(0) Obligation:
Clauses:
init_vars(E1, E2, E1init, E2init) :- ','(find_all_vars(E1, Vars1), ','(find_all_vars(E2, Vars2), ','(intersect(Vars1, Vars2, _X, Notin1, Notin2), ','(init_vars2(Notin1, E1, E1init), init_vars2(Notin2, E2, E2init))))).
find_all_vars(E, Vars) :- ','(find_all_vars2(E, Vars0), sort(Vars0, Vars)).
find_all_vars2([], []).
find_all_vars2(.(=(Vars, _Values), Es), AllVars) :- ','(append(Vars, AllVars1, AllVars), find_all_vars2(Es, AllVars1)).
init_vars2(Notin, E, Einit) :- ','(init_vars3(Notin, E, Einit0), sort(Einit0, Einit)).
init_vars3([], E, E).
init_vars3(.(Var, Vars), E, .(=(.(Var, []), .(unbound, [])), Es)) :- init_vars3(Vars, E, Es).
append([], A, A).
append(.(A, B), C, .(A, D)) :- append(B, C, D).
intersect(As, [], [], [], As) :- !.
intersect([], Bs, [], Bs, []) :- !.
intersect(.(A, As), .(B, Bs), Cs, Ds, Es) :- ;(;(->(=(A, B), ','(=(Cs, .(A, Cs2)), intersect(As, Bs, Cs2, Ds, Es))), ->(@<(A, B), ','(=(Es, .(A, Es2)), intersect(As, .(B, Bs), Cs, Ds, Es2)))), ','(=(Ds, .(B, Ds2)), intersect(.(A, As), Bs, Cs, Ds2, Es))).
get_query(E1, E2) :- ','(=(E1, .(=(X, .(a, [])), .(=(X, .(a, [])), .(=(X, .(a, [])), .(=(X, .(a, [])), []))))), ','(=(E2, .(=(Y, .(a, [])), .(=(Y, .(a, [])), .(=(Y, .(a, [])), .(=(Y, .(a, [])), []))))), ','(=(X, .(5, .(7, .(8, .(3, .(2, .(4, .(1, .(6, .(9, .(15, .(17, .(18, .(13, .(12, .(14, .(11, .(16, .(19, .(25, .(27, .(28, .(23, .(22, .(24, .(21, .(26, .(29, [])))))))))))))))))))))))))))), =(Y, .(15, .(17, .(18, .(13, .(12, .(14, .(11, .(16, .(19, .(35, .(37, .(38, .(33, .(32, .(34, .(5, .(7, .(8, .(3, .(2, .(4, .(1, .(6, .(9, .(31, .(36, .(39, []))))))))))))))))))))))))))))))).
Query: init_vars(g,g,a,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
appendA(.(X1, X2), X3, .(X1, X4)) :- appendA(X2, X3, X4).
find_all_vars2B(.(=(X1, X2), X3), X4) :- appendC(X1, X5, X4).
find_all_vars2B(.(=(X1, X2), X3), X4) :- ','(appendcC(X1, X5, X4), find_all_vars2B(X3, X5)).
appendC(.(X1, X2), X3, .(X1, X4)) :- appendC(X2, X3, X4).
init_varsE(.(=(X1, X2), X3), X4, X5, X6) :- appendA(X1, X7, X8).
init_varsE(.(=(X1, X2), X3), X4, X5, X6) :- ','(appendcA(X1, X7, X8), find_all_vars2B(X3, X7)).
Clauses:
appendcA([], X1, X1).
appendcA(.(X1, X2), X3, .(X1, X4)) :- appendcA(X2, X3, X4).
find_all_vars2cB([], []).
find_all_vars2cB(.(=(X1, X2), X3), X4) :- ','(appendcC(X1, X5, X4), find_all_vars2cB(X3, X5)).
appendcC([], X1, X1).
appendcC(.(X1, X2), X3, .(X1, X4)) :- appendcC(X2, X3, X4).
find_all_vars2cD([], []).
find_all_vars2cD(.(=(X1, X2), X3), X4) :- ','(appendcA(X1, X5, X4), find_all_vars2cB(X3, X5)).
Afs:
init_varsE(x1, x2, x3, x4) = init_varsE(x1, x2)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
init_varsE_in: (b,b,f,f)
appendA_in: (b,f,f)
appendcA_in: (b,f,f)
find_all_vars2B_in: (b,f)
appendC_in: (b,f,f)
appendcC_in: (b,f,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → U6_GGAA(X1, X2, X3, X4, X5, X6, appendA_in_gaa(X1, X7, X8))
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → APPENDA_IN_GAA(X1, X7, X8)
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_in_gaa(X1, X7, X8))
U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_out_gaa(X1, X7, X8)) → U8_GGAA(X1, X2, X3, X4, X5, X6, find_all_vars2B_in_ga(X3, X7))
U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_out_gaa(X1, X7, X8)) → FIND_ALL_VARS2B_IN_GA(X3, X7)
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U2_GA(X1, X2, X3, X4, appendC_in_gaa(X1, X5, X4))
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → APPENDC_IN_GAA(X1, X5, X4)
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U5_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U3_GA(X1, X2, X3, X4, appendcC_in_gaa(X1, X5, X4))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → U4_GA(X1, X2, X3, X4, find_all_vars2B_in_ga(X3, X5))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → FIND_ALL_VARS2B_IN_GA(X3, X5)
The TRS R consists of the following rules:
appendcA_in_gaa([], X1, X1) → appendcA_out_gaa([], X1, X1)
appendcA_in_gaa(.(X1, X2), X3, .(X1, X4)) → U10_gaa(X1, X2, X3, X4, appendcA_in_gaa(X2, X3, X4))
U10_gaa(X1, X2, X3, X4, appendcA_out_gaa(X2, X3, X4)) → appendcA_out_gaa(.(X1, X2), X3, .(X1, X4))
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
appendA_in_gaa(
x1,
x2,
x3) =
appendA_in_gaa(
x1)
appendcA_in_gaa(
x1,
x2,
x3) =
appendcA_in_gaa(
x1)
[] =
[]
appendcA_out_gaa(
x1,
x2,
x3) =
appendcA_out_gaa(
x1)
U10_gaa(
x1,
x2,
x3,
x4,
x5) =
U10_gaa(
x1,
x2,
x5)
find_all_vars2B_in_ga(
x1,
x2) =
find_all_vars2B_in_ga(
x1)
appendC_in_gaa(
x1,
x2,
x3) =
appendC_in_gaa(
x1)
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
INIT_VARSE_IN_GGAA(
x1,
x2,
x3,
x4) =
INIT_VARSE_IN_GGAA(
x1,
x2)
U6_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U6_GGAA(
x1,
x2,
x3,
x4,
x7)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x1,
x2,
x5)
U7_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U7_GGAA(
x1,
x2,
x3,
x4,
x7)
U8_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U8_GGAA(
x1,
x2,
x3,
x4,
x7)
FIND_ALL_VARS2B_IN_GA(
x1,
x2) =
FIND_ALL_VARS2B_IN_GA(
x1)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x1,
x2,
x3,
x5)
APPENDC_IN_GAA(
x1,
x2,
x3) =
APPENDC_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4,
x5) =
U5_GAA(
x1,
x2,
x5)
U3_GA(
x1,
x2,
x3,
x4,
x5) =
U3_GA(
x1,
x2,
x3,
x5)
U4_GA(
x1,
x2,
x3,
x4,
x5) =
U4_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → U6_GGAA(X1, X2, X3, X4, X5, X6, appendA_in_gaa(X1, X7, X8))
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → APPENDA_IN_GAA(X1, X7, X8)
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U1_GAA(X1, X2, X3, X4, appendA_in_gaa(X2, X3, X4))
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
INIT_VARSE_IN_GGAA(.(=(X1, X2), X3), X4, X5, X6) → U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_in_gaa(X1, X7, X8))
U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_out_gaa(X1, X7, X8)) → U8_GGAA(X1, X2, X3, X4, X5, X6, find_all_vars2B_in_ga(X3, X7))
U7_GGAA(X1, X2, X3, X4, X5, X6, appendcA_out_gaa(X1, X7, X8)) → FIND_ALL_VARS2B_IN_GA(X3, X7)
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U2_GA(X1, X2, X3, X4, appendC_in_gaa(X1, X5, X4))
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → APPENDC_IN_GAA(X1, X5, X4)
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → U5_GAA(X1, X2, X3, X4, appendC_in_gaa(X2, X3, X4))
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U3_GA(X1, X2, X3, X4, appendcC_in_gaa(X1, X5, X4))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → U4_GA(X1, X2, X3, X4, find_all_vars2B_in_ga(X3, X5))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → FIND_ALL_VARS2B_IN_GA(X3, X5)
The TRS R consists of the following rules:
appendcA_in_gaa([], X1, X1) → appendcA_out_gaa([], X1, X1)
appendcA_in_gaa(.(X1, X2), X3, .(X1, X4)) → U10_gaa(X1, X2, X3, X4, appendcA_in_gaa(X2, X3, X4))
U10_gaa(X1, X2, X3, X4, appendcA_out_gaa(X2, X3, X4)) → appendcA_out_gaa(.(X1, X2), X3, .(X1, X4))
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
appendA_in_gaa(
x1,
x2,
x3) =
appendA_in_gaa(
x1)
appendcA_in_gaa(
x1,
x2,
x3) =
appendcA_in_gaa(
x1)
[] =
[]
appendcA_out_gaa(
x1,
x2,
x3) =
appendcA_out_gaa(
x1)
U10_gaa(
x1,
x2,
x3,
x4,
x5) =
U10_gaa(
x1,
x2,
x5)
find_all_vars2B_in_ga(
x1,
x2) =
find_all_vars2B_in_ga(
x1)
appendC_in_gaa(
x1,
x2,
x3) =
appendC_in_gaa(
x1)
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
INIT_VARSE_IN_GGAA(
x1,
x2,
x3,
x4) =
INIT_VARSE_IN_GGAA(
x1,
x2)
U6_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U6_GGAA(
x1,
x2,
x3,
x4,
x7)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x1,
x2,
x5)
U7_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U7_GGAA(
x1,
x2,
x3,
x4,
x7)
U8_GGAA(
x1,
x2,
x3,
x4,
x5,
x6,
x7) =
U8_GGAA(
x1,
x2,
x3,
x4,
x7)
FIND_ALL_VARS2B_IN_GA(
x1,
x2) =
FIND_ALL_VARS2B_IN_GA(
x1)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x1,
x2,
x3,
x5)
APPENDC_IN_GAA(
x1,
x2,
x3) =
APPENDC_IN_GAA(
x1)
U5_GAA(
x1,
x2,
x3,
x4,
x5) =
U5_GAA(
x1,
x2,
x5)
U3_GA(
x1,
x2,
x3,
x4,
x5) =
U3_GA(
x1,
x2,
x3,
x5)
U4_GA(
x1,
x2,
x3,
x4,
x5) =
U4_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)
The TRS R consists of the following rules:
appendcA_in_gaa([], X1, X1) → appendcA_out_gaa([], X1, X1)
appendcA_in_gaa(.(X1, X2), X3, .(X1, X4)) → U10_gaa(X1, X2, X3, X4, appendcA_in_gaa(X2, X3, X4))
U10_gaa(X1, X2, X3, X4, appendcA_out_gaa(X2, X3, X4)) → appendcA_out_gaa(.(X1, X2), X3, .(X1, X4))
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendcA_in_gaa(
x1,
x2,
x3) =
appendcA_in_gaa(
x1)
[] =
[]
appendcA_out_gaa(
x1,
x2,
x3) =
appendcA_out_gaa(
x1)
U10_gaa(
x1,
x2,
x3,
x4,
x5) =
U10_gaa(
x1,
x2,
x5)
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
APPENDC_IN_GAA(
x1,
x2,
x3) =
APPENDC_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(8) UsableRulesProof (EQUIVALENT transformation)
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:
APPENDC_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDC_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDC_IN_GAA(
x1,
x2,
x3) =
APPENDC_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (SOUND transformation)
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:
APPENDC_IN_GAA(.(X1, X2)) → APPENDC_IN_GAA(X2)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
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:
- APPENDC_IN_GAA(.(X1, X2)) → APPENDC_IN_GAA(X2)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U3_GA(X1, X2, X3, X4, appendcC_in_gaa(X1, X5, X4))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → FIND_ALL_VARS2B_IN_GA(X3, X5)
The TRS R consists of the following rules:
appendcA_in_gaa([], X1, X1) → appendcA_out_gaa([], X1, X1)
appendcA_in_gaa(.(X1, X2), X3, .(X1, X4)) → U10_gaa(X1, X2, X3, X4, appendcA_in_gaa(X2, X3, X4))
U10_gaa(X1, X2, X3, X4, appendcA_out_gaa(X2, X3, X4)) → appendcA_out_gaa(.(X1, X2), X3, .(X1, X4))
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
appendcA_in_gaa(
x1,
x2,
x3) =
appendcA_in_gaa(
x1)
[] =
[]
appendcA_out_gaa(
x1,
x2,
x3) =
appendcA_out_gaa(
x1)
U10_gaa(
x1,
x2,
x3,
x4,
x5) =
U10_gaa(
x1,
x2,
x5)
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
FIND_ALL_VARS2B_IN_GA(
x1,
x2) =
FIND_ALL_VARS2B_IN_GA(
x1)
U3_GA(
x1,
x2,
x3,
x4,
x5) =
U3_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
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:
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3), X4) → U3_GA(X1, X2, X3, X4, appendcC_in_gaa(X1, X5, X4))
U3_GA(X1, X2, X3, X4, appendcC_out_gaa(X1, X5, X4)) → FIND_ALL_VARS2B_IN_GA(X3, X5)
The TRS R consists of the following rules:
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
=(
x1,
x2) =
=(
x1,
x2)
[] =
[]
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
FIND_ALL_VARS2B_IN_GA(
x1,
x2) =
FIND_ALL_VARS2B_IN_GA(
x1)
U3_GA(
x1,
x2,
x3,
x4,
x5) =
U3_GA(
x1,
x2,
x3,
x5)
We have to consider all (P,R,Pi)-chains
(17) PiDPToQDPProof (SOUND transformation)
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:
FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3)) → U3_GA(X1, X2, X3, appendcC_in_gaa(X1))
U3_GA(X1, X2, X3, appendcC_out_gaa(X1)) → FIND_ALL_VARS2B_IN_GA(X3)
The TRS R consists of the following rules:
appendcC_in_gaa([]) → appendcC_out_gaa([])
appendcC_in_gaa(.(X1, X2)) → U13_gaa(X1, X2, appendcC_in_gaa(X2))
U13_gaa(X1, X2, appendcC_out_gaa(X2)) → appendcC_out_gaa(.(X1, X2))
The set Q consists of the following terms:
appendcC_in_gaa(x0)
U13_gaa(x0, x1, x2)
We have to consider all (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
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:
- U3_GA(X1, X2, X3, appendcC_out_gaa(X1)) → FIND_ALL_VARS2B_IN_GA(X3)
The graph contains the following edges 3 >= 1
- FIND_ALL_VARS2B_IN_GA(.(=(X1, X2), X3)) → U3_GA(X1, X2, X3, appendcC_in_gaa(X1))
The graph contains the following edges 1 > 1, 1 > 2, 1 > 3
(20) YES
(21) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
The TRS R consists of the following rules:
appendcA_in_gaa([], X1, X1) → appendcA_out_gaa([], X1, X1)
appendcA_in_gaa(.(X1, X2), X3, .(X1, X4)) → U10_gaa(X1, X2, X3, X4, appendcA_in_gaa(X2, X3, X4))
U10_gaa(X1, X2, X3, X4, appendcA_out_gaa(X2, X3, X4)) → appendcA_out_gaa(.(X1, X2), X3, .(X1, X4))
appendcC_in_gaa([], X1, X1) → appendcC_out_gaa([], X1, X1)
appendcC_in_gaa(.(X1, X2), X3, .(X1, X4)) → U13_gaa(X1, X2, X3, X4, appendcC_in_gaa(X2, X3, X4))
U13_gaa(X1, X2, X3, X4, appendcC_out_gaa(X2, X3, X4)) → appendcC_out_gaa(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
appendcA_in_gaa(
x1,
x2,
x3) =
appendcA_in_gaa(
x1)
[] =
[]
appendcA_out_gaa(
x1,
x2,
x3) =
appendcA_out_gaa(
x1)
U10_gaa(
x1,
x2,
x3,
x4,
x5) =
U10_gaa(
x1,
x2,
x5)
appendcC_in_gaa(
x1,
x2,
x3) =
appendcC_in_gaa(
x1)
appendcC_out_gaa(
x1,
x2,
x3) =
appendcC_out_gaa(
x1)
U13_gaa(
x1,
x2,
x3,
x4,
x5) =
U13_gaa(
x1,
x2,
x5)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(22) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(23) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2), X3, .(X1, X4)) → APPENDA_IN_GAA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPENDA_IN_GAA(
x1,
x2,
x3) =
APPENDA_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(24) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(25) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APPENDA_IN_GAA(.(X1, X2)) → APPENDA_IN_GAA(X2)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(26) QDPSizeChangeProof (EQUIVALENT transformation)
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:
- APPENDA_IN_GAA(.(X1, X2)) → APPENDA_IN_GAA(X2)
The graph contains the following edges 1 > 1
(27) YES