6.62/2.61 YES 6.84/2.64 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 6.84/2.64 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 6.84/2.64 6.84/2.64 6.84/2.64 Left Termination of the query pattern 6.84/2.64 6.84/2.64 transpose(a,g) 6.84/2.64 6.84/2.64 w.r.t. the given Prolog program could successfully be proven: 6.84/2.64 6.84/2.64 (0) Prolog 6.84/2.64 (1) PrologToPiTRSProof [SOUND, 0 ms] 6.84/2.64 (2) PiTRS 6.84/2.64 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 6.84/2.64 (4) PiDP 6.84/2.64 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 6.84/2.64 (6) AND 6.84/2.64 (7) PiDP 6.84/2.64 (8) UsableRulesProof [EQUIVALENT, 0 ms] 6.84/2.64 (9) PiDP 6.84/2.64 (10) PiDPToQDPProof [SOUND, 0 ms] 6.84/2.64 (11) QDP 6.84/2.64 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 6.84/2.64 (13) YES 6.84/2.64 (14) PiDP 6.84/2.64 (15) UsableRulesProof [EQUIVALENT, 0 ms] 6.84/2.64 (16) PiDP 6.84/2.64 (17) PiDPToQDPProof [SOUND, 0 ms] 6.84/2.64 (18) QDP 6.84/2.64 (19) QDPOrderProof [EQUIVALENT, 34 ms] 6.84/2.64 (20) QDP 6.84/2.64 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 6.84/2.64 (22) TRUE 6.84/2.64 6.84/2.64 6.84/2.64 ---------------------------------------- 6.84/2.64 6.84/2.64 (0) 6.84/2.64 Obligation: 6.84/2.64 Clauses: 6.84/2.64 6.84/2.64 transpose(A, B) :- transpose_aux(A, [], B). 6.84/2.64 transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)). 6.84/2.64 transpose_aux([], X, X). 6.84/2.64 row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As). 6.84/2.64 row2col([], [], [], []). 6.84/2.64 6.84/2.64 6.84/2.64 Query: transpose(a,g) 6.84/2.64 ---------------------------------------- 6.84/2.64 6.84/2.64 (1) PrologToPiTRSProof (SOUND) 6.84/2.64 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 6.84/2.64 6.84/2.64 transpose_in_2: (f,b) 6.84/2.64 6.84/2.64 transpose_aux_in_3: (f,b,b) 6.84/2.64 6.84/2.64 row2col_in_4: (f,b,f,f) 6.84/2.64 6.84/2.64 Transforming Prolog into the following Term Rewriting System: 6.84/2.64 6.84/2.64 Pi-finite rewrite system: 6.84/2.64 The TRS R consists of the following rules: 6.84/2.64 6.84/2.64 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.64 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.64 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.64 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.64 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.64 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.64 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.64 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.64 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.64 6.84/2.64 The argument filtering Pi contains the following mapping: 6.84/2.64 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.64 6.84/2.64 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.64 6.84/2.64 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.64 6.84/2.64 .(x1, x2) = .(x1, x2) 6.84/2.64 6.84/2.64 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.64 6.84/2.64 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.64 6.84/2.64 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.64 6.84/2.64 [] = [] 6.84/2.64 6.84/2.64 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.64 6.84/2.64 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.64 6.84/2.64 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.64 6.84/2.64 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.64 6.84/2.64 6.84/2.64 6.84/2.64 6.84/2.64 6.84/2.64 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 6.84/2.64 6.84/2.64 6.84/2.64 6.84/2.64 ---------------------------------------- 6.84/2.64 6.84/2.64 (2) 6.84/2.64 Obligation: 6.84/2.64 Pi-finite rewrite system: 6.84/2.64 The TRS R consists of the following rules: 6.84/2.64 6.84/2.64 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.64 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.64 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.64 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.64 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.64 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.64 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.64 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.64 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.64 6.84/2.64 The argument filtering Pi contains the following mapping: 6.84/2.64 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.64 6.84/2.64 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.64 6.84/2.64 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.64 6.84/2.64 .(x1, x2) = .(x1, x2) 6.84/2.64 6.84/2.64 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.64 6.84/2.64 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.64 6.84/2.64 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.64 6.84/2.64 [] = [] 6.84/2.64 6.84/2.64 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.64 6.84/2.64 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.64 6.84/2.64 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.64 6.84/2.64 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.64 6.84/2.64 6.84/2.64 6.84/2.64 ---------------------------------------- 6.84/2.64 6.84/2.64 (3) DependencyPairsProof (EQUIVALENT) 6.84/2.64 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 6.84/2.64 Pi DP problem: 6.84/2.64 The TRS P consists of the following rules: 6.84/2.64 6.84/2.64 TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.64 TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, [], B) 6.84/2.64 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.64 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm) 6.84/2.64 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.64 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 6.84/2.64 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.64 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 6.84/2.64 6.84/2.64 The TRS R consists of the following rules: 6.84/2.64 6.84/2.64 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.64 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.64 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.64 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.64 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.64 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.64 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.64 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.64 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.64 6.84/2.64 The argument filtering Pi contains the following mapping: 6.84/2.64 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.64 6.84/2.64 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.64 6.84/2.64 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.64 6.84/2.64 .(x1, x2) = .(x1, x2) 6.84/2.64 6.84/2.64 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.64 6.84/2.64 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.64 6.84/2.64 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.64 6.84/2.64 [] = [] 6.84/2.64 6.84/2.64 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.64 6.84/2.64 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.64 6.84/2.64 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.64 6.84/2.64 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.64 6.84/2.64 TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2) 6.84/2.64 6.84/2.64 U1_AG(x1, x2, x3) = U1_AG(x3) 6.84/2.64 6.84/2.64 TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) 6.84/2.64 6.84/2.64 U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) 6.84/2.64 6.84/2.64 ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) 6.84/2.64 6.84/2.64 U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7) 6.84/2.65 6.84/2.65 U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (4) 6.84/2.65 Obligation: 6.84/2.65 Pi DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 TRANSPOSE_IN_AG(A, B) -> U1_AG(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.65 TRANSPOSE_IN_AG(A, B) -> TRANSPOSE_AUX_IN_AGG(A, [], B) 6.84/2.65 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm) 6.84/2.65 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.65 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 6.84/2.65 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.65 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.65 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.65 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.65 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.65 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.65 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.65 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.65 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.65 6.84/2.65 The argument filtering Pi contains the following mapping: 6.84/2.65 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.65 6.84/2.65 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.65 6.84/2.65 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.65 6.84/2.65 .(x1, x2) = .(x1, x2) 6.84/2.65 6.84/2.65 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.65 6.84/2.65 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.65 6.84/2.65 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.65 6.84/2.65 [] = [] 6.84/2.65 6.84/2.65 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.65 6.84/2.65 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.65 6.84/2.65 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.65 6.84/2.65 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.65 6.84/2.65 TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2) 6.84/2.65 6.84/2.65 U1_AG(x1, x2, x3) = U1_AG(x3) 6.84/2.65 6.84/2.65 TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) 6.84/2.65 6.84/2.65 U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) 6.84/2.65 6.84/2.65 U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7) 6.84/2.65 6.84/2.65 U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (5) DependencyGraphProof (EQUIVALENT) 6.84/2.65 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (6) 6.84/2.65 Complex Obligation (AND) 6.84/2.65 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (7) 6.84/2.65 Obligation: 6.84/2.65 Pi DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.65 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.65 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.65 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.65 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.65 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.65 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.65 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.65 6.84/2.65 The argument filtering Pi contains the following mapping: 6.84/2.65 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.65 6.84/2.65 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.65 6.84/2.65 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.65 6.84/2.65 .(x1, x2) = .(x1, x2) 6.84/2.65 6.84/2.65 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.65 6.84/2.65 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.65 6.84/2.65 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.65 6.84/2.65 [] = [] 6.84/2.65 6.84/2.65 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.65 6.84/2.65 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.65 6.84/2.65 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.65 6.84/2.65 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (8) UsableRulesProof (EQUIVALENT) 6.84/2.65 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (9) 6.84/2.65 Obligation: 6.84/2.65 Pi DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_AGAA(Xs, Cols, Cols1, As) 6.84/2.65 6.84/2.65 R is empty. 6.84/2.65 The argument filtering Pi contains the following mapping: 6.84/2.65 .(x1, x2) = .(x1, x2) 6.84/2.65 6.84/2.65 [] = [] 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (10) PiDPToQDPProof (SOUND) 6.84/2.65 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (11) 6.84/2.65 Obligation: 6.84/2.65 Q DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 ROW2COL_IN_AGAA(.(.(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols) 6.84/2.65 6.84/2.65 R is empty. 6.84/2.65 Q is empty. 6.84/2.65 We have to consider all (P,Q,R)-chains. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (12) QDPSizeChangeProof (EQUIVALENT) 6.84/2.65 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. 6.84/2.65 6.84/2.65 From the DPs we obtained the following set of size-change graphs: 6.84/2.65 *ROW2COL_IN_AGAA(.(.(X, Ys), Cols)) -> ROW2COL_IN_AGAA(Cols) 6.84/2.65 The graph contains the following edges 1 > 1 6.84/2.65 6.84/2.65 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (13) 6.84/2.65 YES 6.84/2.65 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (14) 6.84/2.65 Obligation: 6.84/2.65 Pi DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 6.84/2.65 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 transpose_in_ag(A, B) -> U1_ag(A, B, transpose_aux_in_agg(A, [], B)) 6.84/2.65 transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) -> U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.65 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.65 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.65 U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1)) 6.84/2.65 transpose_aux_in_agg([], X, X) -> transpose_aux_out_agg([], X, X) 6.84/2.65 U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) -> transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs)) 6.84/2.65 U1_ag(A, B, transpose_aux_out_agg(A, [], B)) -> transpose_out_ag(A, B) 6.84/2.65 6.84/2.65 The argument filtering Pi contains the following mapping: 6.84/2.65 transpose_in_ag(x1, x2) = transpose_in_ag(x2) 6.84/2.65 6.84/2.65 U1_ag(x1, x2, x3) = U1_ag(x3) 6.84/2.65 6.84/2.65 transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3) 6.84/2.65 6.84/2.65 .(x1, x2) = .(x1, x2) 6.84/2.65 6.84/2.65 U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6) 6.84/2.65 6.84/2.65 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.65 6.84/2.65 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.65 6.84/2.65 [] = [] 6.84/2.65 6.84/2.65 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.65 6.84/2.65 U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6) 6.84/2.65 6.84/2.65 transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1) 6.84/2.65 6.84/2.65 transpose_out_ag(x1, x2) = transpose_out_ag(x1) 6.84/2.65 6.84/2.65 TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) 6.84/2.65 6.84/2.65 U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (15) UsableRulesProof (EQUIVALENT) 6.84/2.65 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (16) 6.84/2.65 Obligation: 6.84/2.65 Pi DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1) 6.84/2.65 TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) -> U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm)) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As)) 6.84/2.65 U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 6.84/2.65 row2col_in_agaa([], [], [], []) -> row2col_out_agaa([], [], [], []) 6.84/2.65 6.84/2.65 The argument filtering Pi contains the following mapping: 6.84/2.65 .(x1, x2) = .(x1, x2) 6.84/2.65 6.84/2.65 row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2) 6.84/2.65 6.84/2.65 U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7) 6.84/2.65 6.84/2.65 [] = [] 6.84/2.65 6.84/2.65 row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4) 6.84/2.65 6.84/2.65 TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3) 6.84/2.65 6.84/2.65 U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6) 6.84/2.65 6.84/2.65 6.84/2.65 We have to consider all (P,R,Pi)-chains 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (17) PiDPToQDPProof (SOUND) 6.84/2.65 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (18) 6.84/2.65 Obligation: 6.84/2.65 Q DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1) 6.84/2.65 TRANSPOSE_AUX_IN_AGG(X1, .(C, Cs)) -> U2_AGG(row2col_in_agaa(.(C, Cs))) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 row2col_in_agaa(.(.(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) 6.84/2.65 U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As)) 6.84/2.65 row2col_in_agaa([]) -> row2col_out_agaa([], [], []) 6.84/2.65 6.84/2.65 The set Q consists of the following terms: 6.84/2.65 6.84/2.65 row2col_in_agaa(x0) 6.84/2.65 U4_agaa(x0, x1, x2) 6.84/2.65 6.84/2.65 We have to consider all (P,Q,R)-chains. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (19) QDPOrderProof (EQUIVALENT) 6.84/2.65 We use the reduction pair processor [LPAR04,JAR06]. 6.84/2.65 6.84/2.65 6.84/2.65 The following pairs can be oriented strictly and are deleted. 6.84/2.65 6.84/2.65 TRANSPOSE_AUX_IN_AGG(X1, .(C, Cs)) -> U2_AGG(row2col_in_agaa(.(C, Cs))) 6.84/2.65 The remaining pairs can at least be oriented weakly. 6.84/2.65 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 6.84/2.65 6.84/2.65 POL( U2_AGG_1(x_1) ) = 2x_1 + 2 6.84/2.65 POL( row2col_in_agaa_1(x_1) ) = max{0, x_1 - 2} 6.84/2.65 POL( ._2(x_1, x_2) ) = 2x_1 + x_2 + 2 6.84/2.65 POL( U4_agaa_3(x_1, ..., x_3) ) = 2x_1 + 2x_2 + x_3 + 2 6.84/2.65 POL( [] ) = 0 6.84/2.65 POL( row2col_out_agaa_3(x_1, ..., x_3) ) = max{0, x_2 - 1} 6.84/2.65 POL( TRANSPOSE_AUX_IN_AGG_2(x_1, x_2) ) = 2x_2 6.84/2.65 6.84/2.65 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 6.84/2.65 6.84/2.65 row2col_in_agaa(.(.(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) 6.84/2.65 row2col_in_agaa([]) -> row2col_out_agaa([], [], []) 6.84/2.65 U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As)) 6.84/2.65 6.84/2.65 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (20) 6.84/2.65 Obligation: 6.84/2.65 Q DP problem: 6.84/2.65 The TRS P consists of the following rules: 6.84/2.65 6.84/2.65 U2_AGG(row2col_out_agaa(R, Cols1, Accm)) -> TRANSPOSE_AUX_IN_AGG(Accm, Cols1) 6.84/2.65 6.84/2.65 The TRS R consists of the following rules: 6.84/2.65 6.84/2.65 row2col_in_agaa(.(.(X, Ys), Cols)) -> U4_agaa(X, Ys, row2col_in_agaa(Cols)) 6.84/2.65 U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) -> row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As)) 6.84/2.65 row2col_in_agaa([]) -> row2col_out_agaa([], [], []) 6.84/2.65 6.84/2.65 The set Q consists of the following terms: 6.84/2.65 6.84/2.65 row2col_in_agaa(x0) 6.84/2.65 U4_agaa(x0, x1, x2) 6.84/2.65 6.84/2.65 We have to consider all (P,Q,R)-chains. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (21) DependencyGraphProof (EQUIVALENT) 6.84/2.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 6.84/2.65 ---------------------------------------- 6.84/2.65 6.84/2.65 (22) 6.84/2.65 TRUE 6.95/2.68 EOF