(0) Obligation:
Clauses:
app([], Y, Y).
app(X, Y, .(H, Z)) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), app(T, Y, Z)))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Query: app(g,a,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
appA([], T5, T5).
appA(.(T57, T58), T47, .(T57, T48)) :- appA(T58, T47, T48).
Query: appA(g,a,a)
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
appA_in: (b,f,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
[] =
[]
appA_out_gaa(
x1,
x2,
x3) =
appA_out_gaa
.(
x1,
x2) =
.(
x1,
x2)
U1_gaa(
x1,
x2,
x3,
x4,
x5) =
U1_gaa(
x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
[] =
[]
appA_out_gaa(
x1,
x2,
x3) =
appA_out_gaa
.(
x1,
x2) =
.(
x1,
x2)
U1_gaa(
x1,
x2,
x3,
x4,
x5) =
U1_gaa(
x5)
(5) 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_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)
The TRS R consists of the following rules:
appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
[] =
[]
appA_out_gaa(
x1,
x2,
x3) =
appA_out_gaa
.(
x1,
x2) =
.(
x1,
x2)
U1_gaa(
x1,
x2,
x3,
x4,
x5) =
U1_gaa(
x5)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x5)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)
The TRS R consists of the following rules:
appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
[] =
[]
appA_out_gaa(
x1,
x2,
x3) =
appA_out_gaa
.(
x1,
x2) =
.(
x1,
x2)
U1_gaa(
x1,
x2,
x3,
x4,
x5) =
U1_gaa(
x5)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
U1_GAA(
x1,
x2,
x3,
x4,
x5) =
U1_GAA(
x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)
The TRS R consists of the following rules:
appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))
The argument filtering Pi contains the following mapping:
appA_in_gaa(
x1,
x2,
x3) =
appA_in_gaa(
x1)
[] =
[]
appA_out_gaa(
x1,
x2,
x3) =
appA_out_gaa
.(
x1,
x2) =
.(
x1,
x2)
U1_gaa(
x1,
x2,
x3,
x4,
x5) =
U1_gaa(
x5)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(9) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(10) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_GAA(
x1,
x2,
x3) =
APPA_IN_GAA(
x1)
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
APPA_IN_GAA(.(T57, T58)) → APPA_IN_GAA(T58)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) 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_GAA(.(T57, T58)) → APPA_IN_GAA(T58)
The graph contains the following edges 1 > 1
(14) YES