(0) Obligation:
Clauses:
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
Query: app(g,g,a)
(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)
Transformed Prolog program to (Pi-)TRS.
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
appA_in_gga([], T5, T5) → appA_out_gga([], T5, T5)
appA_in_gga(.(T10, []), T19, .(T10, T19)) → appA_out_gga(.(T10, []), T19, .(T10, T19))
appA_in_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_gga(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
U1_gga(T10, T28, T29, T30, T32, appA_out_gga(T29, T30, T32)) → appA_out_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32)))
The argument filtering Pi contains the following mapping:
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
[] =
[]
appA_out_gga(
x1,
x2,
x3) =
appA_out_gga(
x1,
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x1,
x2,
x3,
x4,
x6)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_GGA(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → APPA_IN_GGA(T29, T30, T32)
The TRS R consists of the following rules:
appA_in_gga([], T5, T5) → appA_out_gga([], T5, T5)
appA_in_gga(.(T10, []), T19, .(T10, T19)) → appA_out_gga(.(T10, []), T19, .(T10, T19))
appA_in_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_gga(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
U1_gga(T10, T28, T29, T30, T32, appA_out_gga(T29, T30, T32)) → appA_out_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32)))
The argument filtering Pi contains the following mapping:
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
[] =
[]
appA_out_gga(
x1,
x2,
x3) =
appA_out_gga(
x1,
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x1,
x2,
x3,
x4,
x6)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGA(
x1,
x2,
x3,
x4,
x6)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_GGA(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → APPA_IN_GGA(T29, T30, T32)
The TRS R consists of the following rules:
appA_in_gga([], T5, T5) → appA_out_gga([], T5, T5)
appA_in_gga(.(T10, []), T19, .(T10, T19)) → appA_out_gga(.(T10, []), T19, .(T10, T19))
appA_in_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_gga(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
U1_gga(T10, T28, T29, T30, T32, appA_out_gga(T29, T30, T32)) → appA_out_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32)))
The argument filtering Pi contains the following mapping:
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
[] =
[]
appA_out_gga(
x1,
x2,
x3) =
appA_out_gga(
x1,
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x1,
x2,
x3,
x4,
x6)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
U1_GGA(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_GGA(
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 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → APPA_IN_GGA(T29, T30, T32)
The TRS R consists of the following rules:
appA_in_gga([], T5, T5) → appA_out_gga([], T5, T5)
appA_in_gga(.(T10, []), T19, .(T10, T19)) → appA_out_gga(.(T10, []), T19, .(T10, T19))
appA_in_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → U1_gga(T10, T28, T29, T30, T32, appA_in_gga(T29, T30, T32))
U1_gga(T10, T28, T29, T30, T32, appA_out_gga(T29, T30, T32)) → appA_out_gga(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32)))
The argument filtering Pi contains the following mapping:
appA_in_gga(
x1,
x2,
x3) =
appA_in_gga(
x1,
x2)
[] =
[]
appA_out_gga(
x1,
x2,
x3) =
appA_out_gga(
x1,
x2,
x3)
.(
x1,
x2) =
.(
x1,
x2)
U1_gga(
x1,
x2,
x3,
x4,
x5,
x6) =
U1_gga(
x1,
x2,
x3,
x4,
x6)
APPA_IN_GGA(
x1,
x2,
x3) =
APPA_IN_GGA(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(7) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(T10, .(T28, T29)), T30, .(T10, .(T28, T32))) → APPA_IN_GGA(T29, T30, T32)
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
(9) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APPA_IN_GGA(.(T10, .(T28, T29)), T30) → APPA_IN_GGA(T29, T30)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) 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(.(T10, .(T28, T29)), T30) → APPA_IN_GGA(T29, T30)
The graph contains the following edges 1 > 1, 2 >= 2
(12) YES