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



↳ Prolog
  ↳ PrologToPiTRSProof

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).

We use the technique of [30]. 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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)


Using Dependency Pairs [1,30] 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)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x1, x2, x4, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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

DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
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))), +(*(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)

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)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

DER_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → DER_IN_GA(d(e(X)), DX)
DER_IN_GA(d(e(+(X, Y))), +(DX, DY)) → DER_IN_GA(d(e(X)), DX)
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))), +(*(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)

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)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x2, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x5)
DER_IN_GA(x1, x2)  =  DER_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
              ↳ PiDP

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

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

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


DER_IN_GA(d(e(*(X, Y)))) → DER_IN_GA(d(e(X)))
U3_GA(X, Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
The remaining pairs can at least be oriented weakly.

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))))
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))))
Used ordering: Polynomial interpretation [25]:

POL(*(x1, x2)) = 1 + x1 + x2   
POL(+(x1, x2)) = 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)) = 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)) = 0   
POL(e(x1)) = x1   
POL(t) = 0   

The following usable rules [17] were oriented: none



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
              ↳ PiDP

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

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))))
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))))

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
              ↳ PiDP

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

DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
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))))

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


DER_IN_GA(d(e(+(X, Y)))) → DER_IN_GA(d(e(X)))
DER_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, der_in_ga(d(e(X))))
The remaining pairs can at least be oriented weakly.

U1_GA(Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( d(x1) ) =
/0
0/
+
/10
00/
·x1

M( 0 ) =
/0
0/

M( U2_ga(x1, x2) ) =
/0
0/
+
/00
00/
·x1+
/00
00/
·x2

M( e(x1) ) =
/0
0/
+
/11
00/
·x1

M( *(x1, x2) ) =
/0
0/
+
/00
00/
·x1+
/10
10/
·x2

M( U1_ga(x1, x2) ) =
/0
0/
+
/00
00/
·x1+
/00
00/
·x2

M( U4_ga(x1, ..., x4) ) =
/0
0/
+
/00
00/
·x1+
/00
00/
·x2+
/00
00/
·x3+
/00
00/
·x4

M( const(x1) ) =
/0
0/
+
/00
00/
·x1

M( +(x1, x2) ) =
/1
1/
+
/00
11/
·x1+
/11
11/
·x2

M( t ) =
/0
0/

M( 1 ) =
/0
0/

M( U3_ga(x1, ..., x3) ) =
/0
0/
+
/00
00/
·x1+
/00
00/
·x2+
/00
00/
·x3

M( der_out_ga(x1) ) =
/0
0/
+
/00
00/
·x1

M( der_in_ga(x1) ) =
/0
0/
+
/00
00/
·x1

Tuple symbols:
M( DER_IN_GA(x1) ) = 0+
[1,0]
·x1

M( U1_GA(x1, x2) ) = 0+
[1,1]
·x1+
[0,0]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented: none



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ DependencyGraphProof
              ↳ PiDP

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

U1_GA(Y, der_out_ga(DX)) → DER_IN_GA(d(e(Y)))

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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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.
By using the subterm criterion [20] together with the size-change analysis [32] 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: