(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).
Queries:
app3_b(g,g,g,a).
(1) PrologToDTProblemTransformerProof (SOUND transformation)
Built DT problem from termination graph.
(2) Obligation:
Triples:app7(.(T34, T35), T36, .(T34, T38)) :- app7(T35, T36, T38).
app19(.(T66, T67), T68, .(T66, X85)) :- app19(T67, T68, X85).
app3_b1(T9, [], T18, T13) :- app7(T9, T18, T13).
app3_b1(T9, .(T47, T48), T49, T13) :- app19(T48, T49, X56).
app3_b1(T9, .(T47, T48), T49, T13) :- ','(appc19(T48, T49, T52), app7(T9, .(T47, T52), T13)).
Clauses:
appc7([], T25, T25).
appc7(.(T34, T35), T36, .(T34, T38)) :- appc7(T35, T36, T38).
appc19([], T59, T59).
appc19(.(T66, T67), T68, .(T66, X85)) :- appc19(T67, T68, X85).
Afs:
app3_b1(x1, x2, x3, x4) = app3_b1(x1, x2, x3)
(3) TriplesToPiDPProof (SOUND transformation)
We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:app3_b1_in: (b,b,b,f)
app7_in: (b,b,f)
app19_in: (b,b,f)
appc19_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:
APP3_B1_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, app7_in_gga(T9, T18, T13))
APP3_B1_IN_GGGA(T9, [], T18, T13) → APP7_IN_GGA(T9, T18, T13)
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → APP19_IN_GGA(T48, T49, X56)
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, appc19_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, appc19_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, appc19_out_gga(T48, T49, T52)) → APP7_IN_GGA(T9, .(T47, T52), T13)
The TRS R consists of the following rules:
appc19_in_gga([], T59, T59) → appc19_out_gga([], T59, T59)
appc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appc19_out_gga(T67, T68, X85)) → appc19_out_gga(.(T66, T67), T68, .(T66, X85))
The argument filtering Pi contains the following mapping:
[] = []
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
appc19_in_gga(x1, x2, x3) = appc19_in_gga(x1, x2)
appc19_out_gga(x1, x2, x3) = appc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
APP3_B1_IN_GGGA(x1, x2, x3, x4) = APP3_B1_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x1, x2, x4)
APP7_IN_GGA(x1, x2, x3) = APP7_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)
APP19_IN_GGA(x1, x2, x3) = APP19_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_B1_IN_GGGA(T9, [], T18, T13) → U3_GGGA(T9, T18, T13, app7_in_gga(T9, T18, T13))
APP3_B1_IN_GGGA(T9, [], T18, T13) → APP7_IN_GGA(T9, T18, T13)
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → U1_GGA(T34, T35, T36, T38, app7_in_gga(T35, T36, T38))
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U4_GGGA(T9, T47, T48, T49, T13, app19_in_gga(T48, T49, X56))
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → APP19_IN_GGA(T48, T49, X56)
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → U2_GGA(T66, T67, T68, X85, app19_in_gga(T67, T68, X85))
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
APP3_B1_IN_GGGA(T9, .(T47, T48), T49, T13) → U5_GGGA(T9, T47, T48, T49, T13, appc19_in_gga(T48, T49, T52))
U5_GGGA(T9, T47, T48, T49, T13, appc19_out_gga(T48, T49, T52)) → U6_GGGA(T9, T47, T48, T49, T13, app7_in_gga(T9, .(T47, T52), T13))
U5_GGGA(T9, T47, T48, T49, T13, appc19_out_gga(T48, T49, T52)) → APP7_IN_GGA(T9, .(T47, T52), T13)
The TRS R consists of the following rules:
appc19_in_gga([], T59, T59) → appc19_out_gga([], T59, T59)
appc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appc19_out_gga(T67, T68, X85)) → appc19_out_gga(.(T66, T67), T68, .(T66, X85))
The argument filtering Pi contains the following mapping:
[] = []
app7_in_gga(x1, x2, x3) = app7_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
app19_in_gga(x1, x2, x3) = app19_in_gga(x1, x2)
appc19_in_gga(x1, x2, x3) = appc19_in_gga(x1, x2)
appc19_out_gga(x1, x2, x3) = appc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
APP3_B1_IN_GGGA(x1, x2, x3, x4) = APP3_B1_IN_GGGA(x1, x2, x3)
U3_GGGA(x1, x2, x3, x4) = U3_GGGA(x1, x2, x4)
APP7_IN_GGA(x1, x2, x3) = APP7_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)
APP19_IN_GGA(x1, x2, x3) = APP19_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:
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
The TRS R consists of the following rules:
appc19_in_gga([], T59, T59) → appc19_out_gga([], T59, T59)
appc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appc19_out_gga(T67, T68, X85)) → appc19_out_gga(.(T66, T67), T68, .(T66, X85))
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
appc19_in_gga(x1, x2, x3) = appc19_in_gga(x1, x2)
appc19_out_gga(x1, x2, x3) = appc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
APP19_IN_GGA(x1, x2, x3) = APP19_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:
APP19_IN_GGA(.(T66, T67), T68, .(T66, X85)) → APP19_IN_GGA(T67, T68, X85)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP19_IN_GGA(x1, x2, x3) = APP19_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:
APP19_IN_GGA(.(T66, T67), T68) → APP19_IN_GGA(T67, T68)
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:
- APP19_IN_GGA(.(T66, T67), T68) → APP19_IN_GGA(T67, T68)
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:
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
The TRS R consists of the following rules:
appc19_in_gga([], T59, T59) → appc19_out_gga([], T59, T59)
appc19_in_gga(.(T66, T67), T68, .(T66, X85)) → U9_gga(T66, T67, T68, X85, appc19_in_gga(T67, T68, X85))
U9_gga(T66, T67, T68, X85, appc19_out_gga(T67, T68, X85)) → appc19_out_gga(.(T66, T67), T68, .(T66, X85))
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
appc19_in_gga(x1, x2, x3) = appc19_in_gga(x1, x2)
appc19_out_gga(x1, x2, x3) = appc19_out_gga(x1, x2, x3)
U9_gga(x1, x2, x3, x4, x5) = U9_gga(x1, x2, x3, x5)
APP7_IN_GGA(x1, x2, x3) = APP7_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:
APP7_IN_GGA(.(T34, T35), T36, .(T34, T38)) → APP7_IN_GGA(T35, T36, T38)
R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP7_IN_GGA(x1, x2, x3) = APP7_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:
APP7_IN_GGA(.(T34, T35), T36) → APP7_IN_GGA(T35, T36)
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:
- APP7_IN_GGA(.(T34, T35), T36) → APP7_IN_GGA(T35, T36)
The graph contains the following edges 1 > 1, 2 >= 2