(0) Obligation:
Clauses:fold(X, .(Y, Ys), Z) :- ','(myop(X, Y, V), fold(V, Ys, Z)).
fold(X, [], X).
myop(a, b, c).
Queries:
fold(g,g,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:fold1(a, .(b, []), c).
fold1(T37, [], T37).
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(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)
fold1_in_gga(T37, [], T37) → fold1_out_gga(T37, [], T37)
The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3) = fold1_in_gga(x1, x2)
a = a
.(x1, x2) = .(x1, x2)
b = b
[] = []
fold1_out_gga(x1, x2, x3) = fold1_out_gga(x3)
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(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)
fold1_in_gga(T37, [], T37) → fold1_out_gga(T37, [], T37)
The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3) = fold1_in_gga(x1, x2)
a = a
.(x1, x2) = .(x1, x2)
b = b
[] = []
fold1_out_gga(x1, x2, x3) = fold1_out_gga(x3)
(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(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)
fold1_in_gga(T37, [], T37) → fold1_out_gga(T37, [], T37)
The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3) = fold1_in_gga(x1, x2)
a = a
.(x1, x2) = .(x1, x2)
b = b
[] = []
fold1_out_gga(x1, x2, x3) = fold1_out_gga(x3)
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(a, .(b, []), c) → fold1_out_gga(a, .(b, []), c)
fold1_in_gga(T37, [], T37) → fold1_out_gga(T37, [], T37)
The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3) = fold1_in_gga(x1, x2)
a = a
.(x1, x2) = .(x1, x2)
b = b
[] = []
fold1_out_gga(x1, x2, x3) = fold1_out_gga(x3)
We have to consider all (P,R,Pi)-chains