Left Termination of the query pattern
transpose_in_2(a, g)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PrologToPiTRSProof
Clauses:
transpose(A, B) :- transpose_aux(A, [], B).
transpose_aux(.(R, Rs), X, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)).
transpose_aux([], X, X).
row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As).
row2col([], [], [], []).
Queries:
transpose(a,g).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_in: (f,b)
transpose_aux_in: (f,b,b)
row2col_in: (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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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)
Using Dependency Pairs [1,30] 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, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
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))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6)
U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U1_AG(x1, x2, x3) = U1_AG(x3)
TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2)
U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7)
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
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, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
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))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6)
U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U1_AG(x1, x2, x3) = U1_AG(x3)
TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2)
U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7)
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], 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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ PiDP
Pi DP problem:
The TRS P consists of the following rules:
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
↳ PiDP
Q DP problem:
The TRS P consists of the following rules:
ROW2COL_IN_AGAA(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(Cols)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(Cols)
The graph contains the following edges 1 > 1
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(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, [], B))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
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))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], 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)
.(x1, x2) = .(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)
[] = []
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)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
The TRS R consists of the following rules:
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))
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))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(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)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
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(X, .(C, Cs)) → U2_AGG(row2col_in_agaa(.(C, Cs)))
The TRS R consists of the following rules:
row2col_in_agaa(.(.(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As))
row2col_in_agaa([]) → row2col_out_agaa([], [], [])
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.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
TRANSPOSE_AUX_IN_AGG(X, .(C, Cs)) → U2_AGG(row2col_in_agaa(.(C, Cs)))
The remaining pairs can at least be oriented weakly.
U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( row2col_in_agaa(x1) ) = | | + | | · | x1 |
| M( .(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( U4_agaa(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( row2col_out_agaa(x1, ..., x3) ) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Tuple symbols:
| M( TRANSPOSE_AUX_IN_AGG(x1, x2) ) = | 1 | + | | · | x1 | + | | · | x2 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
row2col_in_agaa(.(.(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As))
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ AND
↳ PiDP
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
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(.(.(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(Ys, Cols1), .([], As))
row2col_in_agaa([]) → row2col_out_agaa([], [], [])
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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.