .([], [])),\nX76 -> X112,\nX110 -> X112,\nT62 -> T65,\nT64 -> T66"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 184,
6.82/2.71	                "to": 188,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 187,
6.82/2.71	                "to": 191,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 191,
6.82/2.71	                "to": 202,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X140, X141), .(.(X140, X142), X143), .(X142, X144), X145, X146) :- row2col(X141, X143, X144, .([], X145), X146).\nand substitutionX140 -> T79,\nX141 -> T83,\nT65 -> .(T79, T83),\nX142 -> T81,\nX143 -> T84,\nT66 -> .(.(T79, T81), T84),\nX144 -> X147,\nX111 -> .(T81, X147),\nX145 -> .([], .([], .([], []))),\nX112 -> X148,\nX146 -> X148,\nT80 -> T83,\nT82 -> T84"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 191,
6.82/2.71	                "to": 203,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 202,
6.82/2.71	                "to": 206,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 206,
6.82/2.71	                "to": 212,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X176, X177), .(.(X176, X178), X179), .(X178, X180), X181, X182) :- row2col(X177, X179, X180, .([], X181), X182).\nand substitutionX176 -> T97,\nX177 -> T101,\nT83 -> .(T97, T101),\nX178 -> T99,\nX179 -> T102,\nT84 -> .(.(T97, T99), T102),\nX180 -> X183,\nX147 -> .(T99, X183),\nX181 -> .([], .([], .([], .([], [])))),\nX148 -> X184,\nX182 -> X184,\nT98 -> T101,\nT100 -> T102"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 206,
6.82/2.71	                "to": 213,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 212,
6.82/2.71	                "to": 214,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 214,
6.82/2.71	                "to": 219,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X212, X213), .(.(X212, X214), X215), .(X214, X216), X217, X218) :- row2col(X213, X215, X216, .([], X217), X218).\nand substitutionX212 -> T115,\nX213 -> T119,\nT101 -> .(T115, T119),\nX214 -> T117,\nX215 -> T120,\nT102 -> .(.(T115, T117), T120),\nX216 -> X219,\nX183 -> .(T117, X219),\nX217 -> .([], .([], .([], .([], .([], []))))),\nX184 -> X220,\nX218 -> X220,\nT116 -> T119,\nT118 -> T120"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 214,
6.82/2.71	                "to": 220,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 219,
6.82/2.71	                "to": 224,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 224,
6.82/2.71	                "to": 234,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X248, X249), .(.(X248, X250), X251), .(X250, X252), X253, X254) :- row2col(X249, X251, X252, .([], X253), X254).\nand substitutionX248 -> T133,\nX249 -> T137,\nT119 -> .(T133, T137),\nX250 -> T135,\nX251 -> T138,\nT120 -> .(.(T133, T135), T138),\nX252 -> X255,\nX219 -> .(T135, X255),\nX253 -> .([], .([], .([], .([], .([], .([], [])))))),\nX220 -> X256,\nX254 -> X256,\nT134 -> T137,\nT136 -> T138"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 224,
6.82/2.71	                "to": 235,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 234,
6.82/2.71	                "to": 239,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 239,
6.82/2.71	                "to": 243,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X284, X285), .(.(X284, X286), X287), .(X286, X288), X289, X290) :- row2col(X285, X287, X288, .([], X289), X290).\nand substitutionX284 -> T151,\nX285 -> T155,\nT137 -> .(T151, T155),\nX286 -> T153,\nX287 -> T156,\nT138 -> .(.(T151, T153), T156),\nX288 -> X291,\nX255 -> .(T153, X291),\nX289 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX256 -> X292,\nX290 -> X292,\nT152 -> T155,\nT154 -> T156"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 239,
6.82/2.71	                "to": 244,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 243,
6.82/2.71	                "to": 245,
6.82/2.71	                "label": "GENERALIZATION\nT161 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT161 is ground"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 245,
6.82/2.71	                "to": 246,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 246,
6.82/2.71	                "to": 247,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X329, X330), .(.(X329, X331), X332), .(X331, X333), X334, X335) :- row2col(X330, X332, X333, .([], X334), X335).\nand substitutionX329 -> T177,\nX330 -> T182,\nT155 -> .(T177, T182),\nX331 -> T179,\nX332 -> T183,\nT156 -> .(.(T177, T179), T183),\nX333 -> X336,\nX291 -> .(T179, X336),\nT161 -> T181,\nX334 -> .([], T181),\nX292 -> X337,\nX335 -> X337,\nT178 -> T182,\nT180 -> T183"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 246,
6.82/2.71	                "to": 248,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 247,
6.82/2.71	                "to": 245,
6.82/2.71	                "label": "INSTANCE with matching:\nT155 -> T182\nT156 -> T183\nX291 -> X336\nT161 -> .([], T181)\nX292 -> X337"
6.82/2.71	            }
6.82/2.71	        ],
6.82/2.71	        "type": "Graph"
6.82/2.71	    }
6.82/2.71	}
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(34)
6.82/2.71	Obligation:
6.82/2.71	Triples:
6.82/2.71	
6.82/2.71	row2colA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) :- row2colA(X2, X4, X5, .([], X6), X7).
6.82/2.71	transpose_auxB(.(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10), X11, .(.(X1, X12), .(.(X2, X13), .(.(X3, X14), .(.(X4, X15), .(.(X5, X16), .(.(X6, X17), .(.(X7, X18), .(.(X8, X19), X20))))))))) :- row2colA(X9, X20, X21, .([], .([], .([], .([], .([], .([], .([], []))))))), X22).
6.82/2.71	
6.82/2.71	Clauses:
6.82/2.71	
6.82/2.71	row2colcA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) :- row2colcA(X2, X4, X5, .([], X6), X7).
6.82/2.71	
6.82/2.71	Afs:
6.82/2.71	
6.82/2.71	transpose_auxB(x1, x2, x3)  =  transpose_auxB(x2)
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(35) TriplesToPiDPProof (SOUND)
6.82/2.71	We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.82/2.71	
6.82/2.71	transpose_auxB_in_3: (f,b,f)
6.82/2.71	
6.82/2.71	row2colA_in_5: (f,f,f,b,f)
6.82/2.71	
6.82/2.71	Transforming TRIPLES into the following Term Rewriting System:
6.82/2.71	
6.82/2.71	Pi DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   TRANSPOSE_AUXB_IN_AGA(.(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10), X11, .(.(X1, X12), .(.(X2, X13), .(.(X3, X14), .(.(X4, X15), .(.(X5, X16), .(.(X6, X17), .(.(X7, X18), .(.(X8, X19), X20))))))))) -> U2_AGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colA_in_aaaga(X9, X20, X21, .([], .([], .([], .([], .([], .([], .([], []))))))), X22))
6.82/2.71	   TRANSPOSE_AUXB_IN_AGA(.(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10), X11, .(.(X1, X12), .(.(X2, X13), .(.(X3, X14), .(.(X4, X15), .(.(X5, X16), .(.(X6, X17), .(.(X7, X18), .(.(X8, X19), X20))))))))) -> ROW2COLA_IN_AAAGA(X9, X20, X21, .([], .([], .([], .([], .([], .([], .([], []))))))), X22)
6.82/2.71	   ROW2COLA_IN_AAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) -> U1_AAAGA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_aaaga(X2, X4, X5, .([], X6), X7))
6.82/2.71	   ROW2COLA_IN_AAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) -> ROW2COLA_IN_AAAGA(X2, X4, X5, .([], X6), X7)
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	The argument filtering Pi contains the following mapping:
6.82/2.71	row2colA_in_aaaga(x1, x2, x3, x4, x5)  =  row2colA_in_aaaga(x4)
6.82/2.71	
6.82/2.71	.(x1, x2)  =  .(x1, x2)
6.82/2.71	
6.82/2.71	[]  =  []
6.82/2.71	
6.82/2.71	TRANSPOSE_AUXB_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUXB_IN_AGA(x2)
6.82/2.71	
6.82/2.71	U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U2_AGA(x11, x21)
6.82/2.71	
6.82/2.71	ROW2COLA_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COLA_IN_AAAGA(x4)
6.82/2.71	
6.82/2.71	U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAGA(x6, x8)
6.82/2.71	
6.82/2.71	
6.82/2.71	We have to consider all (P,R,Pi)-chains
6.82/2.71	
6.82/2.71	
6.82/2.71	Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
6.82/2.71	
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(36)
6.82/2.71	Obligation:
6.82/2.71	Pi DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   TRANSPOSE_AUXB_IN_AGA(.(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10), X11, .(.(X1, X12), .(.(X2, X13), .(.(X3, X14), .(.(X4, X15), .(.(X5, X16), .(.(X6, X17), .(.(X7, X18), .(.(X8, X19), X20))))))))) -> U2_AGA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, row2colA_in_aaaga(X9, X20, X21, .([], .([], .([], .([], .([], .([], .([], []))))))), X22))
6.82/2.71	   TRANSPOSE_AUXB_IN_AGA(.(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10), X11, .(.(X1, X12), .(.(X2, X13), .(.(X3, X14), .(.(X4, X15), .(.(X5, X16), .(.(X6, X17), .(.(X7, X18), .(.(X8, X19), X20))))))))) -> ROW2COLA_IN_AAAGA(X9, X20, X21, .([], .([], .([], .([], .([], .([], .([], []))))))), X22)
6.82/2.71	   ROW2COLA_IN_AAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) -> U1_AAAGA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_aaaga(X2, X4, X5, .([], X6), X7))
6.82/2.71	   ROW2COLA_IN_AAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) -> ROW2COLA_IN_AAAGA(X2, X4, X5, .([], X6), X7)
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	The argument filtering Pi contains the following mapping:
6.82/2.71	row2colA_in_aaaga(x1, x2, x3, x4, x5)  =  row2colA_in_aaaga(x4)
6.82/2.71	
6.82/2.71	.(x1, x2)  =  .(x1, x2)
6.82/2.71	
6.82/2.71	[]  =  []
6.82/2.71	
6.82/2.71	TRANSPOSE_AUXB_IN_AGA(x1, x2, x3)  =  TRANSPOSE_AUXB_IN_AGA(x2)
6.82/2.71	
6.82/2.71	U2_AGA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21)  =  U2_AGA(x11, x21)
6.82/2.71	
6.82/2.71	ROW2COLA_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COLA_IN_AAAGA(x4)
6.82/2.71	
6.82/2.71	U1_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U1_AAAGA(x6, x8)
6.82/2.71	
6.82/2.71	
6.82/2.71	We have to consider all (P,R,Pi)-chains
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(37) DependencyGraphProof (EQUIVALENT)
6.82/2.71	The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(38)
6.82/2.71	Obligation:
6.82/2.71	Pi DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   ROW2COLA_IN_AAAGA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), X6, X7) -> ROW2COLA_IN_AAAGA(X2, X4, X5, .([], X6), X7)
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	The argument filtering Pi contains the following mapping:
6.82/2.71	.(x1, x2)  =  .(x1, x2)
6.82/2.71	
6.82/2.71	[]  =  []
6.82/2.71	
6.82/2.71	ROW2COLA_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COLA_IN_AAAGA(x4)
6.82/2.71	
6.82/2.71	
6.82/2.71	We have to consider all (P,R,Pi)-chains
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(39) PiDPToQDPProof (SOUND)
6.82/2.71	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(40)
6.82/2.71	Obligation:
6.82/2.71	Q DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   ROW2COLA_IN_AAAGA(X6) -> ROW2COLA_IN_AAAGA(.([], X6))
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	Q is empty.
6.82/2.71	We have to consider all (P,Q,R)-chains.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(41) TransformationProof (EQUIVALENT)
6.82/2.71	By instantiating [LPAR04] the rule ROW2COLA_IN_AAAGA(X6) -> ROW2COLA_IN_AAAGA(.([], X6)) we obtained the following new rules [LPAR04]:
6.82/2.71	
6.82/2.71	   (ROW2COLA_IN_AAAGA(.([], z0)) -> ROW2COLA_IN_AAAGA(.([], .([], z0))),ROW2COLA_IN_AAAGA(.([], z0)) -> ROW2COLA_IN_AAAGA(.([], .([], z0))))
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(42)
6.82/2.71	Obligation:
6.82/2.71	Q DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   ROW2COLA_IN_AAAGA(.([], z0)) -> ROW2COLA_IN_AAAGA(.([], .([], z0)))
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	Q is empty.
6.82/2.71	We have to consider all (P,Q,R)-chains.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(43) TransformationProof (EQUIVALENT)
6.82/2.71	By instantiating [LPAR04] the rule ROW2COLA_IN_AAAGA(.([], z0)) -> ROW2COLA_IN_AAAGA(.([], .([], z0))) we obtained the following new rules [LPAR04]:
6.82/2.71	
6.82/2.71	   (ROW2COLA_IN_AAAGA(.([], .([], z0))) -> ROW2COLA_IN_AAAGA(.([], .([], .([], z0)))),ROW2COLA_IN_AAAGA(.([], .([], z0))) -> ROW2COLA_IN_AAAGA(.([], .([], .([], z0)))))
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(44)
6.82/2.71	Obligation:
6.82/2.71	Q DP problem:
6.82/2.71	The TRS P consists of the following rules:
6.82/2.71	
6.82/2.71	   ROW2COLA_IN_AAAGA(.([], .([], z0))) -> ROW2COLA_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.71	
6.82/2.71	R is empty.
6.82/2.71	Q is empty.
6.82/2.71	We have to consider all (P,Q,R)-chains.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(45) NonTerminationLoopProof (COMPLETE)
6.82/2.71	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
6.82/2.71	Found a loop by semiunifying a rule from P directly.
6.82/2.71	
6.82/2.71	s = ROW2COLA_IN_AAAGA(.([], .([], z0))) evaluates to  t =ROW2COLA_IN_AAAGA(.([], .([], .([], z0))))
6.82/2.71	
6.82/2.71	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
6.82/2.71	* Matcher: [z0 / .([], z0)]
6.82/2.71	* Semiunifier: [ ]
6.82/2.71	
6.82/2.71	--------------------------------------------------------------------------------
6.82/2.71	Rewriting sequence
6.82/2.71	
6.82/2.71	The DP semiunifies directly so there is only one rewrite step from ROW2COLA_IN_AAAGA(.([], .([], z0))) to ROW2COLA_IN_AAAGA(.([], .([], .([], z0)))).
6.82/2.71	
6.82/2.71	
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(46)
6.82/2.71	NO
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(47) PrologToIRSwTTransformerProof (SOUND)
6.82/2.71	Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert
6.82/2.71	
6.82/2.71	{
6.82/2.71	    "root": 2,
6.82/2.71	    "program": {
6.82/2.71	        "directives": [],
6.82/2.71	        "clauses": [
6.82/2.71	            [
6.82/2.71	                "(transpose_aux (. R Rs) X1 (. C Cs))",
6.82/2.71	                "(row2col R (. C Cs) Cols1 ([]) Accm)"
6.82/2.71	            ],
6.82/2.71	            [
6.82/2.71	                "(row2col (. X Xs) (. (. X Ys) Cols) (. Ys Cols1) A B)",
6.82/2.71	                "(row2col Xs Cols Cols1 (. ([]) A) B)"
6.82/2.71	            ]
6.82/2.71	        ]
6.82/2.71	    },
6.82/2.71	    "graph": {
6.82/2.71	        "nodes": {
6.82/2.71	            "171": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": -1,
6.82/2.71	                    "scope": -1,
6.82/2.71	                    "term": "(row2col T38 T39 X55 (. ([]) ([])) X56)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X55",
6.82/2.71	                        "X56"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "193": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": -1,
6.82/2.71	                    "scope": -1,
6.82/2.71	                    "term": "(row2col T74 T75 X127 (. ([]) (. ([]) (. ([]) ([])))) X128)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X127",
6.82/2.71	                        "X128"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "type": "Nodes",
6.82/2.71	            "172": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "194": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "195": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": 1,
6.82/2.71	                    "scope": 5,
6.82/2.71	                    "term": "(row2col T74 T75 X127 (. ([]) (. ([]) (. ([]) ([])))) X128)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X127",
6.82/2.71	                        "X128"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "196": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": -1,
6.82/2.71	                    "scope": -1,
6.82/2.71	                    "term": "(row2col T92 T93 X163 (. ([]) (. ([]) (. ([]) (. ([]) ([]))))) X164)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X163",
6.82/2.71	                        "X164"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "175": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": 1,
6.82/2.71	                    "scope": 3,
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6.82/2.71	                        "X19",
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6.82/2.71	                        "X199",
6.82/2.71	                        "X200"
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6.82/2.71	                        "X235",
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6.82/2.71	                        "X307",
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6.82/2.71	                        "X307",
6.82/2.71	                        "X308"
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6.82/2.71	                }],
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6.82/2.71	                        "X235",
6.82/2.71	                        "X236"
6.82/2.71	                    ],
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6.82/2.71	            }
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6.82/2.71	            {
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6.82/2.71	                "to": 39,
6.82/2.71	                "label": "EVAL with clause\ntranspose_aux(.(X14, X15), X16, .(X17, X18)) :- row2col(X14, .(X17, X18), X19, [], X20).\nand substitutionX14 -> T19,\nX15 -> T15,\nT1 -> .(T19, T15),\nT2 -> T16,\nX16 -> T16,\nX17 -> T20,\nX18 -> T21,\nT3 -> .(T20, T21),\nT14 -> T19,\nT17 -> T20,\nT18 -> T21"
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6.82/2.71	            {
6.82/2.71	                "from": 39,
6.82/2.71	                "to": 167,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 167,
6.82/2.71	                "to": 171,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X48, X49), .(.(X48, X50), X51), .(X50, X52), X53, X54) :- row2col(X49, X51, X52, .([], X53), X54).\nand substitutionX48 -> T34,\nX49 -> T38,\nT19 -> .(T34, T38),\nX50 -> T36,\nT20 -> .(T34, T36),\nT21 -> T39,\nX51 -> T39,\nX52 -> X55,\nX19 -> .(T36, X55),\nX53 -> [],\nX20 -> X56,\nX54 -> X56,\nT35 -> T38,\nT37 -> T39"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 167,
6.82/2.71	                "to": 172,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 171,
6.82/2.71	                "to": 175,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 175,
6.82/2.71	                "to": 180,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X84, X85), .(.(X84, X86), X87), .(X86, X88), X89, X90) :- row2col(X85, X87, X88, .([], X89), X90).\nand substitutionX84 -> T52,\nX85 -> T56,\nT38 -> .(T52, T56),\nX86 -> T54,\nX87 -> T57,\nT39 -> .(.(T52, T54), T57),\nX88 -> X91,\nX55 -> .(T54, X91),\nX89 -> .([], []),\nX56 -> X92,\nX90 -> X92,\nT53 -> T56,\nT55 -> T57"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 175,
6.82/2.71	                "to": 181,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 180,
6.82/2.71	                "to": 185,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 185,
6.82/2.71	                "to": 193,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X120, X121), .(.(X120, X122), X123), .(X122, X124), X125, X126) :- row2col(X121, X123, X124, .([], X125), X126).\nand substitutionX120 -> T70,\nX121 -> T74,\nT56 -> .(T70, T74),\nX122 -> T72,\nX123 -> T75,\nT57 -> .(.(T70, T72), T75),\nX124 -> X127,\nX91 -> .(T72, X127),\nX125 -> .([], .([], [])),\nX92 -> X128,\nX126 -> X128,\nT71 -> T74,\nT73 -> T75"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 185,
6.82/2.71	                "to": 194,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 193,
6.82/2.71	                "to": 195,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 195,
6.82/2.71	                "to": 196,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X156, X157), .(.(X156, X158), X159), .(X158, X160), X161, X162) :- row2col(X157, X159, X160, .([], X161), X162).\nand substitutionX156 -> T88,\nX157 -> T92,\nT74 -> .(T88, T92),\nX158 -> T90,\nX159 -> T93,\nT75 -> .(.(T88, T90), T93),\nX160 -> X163,\nX127 -> .(T90, X163),\nX161 -> .([], .([], .([], []))),\nX128 -> X164,\nX162 -> X164,\nT89 -> T92,\nT91 -> T93"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 195,
6.82/2.71	                "to": 197,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 196,
6.82/2.71	                "to": 198,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 198,
6.82/2.71	                "to": 199,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X192, X193), .(.(X192, X194), X195), .(X194, X196), X197, X198) :- row2col(X193, X195, X196, .([], X197), X198).\nand substitutionX192 -> T106,\nX193 -> T110,\nT92 -> .(T106, T110),\nX194 -> T108,\nX195 -> T111,\nT93 -> .(.(T106, T108), T111),\nX196 -> X199,\nX163 -> .(T108, X199),\nX197 -> .([], .([], .([], .([], [])))),\nX164 -> X200,\nX198 -> X200,\nT107 -> T110,\nT109 -> T111"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 198,
6.82/2.71	                "to": 200,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 199,
6.82/2.71	                "to": 201,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 201,
6.82/2.71	                "to": 204,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X228, X229), .(.(X228, X230), X231), .(X230, X232), X233, X234) :- row2col(X229, X231, X232, .([], X233), X234).\nand substitutionX228 -> T124,\nX229 -> T128,\nT110 -> .(T124, T128),\nX230 -> T126,\nX231 -> T129,\nT111 -> .(.(T124, T126), T129),\nX232 -> X235,\nX199 -> .(T126, X235),\nX233 -> .([], .([], .([], .([], .([], []))))),\nX200 -> X236,\nX234 -> X236,\nT125 -> T128,\nT127 -> T129"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 201,
6.82/2.71	                "to": 205,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 204,
6.82/2.71	                "to": 207,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 207,
6.82/2.71	                "to": 215,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X264, X265), .(.(X264, X266), X267), .(X266, X268), X269, X270) :- row2col(X265, X267, X268, .([], X269), X270).\nand substitutionX264 -> T142,\nX265 -> T146,\nT128 -> .(T142, T146),\nX266 -> T144,\nX267 -> T147,\nT129 -> .(.(T142, T144), T147),\nX268 -> X271,\nX235 -> .(T144, X271),\nX269 -> .([], .([], .([], .([], .([], .([], [])))))),\nX236 -> X272,\nX270 -> X272,\nT143 -> T146,\nT145 -> T147"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 207,
6.82/2.71	                "to": 216,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 215,
6.82/2.71	                "to": 217,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 217,
6.82/2.71	                "to": 221,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X300, X301), .(.(X300, X302), X303), .(X302, X304), X305, X306) :- row2col(X301, X303, X304, .([], X305), X306).\nand substitutionX300 -> T160,\nX301 -> T164,\nT146 -> .(T160, T164),\nX302 -> T162,\nX303 -> T165,\nT147 -> .(.(T160, T162), T165),\nX304 -> X307,\nX271 -> .(T162, X307),\nX305 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX272 -> X308,\nX306 -> X308,\nT161 -> T164,\nT163 -> T165"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 217,
6.82/2.71	                "to": 222,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 221,
6.82/2.71	                "to": 226,
6.82/2.71	                "label": "GENERALIZATION\nT170 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT170 is ground"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 226,
6.82/2.71	                "to": 227,
6.82/2.71	                "label": "CASE"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 227,
6.82/2.71	                "to": 236,
6.82/2.71	                "label": "EVAL with clause\nrow2col(.(X345, X346), .(.(X345, X347), X348), .(X347, X349), X350, X351) :- row2col(X346, X348, X349, .([], X350), X351).\nand substitutionX345 -> T186,\nX346 -> T191,\nT164 -> .(T186, T191),\nX347 -> T188,\nX348 -> T192,\nT165 -> .(.(T186, T188), T192),\nX349 -> X352,\nX307 -> .(T188, X352),\nT170 -> T190,\nX350 -> .([], T190),\nX308 -> X353,\nX351 -> X353,\nT187 -> T191,\nT189 -> T192"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 227,
6.82/2.71	                "to": 237,
6.82/2.71	                "label": "EVAL-BACKTRACK"
6.82/2.71	            },
6.82/2.71	            {
6.82/2.71	                "from": 236,
6.82/2.71	                "to": 226,
6.82/2.71	                "label": "INSTANCE with matching:\nT164 -> T191\nT165 -> T192\nX307 -> X352\nT170 -> .([], T190)\nX308 -> X353"
6.82/2.71	            }
6.82/2.71	        ],
6.82/2.71	        "type": "Graph"
6.82/2.71	    }
6.82/2.71	}
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(48)
6.82/2.71	Obligation:
6.82/2.71	Rules:
6.82/2.71	f237_out -> f227_out(T170) :|: TRUE
6.82/2.71	f227_in(x) -> f237_in :|: TRUE
6.82/2.71	f236_out(T190) -> f227_out(T190) :|: TRUE
6.82/2.71	f227_in(x1) -> f236_in(x1) :|: TRUE
6.82/2.71	f226_out(.([], x2)) -> f236_out(x2) :|: TRUE
6.82/2.71	f236_in(x3) -> f226_in(.([], x3)) :|: TRUE
6.82/2.71	f226_in(x4) -> f227_in(x4) :|: TRUE
6.82/2.71	f227_out(x5) -> f226_out(x5) :|: TRUE
6.82/2.71	f7_out(T2) -> f2_out(T2) :|: TRUE
6.82/2.71	f2_in(x6) -> f7_in(x6) :|: TRUE
6.82/2.71	f39_out -> f7_out(T16) :|: TRUE
6.82/2.71	f7_in(x7) -> f52_in :|: TRUE
6.82/2.71	f7_in(x8) -> f39_in :|: TRUE
6.82/2.71	f52_out -> f7_out(x9) :|: TRUE
6.82/2.71	f167_out -> f39_out :|: TRUE
6.82/2.71	f39_in -> f167_in :|: TRUE
6.82/2.71	f167_in -> f171_in :|: TRUE
6.82/2.71	f171_out -> f167_out :|: TRUE
6.82/2.71	f172_out -> f167_out :|: TRUE
6.82/2.71	f167_in -> f172_in :|: TRUE
6.82/2.71	f171_in -> f175_in :|: TRUE
6.82/2.71	f175_out -> f171_out :|: TRUE
6.82/2.71	f175_in -> f181_in :|: TRUE
6.82/2.71	f175_in -> f180_in :|: TRUE
6.82/2.71	f181_out -> f175_out :|: TRUE
6.82/2.71	f180_out -> f175_out :|: TRUE
6.82/2.71	f185_out -> f180_out :|: TRUE
6.82/2.71	f180_in -> f185_in :|: TRUE
6.82/2.71	f185_in -> f193_in :|: TRUE
6.82/2.71	f193_out -> f185_out :|: TRUE
6.82/2.71	f194_out -> f185_out :|: TRUE
6.82/2.71	f185_in -> f194_in :|: TRUE
6.82/2.71	f193_in -> f195_in :|: TRUE
6.82/2.71	f195_out -> f193_out :|: TRUE
6.82/2.71	f196_out -> f195_out :|: TRUE
6.82/2.71	f197_out -> f195_out :|: TRUE
6.82/2.71	f195_in -> f196_in :|: TRUE
6.82/2.71	f195_in -> f197_in :|: TRUE
6.82/2.71	f198_out -> f196_out :|: TRUE
6.82/2.71	f196_in -> f198_in :|: TRUE
6.82/2.71	f200_out -> f198_out :|: TRUE
6.82/2.71	f199_out -> f198_out :|: TRUE
6.82/2.71	f198_in -> f199_in :|: TRUE
6.82/2.71	f198_in -> f200_in :|: TRUE
6.82/2.71	f201_out -> f199_out :|: TRUE
6.82/2.71	f199_in -> f201_in :|: TRUE
6.82/2.71	f205_out -> f201_out :|: TRUE
6.82/2.71	f204_out -> f201_out :|: TRUE
6.82/2.71	f201_in -> f205_in :|: TRUE
6.82/2.71	f201_in -> f204_in :|: TRUE
6.82/2.71	f207_out -> f204_out :|: TRUE
6.82/2.71	f204_in -> f207_in :|: TRUE
6.82/2.71	f207_in -> f215_in :|: TRUE
6.82/2.71	f215_out -> f207_out :|: TRUE
6.82/2.71	f216_out -> f207_out :|: TRUE
6.82/2.71	f207_in -> f216_in :|: TRUE
6.82/2.71	f215_in -> f217_in :|: TRUE
6.82/2.71	f217_out -> f215_out :|: TRUE
6.82/2.71	f217_in -> f221_in :|: TRUE
6.82/2.71	f222_out -> f217_out :|: TRUE
6.82/2.71	f221_out -> f217_out :|: TRUE
6.82/2.71	f217_in -> f222_in :|: TRUE
6.82/2.71	f226_out(.([], .([], .([], .([], .([], .([], .([], [])))))))) -> f221_out :|: TRUE
6.82/2.71	f221_in -> f226_in(.([], .([], .([], .([], .([], .([], .([], [])))))))) :|: TRUE
6.82/2.71	Start term: f2_in(T2)
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(49) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
6.82/2.71	Constructed simple dependency graph.
6.82/2.71	
6.82/2.71	Simplified to the following IRSwTs:
6.82/2.71	
6.82/2.71	intTRSProblem:
6.82/2.71	f227_in(x1) -> f236_in(x1) :|: TRUE
6.82/2.71	f236_in(x3) -> f226_in(.([], x3)) :|: TRUE
6.82/2.71	f226_in(x4) -> f227_in(x4) :|: TRUE
6.82/2.71	
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(50)
6.82/2.71	Obligation:
6.82/2.71	Rules:
6.82/2.71	f227_in(x1) -> f236_in(x1) :|: TRUE
6.82/2.71	f236_in(x3) -> f226_in(.([], x3)) :|: TRUE
6.82/2.71	f226_in(x4) -> f227_in(x4) :|: TRUE
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(51) IntTRSCompressionProof (EQUIVALENT)
6.82/2.71	Compressed rules.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(52)
6.82/2.71	Obligation:
6.82/2.71	Rules:
6.82/2.71	f227_in(x1:0) -> f227_in(.([], x1:0)) :|: TRUE
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(53) IRSFormatTransformerProof (EQUIVALENT)
6.82/2.71	Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(54)
6.82/2.71	Obligation:
6.82/2.71	Rules:
6.82/2.71	f227_in(x1:0) -> f227_in(.([], x1:0)) :|: TRUE
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(55) IRSwTTerminationDigraphProof (EQUIVALENT)
6.82/2.71	Constructed termination digraph!
6.82/2.71	Nodes:
6.82/2.71	(1) f227_in(x1:0) -> f227_in(.([], x1:0)) :|: TRUE
6.82/2.71	
6.82/2.71	Arcs:
6.82/2.71	(1) -> (1)
6.82/2.71	
6.82/2.71	This digraph is fully evaluated!
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(56)
6.82/2.71	Obligation:
6.82/2.71	
6.82/2.71	Termination digraph:
6.82/2.71	Nodes:
6.82/2.71	(1) f227_in(x1:0) -> f227_in(.([], x1:0)) :|: TRUE
6.82/2.71	
6.82/2.71	Arcs:
6.82/2.71	(1) -> (1)
6.82/2.71	
6.82/2.71	This digraph is fully evaluated!
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(57) IRSwTToIntTRSProof (SOUND)
6.82/2.71	Applied path-length measure to transform intTRS with terms to intTRS.
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(58)
6.82/2.71	Obligation:
6.82/2.71	Rules:
6.82/2.71	f227_in(x) -> f227_in(.([], x)) :|: TRUE
6.82/2.71	
6.82/2.71	----------------------------------------
6.82/2.71	
6.82/2.71	(59) PrologToTRSTransformerProof (SOUND)
6.82/2.71	Transformed Prolog program to TRS.
6.82/2.71	
6.82/2.71	{
6.82/2.71	    "root": 3,
6.82/2.71	    "program": {
6.82/2.71	        "directives": [],
6.82/2.71	        "clauses": [
6.82/2.71	            [
6.82/2.71	                "(transpose_aux (. R Rs) X1 (. C Cs))",
6.82/2.71	                "(row2col R (. C Cs) Cols1 ([]) Accm)"
6.82/2.71	            ],
6.82/2.71	            [
6.82/2.71	                "(row2col (. X Xs) (. (. X Ys) Cols) (. Ys Cols1) A B)",
6.82/2.71	                "(row2col Xs Cols Cols1 (. ([]) A) B)"
6.82/2.71	            ]
6.82/2.71	        ]
6.82/2.71	    },
6.82/2.71	    "graph": {
6.82/2.71	        "nodes": {
6.82/2.71	            "190": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "192": {
6.82/2.71	                "goal": [{
6.82/2.71	                    "clause": 1,
6.82/2.71	                    "scope": 4,
6.82/2.71	                    "term": "(row2col T56 T57 X91 (. ([]) (. ([]) ([]))) X92)"
6.82/2.71	                }],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [
6.82/2.71	                        "X91",
6.82/2.71	                        "X92"
6.82/2.71	                    ],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "type": "Nodes",
6.82/2.71	            "250": {
6.82/2.71	                "goal": [],
6.82/2.71	                "kb": {
6.82/2.71	                    "nonunifying": [],
6.82/2.71	                    "intvars": {},
6.82/2.71	                    "arithmetic": {
6.82/2.71	                        "type": "PlainIntegerRelationState",
6.82/2.71	                        "relations": []
6.82/2.71	                    },
6.82/2.71	                    "ground": [],
6.82/2.71	                    "free": [],
6.82/2.71	                    "exprvars": []
6.82/2.71	                }
6.82/2.71	            },
6.82/2.71	            "251": {
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7.14/2.72	                    "ground": [],
7.14/2.72	                    "free": [
7.14/2.72	                        "X235",
7.14/2.72	                        "X236"
7.14/2.72	                    ],
7.14/2.72	                    "exprvars": []
7.14/2.72	                }
7.14/2.72	            },
7.14/2.72	            "228": {
7.14/2.72	                "goal": [{
7.14/2.72	                    "clause": -1,
7.14/2.72	                    "scope": -1,
7.14/2.72	                    "term": "(row2col T92 T93 X163 (. ([]) (. ([]) (. ([]) (. ([]) ([]))))) X164)"
7.14/2.72	                }],
7.14/2.72	                "kb": {
7.14/2.72	                    "nonunifying": [],
7.14/2.72	                    "intvars": {},
7.14/2.72	                    "arithmetic": {
7.14/2.72	                        "type": "PlainIntegerRelationState",
7.14/2.72	                        "relations": []
7.14/2.72	                    },
7.14/2.72	                    "ground": [],
7.14/2.72	                    "free": [
7.14/2.72	                        "X163",
7.14/2.72	                        "X164"
7.14/2.72	                    ],
7.14/2.72	                    "exprvars": []
7.14/2.72	                }
7.14/2.72	            },
7.14/2.72	            "229": {
7.14/2.72	                "goal": [],
7.14/2.72	                "kb": {
7.14/2.72	                    "nonunifying": [],
7.14/2.72	                    "intvars": {},
7.14/2.72	                    "arithmetic": {
7.14/2.72	                        "type": "PlainIntegerRelationState",
7.14/2.72	                        "relations": []
7.14/2.72	                    },
7.14/2.72	                    "ground": [],
7.14/2.72	                    "free": [],
7.14/2.72	                    "exprvars": []
7.14/2.72	                }
7.14/2.72	            }
7.14/2.72	        },
7.14/2.72	        "edges": [
7.14/2.72	            {
7.14/2.72	                "from": 3,
7.14/2.72	                "to": 138,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 138,
7.14/2.72	                "to": 146,
7.14/2.72	                "label": "EVAL with clause\ntranspose_aux(.(X14, X15), X16, .(X17, X18)) :- row2col(X14, .(X17, X18), X19, [], X20).\nand substitutionX14 -> T19,\nX15 -> T15,\nT1 -> .(T19, T15),\nT2 -> T16,\nX16 -> T16,\nX17 -> T20,\nX18 -> T21,\nT3 -> .(T20, T21),\nT14 -> T19,\nT17 -> T20,\nT18 -> T21"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 138,
7.14/2.72	                "to": 147,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 146,
7.14/2.72	                "to": 177,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 177,
7.14/2.72	                "to": 182,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X48, X49), .(.(X48, X50), X51), .(X50, X52), X53, X54) :- row2col(X49, X51, X52, .([], X53), X54).\nand substitutionX48 -> T34,\nX49 -> T38,\nT19 -> .(T34, T38),\nX50 -> T36,\nT20 -> .(T34, T36),\nT21 -> T39,\nX51 -> T39,\nX52 -> X55,\nX19 -> .(T36, X55),\nX53 -> [],\nX20 -> X56,\nX54 -> X56,\nT35 -> T38,\nT37 -> T39"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 177,
7.14/2.72	                "to": 183,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 182,
7.14/2.72	                "to": 186,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 186,
7.14/2.72	                "to": 189,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X84, X85), .(.(X84, X86), X87), .(X86, X88), X89, X90) :- row2col(X85, X87, X88, .([], X89), X90).\nand substitutionX84 -> T52,\nX85 -> T56,\nT38 -> .(T52, T56),\nX86 -> T54,\nX87 -> T57,\nT39 -> .(.(T52, T54), T57),\nX88 -> X91,\nX55 -> .(T54, X91),\nX89 -> .([], []),\nX56 -> X92,\nX90 -> X92,\nT53 -> T56,\nT55 -> T57"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 186,
7.14/2.72	                "to": 190,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 189,
7.14/2.72	                "to": 192,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 192,
7.14/2.72	                "to": 218,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X120, X121), .(.(X120, X122), X123), .(X122, X124), X125, X126) :- row2col(X121, X123, X124, .([], X125), X126).\nand substitutionX120 -> T70,\nX121 -> T74,\nT56 -> .(T70, T74),\nX122 -> T72,\nX123 -> T75,\nT57 -> .(.(T70, T72), T75),\nX124 -> X127,\nX91 -> .(T72, X127),\nX125 -> .([], .([], [])),\nX92 -> X128,\nX126 -> X128,\nT71 -> T74,\nT73 -> T75"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 192,
7.14/2.72	                "to": 223,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 218,
7.14/2.72	                "to": 225,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 225,
7.14/2.72	                "to": 228,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X156, X157), .(.(X156, X158), X159), .(X158, X160), X161, X162) :- row2col(X157, X159, X160, .([], X161), X162).\nand substitutionX156 -> T88,\nX157 -> T92,\nT74 -> .(T88, T92),\nX158 -> T90,\nX159 -> T93,\nT75 -> .(.(T88, T90), T93),\nX160 -> X163,\nX127 -> .(T90, X163),\nX161 -> .([], .([], .([], []))),\nX128 -> X164,\nX162 -> X164,\nT89 -> T92,\nT91 -> T93"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 225,
7.14/2.72	                "to": 229,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 228,
7.14/2.72	                "to": 238,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 238,
7.14/2.72	                "to": 240,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X192, X193), .(.(X192, X194), X195), .(X194, X196), X197, X198) :- row2col(X193, X195, X196, .([], X197), X198).\nand substitutionX192 -> T106,\nX193 -> T110,\nT92 -> .(T106, T110),\nX194 -> T108,\nX195 -> T111,\nT93 -> .(.(T106, T108), T111),\nX196 -> X199,\nX163 -> .(T108, X199),\nX197 -> .([], .([], .([], .([], [])))),\nX164 -> X200,\nX198 -> X200,\nT107 -> T110,\nT109 -> T111"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 238,
7.14/2.72	                "to": 241,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 240,
7.14/2.72	                "to": 242,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 242,
7.14/2.72	                "to": 249,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X228, X229), .(.(X228, X230), X231), .(X230, X232), X233, X234) :- row2col(X229, X231, X232, .([], X233), X234).\nand substitutionX228 -> T124,\nX229 -> T128,\nT110 -> .(T124, T128),\nX230 -> T126,\nX231 -> T129,\nT111 -> .(.(T124, T126), T129),\nX232 -> X235,\nX199 -> .(T126, X235),\nX233 -> .([], .([], .([], .([], .([], []))))),\nX200 -> X236,\nX234 -> X236,\nT125 -> T128,\nT127 -> T129"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 242,
7.14/2.72	                "to": 250,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 249,
7.14/2.72	                "to": 251,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 251,
7.14/2.72	                "to": 252,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X264, X265), .(.(X264, X266), X267), .(X266, X268), X269, X270) :- row2col(X265, X267, X268, .([], X269), X270).\nand substitutionX264 -> T142,\nX265 -> T146,\nT128 -> .(T142, T146),\nX266 -> T144,\nX267 -> T147,\nT129 -> .(.(T142, T144), T147),\nX268 -> X271,\nX235 -> .(T144, X271),\nX269 -> .([], .([], .([], .([], .([], .([], [])))))),\nX236 -> X272,\nX270 -> X272,\nT143 -> T146,\nT145 -> T147"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 251,
7.14/2.72	                "to": 253,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 252,
7.14/2.72	                "to": 254,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 254,
7.14/2.72	                "to": 255,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X300, X301), .(.(X300, X302), X303), .(X302, X304), X305, X306) :- row2col(X301, X303, X304, .([], X305), X306).\nand substitutionX300 -> T160,\nX301 -> T164,\nT146 -> .(T160, T164),\nX302 -> T162,\nX303 -> T165,\nT147 -> .(.(T160, T162), T165),\nX304 -> X307,\nX271 -> .(T162, X307),\nX305 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX272 -> X308,\nX306 -> X308,\nT161 -> T164,\nT163 -> T165"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 254,
7.14/2.72	                "to": 256,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 255,
7.14/2.72	                "to": 257,
7.14/2.72	                "label": "GENERALIZATION\nT170 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT170 is ground"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 257,
7.14/2.72	                "to": 258,
7.14/2.72	                "label": "CASE"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 258,
7.14/2.72	                "to": 259,
7.14/2.72	                "label": "EVAL with clause\nrow2col(.(X345, X346), .(.(X345, X347), X348), .(X347, X349), X350, X351) :- row2col(X346, X348, X349, .([], X350), X351).\nand substitutionX345 -> T186,\nX346 -> T191,\nT164 -> .(T186, T191),\nX347 -> T188,\nX348 -> T192,\nT165 -> .(.(T186, T188), T192),\nX349 -> X352,\nX307 -> .(T188, X352),\nT170 -> T190,\nX350 -> .([], T190),\nX308 -> X353,\nX351 -> X353,\nT187 -> T191,\nT189 -> T192"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 258,
7.14/2.72	                "to": 260,
7.14/2.72	                "label": "EVAL-BACKTRACK"
7.14/2.72	            },
7.14/2.72	            {
7.14/2.72	                "from": 259,
7.14/2.72	                "to": 257,
7.14/2.72	                "label": "INSTANCE with matching:\nT164 -> T191\nT165 -> T192\nX307 -> X352\nT170 -> .([], T190)\nX308 -> X353"
7.14/2.72	            }
7.14/2.72	        ],
7.14/2.72	        "type": "Graph"
7.14/2.72	    }
7.14/2.72	}
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(60)
7.14/2.72	Obligation:
7.14/2.72	Q restricted rewrite system:
7.14/2.72	The TRS R consists of the following rules:
7.14/2.72	
7.14/2.72	   f3_in(T16) -> U1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   U1(f257_out1(X308), T16) -> f3_out1
7.14/2.72	   f257_in(T190) -> U2(f257_in(.([], T190)), T190)
7.14/2.72	   U2(f257_out1(X353), T190) -> f257_out1(X353)
7.14/2.72	
7.14/2.72	Q is empty.
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(61) AAECC Innermost (EQUIVALENT)
7.14/2.72	We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is 
7.14/2.72	   f257_in(T190) -> U2(f257_in(.([], T190)), T190)
7.14/2.72	   U2(f257_out1(X353), T190) -> f257_out1(X353)
7.14/2.72	
7.14/2.72	The TRS R 2 is 
7.14/2.72	   f3_in(T16) -> U1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   U1(f257_out1(X308), T16) -> f3_out1
7.14/2.72	
7.14/2.72	The signature Sigma is {f3_in_1, U1_2, f3_out1}
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(62)
7.14/2.72	Obligation:
7.14/2.72	Q restricted rewrite system:
7.14/2.72	The TRS R consists of the following rules:
7.14/2.72	
7.14/2.72	   f3_in(T16) -> U1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   U1(f257_out1(X308), T16) -> f3_out1
7.14/2.72	   f257_in(T190) -> U2(f257_in(.([], T190)), T190)
7.14/2.72	   U2(f257_out1(X353), T190) -> f257_out1(X353)
7.14/2.72	
7.14/2.72	The set Q consists of the following terms:
7.14/2.72	
7.14/2.72	   f3_in(x0)
7.14/2.72	   U1(f257_out1(x0), x1)
7.14/2.72	   f257_in(x0)
7.14/2.72	   U2(f257_out1(x0), x1)
7.14/2.72	
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(63) DependencyPairsProof (EQUIVALENT)
7.14/2.72	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(64)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F3_IN(T16) -> U1^1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   F3_IN(T16) -> F257_IN(.([], .([], .([], .([], .([], .([], .([], []))))))))
7.14/2.72	   F257_IN(T190) -> U2^1(f257_in(.([], T190)), T190)
7.14/2.72	   F257_IN(T190) -> F257_IN(.([], T190))
7.14/2.72	
7.14/2.72	The TRS R consists of the following rules:
7.14/2.72	
7.14/2.72	   f3_in(T16) -> U1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   U1(f257_out1(X308), T16) -> f3_out1
7.14/2.72	   f257_in(T190) -> U2(f257_in(.([], T190)), T190)
7.14/2.72	   U2(f257_out1(X353), T190) -> f257_out1(X353)
7.14/2.72	
7.14/2.72	The set Q consists of the following terms:
7.14/2.72	
7.14/2.72	   f3_in(x0)
7.14/2.72	   U1(f257_out1(x0), x1)
7.14/2.72	   f257_in(x0)
7.14/2.72	   U2(f257_out1(x0), x1)
7.14/2.72	
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(65) DependencyGraphProof (EQUIVALENT)
7.14/2.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(66)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F257_IN(T190) -> F257_IN(.([], T190))
7.14/2.72	
7.14/2.72	The TRS R consists of the following rules:
7.14/2.72	
7.14/2.72	   f3_in(T16) -> U1(f257_in(.([], .([], .([], .([], .([], .([], .([], [])))))))), T16)
7.14/2.72	   U1(f257_out1(X308), T16) -> f3_out1
7.14/2.72	   f257_in(T190) -> U2(f257_in(.([], T190)), T190)
7.14/2.72	   U2(f257_out1(X353), T190) -> f257_out1(X353)
7.14/2.72	
7.14/2.72	The set Q consists of the following terms:
7.14/2.72	
7.14/2.72	   f3_in(x0)
7.14/2.72	   U1(f257_out1(x0), x1)
7.14/2.72	   f257_in(x0)
7.14/2.72	   U2(f257_out1(x0), x1)
7.14/2.72	
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(67) UsableRulesProof (EQUIVALENT)
7.14/2.72	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.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(68)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F257_IN(T190) -> F257_IN(.([], T190))
7.14/2.72	
7.14/2.72	R is empty.
7.14/2.72	The set Q consists of the following terms:
7.14/2.72	
7.14/2.72	   f3_in(x0)
7.14/2.72	   U1(f257_out1(x0), x1)
7.14/2.72	   f257_in(x0)
7.14/2.72	   U2(f257_out1(x0), x1)
7.14/2.72	
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(69) QReductionProof (EQUIVALENT)
7.14/2.72	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
7.14/2.72	
7.14/2.72	   f3_in(x0)
7.14/2.72	   U1(f257_out1(x0), x1)
7.14/2.72	   f257_in(x0)
7.14/2.72	   U2(f257_out1(x0), x1)
7.14/2.72	
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(70)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F257_IN(T190) -> F257_IN(.([], T190))
7.14/2.72	
7.14/2.72	R is empty.
7.14/2.72	Q is empty.
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(71) TransformationProof (EQUIVALENT)
7.14/2.72	By instantiating [LPAR04] the rule F257_IN(T190) -> F257_IN(.([], T190)) we obtained the following new rules [LPAR04]:
7.14/2.72	
7.14/2.72	   (F257_IN(.([], z0)) -> F257_IN(.([], .([], z0))),F257_IN(.([], z0)) -> F257_IN(.([], .([], z0))))
7.14/2.72	
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(72)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F257_IN(.([], z0)) -> F257_IN(.([], .([], z0)))
7.14/2.72	
7.14/2.72	R is empty.
7.14/2.72	Q is empty.
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(73) TransformationProof (EQUIVALENT)
7.14/2.72	By instantiating [LPAR04] the rule F257_IN(.([], z0)) -> F257_IN(.([], .([], z0))) we obtained the following new rules [LPAR04]:
7.14/2.72	
7.14/2.72	   (F257_IN(.([], .([], z0))) -> F257_IN(.([], .([], .([], z0)))),F257_IN(.([], .([], z0))) -> F257_IN(.([], .([], .([], z0)))))
7.14/2.72	
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(74)
7.14/2.72	Obligation:
7.14/2.72	Q DP problem:
7.14/2.72	The TRS P consists of the following rules:
7.14/2.72	
7.14/2.72	   F257_IN(.([], .([], z0))) -> F257_IN(.([], .([], .([], z0))))
7.14/2.72	
7.14/2.72	R is empty.
7.14/2.72	Q is empty.
7.14/2.72	We have to consider all minimal (P,Q,R)-chains.
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(75) NonTerminationLoopProof (COMPLETE)
7.14/2.72	We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
7.14/2.72	Found a loop by semiunifying a rule from P directly.
7.14/2.72	
7.14/2.72	s = F257_IN(.([], .([], z0))) evaluates to  t =F257_IN(.([], .([], .([], z0))))
7.14/2.72	
7.14/2.72	Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
7.14/2.72	* Matcher: [z0 / .([], z0)]
7.14/2.72	* Semiunifier: [ ]
7.14/2.72	
7.14/2.72	--------------------------------------------------------------------------------
7.14/2.72	Rewriting sequence
7.14/2.72	
7.14/2.72	The DP semiunifies directly so there is only one rewrite step from F257_IN(.([], .([], z0))) to F257_IN(.([], .([], .([], z0)))).
7.14/2.72	
7.14/2.72	
7.14/2.72	
7.14/2.72	
7.14/2.72	----------------------------------------
7.14/2.72	
7.14/2.72	(76)
7.14/2.72	NO
7.14/2.76	EOF