(0) Obligation:
Clauses:
member(X, .(Y, Xs)) :- member(X, Xs).
member(X, .(X, Xs)).
Query: member(a,g)
(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)
Transformed Prolog program to (Pi-)TRS.
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
memberA_in_ag(T25, .(T7, .(T23, T24))) → U1_ag(T25, T7, T23, T24, memberA_in_ag(T25, T24))
memberA_in_ag(T37, .(T7, .(T37, T38))) → memberA_out_ag(T37, .(T7, .(T37, T38)))
memberA_in_ag(T43, .(T43, T44)) → memberA_out_ag(T43, .(T43, T44))
U1_ag(T25, T7, T23, T24, memberA_out_ag(T25, T24)) → memberA_out_ag(T25, .(T7, .(T23, T24)))
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ag(
x1,
x2,
x3,
x4,
x5) =
U1_ag(
x2,
x3,
x4,
x5)
memberA_out_ag(
x1,
x2) =
memberA_out_ag(
x1,
x2)
(3) 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:
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → U1_AG(T25, T7, T23, T24, memberA_in_ag(T25, T24))
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → MEMBERA_IN_AG(T25, T24)
The TRS R consists of the following rules:
memberA_in_ag(T25, .(T7, .(T23, T24))) → U1_ag(T25, T7, T23, T24, memberA_in_ag(T25, T24))
memberA_in_ag(T37, .(T7, .(T37, T38))) → memberA_out_ag(T37, .(T7, .(T37, T38)))
memberA_in_ag(T43, .(T43, T44)) → memberA_out_ag(T43, .(T43, T44))
U1_ag(T25, T7, T23, T24, memberA_out_ag(T25, T24)) → memberA_out_ag(T25, .(T7, .(T23, T24)))
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ag(
x1,
x2,
x3,
x4,
x5) =
U1_ag(
x2,
x3,
x4,
x5)
memberA_out_ag(
x1,
x2) =
memberA_out_ag(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3,
x4,
x5) =
U1_AG(
x2,
x3,
x4,
x5)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → U1_AG(T25, T7, T23, T24, memberA_in_ag(T25, T24))
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → MEMBERA_IN_AG(T25, T24)
The TRS R consists of the following rules:
memberA_in_ag(T25, .(T7, .(T23, T24))) → U1_ag(T25, T7, T23, T24, memberA_in_ag(T25, T24))
memberA_in_ag(T37, .(T7, .(T37, T38))) → memberA_out_ag(T37, .(T7, .(T37, T38)))
memberA_in_ag(T43, .(T43, T44)) → memberA_out_ag(T43, .(T43, T44))
U1_ag(T25, T7, T23, T24, memberA_out_ag(T25, T24)) → memberA_out_ag(T25, .(T7, .(T23, T24)))
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ag(
x1,
x2,
x3,
x4,
x5) =
U1_ag(
x2,
x3,
x4,
x5)
memberA_out_ag(
x1,
x2) =
memberA_out_ag(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
U1_AG(
x1,
x2,
x3,
x4,
x5) =
U1_AG(
x2,
x3,
x4,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → MEMBERA_IN_AG(T25, T24)
The TRS R consists of the following rules:
memberA_in_ag(T25, .(T7, .(T23, T24))) → U1_ag(T25, T7, T23, T24, memberA_in_ag(T25, T24))
memberA_in_ag(T37, .(T7, .(T37, T38))) → memberA_out_ag(T37, .(T7, .(T37, T38)))
memberA_in_ag(T43, .(T43, T44)) → memberA_out_ag(T43, .(T43, T44))
U1_ag(T25, T7, T23, T24, memberA_out_ag(T25, T24)) → memberA_out_ag(T25, .(T7, .(T23, T24)))
The argument filtering Pi contains the following mapping:
memberA_in_ag(
x1,
x2) =
memberA_in_ag(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ag(
x1,
x2,
x3,
x4,
x5) =
U1_ag(
x2,
x3,
x4,
x5)
memberA_out_ag(
x1,
x2) =
memberA_out_ag(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
We have to consider all (P,R,Pi)-chains
(7) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(T25, .(T7, .(T23, T24))) → MEMBERA_IN_AG(T25, T24)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
MEMBERA_IN_AG(
x1,
x2) =
MEMBERA_IN_AG(
x2)
We have to consider all (P,R,Pi)-chains
(9) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MEMBERA_IN_AG(.(T7, .(T23, T24))) → MEMBERA_IN_AG(T24)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) 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:
- MEMBERA_IN_AG(.(T7, .(T23, T24))) → MEMBERA_IN_AG(T24)
The graph contains the following edges 1 > 1
(12) YES