Left Termination of the query pattern slowsort_in_2(a, g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

slowsort(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:

slowsort(a,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
slowsort_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SLOWSORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SLOWSORT_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag
SLOWSORT_IN_AG(x1, x2)  =  SLOWSORT_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x4, x5)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
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)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x3)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)

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

(4) Obligation:

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

SLOWSORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SLOWSORT_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag
SLOWSORT_IN_AG(x1, x2)  =  SLOWSORT_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x4, x5)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)
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)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x3)
U4_G(x1, x2, x3, x4)  =  U4_G(x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U3_G(x1, x2, x3, x4)  =  U3_G(x2, x3, x4)

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

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

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.

(19) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] 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:

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 [POLO]:

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

(20) Obligation:

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))

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.

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(22) TRUE

(23) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)

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

(24) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(25) Obligation:

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

(26) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(27) Obligation:

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.

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

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

(29) FALSE

(30) Obligation:

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

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

The TRS R consists of the following rules:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_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

(31) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(32) Obligation:

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

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

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

(33) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(34) Obligation:

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.

(35) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• PERM_IN_AG(.(U, .(V, []))) → U5_AG(V, delete_in_gaa(U))
  The graph contains the following edges 1 > 1

• U5_AG(V, delete_out_gaa) → PERM_IN_AG(V)
  The graph contains the following edges 1 >= 1

(36) TRUE

(37) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
slowsort_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(38) Obligation:

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

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)

(39) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SLOWSORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SLOWSORT_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)
SLOWSORT_IN_AG(x1, x2)  =  SLOWSORT_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_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)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x2, x3, x4)

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

(40) Obligation:

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

SLOWSORT_IN_AG(X, Y) → U1_AG(X, Y, perm_in_ag(X, Y))
SLOWSORT_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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)
SLOWSORT_IN_AG(x1, x2)  =  SLOWSORT_IN_AG(x2)
U1_AG(x1, x2, x3)  =  U1_AG(x2, x3)
PERM_IN_AG(x1, x2)  =  PERM_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_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)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x2, x3, x4)

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

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 10 less nodes.

(42) Complex Obligation (AND)

(43) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(44) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(45) Obligation:

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

(46) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(47) Obligation:

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.

(48) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• LE_IN_GG(s(X), s(Y)) → LE_IN_GG(X, Y)
  The graph contains the following edges 1 > 1, 2 > 2

(49) TRUE

(50) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)
SORTED_IN_G(x1)  =  SORTED_IN_G(x1)
U3_G(x1, x2, x3, x4)  =  U3_G(x1, x2, x3, x4)

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

(51) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(52) Obligation:

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

(53) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(54) Obligation:

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.

(55) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, 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: Polynomial interpretation [POLO]:

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   

(56) Obligation:

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.

(57) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(58) TRUE

(59) Obligation:

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:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_out_ag(x2)
DELETE_IN_GAA(x1, x2, x3)  =  DELETE_IN_GAA(x1)

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

(60) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(61) Obligation:

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

(62) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(63) Obligation:

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.

(64) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

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:

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

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

(65) FALSE

(66) Obligation:

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

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

The TRS R consists of the following rules:

slowsort_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)) → slowsort_out_ag(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ag(x1, x2)  =  slowsort_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)
slowsort_out_ag(x1, x2)  =  slowsort_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

(67) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(68) Obligation:

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

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

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

(69) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(70) Obligation:

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.

(71) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

• PERM_IN_AG(.(U, .(V, []))) → U5_AG(U, V, delete_in_gaa(U))
  The graph contains the following edges 1 > 1, 1 > 2

• U5_AG(U, V, delete_out_gaa(U)) → PERM_IN_AG(V)
  The graph contains the following edges 2 >= 1

(72) TRUE