Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

tree(nil).
tree(node(L, X, R)) :- ','(tree(L), tree(R)).
s2t(s(X), node(T, Y, T)) :- s2t(X, T).
s2t(s(X), node(nil, Y, T)) :- s2t(X, T).
s2t(s(X), node(T, Y, nil)) :- s2t(X, T).
s2t(s(X), node(nil, Y, nil)).
s2t(0, nil).
goal(X) :- ','(s2t(X, T), tree(T)).

Queries:

goal(g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
goal_in: (b)
s2t_in: (b,f)
tree_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) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

goal_in_g(X) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g


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

GOAL_IN_G(X) → U6_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U3_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U4_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U5_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U6_G(X, s2t_out_ga(X, T)) → U7_G(X, tree_in_g(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(node(L, X, R)) → U1_G(L, X, R, tree_in_g(L))
TREE_IN_G(node(L, X, R)) → TREE_IN_G(L)
U1_G(L, X, R, tree_out_g(L)) → U2_G(L, X, R, tree_in_g(R))
U1_G(L, X, R, tree_out_g(L)) → TREE_IN_G(R)

The TRS R consists of the following rules:

goal_in_g(X) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
U6_G(x1, x2)  =  U6_G(x2)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U7_G(x1, x2)  =  U7_G(x2)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)
TREE_IN_G(x1)  =  TREE_IN_G(x1)

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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

GOAL_IN_G(X) → U6_G(X, s2t_in_ga(X, T))
GOAL_IN_G(X) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → U3_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → U4_GA(X, Y, T, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, nil)) → U5_GA(X, T, Y, s2t_in_ga(X, T))
S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
U6_G(X, s2t_out_ga(X, T)) → U7_G(X, tree_in_g(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_G(T)
TREE_IN_G(node(L, X, R)) → U1_G(L, X, R, tree_in_g(L))
TREE_IN_G(node(L, X, R)) → TREE_IN_G(L)
U1_G(L, X, R, tree_out_g(L)) → U2_G(L, X, R, tree_in_g(R))
U1_G(L, X, R, tree_out_g(L)) → TREE_IN_G(R)

The TRS R consists of the following rules:

goal_in_g(X) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g
U5_GA(x1, x2, x3, x4)  =  U5_GA(x4)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)
U6_G(x1, x2)  =  U6_G(x2)
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x4)
U7_G(x1, x2)  =  U7_G(x2)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)
TREE_IN_G(x1)  =  TREE_IN_G(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 8 less nodes.

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

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

TREE_IN_G(node(L, X, R)) → U1_G(L, X, R, tree_in_g(L))
TREE_IN_G(node(L, X, R)) → TREE_IN_G(L)
U1_G(L, X, R, tree_out_g(L)) → TREE_IN_G(R)

The TRS R consists of the following rules:

goal_in_g(X) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
TREE_IN_G(x1)  =  TREE_IN_G(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

TREE_IN_G(node(L, X, R)) → U1_G(L, X, R, tree_in_g(L))
TREE_IN_G(node(L, X, R)) → TREE_IN_G(L)
U1_G(L, X, R, tree_out_g(L)) → TREE_IN_G(R)

The TRS R consists of the following rules:

tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))

The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x3)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
TREE_IN_G(x1)  =  TREE_IN_G(x1)

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

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

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

U1_G(R, tree_out_g) → TREE_IN_G(R)
TREE_IN_G(node(L, R)) → TREE_IN_G(L)
TREE_IN_G(node(L, R)) → U1_G(R, tree_in_g(L))

The TRS R consists of the following rules:

tree_in_g(nil) → tree_out_g
tree_in_g(node(L, R)) → U1_g(R, tree_in_g(L))
U1_g(R, tree_out_g) → U2_g(tree_in_g(R))
U2_g(tree_out_g) → tree_out_g

The set Q consists of the following terms:

tree_in_g(x0)
U1_g(x0, x1)
U2_g(x0)

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

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



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

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

S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)

The TRS R consists of the following rules:

goal_in_g(X) → U6_g(X, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, T)) → U3_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, T)) → U4_ga(X, Y, T, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(T, Y, nil)) → U5_ga(X, T, Y, s2t_in_ga(X, T))
s2t_in_ga(s(X), node(nil, Y, nil)) → s2t_out_ga(s(X), node(nil, Y, nil))
s2t_in_ga(0, nil) → s2t_out_ga(0, nil)
U5_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, nil))
U4_ga(X, Y, T, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(nil, Y, T))
U3_ga(X, T, Y, s2t_out_ga(X, T)) → s2t_out_ga(s(X), node(T, Y, T))
U6_g(X, s2t_out_ga(X, T)) → U7_g(X, tree_in_g(T))
tree_in_g(nil) → tree_out_g(nil)
tree_in_g(node(L, X, R)) → U1_g(L, X, R, tree_in_g(L))
U1_g(L, X, R, tree_out_g(L)) → U2_g(L, X, R, tree_in_g(R))
U2_g(L, X, R, tree_out_g(R)) → tree_out_g(node(L, X, R))
U7_g(X, tree_out_g(T)) → goal_out_g(X)

The argument filtering Pi contains the following mapping:
goal_in_g(x1)  =  goal_in_g(x1)
U6_g(x1, x2)  =  U6_g(x2)
s2t_in_ga(x1, x2)  =  s2t_in_ga(x1)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x4)
U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)
U5_ga(x1, x2, x3, x4)  =  U5_ga(x4)
s2t_out_ga(x1, x2)  =  s2t_out_ga(x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
goal_out_g(x1)  =  goal_out_g
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

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

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

S2T_IN_GA(s(X), node(T, Y, nil)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(T, Y, T)) → S2T_IN_GA(X, T)
S2T_IN_GA(s(X), node(nil, Y, T)) → S2T_IN_GA(X, T)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(x1)

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

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

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

S2T_IN_GA(s(X)) → S2T_IN_GA(X)

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

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