Left Termination proof of /tmp/tmpR0aQJF/btree.pl

Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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: (f)
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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

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

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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

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_a(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_A(T)
TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → U2_A(L, X, R, tree_in_a(R))
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(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
0 = 0
U7_g(x1, x2) = U7_g(x2)
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(x4)
goal_out_g(x1) = goal_out_g
U5_GA(x1, x2, x3, x4) = U5_GA(x4)
U1_A(x1, x2, x3, x4) = U1_A(x4)
U2_A(x1, x2, x3, x4) = U2_A(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)
TREE_IN_A(x1) = TREE_IN_A
U7_G(x1, x2) = U7_G(x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

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_a(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_A(T)
TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → U2_A(L, X, R, tree_in_a(R))
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(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
0 = 0
U7_g(x1, x2) = U7_g(x2)
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(x4)
goal_out_g(x1) = goal_out_g
U1_A(x1, x2, x3, x4) = U1_A(x4)
TREE_IN_A(x1) = TREE_IN_A

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

  ↳ PrologToPiTRSProof

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

TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))

The argument filtering Pi contains the following mapping:
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(x4)
U1_A(x1, x2, x3, x4) = U1_A(x4)
TREE_IN_A(x1) = TREE_IN_A

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

                        ↳ Narrowing

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(tree_in_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

tree_in_a
U1_a(x0)
U2_a(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TREE_IN_A → U1_A(tree_in_a) at position [0] we obtained the following new rules:

TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

tree_in_a
U1_a(x0)
U2_a(x0)

We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

s = TREE_IN_A evaluates to t =TREE_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TREE_IN_A to TREE_IN_A.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(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
0 = 0
U7_g(x1, x2) = U7_g(x2)
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(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

  ↳ PrologToPiTRSProof

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)
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

  ↳ PrologToPiTRSProof

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:

S2T_IN_GA(s(X)) → S2T_IN_GA(X)
The graph contains the following edges 1 > 1

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: (f)
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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

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

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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

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_a(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_A(T)
TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → U2_A(L, X, R, tree_in_a(R))
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(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(x1, x2)
s2t_in_ga(x1, x2) = s2t_in_ga(x1)
s(x1) = s(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
s2t_out_ga(x1, x2) = s2t_out_ga(x1)
0 = 0
U7_g(x1, x2) = U7_g(x1, x2)
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(x4)
goal_out_g(x1) = goal_out_g(x1)
U5_GA(x1, x2, x3, x4) = U5_GA(x1, x4)
U1_A(x1, x2, x3, x4) = U1_A(x4)
U2_A(x1, x2, x3, x4) = U2_A(x4)
U4_GA(x1, x2, x3, x4) = U4_GA(x1, x4)
U6_G(x1, x2) = U6_G(x1, 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(x1, x4)
TREE_IN_A(x1) = TREE_IN_A
U7_G(x1, x2) = U7_G(x1, x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

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_a(T))
U6_G(X, s2t_out_ga(X, T)) → TREE_IN_A(T)
TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → U2_A(L, X, R, tree_in_a(R))
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(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(x1, x2)
s2t_in_ga(x1, x2) = s2t_in_ga(x1)
s(x1) = s(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x4)
U4_ga(x1, x2, x3, x4) = U4_ga(x1, x4)
U5_ga(x1, x2, x3, x4) = U5_ga(x1, x4)
s2t_out_ga(x1, x2) = s2t_out_ga(x1)
0 = 0
U7_g(x1, x2) = U7_g(x1, x2)
tree_in_a(x1) = tree_in_a
tree_out_a(x1) = tree_out_a
U1_a(x1, x2, x3, x4) = U1_a(x4)
U2_a(x1, x2, x3, x4) = U2_a(x4)
goal_out_g(x1) = goal_out_g(x1)
U1_A(x1, x2, x3, x4) = U1_A(x4)
TREE_IN_A(x1) = TREE_IN_A

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

TREE_IN_A(node(L, X, R)) → U1_A(L, X, R, tree_in_a(L))
TREE_IN_A(node(L, X, R)) → TREE_IN_A(L)
U1_A(L, X, R, tree_out_a(L)) → TREE_IN_A(R)

The TRS R consists of the following rules:

tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

              ↳ PiDP

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

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(tree_in_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

tree_in_a
U1_a(x0)
U2_a(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TREE_IN_A → U1_A(tree_in_a) at position [0] we obtained the following new rules:

TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ NonTerminationProof

              ↳ PiDP

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

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

The set Q consists of the following terms:

tree_in_a
U1_a(x0)
U2_a(x0)

TREE_IN_A → TREE_IN_A
TREE_IN_A → U1_A(U1_a(tree_in_a))
TREE_IN_A → U1_A(tree_out_a)
U1_A(tree_out_a) → TREE_IN_A

The TRS R consists of the following rules:

tree_in_a → tree_out_a
tree_in_a → U1_a(tree_in_a)
U1_a(tree_out_a) → U2_a(tree_in_a)
U2_a(tree_out_a) → tree_out_a

s = TREE_IN_A evaluates to t =TREE_IN_A

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from TREE_IN_A to TREE_IN_A.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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_a(T))
tree_in_a(nil) → tree_out_a(nil)
tree_in_a(node(L, X, R)) → U1_a(L, X, R, tree_in_a(L))
U1_a(L, X, R, tree_out_a(L)) → U2_a(L, X, R, tree_in_a(R))
U2_a(L, X, R, tree_out_a(R)) → tree_out_a(node(L, X, R))
U7_g(X, tree_out_a(T)) → goal_out_g(X)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ 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)

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

S2T_IN_GA(s(X)) → S2T_IN_GA(X)
The graph contains the following edges 1 > 1