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)
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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
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_gg(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → 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)
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
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)
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_ga(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_AG(X, DDX, p_out_ga(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U5_GG(x1, x2, x3)  =  U5_GG(x2, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x4, x5)
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(x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
P_IN_AG(x1, x2)  =  P_IN_AG(x2)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, 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_gg(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → 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)
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
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)
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_ga(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_AG(X, DDX, p_out_ga(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
U5_GG(x1, x2, x3)  =  U5_GG(x2, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x4, x5)
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(x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x5)
P_IN_AG(x1, x2)  =  P_IN_AG(x2)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x4, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 23 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ 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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
              ↳ 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)
e(x1)  =  e
d(x1)  =  d
*(x1, x2)  =  *(x2)
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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)

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

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

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

U3_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(p_in_ga(d))
P_IN_GA(d) → U3_GA(p_in_ga(d))
P_IN_GA(d) → P_IN_GA(d)
U1_GA(p_out_ga(DX)) → P_IN_GA(d)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_GA(d) → U3_GA(p_in_ga(d)) at position [0] we obtained the following new rules:

P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(p_out_ga(const))
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))



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

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

P_IN_GA(d) → U1_GA(p_in_ga(d))
U3_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(p_out_ga(const))
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
U1_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_GA(d) → U1_GA(p_in_ga(d)) at position [0] we obtained the following new rules:

P_IN_GA(d) → U1_GA(p_out_ga(const))
P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

P_IN_GA(d) → U1_GA(p_out_ga(const))
U3_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(p_out_ga(const))
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
U1_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

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_GA(d) → U1_GA(p_out_ga(const))
U3_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(p_out_ga(const))
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
U1_GA(p_out_ga(DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))

The TRS R consists of the following rules:

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


s = P_IN_GA(d) evaluates to t =P_IN_GA(d)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_GA(d) to P_IN_GA(d).





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ 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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ 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
P_IN_GA(x1, x2)  =  P_IN_GA(x1)

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

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

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

P_IN_GA(d) → P_IN_GA(d)

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_GA(d) → P_IN_GA(d)

The TRS R consists of the following rules:none


s = P_IN_GA(d) evaluates to t =P_IN_GA(d)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_GA(d) to P_IN_GA(d).





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ 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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(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)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x4, x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ 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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(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
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
  ↳ PrologToPiTRSProof

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

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

The TRS R consists of the following rules:

p_in_gg(d, const) → p_out_gg
p_in_gg(d, +(DX, DY)) → U1_gg(DY, p_in_gg(d, DX))
p_in_gg(d, +(*(DY), *(DX))) → U3_gg(DY, p_in_gg(d, DX))
U1_gg(DY, p_out_gg) → U2_gg(p_in_gg(d, DY))
U3_gg(DY, p_out_gg) → U4_gg(p_in_gg(d, 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)
U3_gg(x0, x1)
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:


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)
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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
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_gg(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → 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)
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
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)
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_ga(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_AG(X, DDX, p_out_ga(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)
U5_GG(x1, x2, x3)  =  U5_GG(x2, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x2, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x4, x5)
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(x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
P_IN_AG(x1, x2)  =  P_IN_AG(x2)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(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_gg(d(e(X)), DX))
P_IN_AG(d(e(+(X, Y))), +(DX, DY)) → 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)
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_AG(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U1_AG(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → P_IN_GG(d(e(Y)), DY)
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)
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_ga(d(X), DX))
P_IN_AG(d(d(X)), DDX) → P_IN_GA(d(X), DX)
U5_AG(X, DDX, p_out_ga(d(X), DX)) → U6_AG(X, DDX, DX, p_in_gg(d(e(DX)), DDX))
U5_AG(X, DDX, p_out_ga(d(X), DX)) → P_IN_GG(d(e(DX)), DDX)

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)
U5_GG(x1, x2, x3)  =  U5_GG(x2, x3)
U6_GA(x1, x2, x3, x4)  =  U6_GA(x4)
U6_GG(x1, x2, x3, x4)  =  U6_GG(x2, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x4, x5)
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(x3, x4, x5)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U4_GG(x1, x2, x3, x4, x5)  =  U4_GG(x3, x4, x5)
U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)
P_IN_AG(x1, x2)  =  P_IN_AG(x2)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x3, x5)
U6_AG(x1, x2, x3, x4)  =  U6_AG(x2, x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5)  =  U4_AG(x3, x4, x5)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x3, x4, x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)
U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x3, x4, x5)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 23 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ 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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x5)

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

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ 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)
e(x1)  =  e
d(x1)  =  d
*(x1, x2)  =  *(x2)
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(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
              ↳ PiDP
              ↳ PiDP

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

P_IN_GA(d) → U1_GA(p_in_ga(d))
P_IN_GA(d) → U3_GA(p_in_ga(d))
U3_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → P_IN_GA(d)
U1_GA(p_out_ga(d, DX)) → P_IN_GA(d)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_GA(d) → U3_GA(p_in_ga(d)) at position [0] we obtained the following new rules:

P_IN_GA(d) → U3_GA(p_out_ga(d, const))
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))



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

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

P_IN_GA(d) → U3_GA(p_out_ga(d, const))
P_IN_GA(d) → U1_GA(p_in_ga(d))
U3_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))
U1_GA(p_out_ga(d, DX)) → P_IN_GA(d)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule P_IN_GA(d) → U1_GA(p_in_ga(d)) at position [0] we obtained the following new rules:

P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(p_out_ga(d, const))
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP

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

P_IN_GA(d) → U3_GA(p_out_ga(d, const))
U3_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(p_out_ga(d, const))
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
U1_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

p_in_ga(x0)
U1_ga(x0)
U3_ga(x0)
U2_ga(x0, x1)
U4_ga(x0, x1)

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_GA(d) → U3_GA(p_out_ga(d, const))
U3_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(p_out_ga(d, const))
P_IN_GA(d) → P_IN_GA(d)
P_IN_GA(d) → U1_GA(U1_ga(p_in_ga(d)))
P_IN_GA(d) → U1_GA(U3_ga(p_in_ga(d)))
P_IN_GA(d) → U3_GA(U1_ga(p_in_ga(d)))
U1_GA(p_out_ga(d, DX)) → P_IN_GA(d)
P_IN_GA(d) → U3_GA(U3_ga(p_in_ga(d)))

The TRS R consists of the following rules:

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


s = P_IN_GA(d) evaluates to t =P_IN_GA(d)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_GA(d) to P_IN_GA(d).





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ 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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)

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

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

R is empty.
The argument filtering Pi contains the following mapping:
d(x1)  =  d
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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

P_IN_GA(d) → P_IN_GA(d)

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_GA(d) → P_IN_GA(d)

The TRS R consists of the following rules:none


s = P_IN_GA(d) evaluates to t =P_IN_GA(d)

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from P_IN_GA(d) to P_IN_GA(d).





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

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_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))
U1_ag(X, Y, DX, DY, p_out_gg(d(e(X)), DX)) → U2_ag(X, Y, DX, DY, p_in_gg(d(e(Y)), DY))
U2_ag(X, Y, DX, DY, p_out_gg(d(e(Y)), DY)) → p_out_ag(d(e(+(X, Y))), +(DX, DY))
p_in_ag(d(e(*(X, Y))), +(*(X, DY), *(Y, DX))) → U3_ag(X, Y, DY, DX, p_in_gg(d(e(X)), DX))
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_ga(d(X), DX))
U5_ag(X, DDX, p_out_ga(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)

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, 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(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(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(x4, x5)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x3, x5)
U6_gg(x1, x2, x3, x4)  =  U6_gg(x2, x4)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)
U4_ag(x1, x2, x3, x4, x5)  =  U4_ag(x3, x4, x5)
U5_ag(x1, x2, x3)  =  U5_ag(x2, x3)
U6_ag(x1, x2, x3, x4)  =  U6_ag(x2, x4)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x3, x4, x5)

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

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

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)
p_in_gg(x1, x2)  =  p_in_gg(x1, x2)
e(x1)  =  e
d(x1)  =  d
p_out_gg(x1, x2)  =  p_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x3, x4, x5)
*(x1, x2)  =  *(x2)
U3_gg(x1, x2, x3, x4, x5)  =  U3_gg(x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5)  =  U4_gg(x3, x4, x5)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x3, x4, x5)
P_IN_GG(x1, x2)  =  P_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4, x5)  =  U3_GG(x3, x4, x5)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(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
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof

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

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

The TRS R consists of the following rules:

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

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, x1, x2)
U4_gg(x0, x1, x2)

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: