Left Termination proof of /tmp/tmplt9JPz/der-fb.pl

Left Termination of the query pattern p_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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(a,g).

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

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(d(X)), DDX) → U5_GG(X, DDX, p_in_ga(d(X), DX))
P_IN_GG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
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)
U5_GG(X, DDX, p_out_ga(d(X), DX)) → U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_GG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_AG(d(d(X)), DDX) → U5_AG(X, DDX, p_in_aa(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_AA(d(X), DX)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(d(X)), DDX) → U5_AA(X, DDX, p_in_aa(d(X), DX))
P_IN_AA(d(d(X)), DDX) → P_IN_AA(d(X), DX)
U5_AA(X, DDX, p_out_aa(d(X), DX)) → U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_AA(X, DDX, p_out_aa(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U5_AG(X, DDX, p_out_aa(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_aa(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
p_in_ag(x1, x2) = p_in_ag(x2)
const(x1) = const
1 = 1
p_out_ag(x1, x2) = p_out_ag(x1)
0 = 0
+(x1, x2) = +(x1, x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x4, x5)
*(x1, x2) = *(x1, x2)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x5)
p_in_gg(x1, x2) = p_in_gg(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_gg(x1, x2) = p_out_gg
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5)
U5_gg(x1, x2, x3) = U5_gg(x2, x3)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
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)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x5)
U5_ag(x1, x2, x3) = U5_ag(x2, x3)
p_in_aa(x1, x2) = p_in_aa
p_out_aa(x1, x2) = p_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5)
U5_aa(x1, x2, x3) = U5_aa(x3)
U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5)
U6_ag(x1, x2, x3, x4) = U6_ag(x1, x4)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5)
U5_GG(x1, x2, x3) = U5_GG(x2, x3)
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)
U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x3, x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x5)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x4, x5)
U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5)
U6_GA(x1, x2, x3, x4) = U6_GA(x4)
U6_GG(x1, x2, x3, x4) = U6_GG(x4)
U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4)
U5_AG(x1, x2, x3) = U5_AG(x2, x3)
P_IN_GG(x1, x2) = P_IN_GG(x1, x2)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, x5)
P_IN_AG(x1, x2) = P_IN_AG(x2)
U5_AA(x1, x2, x3) = U5_AA(x3)
U6_AG(x1, x2, x3, x4) = U6_AG(x1, x4)
P_IN_AA(x1, x2) = P_IN_AA
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x2, x5)
U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(d(X)), DDX) → U5_GG(X, DDX, p_in_ga(d(X), DX))
P_IN_GG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
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)
U5_GG(X, DDX, p_out_ga(d(X), DX)) → U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_GG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_AG(d(d(X)), DDX) → U5_AG(X, DDX, p_in_aa(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_AA(d(X), DX)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(d(X)), DDX) → U5_AA(X, DDX, p_in_aa(d(X), DX))
P_IN_AA(d(d(X)), DDX) → P_IN_AA(d(X), DX)
U5_AA(X, DDX, p_out_aa(d(X), DX)) → U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_AA(X, DDX, p_out_aa(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U5_AG(X, DDX, p_out_aa(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_aa(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)

The TRS R consists of the following rules:

p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The argument filtering Pi contains the following mapping:
const(x1) = const
1 = 1
0 = 0
+(x1, x2) = +(x1, x2)
*(x1, x2) = *(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_in_aa(x1, x2) = p_in_aa
p_out_aa(x1, x2) = p_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
P_IN_AA(x1, x2) = P_IN_AA

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

                        ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U1_AA(p_in_aa)
P_IN_AA → U3_AA(p_in_aa)

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_AA → U3_AA(p_in_aa) at position [0] we obtained the following new rules:

P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U1_AA(p_in_aa)
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_AA → U1_AA(p_in_aa) at position [0] we obtained the following new rules:

P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

s = P_IN_AA evaluates to t =P_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_AA to P_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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:
const(x1) = const
1 = 1
0 = 0
+(x1, x2) = +(x1, x2)
*(x1, x2) = *(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x2, x5)
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)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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)
p_in_ga(d(e(const))) → p_out_ga(const)
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.

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)))

The remaining pairs can at least be oriented weakly.

P_IN_GA(d(e(*(X, Y)))) → U3_GA(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))))

Used ordering: Polynomial interpretation [25]:

POL(*(x₁, x₂)) = 1 + x₁ + x₂
POL(+(x₁, x₂)) = x₁ + x₂
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)) = 1 + x₂
POL(U3_ga(x₁, x₂, x₃)) = 0
POL(U4_ga(x₁, x₂, x₃, x₄)) = 0
POL(const) = 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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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))))
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)
p_in_ga(d(e(const))) → p_out_ga(const)
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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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)
p_in_ga(d(e(const))) → p_out_ga(const)
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.

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 remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U1_GA(x1, x2) = U1_GA(x1, x2)
p_out_ga(x1) = p_out_ga
P_IN_GA(x1) = P_IN_GA(x1)
d(x1) = x1
e(x1) = x1
+(x1, x2) = +(x1, x2)
p_in_ga(x1) = x1
U1_ga(x1, x2) = U1_ga(x1, x2)
U2_ga(x1, x2) = U2_ga
const = const
*(x1, x2) = x2
U3_ga(x1, x2, x3) = x2
U4_ga(x1, x2, x3, x4) = x2
t = t

Recursive path order with status [2].
Quasi-Precedence:

[+₂, U1_{ga_2]} > U1_GA₂ > P_{IN_GA₁} > [p_{out_ga}, U2_ga, const]
t > [p_{out_ga}, U2_ga, const]

Status:

const: multiset
U1_ga₂: [1,2]
U2_ga: multiset
+₂: [2,1]
t: multiset
U1_GA₂: multiset
P_{IN_GA₁}: multiset
p_{out_ga}: multiset

The following usable rules [17] were oriented:

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ PisEmptyProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

p_in_ga(d(e(t))) → p_out_ga(const)
p_in_ga(d(e(const))) → p_out_ga(const)
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 TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

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

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

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_AA → P_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

P_IN_AA → P_IN_AA

The TRS R consists of the following rules:none

s = P_IN_AA evaluates to t =P_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_AA to P_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The argument filtering Pi contains the following mapping:
const(x1) = const
1 = 1
0 = 0
+(x1, x2) = +(x1, x2)
*(x1, x2) = *(x1, x2)
p_in_gg(x1, x2) = p_in_gg(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_gg(x1, x2) = p_out_gg
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x2, x4, x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x2, x3, x5)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x2, x4, x5)
P_IN_GG(x1, x2) = P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x2, x3, 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

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(Y, DY, p_in_gg(d(e(X)), DX))
U1_GG(Y, DY, p_out_gg) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
U3_GG(Y, DY, p_out_gg) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(Y, DY, p_in_gg(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_gg(d(e(t)), const) → p_out_gg
p_in_gg(d(e(const)), const) → p_out_gg
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(Y, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(Y, DY, p_in_gg(d(e(X)), DX))
U1_gg(Y, DY, p_out_gg) → U2_gg(p_in_gg(d(e(Y)), DY))
U3_gg(Y, DY, p_out_gg) → U4_gg(p_in_gg(d(e(Y)), DY))
U2_gg(p_out_gg) → p_out_gg
U4_gg(p_out_gg) → p_out_gg

The set Q consists of the following terms:

p_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gg(x0)
U4_gg(x0)

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_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(Y, DY, p_in_gg(d(e(X)), DX))
The graph contains the following edges 1 > 1, 2 > 2
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(Y, DY, p_in_gg(d(e(X)), DX))
The graph contains the following edges 1 > 1, 2 > 1, 2 > 2
U3_GG(Y, DY, p_out_gg) → P_IN_GG(d(e(Y)), DY)
The graph contains the following edges 2 >= 2
U1_GG(Y, DY, p_out_gg) → P_IN_GG(d(e(Y)), DY)
The graph contains the following edges 2 >= 2
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
The graph contains the following edges 2 > 2
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
The graph contains the following edges 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

U1_AG(DY, p_out_ag(d(e(X)))) → P_IN_AG(DY)
P_IN_AG(+(DX, DY)) → U1_AG(DY, p_in_ag(DX))
P_IN_AG(+(DX, DY)) → P_IN_AG(DX)

The TRS R consists of the following rules:

p_in_ag(const) → p_out_ag(d(e(t)))
p_in_ag(const) → p_out_ag(d(e(const)))
p_in_ag(+(DX, DY)) → U1_ag(DY, p_in_ag(DX))
p_in_ag(+(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, p_in_gg(d(e(X)), DX))
U1_ag(DY, p_out_ag(d(e(X)))) → U2_ag(X, p_in_ag(DY))
U3_ag(X, Y, DY, p_out_gg) → U4_ag(X, Y, p_in_gg(d(e(Y)), DY))
U2_ag(X, p_out_ag(d(e(Y)))) → p_out_ag(d(e(+(X, Y))))
p_in_gg(d(e(t)), const) → p_out_gg
p_in_gg(d(e(const)), const) → p_out_gg
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(Y, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(Y, DY, p_in_gg(d(e(X)), DX))
U4_ag(X, Y, p_out_gg) → p_out_ag(d(e(*(X, Y))))
U1_gg(Y, DY, p_out_gg) → U2_gg(p_in_gg(d(e(Y)), DY))
U3_gg(Y, DY, p_out_gg) → U4_gg(p_in_gg(d(e(Y)), DY))
U2_gg(p_out_gg) → p_out_gg
U4_gg(p_out_gg) → p_out_gg

The set Q consists of the following terms:

p_in_ag(x0)
U1_ag(x0, x1)
U3_ag(x0, x1, x2, x3)
U2_ag(x0, x1)
p_in_gg(x0, x1)
U4_ag(x0, x1, x2)
U1_gg(x0, x1, x2)
U3_gg(x0, x1, x2)
U2_gg(x0)
U4_gg(x0)

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_AG(+(DX, DY)) → P_IN_AG(DX)
The graph contains the following edges 1 > 1
P_IN_AG(+(DX, DY)) → U1_AG(DY, p_in_ag(DX))
The graph contains the following edges 1 > 1
U1_AG(DY, p_out_ag(d(e(X)))) → P_IN_AG(DY)
The graph contains the following edges 1 >= 1

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

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(d(X)), DDX) → U5_GG(X, DDX, p_in_ga(d(X), DX))
P_IN_GG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
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)
U5_GG(X, DDX, p_out_ga(d(X), DX)) → U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_GG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_AG(d(d(X)), DDX) → U5_AG(X, DDX, p_in_aa(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_AA(d(X), DX)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(d(X)), DDX) → U5_AA(X, DDX, p_in_aa(d(X), DX))
P_IN_AA(d(d(X)), DDX) → P_IN_AA(d(X), DX)
U5_AA(X, DDX, p_out_aa(d(X), DX)) → U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_AA(X, DDX, p_out_aa(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U5_AG(X, DDX, p_out_aa(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_aa(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

The argument filtering Pi contains the following mapping:
p_in_ag(x1, x2) = p_in_ag(x2)
const(x1) = const
1 = 1
p_out_ag(x1, x2) = p_out_ag(x1, x2)
0 = 0
+(x1, x2) = +(x1, x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x4, x5)
*(x1, x2) = *(x1, x2)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x1, x2, x3, x4, x5)
p_in_gg(x1, x2) = p_in_gg(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_gg(x1, x2) = p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5)
U5_gg(x1, x2, x3) = U5_gg(x1, x2, x3)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
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)
U6_gg(x1, x2, x3, x4) = U6_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5)
U5_ag(x1, x2, x3) = U5_ag(x2, x3)
p_in_aa(x1, x2) = p_in_aa
p_out_aa(x1, x2) = p_out_aa(x1, x2)
U1_aa(x1, x2, x3, x4, x5) = U1_aa(x5)
U3_aa(x1, x2, x3, x4, x5) = U3_aa(x5)
U5_aa(x1, x2, x3) = U5_aa(x3)
U6_aa(x1, x2, x3, x4) = U6_aa(x1, x4)
U4_aa(x1, x2, x3, x4, x5) = U4_aa(x1, x4, x5)
U2_aa(x1, x2, x3, x4, x5) = U2_aa(x1, x3, x5)
U6_ag(x1, x2, x3, x4) = U6_ag(x1, x2, x4)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x3, x4, x5)
U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U5_GA(x1, x2, x3) = U5_GA(x1, x3)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x4, x5)
U4_GG(x1, x2, x3, x4, x5) = U4_GG(x1, x2, x3, x4, x5)
U3_AA(x1, x2, x3, x4, x5) = U3_AA(x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x1, x2, x3, x4, x5)
U2_GA(x1, x2, x3, x4, x5) = U2_GA(x1, x2, x3, x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x3, x4, x5)
U2_AA(x1, x2, x3, x4, x5) = U2_AA(x1, x3, x5)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x4, x5)
U4_AA(x1, x2, x3, x4, x5) = U4_AA(x1, x4, x5)
U6_GA(x1, x2, x3, x4) = U6_GA(x1, x4)
U6_GG(x1, x2, x3, x4) = U6_GG(x1, x2, x4)
U6_AA(x1, x2, x3, x4) = U6_AA(x1, x4)
U5_AG(x1, x2, x3) = U5_AG(x2, x3)
P_IN_GG(x1, x2) = P_IN_GG(x1, x2)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x3, x4, x5)
U1_AA(x1, x2, x3, x4, x5) = U1_AA(x5)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5)
P_IN_AG(x1, x2) = P_IN_AG(x2)
U5_AA(x1, x2, x3) = U5_AA(x3)
U6_AG(x1, x2, x3, x4) = U6_AG(x1, x2, x4)
P_IN_AA(x1, x2) = P_IN_AA
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x5)
U2_GG(x1, x2, x3, x4, x5) = U2_GG(x1, x2, x3, x4, x5)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_AG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(d(X)), DDX) → U5_GG(X, DDX, p_in_ga(d(X), DX))
P_IN_GG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
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)
U5_GG(X, DDX, p_out_ga(d(X), DX)) → U6_GG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_GG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_GG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_GG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_AG(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U3_AG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_AG(d(d(X)), DDX) → U5_AG(X, DDX, p_in_aa(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_AA(d(X), DX)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)
P_IN_AA(d(d(X)), DDX) → U5_AA(X, DDX, p_in_aa(d(X), DX))
P_IN_AA(d(d(X)), DDX) → P_IN_AA(d(X), DX)
U5_AA(X, DDX, p_out_aa(d(X), DX)) → U6_AA(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U5_AA(X, DDX, p_out_aa(d(X), DX)) → P_IN_GA(d(e(DX)), DDX)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_AA(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_AA(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U5_AG(X, DDX, p_out_aa(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_aa(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_AA(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → P_IN_AA(d(e(X)), DX)
U1_AA(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
U3_AA(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → P_IN_AA(d(e(Y)), DY)
P_IN_AA(d(e(+(X, Y))), +(DX, DY)) → U1_AA(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
P_IN_AA(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_AA(d(e(X)), DX)

The TRS R consists of the following rules:

p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U1_AA(p_in_aa)
P_IN_AA → U3_AA(p_in_aa)

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_AA → U3_AA(p_in_aa) at position [0] we obtained the following new rules:

P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U1_AA(p_in_aa)
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_AA → U1_AA(p_in_aa) at position [0] we obtained the following new rules:

P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_aa
U1_aa(x0)
U3_aa(x0)
U2_aa(x0, x1, x2)
U4_aa(x0, x1, x2)

P_IN_AA → P_IN_AA
U3_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
U1_AA(p_out_aa(d(e(X)), DX)) → P_IN_AA
P_IN_AA → U3_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U3_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U1_AA(p_out_aa(d(e(const)), const))
P_IN_AA → U1_AA(U1_aa(p_in_aa))
P_IN_AA → U1_AA(U3_aa(p_in_aa))
P_IN_AA → U1_AA(p_out_aa(d(e(t)), const))
P_IN_AA → U3_AA(U1_aa(p_in_aa))
P_IN_AA → U3_AA(U3_aa(p_in_aa))

The TRS R consists of the following rules:

p_in_aa → p_out_aa(d(e(t)), const)
p_in_aa → p_out_aa(d(e(const)), const)
p_in_aa → U1_aa(p_in_aa)
p_in_aa → U3_aa(p_in_aa)
U1_aa(p_out_aa(d(e(X)), DX)) → U2_aa(X, DX, p_in_aa)
U3_aa(p_out_aa(d(e(X)), DX)) → U4_aa(X, DX, p_in_aa)
U2_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U4_aa(X, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

s = P_IN_AA evaluates to t =P_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_AA to P_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ 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_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ 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:
const(x1) = const
1 = 1
0 = 0
+(x1, x2) = +(x1, x2)
*(x1, x2) = *(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5) = U1_ga(x1, x2, x5)
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)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x5)
U1_GA(x1, x2, x3, x4, x5) = U1_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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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.
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)))) → P_IN_GA(d(e(X)))
U3_GA(X, Y, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)))
U1_GA(X, Y, p_out_ga(d(e(X)), DX)) → P_IN_GA(d(e(Y)))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(*(x₁, x₂)) = 1 + x₁ + x₂
POL(+(x₁, x₂)) = 1 + x₁ + x₂
POL(P_IN_GA(x₁)) = x₁
POL(U1_GA(x₁, x₂, x₃)) = 1 + x₂
POL(U1_ga(x₁, x₂, x₃)) = 0
POL(U2_ga(x₁, x₂, x₃, x₄)) = 0
POL(U3_GA(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U3_ga(x₁, x₂, x₃)) = 0
POL(U4_ga(x₁, x₂, x₃, x₄)) = 0
POL(const) = 0
POL(d(x₁)) = x₁
POL(e(x₁)) = x₁
POL(p_in_ga(x₁)) = 0
POL(p_out_ga(x₁, x₂)) = 0
POL(t) = 0

The following usable rules [17] were oriented: none

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_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.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

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

P_IN_AA → P_IN_AA

The TRS R consists of the following rules:none

s = P_IN_AA evaluates to t =P_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_AA to P_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

P_IN_GA(d(d(X))) → P_IN_GA(d(X))

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The argument filtering Pi contains the following mapping:
const(x1) = const
1 = 1
0 = 0
+(x1, x2) = +(x1, x2)
*(x1, x2) = *(x1, x2)
p_in_gg(x1, x2) = p_in_gg(x1, x2)
d(x1) = d(x1)
e(x1) = e(x1)
t = t
p_out_gg(x1, x2) = p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5)
P_IN_GG(x1, x2) = P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, 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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)

The TRS R consists of the following rules:

p_in_gg(d(e(t)), const) → p_out_gg(d(e(t)), const)
p_in_gg(d(e(const)), const) → p_out_gg(d(e(const)), const)
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_gg(x0, x1)
U1_gg(x0, x1, x2, x3, x4)
U3_gg(x0, x1, x2, x3, x4)
U2_gg(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2, x3, x4)

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_GG(d(e(+(X, Y))), +(DX, DY)) → U1_GG(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_GG(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
The graph contains the following edges 1 > 1, 2 > 1, 1 > 2, 2 > 2, 2 > 3, 2 > 4
U3_GG(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
The graph contains the following edges 3 >= 2
U1_GG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
The graph contains the following edges 4 >= 2
P_IN_GG(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → P_IN_GG(d(e(X)), DX)
The graph contains the following edges 2 > 2
P_IN_GG(d(e(+(X, Y))), +(DX, DY)) → P_IN_GG(d(e(X)), DX)
The graph contains the following edges 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
p_in_gg(d(d(X)), DDX) → U5_gg(X, DDX, p_in_ga(d(X), DX))
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))
U5_gg(X, DDX, p_out_ga(d(X), DX)) → U6_gg(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_gg(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_gg(d(d(X)), DDX)
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
p_in_ag(d(d(X)), DDX) → U5_ag(X, DDX, p_in_aa(d(X), DX))
p_in_aa(d(e(t)), const(1)) → p_out_aa(d(e(t)), const(1))
p_in_aa(d(e(const(A))), const(0)) → p_out_aa(d(e(const(A))), const(0))
p_in_aa(d(e(+(X, Y))), +(DX, DY)) → U1_aa(X, Y, DX, DY, p_in_aa(d(e(X)), DX))
p_in_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_aa(X, Y, DY, DX, p_in_aa(d(e(X)), DX))
p_in_aa(d(d(X)), DDX) → U5_aa(X, DDX, p_in_aa(d(X), DX))
U5_aa(X, DDX, p_out_aa(d(X), DX)) → U6_aa(X, DDX, DX, p_in_ga(d(e(DX)), DDX))
U6_aa(X, DDX, DX, p_out_ga(d(e(DX)), DDX)) → p_out_aa(d(d(X)), DDX)
U3_aa(X, Y, DY, DX, p_out_aa(d(e(X)), DX)) → U4_aa(X, Y, DY, DX, p_in_aa(d(e(Y)), DY))
U4_aa(X, Y, DY, DX, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_aa(X, Y, DX, DY, p_out_aa(d(e(X)), DX)) → U2_aa(X, Y, DX, DY, p_in_aa(d(e(Y)), DY))
U2_aa(X, Y, DX, DY, p_out_aa(d(e(Y)), DY)) → p_out_aa(d(e(+(X, Y))), +(DX, DY))
U5_ag(X, DDX, p_out_aa(d(X), DX)) → U6_ag(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U6_ag(X, DDX, DX, p_out_gg(d(e(DX)), DDX)) → p_out_ag(d(d(X)), DDX)
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → P_IN_AG(d(e(X)), DX)
U1_AG(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(d(e(Y)), DY)
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → U1_AG(X, Y, DX, DY, p_in_ag(d(e(X)), DX))

The TRS R consists of the following rules:

p_in_ag(d(e(t)), const(1)) → p_out_ag(d(e(t)), const(1))
p_in_ag(d(e(const(A))), const(0)) → p_out_ag(d(e(const(A))), const(0))
p_in_ag(d(e(+(X, Y))), +(DX, DY)) → U1_ag(X, Y, DX, DY, p_in_ag(d(e(X)), DX))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_ag(X, Y, DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_ag(d(e(Y)), DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_gg(d(e(t)), const(1)) → p_out_gg(d(e(t)), const(1))
p_in_gg(d(e(const(A))), const(0)) → p_out_gg(d(e(const(A))), const(0))
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

P_IN_AG(+(DX, DY)) → U1_AG(DX, DY, p_in_ag(DX))
U1_AG(DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(DY)
P_IN_AG(+(DX, DY)) → P_IN_AG(DX)

The TRS R consists of the following rules:

p_in_ag(const) → p_out_ag(d(e(t)), const)
p_in_ag(const) → p_out_ag(d(e(const)), const)
p_in_ag(+(DX, DY)) → U1_ag(DX, DY, p_in_ag(DX))
p_in_ag(+(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U1_ag(DX, DY, p_out_ag(d(e(X)), DX)) → U2_ag(X, DX, DY, p_in_ag(DY))
U3_ag(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_ag(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_ag(X, DX, DY, p_out_ag(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_gg(d(e(t)), const) → p_out_gg(d(e(t)), const)
p_in_gg(d(e(const)), const) → p_out_gg(d(e(const)), const)
p_in_gg(d(e(+(X, Y))), +(DX, DY)) → U1_gg(X, Y, DX, DY, p_in_gg(d(e(X)), DX))
p_in_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_gg(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
U4_ag(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))
U1_gg(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_gg(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U3_gg(X, Y, DY, DX, p_out_gg(d(e(X)), DX)) → U4_gg(X, Y, DY, DX, p_in_gg(d(e(Y)), DY))
U2_gg(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(+(X, Y))), +(DX, DY))
U4_gg(X, Y, DY, DX, p_out_gg(d(e(Y)), DY)) → p_out_gg(d(e(*(X, Y))), +(*(X, DY), *(Y, DX)))

The set Q consists of the following terms:

p_in_ag(x0)
U1_ag(x0, x1, x2)
U3_ag(x0, x1, x2, x3, x4)
U2_ag(x0, x1, x2, x3)
p_in_gg(x0, x1)
U4_ag(x0, x1, x2, x3, x4)
U1_gg(x0, x1, x2, x3, x4)
U3_gg(x0, x1, x2, x3, x4)
U2_gg(x0, x1, x2, x3, x4)
U4_gg(x0, x1, x2, x3, x4)

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_AG(+(DX, DY)) → P_IN_AG(DX)
The graph contains the following edges 1 > 1
P_IN_AG(+(DX, DY)) → U1_AG(DX, DY, p_in_ag(DX))
The graph contains the following edges 1 > 1, 1 > 2
U1_AG(DX, DY, p_out_ag(d(e(X)), DX)) → P_IN_AG(DY)
The graph contains the following edges 2 >= 1