(0) Obligation:
Clauses:
fold(X, [], X).
fold(X, Y, Z) :- ','(no(empty(Y)), ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z))))).
myop(a, b, c).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).
Query: fold(g,g,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
foldA(T5, [], T5).
foldA(a, .(b, []), c).
Query: foldA(g,g,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:
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
foldA_in_gga(T5, [], T5) → foldA_out_gga(T5, [], T5)
foldA_in_gga(a, .(b, []), c) → foldA_out_gga(a, .(b, []), c)
The argument filtering Pi contains the following mapping:
foldA_in_gga(
x1,
x2,
x3) =
foldA_in_gga(
x1,
x2)
[] =
[]
foldA_out_gga(
x1,
x2,
x3) =
foldA_out_gga(
x3)
a =
a
.(
x1,
x2) =
.(
x1,
x2)
b =
b
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
foldA_in_gga(T5, [], T5) → foldA_out_gga(T5, [], T5)
foldA_in_gga(a, .(b, []), c) → foldA_out_gga(a, .(b, []), c)
The argument filtering Pi contains the following mapping:
foldA_in_gga(
x1,
x2,
x3) =
foldA_in_gga(
x1,
x2)
[] =
[]
foldA_out_gga(
x1,
x2,
x3) =
foldA_out_gga(
x3)
a =
a
.(
x1,
x2) =
.(
x1,
x2)
b =
b
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
P is empty.
The TRS R consists of the following rules:
foldA_in_gga(T5, [], T5) → foldA_out_gga(T5, [], T5)
foldA_in_gga(a, .(b, []), c) → foldA_out_gga(a, .(b, []), c)
The argument filtering Pi contains the following mapping:
foldA_in_gga(
x1,
x2,
x3) =
foldA_in_gga(
x1,
x2)
[] =
[]
foldA_out_gga(
x1,
x2,
x3) =
foldA_out_gga(
x3)
a =
a
.(
x1,
x2) =
.(
x1,
x2)
b =
b
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
P is empty.
The TRS R consists of the following rules:
foldA_in_gga(T5, [], T5) → foldA_out_gga(T5, [], T5)
foldA_in_gga(a, .(b, []), c) → foldA_out_gga(a, .(b, []), c)
The argument filtering Pi contains the following mapping:
foldA_in_gga(
x1,
x2,
x3) =
foldA_in_gga(
x1,
x2)
[] =
[]
foldA_out_gga(
x1,
x2,
x3) =
foldA_out_gga(
x3)
a =
a
.(
x1,
x2) =
.(
x1,
x2)
b =
b
We have to consider all (P,R,Pi)-chains
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,R,Pi) chain.
(8) YES