Left Termination of the query pattern slowsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇔)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 UsableRulesReductionPairsProof (⇔)
        ↳20 QDP
        ↳21 DependencyGraphProof (⇔)
        ↳22 TRUE
      ↳23 PiDP
        ↳24 UsableRulesProof (⇔)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇐)
        ↳27 QDP
        ↳28 QDPSizeChangeProof (⇔)
        ↳29 TRUE
      ↳30 PiDP
        ↳31 UsableRulesProof (⇔)
        ↳32 PiDP
        ↳33 PiDPToQDPProof (⇐)
        ↳34 QDP
        ↳35 UsableRulesReductionPairsProof (⇔)
        ↳36 QDP
        ↳37 PisEmptyProof (⇔)
        ↳38 TRUE
  ↳39 PrologToPiTRSProof (⇐)
    ↳40 PiTRS
    ↳41 DependencyPairsProof (⇔)
    ↳42 PiDP
    ↳43 DependencyGraphProof (⇔)
    ↳44 AND
      ↳45 PiDP
        ↳46 UsableRulesProof (⇔)
        ↳47 PiDP
        ↳48 PiDPToQDPProof (⇔)
        ↳49 QDP
        ↳50 QDPSizeChangeProof (⇔)
        ↳51 TRUE
      ↳52 PiDP
        ↳53 UsableRulesProof (⇔)
        ↳54 PiDP
        ↳55 PiDPToQDPProof (⇔)
        ↳56 QDP
        ↳57 MRRProof (⇔)
        ↳58 QDP
        ↳59 DependencyGraphProof (⇔)
        ↳60 TRUE
      ↳61 PiDP
        ↳62 UsableRulesProof (⇔)
        ↳63 PiDP
        ↳64 PiDPToQDPProof (⇐)
        ↳65 QDP
        ↳66 QDPSizeChangeProof (⇔)
        ↳67 TRUE
      ↳68 PiDP
        ↳69 UsableRulesProof (⇔)
        ↳70 PiDP
        ↳71 PiDPToQDPProof (⇐)
        ↳72 QDP

(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(g,a).

(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: (b,f)
perm_in: (b,f)
delete_in: (f,b,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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)

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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)

(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_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
SLOWSORT_IN_GA(X, Y) → PERM_IN_GA(X, Y)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_AGA(U, .(X, Y), Z)
DELETE_IN_AGA(X, .(Y, Z), W) → U7_AGA(X, Y, Z, W, delete_in_aga(X, Z, W))
DELETE_IN_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_GA(X, Y, U, V, perm_in_ga(Z, V))
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
U1_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, sorted_in_g(Y))
U1_GA(X, Y, perm_out_ga(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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
SLOWSORT_IN_GA(x1, x2)  =  SLOWSORT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U7_AGA(x1, x2, x3, x4, x5)  =  U7_AGA(x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x2, 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_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
SLOWSORT_IN_GA(X, Y) → PERM_IN_GA(X, Y)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_AGA(U, .(X, Y), Z)
DELETE_IN_AGA(X, .(Y, Z), W) → U7_AGA(X, Y, Z, W, delete_in_aga(X, Z, W))
DELETE_IN_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_GA(X, Y, U, V, perm_in_ga(Z, V))
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
U1_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, sorted_in_g(Y))
U1_GA(X, Y, perm_out_ga(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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
SLOWSORT_IN_GA(x1, x2)  =  SLOWSORT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U7_AGA(x1, x2, x3, x4, x5)  =  U7_AGA(x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x2, 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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
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_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)

The TRS R consists of the following rules:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

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_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)

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

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_AGA(.(Y, Z)) → DELETE_IN_AGA(Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(28) 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:

  • DELETE_IN_AGA(.(Y, Z)) → DELETE_IN_AGA(Z)
    The graph contains the following edges 1 > 1

(29) TRUE

(30) Obligation:

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

U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))

The TRS R consists of the following rules:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x2, 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_ga(x1, x2)  =  slowsort_out_ga(x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(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_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))

The TRS R consists of the following rules:

delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(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_GA(delete_out_aga(U, Z)) → PERM_IN_GA(Z)
PERM_IN_GA(.(X, .(Y, []))) → U5_GA(delete_in_aga(.(X, Y)))

The TRS R consists of the following rules:

delete_in_aga(.(X, Y)) → delete_out_aga(X, Y)
delete_in_aga(.(Y, Z)) → U7_aga(delete_in_aga(Z))
U7_aga(delete_out_aga(X, W)) → delete_out_aga(X, W)

The set Q consists of the following terms:

delete_in_aga(x0)
U7_aga(x0)

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

(35) 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:

U5_GA(delete_out_aga(U, Z)) → PERM_IN_GA(Z)
PERM_IN_GA(.(X, .(Y, []))) → U5_GA(delete_in_aga(.(X, Y)))
The following rules are removed from R:

U7_aga(delete_out_aga(X, W)) → delete_out_aga(X, W)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + 2·x1 + x2   
POL(PERM_IN_GA(x1)) = x1   
POL(U5_GA(x1)) = 1 + x1   
POL(U7_aga(x1)) = 1 + x1   
POL([]) = 2   
POL(delete_in_aga(x1)) = x1   
POL(delete_out_aga(x1, x2)) = 1 + 2·x1 + x2   

(36) Obligation:

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

delete_in_aga(.(X, Y)) → delete_out_aga(X, Y)
delete_in_aga(.(Y, Z)) → U7_aga(delete_in_aga(Z))

The set Q consists of the following terms:

delete_in_aga(x0)
U7_aga(x0)

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

(37) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(38) TRUE

(39) 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: (b,f)
perm_in: (b,f)
delete_in: (f,b,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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(40) Obligation:

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

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)

(41) 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_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
SLOWSORT_IN_GA(X, Y) → PERM_IN_GA(X, Y)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_AGA(U, .(X, Y), Z)
DELETE_IN_AGA(X, .(Y, Z), W) → U7_AGA(X, Y, Z, W, delete_in_aga(X, Z, W))
DELETE_IN_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_GA(X, Y, U, V, perm_in_ga(Z, V))
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
U1_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, sorted_in_g(Y))
U1_GA(X, Y, perm_out_ga(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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)
SLOWSORT_IN_GA(x1, x2)  =  SLOWSORT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U7_AGA(x1, x2, x3, x4, x5)  =  U7_AGA(x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, 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

(42) Obligation:

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

SLOWSORT_IN_GA(X, Y) → U1_GA(X, Y, perm_in_ga(X, Y))
SLOWSORT_IN_GA(X, Y) → PERM_IN_GA(X, Y)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → DELETE_IN_AGA(U, .(X, Y), Z)
DELETE_IN_AGA(X, .(Y, Z), W) → U7_AGA(X, Y, Z, W, delete_in_aga(X, Z, W))
DELETE_IN_AGA(X, .(Y, Z), W) → DELETE_IN_AGA(X, Z, W)
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_GA(X, Y, U, V, perm_in_ga(Z, V))
U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
U1_GA(X, Y, perm_out_ga(X, Y)) → U2_GA(X, Y, sorted_in_g(Y))
U1_GA(X, Y, perm_out_ga(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_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)
SLOWSORT_IN_GA(x1, x2)  =  SLOWSORT_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)
U7_AGA(x1, x2, x3, x4, x5)  =  U7_AGA(x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3)  =  U2_GA(x1, 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

(43) DependencyGraphProof (EQUIVALENT transformation)

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

(44) Complex Obligation (AND)

(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)

The TRS R consists of the following rules:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)
LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) 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

(48) PiDPToQDPProof (EQUIVALENT transformation)

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

(49) 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.

(50) 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

(51) TRUE

(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:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, 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

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) 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

(55) PiDPToQDPProof (EQUIVALENT transformation)

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

(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))
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.

(57) 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   

(58) 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.

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(60) TRUE

(61) Obligation:

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

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

The TRS R consists of the following rules:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)
DELETE_IN_AGA(x1, x2, x3)  =  DELETE_IN_AGA(x2)

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

(62) UsableRulesProof (EQUIVALENT transformation)

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

(63) Obligation:

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

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

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

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

(64) PiDPToQDPProof (SOUND transformation)

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

(65) Obligation:

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

DELETE_IN_AGA(.(Y, Z)) → DELETE_IN_AGA(Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(66) 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:

  • DELETE_IN_AGA(.(Y, Z)) → DELETE_IN_AGA(Z)
    The graph contains the following edges 1 > 1

(67) TRUE

(68) Obligation:

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

U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))

The TRS R consists of the following rules:

slowsort_in_ga(X, Y) → U1_ga(X, Y, perm_in_ga(X, Y))
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, .(Y, [])), .(U, .(V, []))) → U5_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)
U5_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → U6_ga(X, Y, U, V, perm_in_ga(Z, V))
U6_ga(X, Y, U, V, perm_out_ga(Z, V)) → perm_out_ga(.(X, .(Y, [])), .(U, .(V, [])))
U1_ga(X, Y, perm_out_ga(X, Y)) → U2_ga(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_ga(X, Y, sorted_out_g(Y)) → slowsort_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
slowsort_in_ga(x1, x2)  =  slowsort_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
perm_in_ga(x1, x2)  =  perm_in_ga(x1)
[]  =  []
perm_out_ga(x1, x2)  =  perm_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U5_ga(x1, x2, x3, x4, x5)  =  U5_ga(x1, x2, x5)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x1, x2, x3, x5)
U2_ga(x1, x2, x3)  =  U2_ga(x1, 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_ga(x1, x2)  =  slowsort_out_ga(x1, x2)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)

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

(69) UsableRulesProof (EQUIVALENT transformation)

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

(70) Obligation:

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

U5_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z, V)
PERM_IN_GA(.(X, .(Y, [])), .(U, .(V, []))) → U5_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), Z))

The TRS R consists of the following rules:

delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(X, .(Y, Z), W) → U7_aga(X, Y, Z, W, delete_in_aga(X, Z, W))
U7_aga(X, Y, Z, W, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
delete_in_aga(x1, x2, x3)  =  delete_in_aga(x2)
delete_out_aga(x1, x2, x3)  =  delete_out_aga(x1, x2, x3)
U7_aga(x1, x2, x3, x4, x5)  =  U7_aga(x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)

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

(71) PiDPToQDPProof (SOUND transformation)

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

(72) Obligation:

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

U5_GA(X, Y, delete_out_aga(U, .(X, Y), Z)) → PERM_IN_GA(Z)
PERM_IN_GA(.(X, .(Y, []))) → U5_GA(X, Y, delete_in_aga(.(X, Y)))

The TRS R consists of the following rules:

delete_in_aga(.(X, Y)) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(.(Y, Z)) → U7_aga(Y, Z, delete_in_aga(Z))
U7_aga(Y, Z, delete_out_aga(X, Z, W)) → delete_out_aga(X, .(Y, Z), W)

The set Q consists of the following terms:

delete_in_aga(x0)
U7_aga(x0, x1, x2)

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