6.72/2.70	YES
6.72/2.72	proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
6.72/2.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.72/2.72	
6.72/2.72	
6.72/2.72	Left Termination of the query pattern
6.72/2.72	
6.72/2.72	transpose(a,g)
6.72/2.72	
6.72/2.72	w.r.t. the given Prolog program could successfully be proven:
6.72/2.72	
6.72/2.72	(0) Prolog
6.72/2.72	(1) PrologToPiTRSProof [SOUND, 0 ms]
6.72/2.72	(2) PiTRS
6.72/2.72	(3) DependencyPairsProof [EQUIVALENT, 15 ms]
6.72/2.72	(4) PiDP
6.72/2.72	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.72	(6) AND
6.72/2.72	    (7) PiDP
6.72/2.72	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.72	        (9) PiDP
6.72/2.72	        (10) PiDPToQDPProof [SOUND, 0 ms]
6.72/2.72	        (11) QDP
6.72/2.72	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.72/2.72	        (13) YES
6.72/2.72	    (14) PiDP
6.72/2.72	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.72/2.72	        (16) PiDP
6.72/2.72	        (17) PiDPToQDPProof [SOUND, 0 ms]
6.72/2.72	        (18) QDP
6.72/2.72	        (19) QDPOrderProof [EQUIVALENT, 53 ms]
6.72/2.72	        (20) QDP
6.72/2.72	        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
6.72/2.72	        (22) TRUE
6.72/2.72	
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(0)
6.72/2.72	Obligation:
6.72/2.72	Clauses:
6.72/2.72	
6.72/2.72	transpose(A, B) :- transpose_aux(A, nil, B).
6.72/2.72	transpose_aux(cons(R, Rs), X1, cons(C, Cs)) :- ','(row2col(R, cons(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)).
6.72/2.72	transpose_aux(nil, X, X).
6.72/2.72	row2col(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) :- row2col(Xs, Cols, Cols1, As).
6.72/2.72	row2col(nil, nil, nil, nil).
6.72/2.72	
6.72/2.72	
6.72/2.72	Query: transpose(a,g)
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(1) PrologToPiTRSProof (SOUND)
6.72/2.72	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.72/2.72	
6.72/2.72	transpose_in_2: (f,b)
6.72/2.72	
6.72/2.72	transpose_aux_in_3: (f,b,b)
6.72/2.72	
6.72/2.72	row2col_in_4: (f,b,f,f)
6.72/2.72	
6.72/2.72	Transforming Prolog into the following Term Rewriting System:
6.72/2.72	
6.72/2.72	Pi-finite rewrite system:
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	
6.72/2.72	
6.72/2.72	
6.72/2.72	
6.72/2.72	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.72/2.72	
6.72/2.72	
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(2)
6.72/2.72	Obligation:
6.72/2.72	Pi-finite rewrite system:
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(3) DependencyPairsProof (EQUIVALENT)
6.72/2.72	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, nil, B)
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm)
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	TRANSPOSE_IN_AG(x1, x2)  =  TRANSPOSE_IN_AG(x2)
6.72/2.72	
6.72/2.72	U1_AG(x1, x2, x3)  =  U1_AG(x3)
6.72/2.72	
6.72/2.72	TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
6.72/2.72	
6.72/2.72	U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
6.72/2.72	
6.72/2.72	ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
6.72/2.72	
6.72/2.72	U4_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x1, x3, x7)
6.72/2.72	
6.72/2.72	U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x1, x6)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(4)
6.72/2.72	Obligation:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, nil, B)
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm)
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	TRANSPOSE_IN_AG(x1, x2)  =  TRANSPOSE_IN_AG(x2)
6.72/2.72	
6.72/2.72	U1_AG(x1, x2, x3)  =  U1_AG(x3)
6.72/2.72	
6.72/2.72	TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
6.72/2.72	
6.72/2.72	U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
6.72/2.72	
6.72/2.72	ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
6.72/2.72	
6.72/2.72	U4_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x1, x3, x7)
6.72/2.72	
6.72/2.72	U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x1, x6)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(5) DependencyGraphProof (EQUIVALENT)
6.72/2.72	The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(6)
6.72/2.72	Complex Obligation (AND)
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(7)
6.72/2.72	Obligation:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(8) UsableRulesProof (EQUIVALENT)
6.72/2.72	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(9)
6.72/2.72	Obligation:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
6.72/2.72	
6.72/2.72	R is empty.
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(10) PiDPToQDPProof (SOUND)
6.72/2.72	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(11)
6.72/2.72	Obligation:
6.72/2.72	Q DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   ROW2COL_IN_AGAA(cons(cons(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols)
6.72/2.72	
6.72/2.72	R is empty.
6.72/2.72	Q is empty.
6.72/2.72	We have to consider all (P,Q,R)-chains.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(12) QDPSizeChangeProof (EQUIVALENT)
6.72/2.72	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
6.72/2.72	
6.72/2.72	From the DPs we obtained the following set of size-change graphs:
6.72/2.72	*ROW2COL_IN_AGAA(cons(cons(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols)
6.72/2.72	The graph contains the following edges 1 > 1
6.72/2.72	
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(13)
6.72/2.72	YES
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(14)
6.72/2.72	Obligation:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B))
6.72/2.72	   transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
6.72/2.72	   transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X)
6.72/2.72	   U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
6.72/2.72	   U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	transpose_in_ag(x1, x2)  =  transpose_in_ag(x2)
6.72/2.72	
6.72/2.72	U1_ag(x1, x2, x3)  =  U1_ag(x3)
6.72/2.72	
6.72/2.72	transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
6.72/2.72	
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x1, x6)
6.72/2.72	
6.72/2.72	transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
6.72/2.72	
6.72/2.72	transpose_out_ag(x1, x2)  =  transpose_out_ag(x1)
6.72/2.72	
6.72/2.72	TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
6.72/2.72	
6.72/2.72	U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(15) UsableRulesProof (EQUIVALENT)
6.72/2.72	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(16)
6.72/2.72	Obligation:
6.72/2.72	Pi DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
6.72/2.72	   U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil)
6.72/2.72	
6.72/2.72	The argument filtering Pi contains the following mapping:
6.72/2.72	cons(x1, x2)  =  cons(x1, x2)
6.72/2.72	
6.72/2.72	row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
6.72/2.72	
6.72/2.72	U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x1, x3, x7)
6.72/2.72	
6.72/2.72	nil  =  nil
6.72/2.72	
6.72/2.72	row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
6.72/2.72	
6.72/2.72	TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
6.72/2.72	
6.72/2.72	U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
6.72/2.72	
6.72/2.72	
6.72/2.72	We have to consider all (P,R,Pi)-chains
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(17) PiDPToQDPProof (SOUND)
6.72/2.72	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(18)
6.72/2.72	Obligation:
6.72/2.72	Q DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(X1, cons(C, Cs)) -> U2_AGG(row2col_in_agaa(cons(C, Cs)))
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols))
6.72/2.72	   U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil)
6.72/2.72	
6.72/2.72	The set Q consists of the following terms:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(x0)
6.72/2.72	   U4_agaa(x0, x1, x2)
6.72/2.72	
6.72/2.72	We have to consider all (P,Q,R)-chains.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(19) QDPOrderProof (EQUIVALENT)
6.72/2.72	We use the reduction pair processor [LPAR04,JAR06].
6.72/2.72	
6.72/2.72	
6.72/2.72	The following pairs can be oriented strictly and are deleted.
6.72/2.72	
6.72/2.72	   TRANSPOSE_AUX_IN_AGG(X1, cons(C, Cs)) -> U2_AGG(row2col_in_agaa(cons(C, Cs)))
6.72/2.72	The remaining pairs can at least be oriented weakly.
6.72/2.72	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
6.72/2.72	
6.72/2.72	POL( U2_AGG_1(x_1) ) = 2x_1 + 2
6.72/2.72	POL( row2col_in_agaa_1(x_1) ) = max{0, x_1 - 2}
6.72/2.72	POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2
6.72/2.72	POL( U4_agaa_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2
6.72/2.72	POL( nil ) = 0
6.72/2.72	POL( row2col_out_agaa_3(x_1, ..., x_3) ) = max{0, x_2 - 1}
6.72/2.72	POL( TRANSPOSE_AUX_IN_AGG_2(x_1, x_2) ) = 2x_2
6.72/2.72	
6.72/2.72	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols))
6.72/2.72	   row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil)
6.72/2.72	   U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	
6.72/2.72	
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(20)
6.72/2.72	Obligation:
6.72/2.72	Q DP problem:
6.72/2.72	The TRS P consists of the following rules:
6.72/2.72	
6.72/2.72	   U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
6.72/2.72	
6.72/2.72	The TRS R consists of the following rules:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols))
6.72/2.72	   U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))
6.72/2.72	   row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil)
6.72/2.72	
6.72/2.72	The set Q consists of the following terms:
6.72/2.72	
6.72/2.72	   row2col_in_agaa(x0)
6.72/2.72	   U4_agaa(x0, x1, x2)
6.72/2.72	
6.72/2.72	We have to consider all (P,Q,R)-chains.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(21) DependencyGraphProof (EQUIVALENT)
6.72/2.72	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
6.72/2.72	----------------------------------------
6.72/2.72	
6.72/2.72	(22)
6.72/2.72	TRUE
7.08/3.13	EOF