Left Termination of the query pattern goal_in_1(g) 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 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 AND
      ↳25 PiDP
        ↳26 UsableRulesProof (⇔)
        ↳27 PiDP
        ↳28 PiDPToQDPProof (⇐)
        ↳29 QDP
        ↳30 QDPSizeChangeProof (⇔)
        ↳31 TRUE
      ↳32 PiDP
        ↳33 UsableRulesProof (⇔)
        ↳34 PiDP
        ↳35 PiDPToQDPProof (⇐)
        ↳36 QDP
        ↳37 QDPSizeChangeProof (⇔)
        ↳38 TRUE

(0) Obligation:

Clauses:

list([]).
list(.(X, XS)) :- list(XS).
s2l(s(X), .(Y, Xs)) :- s2l(X, Xs).
s2l(0, []).
goal(X) :- ','(s2l(X, XS), list(XS)).

Queries:

goal(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:
goal_in: (b)
s2l_in: (b,f)
list_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g

(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:

GOAL_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, list_in_g(XS))
U3_G(X, s2l_out_ga(X, XS)) → LIST_IN_G(XS)
LIST_IN_G(.(X, XS)) → U1_G(X, XS, list_in_g(XS))
LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x3)

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

(4) Obligation:

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

GOAL_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, list_in_g(XS))
U3_G(X, s2l_out_ga(X, XS)) → LIST_IN_G(XS)
LIST_IN_G(.(X, XS)) → U1_G(X, XS, list_in_g(XS))
LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
U4_G(x1, x2)  =  U4_G(x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g
LIST_IN_G(x1)  =  LIST_IN_G(x1)

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:

LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

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

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

(10) PiDPToQDPProof (SOUND 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:

LIST_IN_G(.(XS)) → LIST_IN_G(XS)

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:

  • LIST_IN_G(.(XS)) → LIST_IN_G(XS)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g
U1_g(x1, x2, x3)  =  U1_g(x3)
goal_out_g(x1)  =  goal_out_g
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)

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:

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)

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:

S2L_IN_GA(s(X)) → S2L_IN_GA(X)

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

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2l_in: (b,f)
list_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)

(21) 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:

GOAL_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, list_in_g(XS))
U3_G(X, s2l_out_ga(X, XS)) → LIST_IN_G(XS)
LIST_IN_G(.(X, XS)) → U1_G(X, XS, list_in_g(XS))
LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x2, x3)

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

(22) Obligation:

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

GOAL_IN_G(X) → U3_G(X, s2l_in_ga(X, XS))
GOAL_IN_G(X) → S2L_IN_GA(X, XS)
S2L_IN_GA(s(X), .(Y, Xs)) → U2_GA(X, Y, Xs, s2l_in_ga(X, Xs))
S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)
U3_G(X, s2l_out_ga(X, XS)) → U4_G(X, list_in_g(XS))
U3_G(X, s2l_out_ga(X, XS)) → LIST_IN_G(XS)
LIST_IN_G(.(X, XS)) → U1_G(X, XS, list_in_g(XS))
LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U4_G(x1, x2)  =  U4_G(x1, x2)
LIST_IN_G(x1)  =  LIST_IN_G(x1)
U1_G(x1, x2, x3)  =  U1_G(x2, x3)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)
LIST_IN_G(x1)  =  LIST_IN_G(x1)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

LIST_IN_G(.(X, XS)) → LIST_IN_G(XS)

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

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

LIST_IN_G(.(XS)) → LIST_IN_G(XS)

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

(30) 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:

  • LIST_IN_G(.(XS)) → LIST_IN_G(XS)
    The graph contains the following edges 1 > 1

(31) TRUE

(32) Obligation:

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

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

The TRS R consists of the following rules:

goal_in_g(X) → U3_g(X, s2l_in_ga(X, XS))
s2l_in_ga(s(X), .(Y, Xs)) → U2_ga(X, Y, Xs, s2l_in_ga(X, Xs))
s2l_in_ga(0, []) → s2l_out_ga(0, [])
U2_ga(X, Y, Xs, s2l_out_ga(X, Xs)) → s2l_out_ga(s(X), .(Y, Xs))
U3_g(X, s2l_out_ga(X, XS)) → U4_g(X, list_in_g(XS))
list_in_g([]) → list_out_g([])
list_in_g(.(X, XS)) → U1_g(X, XS, list_in_g(XS))
U1_g(X, XS, list_out_g(XS)) → list_out_g(.(X, XS))
U4_g(X, list_out_g(XS)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
s2l_in_ga(x1, x2)  =  s2l_in_ga(x1)
s(x1)  =  s(x1)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x4)
0  =  0
s2l_out_ga(x1, x2)  =  s2l_out_ga(x1, x2)
.(x1, x2)  =  .(x2)
U4_g(x1, x2)  =  U4_g(x1, x2)
list_in_g(x1)  =  list_in_g(x1)
[]  =  []
list_out_g(x1)  =  list_out_g(x1)
U1_g(x1, x2, x3)  =  U1_g(x2, x3)
goal_out_g(x1)  =  goal_out_g(x1)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)

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

(33) UsableRulesProof (EQUIVALENT transformation)

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

(34) Obligation:

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

S2L_IN_GA(s(X), .(Y, Xs)) → S2L_IN_GA(X, Xs)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L_IN_GA(x1, x2)  =  S2L_IN_GA(x1)

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

(35) PiDPToQDPProof (SOUND transformation)

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

(36) Obligation:

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

S2L_IN_GA(s(X)) → S2L_IN_GA(X)

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

(37) 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:

  • S2L_IN_GA(s(X)) → S2L_IN_GA(X)
    The graph contains the following edges 1 > 1

(38) TRUE