Left Termination of the query pattern sameleaves_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 QDP
        ↳21 UsableRulesProof (⇔)
        ↳22 QDP
        ↳23 QReductionProof (⇔)
        ↳24 QDP
        ↳25 QDPSizeChangeProof (⇔)
        ↳26 YES

(0) Obligation:

Clauses:

sameleaves(leaf(L), leaf(L)).
sameleaves(tree(T1, T2), tree(S1, S2)) :- ','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S))).
getleave(leaf(A), C, A, C).
getleave(tree(A, B), C, L, O) :- getleave(A, tree(B, C), L, O).

Queries:

sameleaves(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

getleave14(tree(T52, T53), T54, T55, X72) :- getleave14(T52, tree(T53, T54), T55, X72).
p7(leaf(T23), T24, T23, T24, T13, T14, X18) :- getleave14(T13, T14, T23, X18).
p7(leaf(T23), T24, T23, T24, T13, T14, T29) :- ','(getleavec14(T13, T14, T23, T29), sameleaves1(T24, T29)).
p7(tree(T70, T71), T72, X103, X104, T13, T14, X18) :- p7(T70, tree(T71, T72), X103, X104, T13, T14, X18).
sameleaves1(tree(T11, T12), tree(T13, T14)) :- p7(T11, T12, X16, X17, T13, T14, X18).

Clauses:

sameleavesc1(leaf(T6), leaf(T6)).
sameleavesc1(tree(T11, T12), tree(T13, T14)) :- qc7(T11, T12, X16, X17, T13, T14, X18).
getleavec14(leaf(T42), T43, T42, T43).
getleavec14(tree(T52, T53), T54, T55, X72) :- getleavec14(T52, tree(T53, T54), T55, X72).
qc7(leaf(T23), T24, T23, T24, T13, T14, T29) :- ','(getleavec14(T13, T14, T23, T29), sameleavesc1(T24, T29)).
qc7(tree(T70, T71), T72, X103, X104, T13, T14, X18) :- qc7(T70, tree(T71, T72), X103, X104, T13, T14, X18).

Afs:

sameleaves1(x1, x2)  =  sameleaves1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
sameleaves1_in: (b,b)
p7_in: (b,b,f,f,b,b,f)
getleave14_in: (b,b,b,f)
getleavec14_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → U6_GG(T11, T12, T13, T14, p7_in_ggaagga(T11, T12, X16, X17, T13, T14, X18))
SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → P7_IN_GGAAGGA(T11, T12, X16, X17, T13, T14, X18)
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, X18) → U2_GGAAGGA(T23, T24, T13, T14, X18, getleave14_in_ggga(T13, T14, T23, X18))
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, X18) → GETLEAVE14_IN_GGGA(T13, T14, T23, X18)
GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → U1_GGGA(T52, T53, T54, T55, X72, getleave14_in_ggga(T52, tree(T53, T54), T55, X72))
GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55, X72)
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, T29) → U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_in_ggga(T13, T14, T23, T29))
U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_out_ggga(T13, T14, T23, T29)) → U4_GGAAGGA(T23, T24, T13, T14, T29, sameleaves1_in_gg(T24, T29))
U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_out_ggga(T13, T14, T23, T29)) → SAMELEAVES1_IN_GG(T24, T29)
P7_IN_GGAAGGA(tree(T70, T71), T72, X103, X104, T13, T14, X18) → U5_GGAAGGA(T70, T71, T72, X103, X104, T13, T14, X18, p7_in_ggaagga(T70, tree(T71, T72), X103, X104, T13, T14, X18))
P7_IN_GGAAGGA(tree(T70, T71), T72, X103, X104, T13, T14, X18) → P7_IN_GGAAGGA(T70, tree(T71, T72), X103, X104, T13, T14, X18)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42, T43) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55, X72) → U9_ggga(T52, T53, T54, T55, X72, getleavec14_in_ggga(T52, tree(T53, T54), T55, X72))
U9_ggga(T52, T53, T54, T55, X72, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The argument filtering Pi contains the following mapping:
sameleaves1_in_gg(x1, x2)  =  sameleaves1_in_gg(x1, x2)
tree(x1, x2)  =  tree(x1, x2)
p7_in_ggaagga(x1, x2, x3, x4, x5, x6, x7)  =  p7_in_ggaagga(x1, x2, x5, x6)
leaf(x1)  =  leaf(x1)
getleave14_in_ggga(x1, x2, x3, x4)  =  getleave14_in_ggga(x1, x2, x3)
getleavec14_in_ggga(x1, x2, x3, x4)  =  getleavec14_in_ggga(x1, x2, x3)
getleavec14_out_ggga(x1, x2, x3, x4)  =  getleavec14_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVES1_IN_GG(x1, x2)  =  SAMELEAVES1_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
P7_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  P7_IN_GGAAGGA(x1, x2, x5, x6)
U2_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGA(x1, x2, x3, x4, x6)
GETLEAVE14_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE14_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)
U4_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_GGAAGGA(x1, x2, x3, x6, x7, x9)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → U6_GG(T11, T12, T13, T14, p7_in_ggaagga(T11, T12, X16, X17, T13, T14, X18))
SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → P7_IN_GGAAGGA(T11, T12, X16, X17, T13, T14, X18)
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, X18) → U2_GGAAGGA(T23, T24, T13, T14, X18, getleave14_in_ggga(T13, T14, T23, X18))
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, X18) → GETLEAVE14_IN_GGGA(T13, T14, T23, X18)
GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → U1_GGGA(T52, T53, T54, T55, X72, getleave14_in_ggga(T52, tree(T53, T54), T55, X72))
GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55, X72)
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, T29) → U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_in_ggga(T13, T14, T23, T29))
U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_out_ggga(T13, T14, T23, T29)) → U4_GGAAGGA(T23, T24, T13, T14, T29, sameleaves1_in_gg(T24, T29))
U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_out_ggga(T13, T14, T23, T29)) → SAMELEAVES1_IN_GG(T24, T29)
P7_IN_GGAAGGA(tree(T70, T71), T72, X103, X104, T13, T14, X18) → U5_GGAAGGA(T70, T71, T72, X103, X104, T13, T14, X18, p7_in_ggaagga(T70, tree(T71, T72), X103, X104, T13, T14, X18))
P7_IN_GGAAGGA(tree(T70, T71), T72, X103, X104, T13, T14, X18) → P7_IN_GGAAGGA(T70, tree(T71, T72), X103, X104, T13, T14, X18)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42, T43) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55, X72) → U9_ggga(T52, T53, T54, T55, X72, getleavec14_in_ggga(T52, tree(T53, T54), T55, X72))
U9_ggga(T52, T53, T54, T55, X72, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The argument filtering Pi contains the following mapping:
sameleaves1_in_gg(x1, x2)  =  sameleaves1_in_gg(x1, x2)
tree(x1, x2)  =  tree(x1, x2)
p7_in_ggaagga(x1, x2, x3, x4, x5, x6, x7)  =  p7_in_ggaagga(x1, x2, x5, x6)
leaf(x1)  =  leaf(x1)
getleave14_in_ggga(x1, x2, x3, x4)  =  getleave14_in_ggga(x1, x2, x3)
getleavec14_in_ggga(x1, x2, x3, x4)  =  getleavec14_in_ggga(x1, x2, x3)
getleavec14_out_ggga(x1, x2, x3, x4)  =  getleavec14_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVES1_IN_GG(x1, x2)  =  SAMELEAVES1_IN_GG(x1, x2)
U6_GG(x1, x2, x3, x4, x5)  =  U6_GG(x1, x2, x3, x4, x5)
P7_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  P7_IN_GGAAGGA(x1, x2, x5, x6)
U2_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U2_GGAAGGA(x1, x2, x3, x4, x6)
GETLEAVE14_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE14_IN_GGGA(x1, x2, x3)
U1_GGGA(x1, x2, x3, x4, x5, x6)  =  U1_GGGA(x1, x2, x3, x4, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)
U4_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAGGA(x1, x2, x3, x4, x5, x6)
U5_GGAAGGA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U5_GGAAGGA(x1, x2, x3, x6, x7, x9)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55, X72)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42, T43) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55, X72) → U9_ggga(T52, T53, T54, T55, X72, getleavec14_in_ggga(T52, tree(T53, T54), T55, X72))
U9_ggga(T52, T53, T54, T55, X72, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
leaf(x1)  =  leaf(x1)
getleavec14_in_ggga(x1, x2, x3, x4)  =  getleavec14_in_ggga(x1, x2, x3)
getleavec14_out_ggga(x1, x2, x3, x4)  =  getleavec14_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
GETLEAVE14_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE14_IN_GGGA(x1, x2, x3)

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:

GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55, X72) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55, X72)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
GETLEAVE14_IN_GGGA(x1, x2, x3, x4)  =  GETLEAVE14_IN_GGGA(x1, x2, x3)

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:

GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

  • GETLEAVE14_IN_GGGA(tree(T52, T53), T54, T55) → GETLEAVE14_IN_GGGA(T52, tree(T53, T54), T55)
    The graph contains the following edges 1 > 1, 3 >= 3

(13) YES

(14) Obligation:

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

SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → P7_IN_GGAAGGA(T11, T12, X16, X17, T13, T14, X18)
P7_IN_GGAAGGA(leaf(T23), T24, T23, T24, T13, T14, T29) → U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_in_ggga(T13, T14, T23, T29))
U3_GGAAGGA(T23, T24, T13, T14, T29, getleavec14_out_ggga(T13, T14, T23, T29)) → SAMELEAVES1_IN_GG(T24, T29)
P7_IN_GGAAGGA(tree(T70, T71), T72, X103, X104, T13, T14, X18) → P7_IN_GGAAGGA(T70, tree(T71, T72), X103, X104, T13, T14, X18)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42, T43) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55, X72) → U9_ggga(T52, T53, T54, T55, X72, getleavec14_in_ggga(T52, tree(T53, T54), T55, X72))
U9_ggga(T52, T53, T54, T55, X72, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The argument filtering Pi contains the following mapping:
tree(x1, x2)  =  tree(x1, x2)
leaf(x1)  =  leaf(x1)
getleavec14_in_ggga(x1, x2, x3, x4)  =  getleavec14_in_ggga(x1, x2, x3)
getleavec14_out_ggga(x1, x2, x3, x4)  =  getleavec14_out_ggga(x1, x2, x3, x4)
U9_ggga(x1, x2, x3, x4, x5, x6)  =  U9_ggga(x1, x2, x3, x4, x6)
SAMELEAVES1_IN_GG(x1, x2)  =  SAMELEAVES1_IN_GG(x1, x2)
P7_IN_GGAAGGA(x1, x2, x3, x4, x5, x6, x7)  =  P7_IN_GGAAGGA(x1, x2, x5, x6)
U3_GGAAGGA(x1, x2, x3, x4, x5, x6)  =  U3_GGAAGGA(x1, x2, x3, x4, x6)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → P7_IN_GGAAGGA(T11, T12, T13, T14)
P7_IN_GGAAGGA(leaf(T23), T24, T13, T14) → U3_GGAAGGA(T23, T24, T13, T14, getleavec14_in_ggga(T13, T14, T23))
U3_GGAAGGA(T23, T24, T13, T14, getleavec14_out_ggga(T13, T14, T23, T29)) → SAMELEAVES1_IN_GG(T24, T29)
P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55) → U9_ggga(T52, T53, T54, T55, getleavec14_in_ggga(T52, tree(T53, T54), T55))
U9_ggga(T52, T53, T54, T55, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The set Q consists of the following terms:

getleavec14_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SAMELEAVES1_IN_GG(tree(T11, T12), tree(T13, T14)) → P7_IN_GGAAGGA(T11, T12, T13, T14)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(P7_IN_GGAAGGA(x1, x2, x3, x4)) = x1 + x2   
POL(SAMELEAVES1_IN_GG(x1, x2)) = x1   
POL(U3_GGAAGGA(x1, x2, x3, x4, x5)) = x2   
POL(U9_ggga(x1, x2, x3, x4, x5)) = 0   
POL(getleavec14_in_ggga(x1, x2, x3)) = 0   
POL(getleavec14_out_ggga(x1, x2, x3, x4)) = 0   
POL(leaf(x1)) = 0   
POL(tree(x1, x2)) = 1 + x1 + x2   

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

P7_IN_GGAAGGA(leaf(T23), T24, T13, T14) → U3_GGAAGGA(T23, T24, T13, T14, getleavec14_in_ggga(T13, T14, T23))
U3_GGAAGGA(T23, T24, T13, T14, getleavec14_out_ggga(T13, T14, T23, T29)) → SAMELEAVES1_IN_GG(T24, T29)
P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55) → U9_ggga(T52, T53, T54, T55, getleavec14_in_ggga(T52, tree(T53, T54), T55))
U9_ggga(T52, T53, T54, T55, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The set Q consists of the following terms:

getleavec14_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(20) Obligation:

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

P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)

The TRS R consists of the following rules:

getleavec14_in_ggga(leaf(T42), T43, T42) → getleavec14_out_ggga(leaf(T42), T43, T42, T43)
getleavec14_in_ggga(tree(T52, T53), T54, T55) → U9_ggga(T52, T53, T54, T55, getleavec14_in_ggga(T52, tree(T53, T54), T55))
U9_ggga(T52, T53, T54, T55, getleavec14_out_ggga(T52, tree(T53, T54), T55, X72)) → getleavec14_out_ggga(tree(T52, T53), T54, T55, X72)

The set Q consists of the following terms:

getleavec14_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(21) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(22) Obligation:

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

P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)

R is empty.
The set Q consists of the following terms:

getleavec14_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

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

(23) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

getleavec14_in_ggga(x0, x1, x2)
U9_ggga(x0, x1, x2, x3, x4)

(24) Obligation:

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

P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)

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

(25) 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:

  • P7_IN_GGAAGGA(tree(T70, T71), T72, T13, T14) → P7_IN_GGAAGGA(T70, tree(T71, T72), T13, T14)
    The graph contains the following edges 1 > 1, 3 >= 3, 4 >= 4

(26) YES