(0) Obligation:
Clauses:
list([]).
list(.(X1, Ts)) :- list(Ts).
Query: list(g)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
listA([]).
listA(.(T4, [])).
listA(.(T4, .(T10, T11))) :- listA(T11).
Query: listA(g)
(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:
listA_in: (b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))
The argument filtering Pi contains the following mapping:
listA_in_g(
x1) =
listA_in_g(
x1)
[] =
[]
listA_out_g(
x1) =
listA_out_g
.(
x1,
x2) =
.(
x1,
x2)
U1_g(
x1,
x2,
x3,
x4) =
U1_g(
x4)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))
The argument filtering Pi contains the following mapping:
listA_in_g(
x1) =
listA_in_g(
x1)
[] =
[]
listA_out_g(
x1) =
listA_out_g
.(
x1,
x2) =
.(
x1,
x2)
U1_g(
x1,
x2,
x3,
x4) =
U1_g(
x4)
(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:
LISTA_IN_G(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, listA_in_g(T11))
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
The TRS R consists of the following rules:
listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))
The argument filtering Pi contains the following mapping:
listA_in_g(
x1) =
listA_in_g(
x1)
[] =
[]
listA_out_g(
x1) =
listA_out_g
.(
x1,
x2) =
.(
x1,
x2)
U1_g(
x1,
x2,
x3,
x4) =
U1_g(
x4)
LISTA_IN_G(
x1) =
LISTA_IN_G(
x1)
U1_G(
x1,
x2,
x3,
x4) =
U1_G(
x4)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
LISTA_IN_G(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, listA_in_g(T11))
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
The TRS R consists of the following rules:
listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))
The argument filtering Pi contains the following mapping:
listA_in_g(
x1) =
listA_in_g(
x1)
[] =
[]
listA_out_g(
x1) =
listA_out_g
.(
x1,
x2) =
.(
x1,
x2)
U1_g(
x1,
x2,
x3,
x4) =
U1_g(
x4)
LISTA_IN_G(
x1) =
LISTA_IN_G(
x1)
U1_G(
x1,
x2,
x3,
x4) =
U1_G(
x4)
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:
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
The TRS R consists of the following rules:
listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))
The argument filtering Pi contains the following mapping:
listA_in_g(
x1) =
listA_in_g(
x1)
[] =
[]
listA_out_g(
x1) =
listA_out_g
.(
x1,
x2) =
.(
x1,
x2)
U1_g(
x1,
x2,
x3,
x4) =
U1_g(
x4)
LISTA_IN_G(
x1) =
LISTA_IN_G(
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:
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (EQUIVALENT 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:
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
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:
- LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
The graph contains the following edges 1 > 1
(14) YES