(0) Obligation:
Clauses:
suffix(Xs, Ys) :- app(X1, Xs, Ys).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
Query: suffix(a,g)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
appA([], T12, T12).
appA(.(T20, X29), T22, .(T20, T21)) :- appA(X29, T22, T21).
suffixB(T7, T6) :- appA(X6, T7, T6).
Query: suffixB(a,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:
suffixB_in: (f,b)
appA_in: (f,f,b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
suffixB_in_ag(T7, T6) → U2_ag(T7, T6, appA_in_aag(X6, T7, T6))
appA_in_aag([], T12, T12) → appA_out_aag([], T12, T12)
appA_in_aag(.(T20, X29), T22, .(T20, T21)) → U1_aag(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
U1_aag(T20, X29, T22, T21, appA_out_aag(X29, T22, T21)) → appA_out_aag(.(T20, X29), T22, .(T20, T21))
U2_ag(T7, T6, appA_out_aag(X6, T7, T6)) → suffixB_out_ag(T7, T6)
The argument filtering Pi contains the following mapping:
suffixB_in_ag(
x1,
x2) =
suffixB_in_ag(
x2)
U2_ag(
x1,
x2,
x3) =
U2_ag(
x3)
appA_in_aag(
x1,
x2,
x3) =
appA_in_aag(
x3)
appA_out_aag(
x1,
x2,
x3) =
appA_out_aag(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_aag(
x1,
x2,
x3,
x4,
x5) =
U1_aag(
x1,
x5)
suffixB_out_ag(
x1,
x2) =
suffixB_out_ag(
x1)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
suffixB_in_ag(T7, T6) → U2_ag(T7, T6, appA_in_aag(X6, T7, T6))
appA_in_aag([], T12, T12) → appA_out_aag([], T12, T12)
appA_in_aag(.(T20, X29), T22, .(T20, T21)) → U1_aag(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
U1_aag(T20, X29, T22, T21, appA_out_aag(X29, T22, T21)) → appA_out_aag(.(T20, X29), T22, .(T20, T21))
U2_ag(T7, T6, appA_out_aag(X6, T7, T6)) → suffixB_out_ag(T7, T6)
The argument filtering Pi contains the following mapping:
suffixB_in_ag(
x1,
x2) =
suffixB_in_ag(
x2)
U2_ag(
x1,
x2,
x3) =
U2_ag(
x3)
appA_in_aag(
x1,
x2,
x3) =
appA_in_aag(
x3)
appA_out_aag(
x1,
x2,
x3) =
appA_out_aag(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_aag(
x1,
x2,
x3,
x4,
x5) =
U1_aag(
x1,
x5)
suffixB_out_ag(
x1,
x2) =
suffixB_out_ag(
x1)
(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:
SUFFIXB_IN_AG(T7, T6) → U2_AG(T7, T6, appA_in_aag(X6, T7, T6))
SUFFIXB_IN_AG(T7, T6) → APPA_IN_AAG(X6, T7, T6)
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → U1_AAG(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPA_IN_AAG(X29, T22, T21)
The TRS R consists of the following rules:
suffixB_in_ag(T7, T6) → U2_ag(T7, T6, appA_in_aag(X6, T7, T6))
appA_in_aag([], T12, T12) → appA_out_aag([], T12, T12)
appA_in_aag(.(T20, X29), T22, .(T20, T21)) → U1_aag(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
U1_aag(T20, X29, T22, T21, appA_out_aag(X29, T22, T21)) → appA_out_aag(.(T20, X29), T22, .(T20, T21))
U2_ag(T7, T6, appA_out_aag(X6, T7, T6)) → suffixB_out_ag(T7, T6)
The argument filtering Pi contains the following mapping:
suffixB_in_ag(
x1,
x2) =
suffixB_in_ag(
x2)
U2_ag(
x1,
x2,
x3) =
U2_ag(
x3)
appA_in_aag(
x1,
x2,
x3) =
appA_in_aag(
x3)
appA_out_aag(
x1,
x2,
x3) =
appA_out_aag(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_aag(
x1,
x2,
x3,
x4,
x5) =
U1_aag(
x1,
x5)
suffixB_out_ag(
x1,
x2) =
suffixB_out_ag(
x1)
SUFFIXB_IN_AG(
x1,
x2) =
SUFFIXB_IN_AG(
x2)
U2_AG(
x1,
x2,
x3) =
U2_AG(
x3)
APPA_IN_AAG(
x1,
x2,
x3) =
APPA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4,
x5) =
U1_AAG(
x1,
x5)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
SUFFIXB_IN_AG(T7, T6) → U2_AG(T7, T6, appA_in_aag(X6, T7, T6))
SUFFIXB_IN_AG(T7, T6) → APPA_IN_AAG(X6, T7, T6)
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → U1_AAG(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPA_IN_AAG(X29, T22, T21)
The TRS R consists of the following rules:
suffixB_in_ag(T7, T6) → U2_ag(T7, T6, appA_in_aag(X6, T7, T6))
appA_in_aag([], T12, T12) → appA_out_aag([], T12, T12)
appA_in_aag(.(T20, X29), T22, .(T20, T21)) → U1_aag(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
U1_aag(T20, X29, T22, T21, appA_out_aag(X29, T22, T21)) → appA_out_aag(.(T20, X29), T22, .(T20, T21))
U2_ag(T7, T6, appA_out_aag(X6, T7, T6)) → suffixB_out_ag(T7, T6)
The argument filtering Pi contains the following mapping:
suffixB_in_ag(
x1,
x2) =
suffixB_in_ag(
x2)
U2_ag(
x1,
x2,
x3) =
U2_ag(
x3)
appA_in_aag(
x1,
x2,
x3) =
appA_in_aag(
x3)
appA_out_aag(
x1,
x2,
x3) =
appA_out_aag(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_aag(
x1,
x2,
x3,
x4,
x5) =
U1_aag(
x1,
x5)
suffixB_out_ag(
x1,
x2) =
suffixB_out_ag(
x1)
SUFFIXB_IN_AG(
x1,
x2) =
SUFFIXB_IN_AG(
x2)
U2_AG(
x1,
x2,
x3) =
U2_AG(
x3)
APPA_IN_AAG(
x1,
x2,
x3) =
APPA_IN_AAG(
x3)
U1_AAG(
x1,
x2,
x3,
x4,
x5) =
U1_AAG(
x1,
x5)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPA_IN_AAG(X29, T22, T21)
The TRS R consists of the following rules:
suffixB_in_ag(T7, T6) → U2_ag(T7, T6, appA_in_aag(X6, T7, T6))
appA_in_aag([], T12, T12) → appA_out_aag([], T12, T12)
appA_in_aag(.(T20, X29), T22, .(T20, T21)) → U1_aag(T20, X29, T22, T21, appA_in_aag(X29, T22, T21))
U1_aag(T20, X29, T22, T21, appA_out_aag(X29, T22, T21)) → appA_out_aag(.(T20, X29), T22, .(T20, T21))
U2_ag(T7, T6, appA_out_aag(X6, T7, T6)) → suffixB_out_ag(T7, T6)
The argument filtering Pi contains the following mapping:
suffixB_in_ag(
x1,
x2) =
suffixB_in_ag(
x2)
U2_ag(
x1,
x2,
x3) =
U2_ag(
x3)
appA_in_aag(
x1,
x2,
x3) =
appA_in_aag(
x3)
appA_out_aag(
x1,
x2,
x3) =
appA_out_aag(
x1,
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_aag(
x1,
x2,
x3,
x4,
x5) =
U1_aag(
x1,
x5)
suffixB_out_ag(
x1,
x2) =
suffixB_out_ag(
x1)
APPA_IN_AAG(
x1,
x2,
x3) =
APPA_IN_AAG(
x3)
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:
APPA_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPA_IN_AAG(X29, T22, T21)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
APPA_IN_AAG(
x1,
x2,
x3) =
APPA_IN_AAG(
x3)
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (SOUND 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:
APPA_IN_AAG(.(T20, T21)) → APPA_IN_AAG(T21)
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:
- APPA_IN_AAG(.(T20, T21)) → APPA_IN_AAG(T21)
The graph contains the following edges 1 > 1
(14) YES