Left Termination of the query pattern der_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPOrderProof (⇔)
        ↳13 QDP
        ↳14 DependencyGraphProof (⇔)
        ↳15 QDP
      ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
  ↳19 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 AND
      ↳25 PiDP
        ↳26 UsableRulesProof (⇔)
        ↳27 PiDP
        ↳28 PiDPToQDPProof (⇐)
        ↳29 QDP
        ↳30 QDPOrderProof (⇔)
        ↳31 QDP
        ↳32 DependencyGraphProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP
        ↳39 QDPSizeChangeProof (⇔)
        ↳40 TRUE

(0) Obligation:

Clauses:

der(d(e(t)), const(1)).
der(d(e(const(A))), const(0)).
der(d(e(+(X, Y))), +(DX, DY)) :- ','(der(d(e(X)), DX), der(d(e(Y)), DY)).
der(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) :- ','(der(d(e(X)), DX), der(d(e(Y)), DY)).
der(d(d(X)), DDX) :- ','(der(d(X), DX), der(d(e(DX)), DDX)).

Queries:

der(g,a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
der_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)

(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:

DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, der_in_ga(d(X), DX))
DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)
U5_GA(X, DDX, der_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, der_out_ga(d(X), DX)) → DER_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)

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

(4) Obligation:

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

DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, der_in_ga(d(X), DX))
DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)
U5_GA(X, DDX, der_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, der_out_ga(d(X), DX)) → DER_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

U1_GA(Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, der_in_ga(d(e(X))))
DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
DER_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, der_in_ga(d(e(X))))
U3_GA(X, Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))

The TRS R consists of the following rules:

der_in_ga(d(e(t))) → der_out_ga(const(1))
der_in_ga(d(e(const(A)))) → der_out_ga(const(0))
der_in_ga(d(e(+(X, Y)))) → U1_ga(Y, der_in_ga(d(e(X))))
der_in_ga(d(e(*(X, Y)))) → U3_ga(X, Y, der_in_ga(d(e(X))))
U1_ga(Y, der_out_ga(DX)) → U2_ga(DX, der_in_ga(d(e(Y))))
U3_ga(X, Y, der_out_ga(DX)) → U4_ga(X, Y, DX, der_in_ga(d(e(Y))))
U2_ga(DX, der_out_ga(DY)) → der_out_ga(+(DX, DY))
U4_ga(X, Y, DX, der_out_ga(DY)) → der_out_ga(+(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

der_in_ga(x0)
U1_ga(x0, x1)
U3_ga(x0, x1, x2)
U2_ga(x0, x1)
U4_ga(x0, x1, x2, x3)

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DER_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, der_in_ga(d(e(X))))
DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(*(x1, x2)) = x1 + x2   
POL(+(x1, x2)) = 1 + x1 + x2   
POL(0) = 0   
POL(1) = 0   
POL(DER_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2)) = x1   
POL(U1_ga(x1, x2)) = 0   
POL(U2_ga(x1, x2)) = 0   
POL(U3_GA(x1, x2, x3)) = x2   
POL(U3_ga(x1, x2, x3)) = 0   
POL(U4_ga(x1, x2, x3, x4)) = 0   
POL(const(x1)) = 0   
POL(d(x1)) = x1   
POL(der_in_ga(x1)) = 0   
POL(der_out_ga(x1)) = 0   
POL(e(x1)) = x1   
POL(t) = 0   

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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

U1_GA(Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, der_in_ga(d(e(X))))
U3_GA(X, Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))

The TRS R consists of the following rules:

der_in_ga(d(e(t))) → der_out_ga(const(1))
der_in_ga(d(e(const(A)))) → der_out_ga(const(0))
der_in_ga(d(e(+(X, Y)))) → U1_ga(Y, der_in_ga(d(e(X))))
der_in_ga(d(e(*(X, Y)))) → U3_ga(X, Y, der_in_ga(d(e(X))))
U1_ga(Y, der_out_ga(DX)) → U2_ga(DX, der_in_ga(d(e(Y))))
U3_ga(X, Y, der_out_ga(DX)) → U4_ga(X, Y, DX, der_in_ga(d(e(Y))))
U2_ga(DX, der_out_ga(DY)) → der_out_ga(+(DX, DY))
U4_ga(X, Y, DX, der_out_ga(DY)) → der_out_ga(+(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

der_in_ga(x0)
U1_ga(x0, x1)
U3_ga(x0, x1, x2)
U2_ga(x0, x1)
U4_ga(x0, x1, x2, x3)

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

(14) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(15) Obligation:

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

U3_GA(X, Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, der_in_ga(d(e(X))))
DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))

The TRS R consists of the following rules:

der_in_ga(d(e(t))) → der_out_ga(const(1))
der_in_ga(d(e(const(A)))) → der_out_ga(const(0))
der_in_ga(d(e(+(X, Y)))) → U1_ga(Y, der_in_ga(d(e(X))))
der_in_ga(d(e(*(X, Y)))) → U3_ga(X, Y, der_in_ga(d(e(X))))
U1_ga(Y, der_out_ga(DX)) → U2_ga(DX, der_in_ga(d(e(Y))))
U3_ga(X, Y, der_out_ga(DX)) → U4_ga(X, Y, DX, der_in_ga(d(e(Y))))
U2_ga(DX, der_out_ga(DY)) → der_out_ga(+(DX, DY))
U4_ga(X, Y, DX, der_out_ga(DY)) → der_out_ga(+(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

der_in_ga(x0)
U1_ga(x0, x1)
U3_ga(x0, x1, x2)
U2_ga(x0, x1)
U4_ga(x0, x1, x2, x3)

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

(16) Obligation:

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

DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(18) Obligation:

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

DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)

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

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

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
der_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)

(21) 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:

DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, der_in_ga(d(X), DX))
DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)
U5_GA(X, DDX, der_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, der_out_ga(d(X), DX)) → DER_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(22) Obligation:

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

DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, der_in_ga(d(X), DX))
DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)
U5_GA(X, DDX, der_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, der_out_ga(d(X), DX)) → DER_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x1, x4)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)

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

(26) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(27) Obligation:

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

U1_GA(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U3_GA(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)), DY)
DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)

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

(28) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(29) Obligation:

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

U1_GA(X, Y, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(+(X, Y)))) → U1_GA(X, Y, der_in_ga(d(e(X))))
DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
DER_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, der_in_ga(d(e(X))))
U3_GA(X, Y, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))

The TRS R consists of the following rules:

der_in_ga(d(e(t))) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A)))) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y)))) → U1_ga(X, Y, der_in_ga(d(e(X))))
der_in_ga(d(e(*(X, Y)))) → U3_ga(X, Y, der_in_ga(d(e(X))))
U1_ga(X, Y, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, der_in_ga(d(e(Y))))
U3_ga(X, Y, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DX, der_in_ga(d(e(Y))))
U2_ga(X, Y, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))
U4_ga(X, Y, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

der_in_ga(x0)
U1_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U2_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DER_IN_GA(d(e(+(X, Y)))) → U1_GA(X, Y, der_in_ga(d(e(X))))
DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
U3_GA(X, Y, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(*(x1, x2)) = 1 + x1 + x2   
POL(+(x1, x2)) = 1 + x1 + x2   
POL(0) = 0   
POL(1) = 0   
POL(DER_IN_GA(x1)) = x1   
POL(U1_GA(x1, x2, x3)) = x2   
POL(U1_ga(x1, x2, x3)) = 0   
POL(U2_ga(x1, x2, x3, x4)) = 0   
POL(U3_GA(x1, x2, x3)) = 1 + x2   
POL(U3_ga(x1, x2, x3)) = 0   
POL(U4_ga(x1, x2, x3, x4)) = 0   
POL(const(x1)) = 0   
POL(d(x1)) = x1   
POL(der_in_ga(x1)) = 0   
POL(der_out_ga(x1, x2)) = 0   
POL(e(x1)) = x1   
POL(t) = 0   

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

U1_GA(X, Y, der_out_ga(d(e(X)), DX)) → DER_IN_GA(d(e(Y)))
DER_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, der_in_ga(d(e(X))))

The TRS R consists of the following rules:

der_in_ga(d(e(t))) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A)))) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y)))) → U1_ga(X, Y, der_in_ga(d(e(X))))
der_in_ga(d(e(*(X, Y)))) → U3_ga(X, Y, der_in_ga(d(e(X))))
U1_ga(X, Y, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, der_in_ga(d(e(Y))))
U3_ga(X, Y, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DX, der_in_ga(d(e(Y))))
U2_ga(X, Y, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))
U4_ga(X, Y, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

der_in_ga(x0)
U1_ga(x0, x1, x2)
U3_ga(x0, x1, x2)
U2_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)

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

(32) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(33) TRUE

(34) Obligation:

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

DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)

The TRS R consists of the following rules:

der_in_ga(d(e(t)), const(1)) → der_out_ga(d(e(t)), const(1))
der_in_ga(d(e(const(A))), const(0)) → der_out_ga(d(e(const(A))), const(0))
der_in_ga(d(e(+(X, Y))), +(DX, DY)) → U1_ga(X, Y, DX, DY, der_in_ga(d(e(X)), DX))
der_in_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ga(X, Y, DY, DX, der_in_ga(d(e(X)), DX))
der_in_ga(d(d(X)), DDX) → U5_ga(X, DDX, der_in_ga(d(X), DX))
U5_ga(X, DDX, der_out_ga(d(X), DX)) → U6_ga(X, DDX, DX, der_in_ga(d(e(DX)), DDX))
U6_ga(X, DDX, DX, der_out_ga(d(e(DX)), DDX)) → der_out_ga(d(d(X)), DDX)
U3_ga(X, Y, DY, DX, der_out_ga(d(e(X)), DX)) → U4_ga(X, Y, DY, DX, der_in_ga(d(e(Y)), DY))
U4_ga(X, Y, DY, DX, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_ga(X, Y, DX, DY, der_out_ga(d(e(X)), DX)) → U2_ga(X, Y, DX, DY, der_in_ga(d(e(Y)), DY))
U2_ga(X, Y, DX, DY, der_out_ga(d(e(Y)), DY)) → der_out_ga(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
der_in_ga(x1, x2)  =  der_in_ga(x1)
d(x1)  =  d(x1)
e(x1)  =  e(x1)
t  =  t
der_out_ga(x1, x2)  =  der_out_ga(x1, x2)
const(x1)  =  const(x1)
+(x1, x2)  =  +(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
*(x1, x2)  =  *(x1, x2)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x5)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U6_ga(x1, x2, x3, x4)  =  U6_ga(x1, x4)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x1, x2, x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

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

(35) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(36) Obligation:

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

DER_IN_GA(d(d(X)), DDX) → DER_IN_GA(d(X), DX)

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

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

(37) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(38) Obligation:

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

DER_IN_GA(d(d(X))) → DER_IN_GA(d(X))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(39) 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:

  • DER_IN_GA(d(d(X))) → DER_IN_GA(d(X))
    The graph contains the following edges 1 > 1

(40) TRUE