/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern transpose(a,g) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 0 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 12 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPOrderProof [EQUIVALENT, 42 ms] (20) QDP (21) DependencyGraphProof [EQUIVALENT, 0 ms] (22) TRUE ---------------------------------------- (0) Obligation: Clauses: transpose(A, B) :- transpose_aux(A, nil, B). transpose_aux(cons(R, Rs), X1, cons(C, Cs)) :- ','(row2col(R, cons(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)). transpose_aux(nil, X, X). row2col(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) :- row2col(Xs, Cols, Cols1, As). row2col(nil, nil, nil, nil). Query: transpose(a,g) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: transpose_in_2: (f,b) transpose_aux_in_3: (f,b,b) row2col_in_4: (f,b,f,f) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, nil, B)) TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, nil, B) 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)) TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm) 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)) ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 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)) U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x3) TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7) U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, nil, B)) TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, nil, B) 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)) TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) -> ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm) 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)) ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 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)) U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2) U1_AG(x1, x2, x3) = U1_AG(x3) TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7) U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) R is empty. The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) nil = nil ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: ROW2COL_IN_AGAA(cons(cons(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *ROW2COL_IN_AGAA(cons(cons(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 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)) The TRS R consists of the following rules: transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, nil, B)) 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) 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)) 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)) transpose_aux_in_agg(nil, X, X) -> transpose_aux_out_agg(nil, X, X) 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)) U1_ag(A, B, transpose_aux_out_agg(A, nil, B)) -> transpose_out_ag(A, B) The argument filtering Pi contains the following mapping: transpose_in_ag(x1, x2) = transpose_in_ag(x2) U1_ag(x1, x2, x3) = U1_ag(x3) transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) cons(x1, x2) = cons(x1, x2) U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) transpose_out_ag(x1, x2) = transpose_out_ag(x1) TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 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)) The TRS R consists of the following rules: 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)) 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)) row2col_in_agaa(nil, nil, nil, nil) -> row2col_out_agaa(nil, nil, nil, nil) The argument filtering Pi contains the following mapping: cons(x1, x2) = cons(x1, x2) row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) nil = nil row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1) TRANSPOSE_AUX_IN_AGG(X1, cons(C, Cs)) -> U2_AGG(row2col_in_agaa(cons(C, Cs))) The TRS R consists of the following rules: row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As)) row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil) The set Q consists of the following terms: row2col_in_agaa(x0) U4_agaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. TRANSPOSE_AUX_IN_AGG(X1, cons(C, Cs)) -> U2_AGG(row2col_in_agaa(cons(C, Cs))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( U2_AGG_1(x_1) ) = 2x_1 + 2 POL( row2col_in_agaa_1(x_1) ) = max{0, x_1 - 2} POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( U4_agaa_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 POL( nil ) = 0 POL( row2col_out_agaa_3(x_1, ..., x_3) ) = max{0, x_2 - 1} POL( TRANSPOSE_AUX_IN_AGG_2(x_1, x_2) ) = 2x_2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil) U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As)) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1) The TRS R consists of the following rules: row2col_in_agaa(cons(cons(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As)) row2col_in_agaa(nil) -> row2col_out_agaa(nil, nil, nil) The set Q consists of the following terms: row2col_in_agaa(x0) U4_agaa(x0, x1, x2) We have to consider all (P,Q,R)-chains. ---------------------------------------- (21) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (22) TRUE