4.18/1.81 YES 4.25/2.42 proof of /export/starexec/sandbox/benchmark/theBenchmark.pl 4.25/2.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.25/2.42 4.25/2.42 4.25/2.42 Left Termination of the query pattern 4.25/2.42 4.25/2.42 transpose(g,g) 4.25/2.42 4.25/2.42 w.r.t. the given Prolog program could successfully be proven: 4.25/2.42 4.25/2.42 (0) Prolog 4.25/2.42 (1) PrologToPiTRSProof [SOUND, 18 ms] 4.25/2.42 (2) PiTRS 4.25/2.42 (3) DependencyPairsProof [EQUIVALENT, 0 ms] 4.25/2.42 (4) PiDP 4.25/2.42 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 4.25/2.42 (6) AND 4.25/2.42 (7) PiDP 4.25/2.42 (8) UsableRulesProof [EQUIVALENT, 0 ms] 4.25/2.42 (9) PiDP 4.25/2.42 (10) PiDPToQDPProof [SOUND, 6 ms] 4.25/2.42 (11) QDP 4.25/2.42 (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.25/2.42 (13) YES 4.25/2.42 (14) PiDP 4.25/2.42 (15) UsableRulesProof [EQUIVALENT, 0 ms] 4.25/2.42 (16) PiDP 4.25/2.42 (17) PiDPToQDPProof [SOUND, 0 ms] 4.25/2.42 (18) QDP 4.25/2.42 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 4.25/2.42 (20) YES 4.25/2.42 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (0) 4.25/2.42 Obligation: 4.25/2.42 Clauses: 4.25/2.42 4.25/2.42 transpose(A, B) :- transpose_aux(A, [], B). 4.25/2.42 transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)). 4.25/2.42 transpose_aux([], X, X). 4.25/2.42 row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As). 4.25/2.42 row2col([], [], [], []). 4.25/2.42 4.25/2.42 4.25/2.42 Query: transpose(g,g) 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (1) PrologToPiTRSProof (SOUND) 4.25/2.42 We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: 4.25/2.42 4.25/2.42 transpose_in_2: (b,b) 4.25/2.42 4.25/2.42 transpose_aux_in_3: (b,b,b) 4.25/2.42 4.25/2.42 row2col_in_4: (b,b,f,f) 4.25/2.42 4.25/2.42 Transforming Prolog into the following Term Rewriting System: 4.25/2.42 4.25/2.42 Pi-finite rewrite system: 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 4.25/2.42 4.25/2.42 4.25/2.42 4.25/2.42 Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog 4.25/2.42 4.25/2.42 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (2) 4.25/2.42 Obligation: 4.25/2.42 Pi-finite rewrite system: 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (3) DependencyPairsProof (EQUIVALENT) 4.25/2.42 Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 TRANSPOSE_IN_GG(A, B) -> U1_GG(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 TRANSPOSE_IN_GG(A, B) -> TRANSPOSE_AUX_IN_GGG(A, [], B) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm) 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_GGAA(Xs, Cols, Cols1, As) 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 TRANSPOSE_IN_GG(x1, x2) = TRANSPOSE_IN_GG(x1, x2) 4.25/2.42 4.25/2.42 U1_GG(x1, x2, x3) = U1_GG(x3) 4.25/2.42 4.25/2.42 TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, x3) 4.25/2.42 4.25/2.42 U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6) 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(x1, x2) 4.25/2.42 4.25/2.42 U4_GGAA(x1, x2, x3, x4, x5, x6, x7) = U4_GGAA(x3, x7) 4.25/2.42 4.25/2.42 U3_GGG(x1, x2, x3, x4, x5, x6) = U3_GGG(x6) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (4) 4.25/2.42 Obligation: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 TRANSPOSE_IN_GG(A, B) -> U1_GG(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 TRANSPOSE_IN_GG(A, B) -> TRANSPOSE_AUX_IN_GGG(A, [], B) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm) 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_GGAA(Xs, Cols, Cols1, As) 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 TRANSPOSE_IN_GG(x1, x2) = TRANSPOSE_IN_GG(x1, x2) 4.25/2.42 4.25/2.42 U1_GG(x1, x2, x3) = U1_GG(x3) 4.25/2.42 4.25/2.42 TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, x3) 4.25/2.42 4.25/2.42 U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6) 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(x1, x2) 4.25/2.42 4.25/2.42 U4_GGAA(x1, x2, x3, x4, x5, x6, x7) = U4_GGAA(x3, x7) 4.25/2.42 4.25/2.42 U3_GGG(x1, x2, x3, x4, x5, x6) = U3_GGG(x6) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (5) DependencyGraphProof (EQUIVALENT) 4.25/2.42 The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (6) 4.25/2.42 Complex Obligation (AND) 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (7) 4.25/2.42 Obligation: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_GGAA(Xs, Cols, Cols1, As) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(x1, x2) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (8) UsableRulesProof (EQUIVALENT) 4.25/2.42 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (9) 4.25/2.42 Obligation: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> ROW2COL_IN_GGAA(Xs, Cols, Cols1, As) 4.25/2.42 4.25/2.42 R is empty. 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(x1, x2) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (10) PiDPToQDPProof (SOUND) 4.25/2.42 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (11) 4.25/2.42 Obligation: 4.25/2.42 Q DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) -> ROW2COL_IN_GGAA(Xs, Cols) 4.25/2.42 4.25/2.42 R is empty. 4.25/2.42 Q is empty. 4.25/2.42 We have to consider all (P,Q,R)-chains. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (12) QDPSizeChangeProof (EQUIVALENT) 4.25/2.42 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. 4.25/2.42 4.25/2.42 From the DPs we obtained the following set of size-change graphs: 4.25/2.42 *ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) -> ROW2COL_IN_GGAA(Xs, Cols) 4.25/2.42 The graph contains the following edges 1 > 1, 2 > 2 4.25/2.42 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (13) 4.25/2.42 YES 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (14) 4.25/2.42 Obligation: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 transpose_in_gg(A, B) -> U1_gg(A, B, transpose_aux_in_ggg(A, [], B)) 4.25/2.42 transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) -> U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1)) 4.25/2.42 transpose_aux_in_ggg([], X, X) -> transpose_aux_out_ggg([], X, X) 4.25/2.42 U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) -> transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs)) 4.25/2.42 U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) -> transpose_out_gg(A, B) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2) 4.25/2.42 4.25/2.42 U1_gg(x1, x2, x3) = U1_gg(x3) 4.25/2.42 4.25/2.42 transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3) 4.25/2.42 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6) 4.25/2.42 4.25/2.42 transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg 4.25/2.42 4.25/2.42 transpose_out_gg(x1, x2) = transpose_out_gg 4.25/2.42 4.25/2.42 TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, x3) 4.25/2.42 4.25/2.42 U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (15) UsableRulesProof (EQUIVALENT) 4.25/2.42 For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (16) 4.25/2.42 Obligation: 4.25/2.42 Pi DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm)) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) -> U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As)) 4.25/2.42 U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) -> row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) 4.25/2.42 row2col_in_ggaa([], [], [], []) -> row2col_out_ggaa([], [], [], []) 4.25/2.42 4.25/2.42 The argument filtering Pi contains the following mapping: 4.25/2.42 .(x1, x2) = .(x1, x2) 4.25/2.42 4.25/2.42 row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2) 4.25/2.42 4.25/2.42 U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7) 4.25/2.42 4.25/2.42 [] = [] 4.25/2.42 4.25/2.42 row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4) 4.25/2.42 4.25/2.42 TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, x3) 4.25/2.42 4.25/2.42 U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6) 4.25/2.42 4.25/2.42 4.25/2.42 We have to consider all (P,R,Pi)-chains 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (17) PiDPToQDPProof (SOUND) 4.25/2.42 Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (18) 4.25/2.42 Obligation: 4.25/2.42 Q DP problem: 4.25/2.42 The TRS P consists of the following rules: 4.25/2.42 4.25/2.42 U2_GGG(Rs, row2col_out_ggaa(Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(Rs, row2col_in_ggaa(R, .(C, Cs))) 4.25/2.42 4.25/2.42 The TRS R consists of the following rules: 4.25/2.42 4.25/2.42 row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols)) -> U4_ggaa(Ys, row2col_in_ggaa(Xs, Cols)) 4.25/2.42 U4_ggaa(Ys, row2col_out_ggaa(Cols1, As)) -> row2col_out_ggaa(.(Ys, Cols1), .([], As)) 4.25/2.42 row2col_in_ggaa([], []) -> row2col_out_ggaa([], []) 4.25/2.42 4.25/2.42 The set Q consists of the following terms: 4.25/2.42 4.25/2.42 row2col_in_ggaa(x0, x1) 4.25/2.42 U4_ggaa(x0, x1) 4.25/2.42 4.25/2.42 We have to consider all (P,Q,R)-chains. 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (19) QDPSizeChangeProof (EQUIVALENT) 4.25/2.42 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. 4.25/2.42 4.25/2.42 From the DPs we obtained the following set of size-change graphs: 4.25/2.42 *TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) -> U2_GGG(Rs, row2col_in_ggaa(R, .(C, Cs))) 4.25/2.42 The graph contains the following edges 1 > 1 4.25/2.42 4.25/2.42 4.25/2.42 *U2_GGG(Rs, row2col_out_ggaa(Cols1, Accm)) -> TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1) 4.25/2.42 The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3 4.25/2.42 4.25/2.42 4.25/2.42 ---------------------------------------- 4.25/2.42 4.25/2.42 (20) 4.25/2.42 YES 4.31/2.49 EOF