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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 85 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 139 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 PiDP
↳7 UsableRulesProof (⇔, 0 ms)
↳8 PiDP
↳9 PiDPToQDPProof (⇔, 6 ms)
↳10 QDP
↳11 QDPSizeChangeProof (⇔, 0 ms)
↳12 YES

(0) Obligation:

Clauses:

dis(or(B1, B2)) :- ','(con(B1), dis(B2)).
dis(B) :- con(B).
con(and(B1, B2)) :- ','(dis(B1), con(B2)).
con(B) :- bool(B).
bool(0).
bool(1).

Query: dis(g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

conB(and(X1, X2)) :- pC(X1, X2).
pC(X1, X2) :- disA(X1).
pC(X1, X2) :- ','(discA(X1), conB(X2)).
disA(or(and(X1, X2), X3)) :- disA(X1).
disA(or(and(X1, X2), X3)) :- ','(discA(X1), conB(X2)).
disA(or(and(X1, X2), X3)) :- ','(discA(X1), ','(concB(X2), disA(X3))).
disA(or(0, X1)) :- disA(X1).
disA(or(1, X1)) :- disA(X1).
disA(or(X1, X2)) :- conB(or(X1, X2)).
disA(and(X1, X2)) :- pC(X1, X2).

Clauses:

discA(or(and(X1, X2), X3)) :- ','(discA(X1), ','(concB(X2), discA(X3))).
discA(or(0, X1)) :- discA(X1).
discA(or(1, X1)) :- discA(X1).
discA(or(X1, X2)) :- concB(or(X1, X2)).
discA(and(X1, X2)) :- qcC(X1, X2).
discA(0).
discA(1).
concB(and(X1, X2)) :- qcC(X1, X2).
concB(0).
concB(1).
qcC(X1, X2) :- ','(discA(X1), concB(X2)).

Afs:

disA(x1)  =  disA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
disA_in: (b)
discA_in: (b)
concB_in: (b)
qcC_in: (b,b)
conB_in: (b)
pC_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

DISA_IN_G(or(and(X1, X2), X3)) → U5_G(X1, X2, X3, disA_in_g(X1))
DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
U6_G(X1, X2, X3, discA_out_g(X1)) → U7_G(X1, X2, X3, conB_in_g(X2))
U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
CONB_IN_G(and(X1, X2)) → U1_G(X1, X2, pC_in_gg(X1, X2))
CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U2_GG(X1, X2, disA_in_g(X1))
PC_IN_GG(X1, X2) → DISA_IN_G(X1)
DISA_IN_G(or(0, X1)) → U10_G(X1, disA_in_g(X1))
DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(1, X1)) → U11_G(X1, disA_in_g(X1))
DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(X1, X2)) → U12_G(X1, X2, conB_in_g(or(X1, X2)))
DISA_IN_G(or(X1, X2)) → CONB_IN_G(or(X1, X2))
DISA_IN_G(and(X1, X2)) → U13_G(X1, X2, pC_in_gg(X1, X2))
DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
U3_GG(X1, X2, discA_out_g(X1)) → U4_GG(X1, X2, conB_in_g(X2))
U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
U8_G(X1, X2, X3, concB_out_g(X2)) → U9_G(X1, X2, X3, disA_in_g(X3))
U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)

The TRS R consists of the following rules:

discA_in_g(or(and(X1, X2), X3)) → U15_g(X1, X2, X3, discA_in_g(X1))
discA_in_g(or(0, X1)) → U18_g(X1, discA_in_g(X1))
discA_in_g(or(1, X1)) → U19_g(X1, discA_in_g(X1))
discA_in_g(or(X1, X2)) → U20_g(X1, X2, concB_in_g(or(X1, X2)))
concB_in_g(and(X1, X2)) → U22_g(X1, X2, qcC_in_gg(X1, X2))
qcC_in_gg(X1, X2) → U23_gg(X1, X2, discA_in_g(X1))
discA_in_g(and(X1, X2)) → U21_g(X1, X2, qcC_in_gg(X1, X2))
U21_g(X1, X2, qcC_out_gg(X1, X2)) → discA_out_g(and(X1, X2))
discA_in_g(0) → discA_out_g(0)
discA_in_g(1) → discA_out_g(1)
U23_gg(X1, X2, discA_out_g(X1)) → U24_gg(X1, X2, concB_in_g(X2))
concB_in_g(0) → concB_out_g(0)
concB_in_g(1) → concB_out_g(1)
U24_gg(X1, X2, concB_out_g(X2)) → qcC_out_gg(X1, X2)
U22_g(X1, X2, qcC_out_gg(X1, X2)) → concB_out_g(and(X1, X2))
U20_g(X1, X2, concB_out_g(or(X1, X2))) → discA_out_g(or(X1, X2))
U19_g(X1, discA_out_g(X1)) → discA_out_g(or(1, X1))
U18_g(X1, discA_out_g(X1)) → discA_out_g(or(0, X1))
U15_g(X1, X2, X3, discA_out_g(X1)) → U16_g(X1, X2, X3, concB_in_g(X2))
U16_g(X1, X2, X3, concB_out_g(X2)) → U17_g(X1, X2, X3, discA_in_g(X3))
U17_g(X1, X2, X3, discA_out_g(X3)) → discA_out_g(or(and(X1, X2), X3))

Pi is empty.
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:

DISA_IN_G(or(and(X1, X2), X3)) → U5_G(X1, X2, X3, disA_in_g(X1))
DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
U6_G(X1, X2, X3, discA_out_g(X1)) → U7_G(X1, X2, X3, conB_in_g(X2))
U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
CONB_IN_G(and(X1, X2)) → U1_G(X1, X2, pC_in_gg(X1, X2))
CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U2_GG(X1, X2, disA_in_g(X1))
PC_IN_GG(X1, X2) → DISA_IN_G(X1)
DISA_IN_G(or(0, X1)) → U10_G(X1, disA_in_g(X1))
DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(1, X1)) → U11_G(X1, disA_in_g(X1))
DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(X1, X2)) → U12_G(X1, X2, conB_in_g(or(X1, X2)))
DISA_IN_G(or(X1, X2)) → CONB_IN_G(or(X1, X2))
DISA_IN_G(and(X1, X2)) → U13_G(X1, X2, pC_in_gg(X1, X2))
DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
U3_GG(X1, X2, discA_out_g(X1)) → U4_GG(X1, X2, conB_in_g(X2))
U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
U8_G(X1, X2, X3, concB_out_g(X2)) → U9_G(X1, X2, X3, disA_in_g(X3))
U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)

The TRS R consists of the following rules:

discA_in_g(or(and(X1, X2), X3)) → U15_g(X1, X2, X3, discA_in_g(X1))
discA_in_g(or(0, X1)) → U18_g(X1, discA_in_g(X1))
discA_in_g(or(1, X1)) → U19_g(X1, discA_in_g(X1))
discA_in_g(or(X1, X2)) → U20_g(X1, X2, concB_in_g(or(X1, X2)))
concB_in_g(and(X1, X2)) → U22_g(X1, X2, qcC_in_gg(X1, X2))
qcC_in_gg(X1, X2) → U23_gg(X1, X2, discA_in_g(X1))
discA_in_g(and(X1, X2)) → U21_g(X1, X2, qcC_in_gg(X1, X2))
U21_g(X1, X2, qcC_out_gg(X1, X2)) → discA_out_g(and(X1, X2))
discA_in_g(0) → discA_out_g(0)
discA_in_g(1) → discA_out_g(1)
U23_gg(X1, X2, discA_out_g(X1)) → U24_gg(X1, X2, concB_in_g(X2))
concB_in_g(0) → concB_out_g(0)
concB_in_g(1) → concB_out_g(1)
U24_gg(X1, X2, concB_out_g(X2)) → qcC_out_gg(X1, X2)
U22_g(X1, X2, qcC_out_gg(X1, X2)) → concB_out_g(and(X1, X2))
U20_g(X1, X2, concB_out_g(or(X1, X2))) → discA_out_g(or(X1, X2))
U19_g(X1, discA_out_g(X1)) → discA_out_g(or(1, X1))
U18_g(X1, discA_out_g(X1)) → discA_out_g(or(0, X1))
U15_g(X1, X2, X3, discA_out_g(X1)) → U16_g(X1, X2, X3, concB_in_g(X2))
U16_g(X1, X2, X3, concB_out_g(X2)) → U17_g(X1, X2, X3, discA_in_g(X3))
U17_g(X1, X2, X3, discA_out_g(X3)) → discA_out_g(or(and(X1, X2), X3))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 11 less nodes.

(6) Obligation:

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

DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → DISA_IN_G(X1)
DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)

The TRS R consists of the following rules:

discA_in_g(or(and(X1, X2), X3)) → U15_g(X1, X2, X3, discA_in_g(X1))
discA_in_g(or(0, X1)) → U18_g(X1, discA_in_g(X1))
discA_in_g(or(1, X1)) → U19_g(X1, discA_in_g(X1))
discA_in_g(or(X1, X2)) → U20_g(X1, X2, concB_in_g(or(X1, X2)))
concB_in_g(and(X1, X2)) → U22_g(X1, X2, qcC_in_gg(X1, X2))
qcC_in_gg(X1, X2) → U23_gg(X1, X2, discA_in_g(X1))
discA_in_g(and(X1, X2)) → U21_g(X1, X2, qcC_in_gg(X1, X2))
U21_g(X1, X2, qcC_out_gg(X1, X2)) → discA_out_g(and(X1, X2))
discA_in_g(0) → discA_out_g(0)
discA_in_g(1) → discA_out_g(1)
U23_gg(X1, X2, discA_out_g(X1)) → U24_gg(X1, X2, concB_in_g(X2))
concB_in_g(0) → concB_out_g(0)
concB_in_g(1) → concB_out_g(1)
U24_gg(X1, X2, concB_out_g(X2)) → qcC_out_gg(X1, X2)
U22_g(X1, X2, qcC_out_gg(X1, X2)) → concB_out_g(and(X1, X2))
U20_g(X1, X2, concB_out_g(or(X1, X2))) → discA_out_g(or(X1, X2))
U19_g(X1, discA_out_g(X1)) → discA_out_g(or(1, X1))
U18_g(X1, discA_out_g(X1)) → discA_out_g(or(0, X1))
U15_g(X1, X2, X3, discA_out_g(X1)) → U16_g(X1, X2, X3, concB_in_g(X2))
U16_g(X1, X2, X3, concB_out_g(X2)) → U17_g(X1, X2, X3, discA_in_g(X3))
U17_g(X1, X2, X3, discA_out_g(X3)) → discA_out_g(or(and(X1, X2), X3))

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → DISA_IN_G(X1)
DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)

The TRS R consists of the following rules:

discA_in_g(or(and(X1, X2), X3)) → U15_g(X1, X2, X3, discA_in_g(X1))
discA_in_g(or(0, X1)) → U18_g(X1, discA_in_g(X1))
discA_in_g(or(1, X1)) → U19_g(X1, discA_in_g(X1))
discA_in_g(or(X1, X2)) → U20_g(X1, X2, concB_in_g(or(X1, X2)))
discA_in_g(and(X1, X2)) → U21_g(X1, X2, qcC_in_gg(X1, X2))
discA_in_g(0) → discA_out_g(0)
discA_in_g(1) → discA_out_g(1)
concB_in_g(and(X1, X2)) → U22_g(X1, X2, qcC_in_gg(X1, X2))
concB_in_g(0) → concB_out_g(0)
concB_in_g(1) → concB_out_g(1)
U15_g(X1, X2, X3, discA_out_g(X1)) → U16_g(X1, X2, X3, concB_in_g(X2))
U18_g(X1, discA_out_g(X1)) → discA_out_g(or(0, X1))
U19_g(X1, discA_out_g(X1)) → discA_out_g(or(1, X1))
U21_g(X1, X2, qcC_out_gg(X1, X2)) → discA_out_g(and(X1, X2))
U22_g(X1, X2, qcC_out_gg(X1, X2)) → concB_out_g(and(X1, X2))
U16_g(X1, X2, X3, concB_out_g(X2)) → U17_g(X1, X2, X3, discA_in_g(X3))
qcC_in_gg(X1, X2) → U23_gg(X1, X2, discA_in_g(X1))
U17_g(X1, X2, X3, discA_out_g(X3)) → discA_out_g(or(and(X1, X2), X3))
U23_gg(X1, X2, discA_out_g(X1)) → U24_gg(X1, X2, concB_in_g(X2))
U24_gg(X1, X2, concB_out_g(X2)) → qcC_out_gg(X1, X2)

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → DISA_IN_G(X1)
DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)

The TRS R consists of the following rules:

discA_in_g(or(and(X1, X2), X3)) → U15_g(X1, X2, X3, discA_in_g(X1))
discA_in_g(or(0, X1)) → U18_g(X1, discA_in_g(X1))
discA_in_g(or(1, X1)) → U19_g(X1, discA_in_g(X1))
discA_in_g(or(X1, X2)) → U20_g(X1, X2, concB_in_g(or(X1, X2)))
discA_in_g(and(X1, X2)) → U21_g(X1, X2, qcC_in_gg(X1, X2))
discA_in_g(0) → discA_out_g(0)
discA_in_g(1) → discA_out_g(1)
concB_in_g(and(X1, X2)) → U22_g(X1, X2, qcC_in_gg(X1, X2))
concB_in_g(0) → concB_out_g(0)
concB_in_g(1) → concB_out_g(1)
U15_g(X1, X2, X3, discA_out_g(X1)) → U16_g(X1, X2, X3, concB_in_g(X2))
U18_g(X1, discA_out_g(X1)) → discA_out_g(or(0, X1))
U19_g(X1, discA_out_g(X1)) → discA_out_g(or(1, X1))
U21_g(X1, X2, qcC_out_gg(X1, X2)) → discA_out_g(and(X1, X2))
U22_g(X1, X2, qcC_out_gg(X1, X2)) → concB_out_g(and(X1, X2))
U16_g(X1, X2, X3, concB_out_g(X2)) → U17_g(X1, X2, X3, discA_in_g(X3))
qcC_in_gg(X1, X2) → U23_gg(X1, X2, discA_in_g(X1))
U17_g(X1, X2, X3, discA_out_g(X3)) → discA_out_g(or(and(X1, X2), X3))
U23_gg(X1, X2, discA_out_g(X1)) → U24_gg(X1, X2, concB_in_g(X2))
U24_gg(X1, X2, concB_out_g(X2)) → qcC_out_gg(X1, X2)

The set Q consists of the following terms:

discA_in_g(x0)
concB_in_g(x0)
U15_g(x0, x1, x2, x3)
U18_g(x0, x1)
U19_g(x0, x1)
U21_g(x0, x1, x2)
U22_g(x0, x1, x2)
U16_g(x0, x1, x2, x3)
qcC_in_gg(x0, x1)
U17_g(x0, x1, x2, x3)
U23_gg(x0, x1, x2)
U24_gg(x0, x1, x2)

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

(11) 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:

  • CONB_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
    The graph contains the following edges 1 > 1, 1 > 2

  • DISA_IN_G(or(and(X1, X2), X3)) → U6_G(X1, X2, X3, discA_in_g(X1))
    The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

  • U6_G(X1, X2, X3, discA_out_g(X1)) → CONB_IN_G(X2)
    The graph contains the following edges 2 >= 1

  • U6_G(X1, X2, X3, discA_out_g(X1)) → U8_G(X1, X2, X3, concB_in_g(X2))
    The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 3 >= 3

  • PC_IN_GG(X1, X2) → DISA_IN_G(X1)
    The graph contains the following edges 1 >= 1

  • PC_IN_GG(X1, X2) → U3_GG(X1, X2, discA_in_g(X1))
    The graph contains the following edges 1 >= 1, 2 >= 2

  • U3_GG(X1, X2, discA_out_g(X1)) → CONB_IN_G(X2)
    The graph contains the following edges 2 >= 1

  • DISA_IN_G(and(X1, X2)) → PC_IN_GG(X1, X2)
    The graph contains the following edges 1 > 1, 1 > 2

  • U8_G(X1, X2, X3, concB_out_g(X2)) → DISA_IN_G(X3)
    The graph contains the following edges 3 >= 1

  • DISA_IN_G(or(and(X1, X2), X3)) → DISA_IN_G(X1)
    The graph contains the following edges 1 > 1

  • DISA_IN_G(or(0, X1)) → DISA_IN_G(X1)
    The graph contains the following edges 1 > 1

  • DISA_IN_G(or(1, X1)) → DISA_IN_G(X1)
    The graph contains the following edges 1 > 1

(12) YES