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

(0) Obligation:

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).

(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)
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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)

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) → 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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)

(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) → 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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)
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)
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)
U7_G(x1, x2)  =  U7_G(x1, x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x1, x3, x4)

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) → 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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)
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)
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)
U7_G(x1, x2)  =  U7_G(x1, x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x1, x3, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

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

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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x3, x4)

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:

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

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(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x3, x4)

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:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

tree_in_g(x0)
U1_g(x0, x1, x2)
U2_g(x0, x1, x2)

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:

  • TREE_IN_G(node(L, R)) → U1_G(L, R, tree_in_g(L))
    The graph contains the following edges 1 > 1, 1 > 2

  • TREE_IN_G(node(L, R)) → TREE_IN_G(L)
    The graph contains the following edges 1 > 1

  • U1_G(L, R, tree_out_g(L)) → TREE_IN_G(R)
    The graph contains the following edges 2 >= 1

(13) TRUE

(14) Obligation:

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

S2T_IN_GA(s(X), node(nil, Y, T)) → 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(T, Y, nil)) → 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(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, x2)
node(x1, x2, x3)  =  node(x1, x3)
0  =  0
U7_g(x1, x2)  =  U7_g(x1, x2)
tree_in_g(x1)  =  tree_in_g(x1)
nil  =  nil
tree_out_g(x1)  =  tree_out_g(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x3, x4)
goal_out_g(x1)  =  goal_out_g(x1)
S2T_IN_GA(x1, x2)  =  S2T_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:

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

(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:

S2T_IN_GA(s(X)) → S2T_IN_GA(X)

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

(19) 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:

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

(20) TRUE

(21) 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)
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

(22) Obligation:

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

(23) 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) → 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
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(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)
U7_G(x1, x2)  =  U7_G(x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(24) Obligation:

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
GOAL_IN_G(x1)  =  GOAL_IN_G(x1)
U6_G(x1, x2)  =  U6_G(x2)
S2T_IN_GA(x1, x2)  =  S2T_IN_GA(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)
U7_G(x1, x2)  =  U7_G(x2)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 8 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

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

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
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

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

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)
TREE_IN_G(x1)  =  TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

(32) 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:

  • TREE_IN_G(node(L, R)) → U1_G(R, tree_in_g(L))
    The graph contains the following edges 1 > 1

  • TREE_IN_G(node(L, R)) → TREE_IN_G(L)
    The graph contains the following edges 1 > 1

  • U1_G(R, tree_out_g) → TREE_IN_G(R)
    The graph contains the following edges 1 >= 1

(33) TRUE

(34) Obligation:

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

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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.