MAYBE Left Termination proof of ../tpdb/LP/talp/talp/slowsort-oi.pl
Left Termination of the query pattern sort_in_2(a, g) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof

Clauses:

sort(X, Y) :- ','(perm(X, Y), sorted(Y)).
sorted([]).
sorted(.(X, [])).
sorted(.(X, .(Y, Z))) :- ','(le(X, Y), sorted(.(Y, Z))).
perm([], []).
perm(.(X, .(Y, [])), .(U, .(V, []))) :- ','(delete(U, .(X, Y), Z), perm(Z, V)).
delete(X, .(X, Y), Y).
delete(X, .(Y, Z), W) :- delete(X, Z, W).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).

Queries:

sort(a,g).

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

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag

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:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SORT_IN_AG(X, Y) → PERM_IN_AG(X, Y)
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_GAA(U, .(X, Y), Z)
DELETE_IN_GAA(X, .(Y, Z), W) → U7_GAA(X, Y, Z, W, delete_in_gaa(X, Z, W))
DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_AG(X, Y, U, V, perm_in_ag(Z, V))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)
U1_AG(X, Y, perm_out_ag(X, Y)) → U2_AG(X, Y, sorted_in_g(Y))
U1_AG(X, Y, perm_out_ag(X, Y)) → SORTED_IN_G(Y)
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))
SORTED_IN_G(.(X, .(Y, Z))) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U3_G(X, Y, Z, le_out_gg(X, Y)) → U4_G(X, Y, Z, sorted_in_g(.(Y, Z)))
U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x4, x5)
U7_GAA(x1, x2, x3, x4, x5)  =  U7_GAA(x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
SORT_IN_AG(x1, x2)  =  SORT_IN_AG(x2)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U8_GG(x1, x2, x3)  =  U8_GG(x3)

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:

SORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SORT_IN_AG(X, Y) → PERM_IN_AG(X, Y)
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_GAA(U, .(X, Y), Z)
DELETE_IN_GAA(X, .(Y, Z), W) → U7_GAA(X, Y, Z, W, delete_in_gaa(X, Z, W))
DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_AG(X, Y, U, V, perm_in_ag(Z, V))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)
U1_AG(X, Y, perm_out_ag(X, Y)) → U2_AG(X, Y, sorted_in_g(Y))
U1_AG(X, Y, perm_out_ag(X, Y)) → SORTED_IN_G(Y)
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))
SORTED_IN_G(.(X, .(Y, Z))) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U3_G(X, Y, Z, le_out_gg(X, Y)) → U4_G(X, Y, Z, sorted_in_g(.(Y, Z)))
U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x3)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x4, x5)
U7_GAA(x1, x2, x3, x4, x5)  =  U7_GAA(x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x5)
SORT_IN_AG(x1, x2)  =  SORT_IN_AG(x2)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U8_GG(x1, x2, x3)  =  U8_GG(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 10 less nodes.

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

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
LE_IN_GG(x1, x2)  =  LE_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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Pi is empty.
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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
SORTED_IN_G(x1)  =  SORTED_IN_G(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
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
SORTED_IN_G(x1)  =  SORTED_IN_G(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
                        ↳ UsableRulesReductionPairsProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

SORTED_IN_G(.(X, .(Y, Z))) → U3_G(Y, Z, le_in_gg(X, Y))
U3_G(Y, Z, le_out_gg) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_gg(le_out_gg) → le_out_gg

The set Q consists of the following terms:

le_in_gg(x0, x1)
U8_gg(x0)

We have to consider all (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

SORTED_IN_G(.(X, .(Y, Z))) → U3_G(Y, Z, le_in_gg(X, Y))
U3_G(Y, Z, le_out_gg) → SORTED_IN_G(.(Y, Z))
The following rules are removed from R:

le_in_gg(s(X), s(Y)) → U8_gg(le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg
le_in_gg(0, 0) → le_out_gg
U8_gg(le_out_gg) → le_out_gg
Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(0) = 1   
POL(SORTED_IN_G(x1)) = 1 + x1   
POL(U3_G(x1, x2, x3)) = 2·x1 + 2·x2 + 2·x3   
POL(U8_gg(x1)) = 2 + x1   
POL(le_in_gg(x1, x2)) = x1 + x2   
POL(le_out_gg) = 2   
POL(s(x1)) = 1 + 2·x1   



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ UsableRulesReductionPairsProof
                          ↳ QDP
                            ↳ PisEmptyProof
              ↳ PiDP
              ↳ PiDP
  ↳ PrologToPiTRSProof

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

le_in_gg(x0, x1)
U8_gg(x0)

We have to consider all (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

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

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

DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)

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

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

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

DELETE_IN_GAA(X) → DELETE_IN_GAA(X)

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:

DELETE_IN_GAA(X) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:none


s = DELETE_IN_GAA(X) evaluates to t =DELETE_IN_GAA(X)

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 DELETE_IN_GAA(X) to DELETE_IN_GAA(X).





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

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

PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x5)
U2_ag(x1, x2, x3)  =  U2_ag(x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g
U3_g(x1, x2, x3, x4)  =  U3_g(x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg
U4_g(x1, x2, x3, x4)  =  U4_g(x4)
sort_out_ag(x1, x2)  =  sort_out_ag
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
  ↳ PrologToPiTRSProof

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

PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)

The TRS R consists of the following rules:

delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
  ↳ PrologToPiTRSProof

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

U5_AG(V, delete_out_gaa) → PERM_IN_AG(V)
PERM_IN_AG(.(U, .(V, []))) → U5_AG(V, delete_in_gaa(U))

The TRS R consists of the following rules:

delete_in_gaa(X) → delete_out_gaa
delete_in_gaa(X) → U7_gaa(delete_in_gaa(X))
U7_gaa(delete_out_gaa) → delete_out_gaa

The set Q consists of the following terms:

delete_in_gaa(x0)
U7_gaa(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:
sort_in: (f,b)
perm_in: (f,b)
delete_in: (b,f,f)
sorted_in: (b)
le_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)

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:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SORT_IN_AG(X, Y) → PERM_IN_AG(X, Y)
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_GAA(U, .(X, Y), Z)
DELETE_IN_GAA(X, .(Y, Z), W) → U7_GAA(X, Y, Z, W, delete_in_gaa(X, Z, W))
DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_AG(X, Y, U, V, perm_in_ag(Z, V))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)
U1_AG(X, Y, perm_out_ag(X, Y)) → U2_AG(X, Y, sorted_in_g(Y))
U1_AG(X, Y, perm_out_ag(X, Y)) → SORTED_IN_G(Y)
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))
SORTED_IN_G(.(X, .(Y, Z))) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U3_G(X, Y, Z, le_out_gg(X, Y)) → U4_G(X, Y, Z, sorted_in_g(.(Y, Z)))
U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
U7_GAA(x1, x2, x3, x4, x5)  =  U7_GAA(x1, x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
SORT_IN_AG(x1, x2)  =  SORT_IN_AG(x2)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x2, x3, x4)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)

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:

SORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SORT_IN_AG(X, Y) → PERM_IN_AG(X, Y)
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_GAA(U, .(X, Y), Z)
DELETE_IN_GAA(X, .(Y, Z), W) → U7_GAA(X, Y, Z, W, delete_in_gaa(X, Z, W))
DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_AG(X, Y, U, V, perm_in_ag(Z, V))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)
U1_AG(X, Y, perm_out_ag(X, Y)) → U2_AG(X, Y, sorted_in_g(Y))
U1_AG(X, Y, perm_out_ag(X, Y)) → SORTED_IN_G(Y)
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))
SORTED_IN_G(.(X, .(Y, Z))) → LE_IN_GG(X, Y)
LE_IN_GG(s(X), s(Y)) → U8_GG(X, Y, le_in_gg(X, Y))
LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
U3_G(X, Y, Z, le_out_gg(X, Y)) → U4_G(X, Y, Z, sorted_in_g(.(Y, Z)))
U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U2_AG(x1, x2, x3)  =  U2_AG(x2, x3)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
U7_GAA(x1, x2, x3, x4, x5)  =  U7_GAA(x1, x5)
U6_AG(x1, x2, x3, x4, x5)  =  U6_AG(x3, x4, x5)
SORT_IN_AG(x1, x2)  =  SORT_IN_AG(x2)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x2, x3, x4)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 4 SCCs with 10 less nodes.

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

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
LE_IN_GG(x1, x2)  =  LE_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
              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Pi is empty.
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
                        ↳ QDPSizeChangeProof
              ↳ PiDP
              ↳ PiDP
              ↳ PiDP

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

LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)
SORTED_IN_G(x1)  =  SORTED_IN_G(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
              ↳ PiDP

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

Pi is empty.
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
                        ↳ RuleRemovalProof
              ↳ PiDP
              ↳ PiDP

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))
SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

le_in_gg(x0, x1)
U8_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

SORTED_IN_G(.(X, .(Y, Z))) → U3_G(X, Y, Z, le_in_gg(X, Y))


Used ordering: POLO with Polynomial interpretation [25]:

POL(.(x1, x2)) = 1 + 2·x1 + x2   
POL(0) = 0   
POL(SORTED_IN_G(x1)) = 2·x1   
POL(U3_G(x1, x2, x3, x4)) = 2 + 2·x1 + 2·x2 + 2·x3 + x4   
POL(U8_gg(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(le_in_gg(x1, x2)) = x1 + 2·x2   
POL(le_out_gg(x1, x2)) = x1 + 2·x2   
POL(s(x1)) = 2·x1   



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

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

U3_G(X, Y, Z, le_out_gg(X, Y)) → SORTED_IN_G(.(Y, Z))

The TRS R consists of the following rules:

le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

le_in_gg(x0, x1)
U8_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.

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

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

DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

DELETE_IN_GAA(X, .(Y, Z), W) → DELETE_IN_GAA(X, Z, W)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ NonTerminationProof
              ↳ PiDP

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

DELETE_IN_GAA(X) → DELETE_IN_GAA(X)

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:

DELETE_IN_GAA(X) → DELETE_IN_GAA(X)

The TRS R consists of the following rules:none


s = DELETE_IN_GAA(X) evaluates to t =DELETE_IN_GAA(X)

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 DELETE_IN_GAA(X) to DELETE_IN_GAA(X).





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

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

PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)

The TRS R consists of the following rules:

sort_in_ag(X, Y) → U1_ag(X, Y, perm_in_ag(X, Y))
perm_in_ag([], []) → perm_out_ag([], [])
perm_in_ag(.(X, .(Y, [])), .(U, .(V, []))) → U5_ag(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)
U5_ag(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → U6_ag(X, Y, U, V, perm_in_ag(Z, V))
U6_ag(X, Y, U, V, perm_out_ag(Z, V)) → perm_out_ag(.(X, .(Y, [])), .(U, .(V, [])))
U1_ag(X, Y, perm_out_ag(X, Y)) → U2_ag(X, Y, sorted_in_g(Y))
sorted_in_g([]) → sorted_out_g([])
sorted_in_g(.(X, [])) → sorted_out_g(.(X, []))
sorted_in_g(.(X, .(Y, Z))) → U3_g(X, Y, Z, le_in_gg(X, Y))
le_in_gg(s(X), s(Y)) → U8_gg(X, Y, le_in_gg(X, Y))
le_in_gg(0, s(X)) → le_out_gg(0, s(X))
le_in_gg(0, 0) → le_out_gg(0, 0)
U8_gg(X, Y, le_out_gg(X, Y)) → le_out_gg(s(X), s(Y))
U3_g(X, Y, Z, le_out_gg(X, Y)) → U4_g(X, Y, Z, sorted_in_g(.(Y, Z)))
U4_g(X, Y, Z, sorted_out_g(.(Y, Z))) → sorted_out_g(.(X, .(Y, Z)))
U2_ag(X, Y, sorted_out_g(Y)) → sort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
sort_in_ag(x1, x2)  =  sort_in_ag(x2)
U1_ag(x1, x2, x3)  =  U1_ag(x2, x3)
perm_in_ag(x1, x2)  =  perm_in_ag(x2)
[]  =  []
perm_out_ag(x1, x2)  =  perm_out_ag(x2)
.(x1, x2)  =  .(x1, x2)
U5_ag(x1, x2, x3, x4, x5)  =  U5_ag(x3, x4, x5)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
U6_ag(x1, x2, x3, x4, x5)  =  U6_ag(x3, x4, x5)
U2_ag(x1, x2, x3)  =  U2_ag(x2, x3)
sorted_in_g(x1)  =  sorted_in_g(x1)
sorted_out_g(x1)  =  sorted_out_g(x1)
U3_g(x1, x2, x3, x4)  =  U3_g(x1, x2, x3, x4)
le_in_gg(x1, x2)  =  le_in_gg(x1, x2)
s(x1)  =  s(x1)
U8_gg(x1, x2, x3)  =  U8_gg(x1, x2, x3)
0  =  0
le_out_gg(x1, x2)  =  le_out_gg(x1, x2)
U4_g(x1, x2, x3, x4)  =  U4_g(x1, x2, x3, x4)
sort_out_ag(x1, x2)  =  sort_out_ag(x2)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(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
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

PERM_IN_AG(.(X, .(Y, [])), .(U, .(V, []))) → U5_AG(X, Y, U, V, delete_in_gaa(U, .(X, Y), Z))
U5_AG(X, Y, U, V, delete_out_gaa(U, .(X, Y), Z)) → PERM_IN_AG(Z, V)

The TRS R consists of the following rules:

delete_in_gaa(X, .(X, Y), Y) → delete_out_gaa(X, .(X, Y), Y)
delete_in_gaa(X, .(Y, Z), W) → U7_gaa(X, Y, Z, W, delete_in_gaa(X, Z, W))
U7_gaa(X, Y, Z, W, delete_out_gaa(X, Z, W)) → delete_out_gaa(X, .(Y, Z), W)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
delete_in_gaa(x1, x2, x3)  =  delete_in_gaa(x1)
delete_out_gaa(x1, x2, x3)  =  delete_out_gaa(x1)
U7_gaa(x1, x2, x3, x4, x5)  =  U7_gaa(x1, x5)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)

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

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

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

U5_AG(U, V, delete_out_gaa(U)) → PERM_IN_AG(V)
PERM_IN_AG(.(U, .(V, []))) → U5_AG(U, V, delete_in_gaa(U))

The TRS R consists of the following rules:

delete_in_gaa(X) → delete_out_gaa(X)
delete_in_gaa(X) → U7_gaa(X, delete_in_gaa(X))
U7_gaa(X, delete_out_gaa(X)) → delete_out_gaa(X)

The set Q consists of the following terms:

delete_in_gaa(x0)
U7_gaa(x0, x1)

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: