(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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

suffixA_in_ag(T7, T6) → U1_ag(T7, T6, appB_in_aag(X6, T7, T6))
appB_in_aag([], T12, T12) → appB_out_aag([], T12, T12)
appB_in_aag(.(T20, X29), T22, .(T20, T21)) → U2_aag(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
U2_aag(T20, X29, T22, T21, appB_out_aag(X29, T22, T21)) → appB_out_aag(.(T20, X29), T22, .(T20, T21))
U1_ag(T7, T6, appB_out_aag(X6, T7, T6)) → suffixA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
suffixA_in_ag(x1, x2)  =  suffixA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
appB_in_aag(x1, x2, x3)  =  appB_in_aag(x3)
appB_out_aag(x1, x2, x3)  =  appB_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
suffixA_out_ag(x1, x2)  =  suffixA_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:

SUFFIXA_IN_AG(T7, T6) → U1_AG(T7, T6, appB_in_aag(X6, T7, T6))
SUFFIXA_IN_AG(T7, T6) → APPB_IN_AAG(X6, T7, T6)
APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → U2_AAG(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPB_IN_AAG(X29, T22, T21)

The TRS R consists of the following rules:

suffixA_in_ag(T7, T6) → U1_ag(T7, T6, appB_in_aag(X6, T7, T6))
appB_in_aag([], T12, T12) → appB_out_aag([], T12, T12)
appB_in_aag(.(T20, X29), T22, .(T20, T21)) → U2_aag(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
U2_aag(T20, X29, T22, T21, appB_out_aag(X29, T22, T21)) → appB_out_aag(.(T20, X29), T22, .(T20, T21))
U1_ag(T7, T6, appB_out_aag(X6, T7, T6)) → suffixA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
suffixA_in_ag(x1, x2)  =  suffixA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
appB_in_aag(x1, x2, x3)  =  appB_in_aag(x3)
appB_out_aag(x1, x2, x3)  =  appB_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
suffixA_out_ag(x1, x2)  =  suffixA_out_ag(x1, x2)
SUFFIXA_IN_AG(x1, x2)  =  SUFFIXA_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
APPB_IN_AAG(x1, x2, x3)  =  APPB_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, x4, x5)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUFFIXA_IN_AG(T7, T6) → U1_AG(T7, T6, appB_in_aag(X6, T7, T6))
SUFFIXA_IN_AG(T7, T6) → APPB_IN_AAG(X6, T7, T6)
APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → U2_AAG(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPB_IN_AAG(X29, T22, T21)

The TRS R consists of the following rules:

suffixA_in_ag(T7, T6) → U1_ag(T7, T6, appB_in_aag(X6, T7, T6))
appB_in_aag([], T12, T12) → appB_out_aag([], T12, T12)
appB_in_aag(.(T20, X29), T22, .(T20, T21)) → U2_aag(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
U2_aag(T20, X29, T22, T21, appB_out_aag(X29, T22, T21)) → appB_out_aag(.(T20, X29), T22, .(T20, T21))
U1_ag(T7, T6, appB_out_aag(X6, T7, T6)) → suffixA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
suffixA_in_ag(x1, x2)  =  suffixA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
appB_in_aag(x1, x2, x3)  =  appB_in_aag(x3)
appB_out_aag(x1, x2, x3)  =  appB_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
suffixA_out_ag(x1, x2)  =  suffixA_out_ag(x1, x2)
SUFFIXA_IN_AG(x1, x2)  =  SUFFIXA_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
APPB_IN_AAG(x1, x2, x3)  =  APPB_IN_AAG(x3)
U2_AAG(x1, x2, x3, x4, x5)  =  U2_AAG(x1, 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 3 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPB_IN_AAG(X29, T22, T21)

The TRS R consists of the following rules:

suffixA_in_ag(T7, T6) → U1_ag(T7, T6, appB_in_aag(X6, T7, T6))
appB_in_aag([], T12, T12) → appB_out_aag([], T12, T12)
appB_in_aag(.(T20, X29), T22, .(T20, T21)) → U2_aag(T20, X29, T22, T21, appB_in_aag(X29, T22, T21))
U2_aag(T20, X29, T22, T21, appB_out_aag(X29, T22, T21)) → appB_out_aag(.(T20, X29), T22, .(T20, T21))
U1_ag(T7, T6, appB_out_aag(X6, T7, T6)) → suffixA_out_ag(T7, T6)

The argument filtering Pi contains the following mapping:
suffixA_in_ag(x1, x2)  =  suffixA_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
appB_in_aag(x1, x2, x3)  =  appB_in_aag(x3)
appB_out_aag(x1, x2, x3)  =  appB_out_aag(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_aag(x1, x2, x3, x4, x5)  =  U2_aag(x1, x4, x5)
suffixA_out_ag(x1, x2)  =  suffixA_out_ag(x1, x2)
APPB_IN_AAG(x1, x2, x3)  =  APPB_IN_AAG(x3)

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:

APPB_IN_AAG(.(T20, X29), T22, .(T20, T21)) → APPB_IN_AAG(X29, T22, T21)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPB_IN_AAG(x1, x2, x3)  =  APPB_IN_AAG(x3)

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:

APPB_IN_AAG(.(T20, T21)) → APPB_IN_AAG(T21)

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:

  • APPB_IN_AAG(.(T20, T21)) → APPB_IN_AAG(T21)
    The graph contains the following edges 1 > 1

(12) YES