Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

append2(parts([], Y), is(sum(Y))).
append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z))).
append1(parts([], Y), is(sum(Y))).
append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z))).
perm([], []).
perm(L, .(H, T)) :- ','(append2(parts(V, .(H, U)), is(sum(L))), ','(append1(parts(V, U), is(sum(W))), perm(W, T))).

Queries:

perm(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f) (f,f)
append2_in: (b,b)
append1_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(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:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_GG(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GG(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_aa(W, T))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_AA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_AA(L, H, T, perm_in_aa(W, T))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_GG(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GG(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_aa(W, T))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_AA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_AA(L, H, T, perm_in_aa(W, T))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 11 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:

APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)

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:

APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)

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
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

APPEND1_IN_GG(parts, is(sum)) → APPEND1_IN_GG(parts, is(sum))

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:

APPEND1_IN_GG(parts, is(sum)) → APPEND1_IN_GG(parts, is(sum))

The TRS R consists of the following rules:none


s = APPEND1_IN_GG(parts, is(sum)) evaluates to t =APPEND1_IN_GG(parts, is(sum))

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 APPEND1_IN_GG(parts, is(sum)) to APPEND1_IN_GG(parts, is(sum)).





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

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

APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)

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:

APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)

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:

APPEND2_IN_GG(parts, is(sum)) → APPEND2_IN_GG(parts, is(sum))

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:

APPEND2_IN_GG(parts, is(sum)) → APPEND2_IN_GG(parts, is(sum))

The TRS R consists of the following rules:none


s = APPEND2_IN_GG(parts, is(sum)) evaluates to t =APPEND2_IN_GG(parts, is(sum))

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 APPEND2_IN_GG(parts, is(sum)) to APPEND2_IN_GG(parts, is(sum)).





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

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

U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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:

U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))

The argument filtering Pi contains the following mapping:
[]  =  []
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum)))
U4_AA(append1_out_gg) → PERM_IN_AA
U3_AA(append2_out_gg) → U4_AA(append1_in_gg(parts, is(sum)))

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg) → append1_out_gg
U1_gg(append2_out_gg) → append2_out_gg

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U3_AA(append2_out_gg) → U4_AA(append1_in_gg(parts, is(sum))) at position [0] we obtained the following new rules:

U3_AA(append2_out_gg) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))
U3_AA(append2_out_gg) → U4_AA(append1_out_gg)



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

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

PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum)))
U3_AA(append2_out_gg) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))
U4_AA(append1_out_gg) → PERM_IN_AA
U3_AA(append2_out_gg) → U4_AA(append1_out_gg)

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg) → append1_out_gg
U1_gg(append2_out_gg) → append2_out_gg

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum))) at position [0] we obtained the following new rules:

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
PERM_IN_AA → U3_AA(append2_out_gg)



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

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

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
PERM_IN_AA → U3_AA(append2_out_gg)
U4_AA(append1_out_gg) → PERM_IN_AA
U3_AA(append2_out_gg) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))
U3_AA(append2_out_gg) → U4_AA(append1_out_gg)

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg) → append1_out_gg
U1_gg(append2_out_gg) → append2_out_gg

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
PERM_IN_AA → U3_AA(append2_out_gg)
U4_AA(append1_out_gg) → PERM_IN_AA
U3_AA(append2_out_gg) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))
U3_AA(append2_out_gg) → U4_AA(append1_out_gg)

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg) → append1_out_gg
U1_gg(append2_out_gg) → append2_out_gg


s = U3_AA(append2_out_gg) evaluates to t =U3_AA(append2_out_gg)

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



Rewriting sequence

U3_AA(append2_out_gg) → U4_AA(append1_out_gg)
with rule U3_AA(append2_out_gg) → U4_AA(append1_out_gg) at position [] and matcher [ ]

U4_AA(append1_out_gg) → PERM_IN_AA
with rule U4_AA(append1_out_gg) → PERM_IN_AA at position [] and matcher [ ]

PERM_IN_AA → U3_AA(append2_out_gg)
with rule PERM_IN_AA → U3_AA(append2_out_gg)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f) (f,f)
append2_in: (b,b)
append1_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(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:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(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:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_GG(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GG(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_aa(W, T))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_AA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_AA(L, H, T, perm_in_aa(W, T))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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:

PERM_IN_GA(L, .(H, T)) → U3_GA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_GA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_GG(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_GA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_GA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_GG(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_GA(L, H, T, perm_in_aa(W, T))
U4_GA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
PERM_IN_AA(L, .(H, T)) → APPEND2_IN_GG(parts(V, .(H, U)), is(sum(L)))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → APPEND1_IN_GG(parts(V, U), is(sum(W)))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_AA(L, H, T, perm_in_aa(W, T))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U5_GA(x1, x2, x3, x4)  =  U5_GA(x1, x4)
U5_AA(x1, x2, x3, x4)  =  U5_AA(x4)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U4_GA(x1, x2, x3, x4, x5, x6)  =  U4_GA(x1, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U2_GG(x1, x2, x3, x4, x5)  =  U2_GG(x5)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 3 SCCs with 11 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:

APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)

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:

APPEND1_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND1_IN_GG(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
APPEND1_IN_GG(x1, x2)  =  APPEND1_IN_GG(x1, x2)

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
                        ↳ NonTerminationProof
              ↳ PiDP
              ↳ PiDP

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

APPEND1_IN_GG(parts, is(sum)) → APPEND1_IN_GG(parts, is(sum))

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:

APPEND1_IN_GG(parts, is(sum)) → APPEND1_IN_GG(parts, is(sum))

The TRS R consists of the following rules:none


s = APPEND1_IN_GG(parts, is(sum)) evaluates to t =APPEND1_IN_GG(parts, is(sum))

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 APPEND1_IN_GG(parts, is(sum)) to APPEND1_IN_GG(parts, is(sum)).





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

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

APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)

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:

APPEND2_IN_GG(parts(.(H, X), Y), is(sum(.(H, Z)))) → APPEND2_IN_GG(parts(X, Y), is(sum(Z)))

R is empty.
The argument filtering Pi contains the following mapping:
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
APPEND2_IN_GG(x1, x2)  =  APPEND2_IN_GG(x1, x2)

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:

APPEND2_IN_GG(parts, is(sum)) → APPEND2_IN_GG(parts, is(sum))

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:

APPEND2_IN_GG(parts, is(sum)) → APPEND2_IN_GG(parts, is(sum))

The TRS R consists of the following rules:none


s = APPEND2_IN_GG(parts, is(sum)) evaluates to t =APPEND2_IN_GG(parts, is(sum))

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 APPEND2_IN_GG(parts, is(sum)) to APPEND2_IN_GG(parts, is(sum)).





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

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

U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(L, .(H, T)) → U3_ga(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U3_ga(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_ga(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U4_ga(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_ga(L, H, T, perm_in_aa(W, T))
perm_in_aa([], []) → perm_out_aa([], [])
perm_in_aa(L, .(H, T)) → U3_aa(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U3_aa(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_aa(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
U4_aa(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → U5_aa(L, H, T, perm_in_aa(W, T))
U5_aa(L, H, T, perm_out_aa(W, T)) → perm_out_aa(L, .(H, T))
U5_ga(L, H, T, perm_out_aa(W, T)) → perm_out_ga(L, .(H, T))

The argument filtering Pi contains the following mapping:
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x4)
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
U4_ga(x1, x2, x3, x4, x5, x6)  =  U4_ga(x1, x6)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x1, x4)
perm_in_aa(x1, x2)  =  perm_in_aa
perm_out_aa(x1, x2)  =  perm_out_aa(x2)
U3_aa(x1, x2, x3, x4)  =  U3_aa(x4)
U4_aa(x1, x2, x3, x4, x5, x6)  =  U4_aa(x6)
U5_aa(x1, x2, x3, x4)  =  U5_aa(x4)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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:

U3_AA(L, H, T, append2_out_gg(parts(V, .(H, U)), is(sum(L)))) → U4_AA(L, H, T, V, U, append1_in_gg(parts(V, U), is(sum(W))))
PERM_IN_AA(L, .(H, T)) → U3_AA(L, H, T, append2_in_gg(parts(V, .(H, U)), is(sum(L))))
U4_AA(L, H, T, V, U, append1_out_gg(parts(V, U), is(sum(W)))) → PERM_IN_AA(W, T)

The TRS R consists of the following rules:

append1_in_gg(parts([], Y), is(sum(Y))) → append1_out_gg(parts([], Y), is(sum(Y)))
append1_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U2_gg(H, X, Y, Z, append1_in_gg(parts(X, Y), is(sum(Z))))
append2_in_gg(parts([], Y), is(sum(Y))) → append2_out_gg(parts([], Y), is(sum(Y)))
append2_in_gg(parts(.(H, X), Y), is(sum(.(H, Z)))) → U1_gg(H, X, Y, Z, append2_in_gg(parts(X, Y), is(sum(Z))))
U2_gg(H, X, Y, Z, append1_out_gg(parts(X, Y), is(sum(Z)))) → append1_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))
U1_gg(H, X, Y, Z, append2_out_gg(parts(X, Y), is(sum(Z)))) → append2_out_gg(parts(.(H, X), Y), is(sum(.(H, Z))))

The argument filtering Pi contains the following mapping:
[]  =  []
append2_in_gg(x1, x2)  =  append2_in_gg(x1, x2)
parts(x1, x2)  =  parts
.(x1, x2)  =  .
is(x1)  =  is(x1)
sum(x1)  =  sum
append2_out_gg(x1, x2)  =  append2_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x5)
append1_in_gg(x1, x2)  =  append1_in_gg(x1, x2)
append1_out_gg(x1, x2)  =  append1_out_gg(x1, x2)
U2_gg(x1, x2, x3, x4, x5)  =  U2_gg(x5)
U4_AA(x1, x2, x3, x4, x5, x6)  =  U4_AA(x6)
PERM_IN_AA(x1, x2)  =  PERM_IN_AA
U3_AA(x1, x2, x3, x4)  =  U3_AA(x4)

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
                        ↳ Narrowing

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

U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_in_gg(parts, is(sum)))
U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA
PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum)))

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg(parts, is(sum))
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg(parts, is(sum))
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg(parts, is(sum))) → append1_out_gg(parts, is(sum))
U1_gg(append2_out_gg(parts, is(sum))) → append2_out_gg(parts, is(sum))

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_in_gg(parts, is(sum))) at position [0] we obtained the following new rules:

U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))



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

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

U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))
U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))
PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum)))

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg(parts, is(sum))
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg(parts, is(sum))
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg(parts, is(sum))) → append1_out_gg(parts, is(sum))
U1_gg(append2_out_gg(parts, is(sum))) → append2_out_gg(parts, is(sum))

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule PERM_IN_AA → U3_AA(append2_in_gg(parts, is(sum))) at position [0] we obtained the following new rules:

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
PERM_IN_AA → U3_AA(append2_out_gg(parts, is(sum)))



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

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

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))
PERM_IN_AA → U3_AA(append2_out_gg(parts, is(sum)))
U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg(parts, is(sum))
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg(parts, is(sum))
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg(parts, is(sum))) → append1_out_gg(parts, is(sum))
U1_gg(append2_out_gg(parts, is(sum))) → append2_out_gg(parts, is(sum))

The set Q consists of the following terms:

append1_in_gg(x0, x1)
append2_in_gg(x0, x1)
U2_gg(x0)
U1_gg(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

PERM_IN_AA → U3_AA(U1_gg(append2_in_gg(parts, is(sum))))
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))
PERM_IN_AA → U3_AA(append2_out_gg(parts, is(sum)))
U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA
U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(U2_gg(append1_in_gg(parts, is(sum))))

The TRS R consists of the following rules:

append1_in_gg(parts, is(sum)) → append1_out_gg(parts, is(sum))
append1_in_gg(parts, is(sum)) → U2_gg(append1_in_gg(parts, is(sum)))
append2_in_gg(parts, is(sum)) → append2_out_gg(parts, is(sum))
append2_in_gg(parts, is(sum)) → U1_gg(append2_in_gg(parts, is(sum)))
U2_gg(append1_out_gg(parts, is(sum))) → append1_out_gg(parts, is(sum))
U1_gg(append2_out_gg(parts, is(sum))) → append2_out_gg(parts, is(sum))


s = U4_AA(append1_out_gg(parts, is(sum))) evaluates to t =U4_AA(append1_out_gg(parts, is(sum)))

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



Rewriting sequence

U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA
with rule U4_AA(append1_out_gg(parts, is(sum))) → PERM_IN_AA at position [] and matcher [ ]

PERM_IN_AA → U3_AA(append2_out_gg(parts, is(sum)))
with rule PERM_IN_AA → U3_AA(append2_out_gg(parts, is(sum))) at position [] and matcher [ ]

U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))
with rule U3_AA(append2_out_gg(parts, is(sum))) → U4_AA(append1_out_gg(parts, is(sum)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.