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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 188 ms)
↳2 TRIPLES
↳3 UndefinedPredicateInTriplesTransformerProof (⇒, 0 ms)
↳4 TRIPLES
↳5 TriplesToPiDPProof (⇒, 80 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔, 0 ms)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇒, 0 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔, 0 ms)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇒, 0 ms)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔, 0 ms)
        ↳22 YES

(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)).

Query: goal(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

s2tA(s(X1), node(X2, X3, X2)) :- s2tA(X1, X2).
s2tA(s(X1), node(nil, X2, X3)) :- s2tA(X1, X3).
s2tA(s(X1), node(X2, X3, nil)) :- s2tA(X1, X2).
treeB(node(X1, X2, X3)) :- treeB(X1).
treeB(node(X1, X2, X3)) :- ','(treecB(X1), treeB(X3)).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(X2)).
goalD(s(X1)) :- ','(s2tcA(X1, X2), ','(treecB(X2), treeB(X2))).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(node(nil, X3, X2))).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(node(X2, X3, nil))).
goalD(s(X1)) :- treeC.
goalD(s(X1)) :- ','(treecC, treeC).
goalD(0) :- treeC.

Clauses:

s2tcA(s(X1), node(X2, X3, X2)) :- s2tcA(X1, X2).
s2tcA(s(X1), node(nil, X2, X3)) :- s2tcA(X1, X3).
s2tcA(s(X1), node(X2, X3, nil)) :- s2tcA(X1, X2).
s2tcA(s(X1), node(nil, X2, nil)).
s2tcA(0, nil).
treecB(nil).
treecB(node(X1, X2, X3)) :- ','(treecB(X1), treecB(X3)).
treecC.

Afs:

goalD(x1)  =  goalD(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

s2tA(s(X1), node(X2, X3, X2)) :- s2tA(X1, X2).
s2tA(s(X1), node(nil, X2, X3)) :- s2tA(X1, X3).
s2tA(s(X1), node(X2, X3, nil)) :- s2tA(X1, X2).
treeB(node(X1, X2, X3)) :- treeB(X1).
treeB(node(X1, X2, X3)) :- ','(treecB(X1), treeB(X3)).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(X2)).
goalD(s(X1)) :- ','(s2tcA(X1, X2), ','(treecB(X2), treeB(X2))).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(node(nil, X3, X2))).
goalD(s(X1)) :- s2tA(X1, X2).
goalD(s(X1)) :- ','(s2tcA(X1, X2), treeB(node(X2, X3, nil))).

Clauses:

s2tcA(s(X1), node(X2, X3, X2)) :- s2tcA(X1, X2).
s2tcA(s(X1), node(nil, X2, X3)) :- s2tcA(X1, X3).
s2tcA(s(X1), node(X2, X3, nil)) :- s2tcA(X1, X2).
s2tcA(s(X1), node(nil, X2, nil)).
s2tcA(0, nil).
treecB(nil).
treecB(node(X1, X2, X3)) :- ','(treecB(X1), treecB(X3)).
treecC.

Afs:

goalD(x1)  =  goalD(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
goalD_in: (b)
s2tA_in: (b,f)
s2tcA_in: (b,f)
treeB_in: (b)
treecB_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOALD_IN_G(s(X1)) → U7_G(X1, s2tA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → U1_GA(X1, X2, X3, s2tA_in_ga(X1, X2))
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(nil, X2, X3)) → U2_GA(X1, X2, X3, s2tA_in_ga(X1, X3))
S2TA_IN_GA(s(X1), node(nil, X2, X3)) → S2TA_IN_GA(X1, X3)
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → U3_GA(X1, X2, X3, s2tA_in_ga(X1, X2))
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → S2TA_IN_GA(X1, X2)
GOALD_IN_G(s(X1)) → U8_G(X1, s2tcA_in_ga(X1, X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → U9_G(X1, treeB_in_g(X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(X2)
TREEB_IN_G(node(X1, X2, X3)) → U4_G(X1, X2, X3, treeB_in_g(X1))
TREEB_IN_G(node(X1, X2, X3)) → TREEB_IN_G(X1)
TREEB_IN_G(node(X1, X2, X3)) → U5_G(X1, X2, X3, treecB_in_g(X1))
U5_G(X1, X2, X3, treecB_out_g(X1)) → U6_G(X1, X2, X3, treeB_in_g(X3))
U5_G(X1, X2, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
U8_G(X1, s2tcA_out_ga(X1, X2)) → U10_G(X1, X2, treecB_in_g(X2))
U10_G(X1, X2, treecB_out_g(X2)) → U11_G(X1, treeB_in_g(X2))
U10_G(X1, X2, treecB_out_g(X2)) → TREEB_IN_G(X2)
U8_G(X1, s2tcA_out_ga(X1, X2)) → U12_G(X1, treeB_in_g(node(nil, X3, X2)))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(node(nil, X3, X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → U13_G(X1, treeB_in_g(node(X2, X3, nil)))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(node(X2, X3, nil))

The TRS R consists of the following rules:

s2tcA_in_ga(s(X1), node(X2, X3, X2)) → U15_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, X3)) → U16_ga(X1, X2, X3, s2tcA_in_ga(X1, X3))
s2tcA_in_ga(s(X1), node(X2, X3, nil)) → U17_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, nil)) → s2tcA_out_ga(s(X1), node(nil, X2, nil))
s2tcA_in_ga(0, nil) → s2tcA_out_ga(0, nil)
U17_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, nil))
U16_ga(X1, X2, X3, s2tcA_out_ga(X1, X3)) → s2tcA_out_ga(s(X1), node(nil, X2, X3))
U15_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, X2))
treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X2, X3)) → U18_g(X1, X2, X3, treecB_in_g(X1))
U18_g(X1, X2, X3, treecB_out_g(X1)) → U19_g(X1, X2, X3, treecB_in_g(X3))
U19_g(X1, X2, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tcA_in_ga(x1, x2)  =  s2tcA_in_ga(x1)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x1, x4)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
U17_ga(x1, x2, x3, x4)  =  U17_ga(x1, x4)
s2tcA_out_ga(x1, x2)  =  s2tcA_out_ga(x1, x2)
0  =  0
treeB_in_g(x1)  =  treeB_in_g(x1)
treecB_in_g(x1)  =  treecB_in_g(x1)
nil  =  nil
treecB_out_g(x1)  =  treecB_out_g(x1)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x3, x4)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x3, x4)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
TREEB_IN_G(x1)  =  TREEB_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x3, x4)
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x3, x4)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x3, x4)
U10_G(x1, x2, x3)  =  U10_G(x1, x2, x3)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

GOALD_IN_G(s(X1)) → U7_G(X1, s2tA_in_ga(X1, X2))
GOALD_IN_G(s(X1)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → U1_GA(X1, X2, X3, s2tA_in_ga(X1, X2))
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(nil, X2, X3)) → U2_GA(X1, X2, X3, s2tA_in_ga(X1, X3))
S2TA_IN_GA(s(X1), node(nil, X2, X3)) → S2TA_IN_GA(X1, X3)
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → U3_GA(X1, X2, X3, s2tA_in_ga(X1, X2))
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → S2TA_IN_GA(X1, X2)
GOALD_IN_G(s(X1)) → U8_G(X1, s2tcA_in_ga(X1, X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → U9_G(X1, treeB_in_g(X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(X2)
TREEB_IN_G(node(X1, X2, X3)) → U4_G(X1, X2, X3, treeB_in_g(X1))
TREEB_IN_G(node(X1, X2, X3)) → TREEB_IN_G(X1)
TREEB_IN_G(node(X1, X2, X3)) → U5_G(X1, X2, X3, treecB_in_g(X1))
U5_G(X1, X2, X3, treecB_out_g(X1)) → U6_G(X1, X2, X3, treeB_in_g(X3))
U5_G(X1, X2, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
U8_G(X1, s2tcA_out_ga(X1, X2)) → U10_G(X1, X2, treecB_in_g(X2))
U10_G(X1, X2, treecB_out_g(X2)) → U11_G(X1, treeB_in_g(X2))
U10_G(X1, X2, treecB_out_g(X2)) → TREEB_IN_G(X2)
U8_G(X1, s2tcA_out_ga(X1, X2)) → U12_G(X1, treeB_in_g(node(nil, X3, X2)))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(node(nil, X3, X2))
U8_G(X1, s2tcA_out_ga(X1, X2)) → U13_G(X1, treeB_in_g(node(X2, X3, nil)))
U8_G(X1, s2tcA_out_ga(X1, X2)) → TREEB_IN_G(node(X2, X3, nil))

The TRS R consists of the following rules:

s2tcA_in_ga(s(X1), node(X2, X3, X2)) → U15_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, X3)) → U16_ga(X1, X2, X3, s2tcA_in_ga(X1, X3))
s2tcA_in_ga(s(X1), node(X2, X3, nil)) → U17_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, nil)) → s2tcA_out_ga(s(X1), node(nil, X2, nil))
s2tcA_in_ga(0, nil) → s2tcA_out_ga(0, nil)
U17_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, nil))
U16_ga(X1, X2, X3, s2tcA_out_ga(X1, X3)) → s2tcA_out_ga(s(X1), node(nil, X2, X3))
U15_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, X2))
treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X2, X3)) → U18_g(X1, X2, X3, treecB_in_g(X1))
U18_g(X1, X2, X3, treecB_out_g(X1)) → U19_g(X1, X2, X3, treecB_in_g(X3))
U19_g(X1, X2, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
s2tA_in_ga(x1, x2)  =  s2tA_in_ga(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tcA_in_ga(x1, x2)  =  s2tcA_in_ga(x1)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x1, x4)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
U17_ga(x1, x2, x3, x4)  =  U17_ga(x1, x4)
s2tcA_out_ga(x1, x2)  =  s2tcA_out_ga(x1, x2)
0  =  0
treeB_in_g(x1)  =  treeB_in_g(x1)
treecB_in_g(x1)  =  treecB_in_g(x1)
nil  =  nil
treecB_out_g(x1)  =  treecB_out_g(x1)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x3, x4)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x3, x4)
GOALD_IN_G(x1)  =  GOALD_IN_G(x1)
U7_G(x1, x2)  =  U7_G(x1, x2)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U8_G(x1, x2)  =  U8_G(x1, x2)
U9_G(x1, x2)  =  U9_G(x1, x2)
TREEB_IN_G(x1)  =  TREEB_IN_G(x1)
U4_G(x1, x2, x3, x4)  =  U4_G(x1, x3, x4)
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x3, x4)
U6_G(x1, x2, x3, x4)  =  U6_G(x1, x3, x4)
U10_G(x1, x2, x3)  =  U10_G(x1, x2, x3)
U11_G(x1, x2)  =  U11_G(x1, x2)
U12_G(x1, x2)  =  U12_G(x1, x2)
U13_G(x1, x2)  =  U13_G(x1, x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 17 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

TREEB_IN_G(node(X1, X2, X3)) → U5_G(X1, X2, X3, treecB_in_g(X1))
U5_G(X1, X2, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
TREEB_IN_G(node(X1, X2, X3)) → TREEB_IN_G(X1)

The TRS R consists of the following rules:

s2tcA_in_ga(s(X1), node(X2, X3, X2)) → U15_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, X3)) → U16_ga(X1, X2, X3, s2tcA_in_ga(X1, X3))
s2tcA_in_ga(s(X1), node(X2, X3, nil)) → U17_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, nil)) → s2tcA_out_ga(s(X1), node(nil, X2, nil))
s2tcA_in_ga(0, nil) → s2tcA_out_ga(0, nil)
U17_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, nil))
U16_ga(X1, X2, X3, s2tcA_out_ga(X1, X3)) → s2tcA_out_ga(s(X1), node(nil, X2, X3))
U15_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, X2))
treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X2, X3)) → U18_g(X1, X2, X3, treecB_in_g(X1))
U18_g(X1, X2, X3, treecB_out_g(X1)) → U19_g(X1, X2, X3, treecB_in_g(X3))
U19_g(X1, X2, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tcA_in_ga(x1, x2)  =  s2tcA_in_ga(x1)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x1, x4)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
U17_ga(x1, x2, x3, x4)  =  U17_ga(x1, x4)
s2tcA_out_ga(x1, x2)  =  s2tcA_out_ga(x1, x2)
0  =  0
treecB_in_g(x1)  =  treecB_in_g(x1)
nil  =  nil
treecB_out_g(x1)  =  treecB_out_g(x1)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x3, x4)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x3, x4)
TREEB_IN_G(x1)  =  TREEB_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x3, x4)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

TREEB_IN_G(node(X1, X2, X3)) → U5_G(X1, X2, X3, treecB_in_g(X1))
U5_G(X1, X2, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
TREEB_IN_G(node(X1, X2, X3)) → TREEB_IN_G(X1)

The TRS R consists of the following rules:

treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X2, X3)) → U18_g(X1, X2, X3, treecB_in_g(X1))
U18_g(X1, X2, X3, treecB_out_g(X1)) → U19_g(X1, X2, X3, treecB_in_g(X3))
U19_g(X1, X2, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X2, X3))

The argument filtering Pi contains the following mapping:
node(x1, x2, x3)  =  node(x1, x3)
treecB_in_g(x1)  =  treecB_in_g(x1)
nil  =  nil
treecB_out_g(x1)  =  treecB_out_g(x1)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x3, x4)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x3, x4)
TREEB_IN_G(x1)  =  TREEB_IN_G(x1)
U5_G(x1, x2, x3, x4)  =  U5_G(x1, x3, x4)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

TREEB_IN_G(node(X1, X3)) → U5_G(X1, X3, treecB_in_g(X1))
U5_G(X1, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
TREEB_IN_G(node(X1, X3)) → TREEB_IN_G(X1)

The TRS R consists of the following rules:

treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X3)) → U18_g(X1, X3, treecB_in_g(X1))
U18_g(X1, X3, treecB_out_g(X1)) → U19_g(X1, X3, treecB_in_g(X3))
U19_g(X1, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X3))

The set Q consists of the following terms:

treecB_in_g(x0)
U18_g(x0, x1, x2)
U19_g(x0, x1, x2)

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

(14) 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:

  • U5_G(X1, X3, treecB_out_g(X1)) → TREEB_IN_G(X3)
    The graph contains the following edges 2 >= 1

  • TREEB_IN_G(node(X1, X3)) → TREEB_IN_G(X1)
    The graph contains the following edges 1 > 1

  • TREEB_IN_G(node(X1, X3)) → U5_G(X1, X3, treecB_in_g(X1))
    The graph contains the following edges 1 > 1, 1 > 2

(15) YES

(16) Obligation:

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

S2TA_IN_GA(s(X1), node(nil, X2, X3)) → S2TA_IN_GA(X1, X3)
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → S2TA_IN_GA(X1, X2)

The TRS R consists of the following rules:

s2tcA_in_ga(s(X1), node(X2, X3, X2)) → U15_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, X3)) → U16_ga(X1, X2, X3, s2tcA_in_ga(X1, X3))
s2tcA_in_ga(s(X1), node(X2, X3, nil)) → U17_ga(X1, X2, X3, s2tcA_in_ga(X1, X2))
s2tcA_in_ga(s(X1), node(nil, X2, nil)) → s2tcA_out_ga(s(X1), node(nil, X2, nil))
s2tcA_in_ga(0, nil) → s2tcA_out_ga(0, nil)
U17_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, nil))
U16_ga(X1, X2, X3, s2tcA_out_ga(X1, X3)) → s2tcA_out_ga(s(X1), node(nil, X2, X3))
U15_ga(X1, X2, X3, s2tcA_out_ga(X1, X2)) → s2tcA_out_ga(s(X1), node(X2, X3, X2))
treecB_in_g(nil) → treecB_out_g(nil)
treecB_in_g(node(X1, X2, X3)) → U18_g(X1, X2, X3, treecB_in_g(X1))
U18_g(X1, X2, X3, treecB_out_g(X1)) → U19_g(X1, X2, X3, treecB_in_g(X3))
U19_g(X1, X2, X3, treecB_out_g(X3)) → treecB_out_g(node(X1, X2, X3))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
s2tcA_in_ga(x1, x2)  =  s2tcA_in_ga(x1)
U15_ga(x1, x2, x3, x4)  =  U15_ga(x1, x4)
U16_ga(x1, x2, x3, x4)  =  U16_ga(x1, x4)
U17_ga(x1, x2, x3, x4)  =  U17_ga(x1, x4)
s2tcA_out_ga(x1, x2)  =  s2tcA_out_ga(x1, x2)
0  =  0
treecB_in_g(x1)  =  treecB_in_g(x1)
nil  =  nil
treecB_out_g(x1)  =  treecB_out_g(x1)
U18_g(x1, x2, x3, x4)  =  U18_g(x1, x3, x4)
U19_g(x1, x2, x3, x4)  =  U19_g(x1, x3, x4)
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

S2TA_IN_GA(s(X1), node(nil, X2, X3)) → S2TA_IN_GA(X1, X3)
S2TA_IN_GA(s(X1), node(X2, X3, X2)) → S2TA_IN_GA(X1, X2)
S2TA_IN_GA(s(X1), node(X2, X3, nil)) → S2TA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
node(x1, x2, x3)  =  node(x1, x3)
nil  =  nil
S2TA_IN_GA(x1, x2)  =  S2TA_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

S2TA_IN_GA(s(X1)) → S2TA_IN_GA(X1)

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

(21) 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:

  • S2TA_IN_GA(s(X1)) → S2TA_IN_GA(X1)
    The graph contains the following edges 1 > 1

(22) YES