(0) Obligation:

Clauses:

fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(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).
eq(X, X).

Queries:

fold(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

fold1(T7, [], T7).
fold1(a, .(b, []), c).

Queries:

fold1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. 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:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_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:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_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:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_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:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_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) TRUE