(0) Obligation:
Clauses:
app3_a(Xs, Ys, Zs, Us) :- ','(app(Xs, Ys, Vs), app(Vs, Zs, Us)).
app3_b(Xs, Ys, Zs, Us) :- ','(app(Ys, Zs, Vs), app(Xs, Vs, Us)).
app([], Ys, Ys).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
Query: app3_b(g,g,g,a)
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph DT10.
(2) Obligation:
Triples:
appA(.(X1, X2), X3, .(X1, X4)) :- appA(X2, X3, X4).
appB(.(X1, X2), X3, .(X1, X4)) :- appB(X2, X3, X4).
app3_bC(X1, [], X2, X3) :- appA(X1, X2, X3).
app3_bC(X1, .(X2, X3), X4, X5) :- appB(X3, X4, X6).
app3_bC(X1, .(X2, X3), X4, X5) :- ','(appcB(X3, X4, X6), appA(X1, .(X2, X6), X5)).
Clauses:
appcA([], X1, X1).
appcA(.(X1, X2), X3, .(X1, X4)) :- appcA(X2, X3, X4).
appcB([], X1, X1).
appcB(.(X1, X2), X3, .(X1, X4)) :- appcB(X2, X3, X4).
Afs:
app3_bC(x1, x2, x3, x4) = app3_bC(x1, x2, x3)
(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:
app3_bC_in: (b,b,b,f)
appA_in: (b,b,f)
appB_in: (b,b,f)
appcB_in: (b,b,f)
Transforming
TRIPLES into the following
Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APP3_BC_IN_GGGA(X1, [], X2, X3) → U3_GGGA(X1, X2, X3, appA_in_gga(X1, X2, X3))
APP3_BC_IN_GGGA(X1, [], X2, X3) → APPA_IN_GGA(X1, X2, X3)
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U1_GGA(X1, X2, X3, X4, appA_in_gga(X2, X3, X4))
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GGA(X2, X3, X4)
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → U4_GGGA(X1, X2, X3, X4, X5, appB_in_gga(X3, X4, X6))
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → APPB_IN_GGA(X3, X4, X6)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, appcB_in_gga(X3, X4, X6))
U5_GGGA(X1, X2, X3, X4, X5, appcB_out_gga(X3, X4, X6)) → U6_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X1, .(X2, X6), X5))
U5_GGGA(X1, X2, X3, X4, X5, appcB_out_gga(X3, X4, X6)) → APPA_IN_GGA(X1, .(X2, X6), X5)
The TRS R consists of the following rules:
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
appB_in_gga(
x1,
x2,
x3) =
appB_in_gga(
x1,
x2)
appcB_in_gga(
x1,
x2,
x3) =
appcB_in_gga(
x1,
x2)
appcB_out_gga(
x1,
x2,
x3) =
appcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APP3_BC_IN_GGGA(
x1,
x2,
x3,
x4) =
APP3_BC_IN_GGGA(
x1,
x2,
x3)
U3_GGGA(
x1,
x2,
x3,
x4) =
U3_GGGA(
x1,
x2,
x4)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GGGA(
x1,
x2,
x3,
x4,
x6)
APPB_IN_GGA(
x1,
x2,
x3) =
APPB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U5_GGGA(
x1,
x2,
x3,
x4,
x6)
U6_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGGA(
x1,
x2,
x3,
x4,
x6)
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:
APP3_BC_IN_GGGA(X1, [], X2, X3) → U3_GGGA(X1, X2, X3, appA_in_gga(X1, X2, X3))
APP3_BC_IN_GGGA(X1, [], X2, X3) → APPA_IN_GGA(X1, X2, X3)
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U1_GGA(X1, X2, X3, X4, appA_in_gga(X2, X3, X4))
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GGA(X2, X3, X4)
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → U4_GGGA(X1, X2, X3, X4, X5, appB_in_gga(X3, X4, X6))
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → APPB_IN_GGA(X3, X4, X6)
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → U2_GGA(X1, X2, X3, X4, appB_in_gga(X2, X3, X4))
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
APP3_BC_IN_GGGA(X1, .(X2, X3), X4, X5) → U5_GGGA(X1, X2, X3, X4, X5, appcB_in_gga(X3, X4, X6))
U5_GGGA(X1, X2, X3, X4, X5, appcB_out_gga(X3, X4, X6)) → U6_GGGA(X1, X2, X3, X4, X5, appA_in_gga(X1, .(X2, X6), X5))
U5_GGGA(X1, X2, X3, X4, X5, appcB_out_gga(X3, X4, X6)) → APPA_IN_GGA(X1, .(X2, X6), X5)
The TRS R consists of the following rules:
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
appB_in_gga(
x1,
x2,
x3) =
appB_in_gga(
x1,
x2)
appcB_in_gga(
x1,
x2,
x3) =
appcB_in_gga(
x1,
x2)
appcB_out_gga(
x1,
x2,
x3) =
appcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APP3_BC_IN_GGGA(
x1,
x2,
x3,
x4) =
APP3_BC_IN_GGGA(
x1,
x2,
x3)
U3_GGGA(
x1,
x2,
x3,
x4) =
U3_GGGA(
x1,
x2,
x4)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5) =
U1_GGA(
x1,
x2,
x3,
x5)
U4_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U4_GGGA(
x1,
x2,
x3,
x4,
x6)
APPB_IN_GGA(
x1,
x2,
x3) =
APPB_IN_GGA(
x1,
x2)
U2_GGA(
x1,
x2,
x3,
x4,
x5) =
U2_GGA(
x1,
x2,
x3,
x5)
U5_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U5_GGGA(
x1,
x2,
x3,
x4,
x6)
U6_GGGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U6_GGGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
The TRS R consists of the following rules:
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x1,
x2)
appcB_in_gga(
x1,
x2,
x3) =
appcB_in_gga(
x1,
x2)
appcB_out_gga(
x1,
x2,
x3) =
appcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPB_IN_GGA(
x1,
x2,
x3) =
APPB_IN_GGA(
x1,
x2)
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:
APPB_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPB_IN_GGA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPB_IN_GGA(
x1,
x2,
x3) =
APPB_IN_GGA(
x1,
x2)
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:
APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)
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:
- APPB_IN_GGA(.(X1, X2), X3) → APPB_IN_GGA(X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GGA(X2, X3, X4)
The TRS R consists of the following rules:
appcB_in_gga([], X1, X1) → appcB_out_gga([], X1, X1)
appcB_in_gga(.(X1, X2), X3, .(X1, X4)) → U9_gga(X1, X2, X3, X4, appcB_in_gga(X2, X3, X4))
U9_gga(X1, X2, X3, X4, appcB_out_gga(X2, X3, X4)) → appcB_out_gga(.(X1, X2), X3, .(X1, X4))
The argument filtering Pi contains the following mapping:
[] =
[]
.(
x1,
x2) =
.(
x1,
x2)
appcB_in_gga(
x1,
x2,
x3) =
appcB_in_gga(
x1,
x2)
appcB_out_gga(
x1,
x2,
x3) =
appcB_out_gga(
x1,
x2,
x3)
U9_gga(
x1,
x2,
x3,
x4,
x5) =
U9_gga(
x1,
x2,
x3,
x5)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
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:
APPA_IN_GGA(.(X1, X2), X3, .(X1, X4)) → APPA_IN_GGA(X2, X3, X4)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
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:
APPA_IN_GGA(.(X1, X2), X3) → APPA_IN_GGA(X2, X3)
R is empty.
Q is empty.
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:
- APPA_IN_GGA(.(X1, X2), X3) → APPA_IN_GGA(X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2
(20) YES