Left Termination proof of /tmp/tmpyRsAU9/der-bf.pl

Left Termination of the query pattern p_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

p(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

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

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:

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

P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, p_in_ga(d(X), DX))
P_IN_GA(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_GA(X, DDX, p_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, p_out_ga(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2) = p_in_ga(x1)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_ga(x1, x2) = p_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)
P_IN_GA(x1, x2) = P_IN_GA(x1)
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)

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:

P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(d(X)), DDX) → U5_GA(X, DDX, p_in_ga(d(X), DX))
P_IN_GA(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_GA(X, DDX, p_out_ga(d(X), DX)) → U6_GA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_GA(X, DDX, p_out_ga(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → U4_GA(X, Y, DY, DX, p_in_ga(d(e(Y)), DY))
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → U2_GA(X, Y, DX, DY, p_in_ga(d(e(Y)), DY))
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX))
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2) = p_in_ga(x1)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_ga(x1, x2) = p_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)
P_IN_GA(x1, x2) = P_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
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:

P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GA(X, Y, DY, DX, p_in_ga(d(e(X)), DX))
P_IN_GA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GA(d(e(X)), DX)
P_IN_GA(d(e(+(X, Y))), +(DX, DY)) → U1_GA(X, Y, DX, DY, p_in_ga(d(e(X)), DX))
U1_GA(X, Y, DX, DY, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)
U3_GA(X, Y, DY, DX, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)), DY)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2) = p_in_ga(x1)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_ga(x1, x2) = p_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)
P_IN_GA(x1, x2) = P_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
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:

P_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, p_in_ga(d(e(X))))
U3_GA(X, Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
P_IN_GA(d(e(*(X, Y)))) → P_IN_GA(d(e(X)))
U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
P_IN_GA(d(e(+(X, Y)))) → P_IN_GA(d(e(X)))
P_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, p_in_ga(d(e(X))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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.

P_IN_GA(d(e(*(X, Y)))) → U3_GA(X, Y, p_in_ga(d(e(X))))
P_IN_GA(d(e(*(X, Y)))) → P_IN_GA(d(e(X)))

The remaining pairs can at least be oriented weakly.

U3_GA(X, Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
P_IN_GA(d(e(+(X, Y)))) → P_IN_GA(d(e(X)))
P_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, p_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(P_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(e(x₁)) = x₁
POL(p_in_ga(x₁)) = 0
POL(p_out_ga(x₁)) = 0
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:

U3_GA(X, Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
P_IN_GA(d(e(+(X, Y)))) → P_IN_GA(d(e(X)))
P_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, p_in_ga(d(e(X))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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:

U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))
P_IN_GA(d(e(+(X, Y)))) → P_IN_GA(d(e(X)))
P_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, p_in_ga(d(e(X))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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.

P_IN_GA(d(e(+(X, Y)))) → P_IN_GA(d(e(X)))
P_IN_GA(d(e(+(X, Y)))) → U1_GA(Y, p_in_ga(d(e(X))))

The remaining pairs can at least be oriented weakly.

U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))

Used ordering: Polynomial interpretation [25]:

POL(*(x₁, x₂)) = 0
POL(+(x₁, x₂)) = 1 + x₁ + x₂
POL(0) = 0
POL(1) = 0
POL(P_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)) = 0
POL(U4_ga(x₁, x₂, x₃, x₄)) = 0
POL(const(x₁)) = 0
POL(d(x₁)) = x₁
POL(e(x₁)) = x₁
POL(p_in_ga(x₁)) = 0
POL(p_out_ga(x₁)) = 0
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

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

              ↳ PiDP

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

U1_GA(Y, p_out_ga(DX)) → P_IN_GA(d(e(Y)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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:

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

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
p_in_ga(x1, x2) = p_in_ga(x1)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_ga(x1, x2) = p_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)
P_IN_GA(x1, x2) = P_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:

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

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

P_IN_GA(d(d(X))) → P_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:

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