Left Termination of the query pattern less_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 CutEliminatorProof (⇐)
↳2 Prolog
  ↳3 PrologToPiTRSProof (⇐)
    ↳4 PiTRS
    ↳5 DependencyPairsProof (⇔)
    ↳6 PiDP
    ↳7 DependencyGraphProof (⇔)
    ↳8 PiDP
    ↳9 UsableRulesProof (⇔)
    ↳10 PiDP
    ↳11 PiDPToQDPProof (⇐)
    ↳12 QDP
    ↳13 UsableRulesReductionPairsProof (⇔)
    ↳14 QDP
    ↳15 Rewriting (⇔)
    ↳16 QDP
    ↳17 UsableRulesProof (⇔)
    ↳18 QDP
    ↳19 QReductionProof (⇔)
    ↳20 QDP
    ↳21 Narrowing (⇐)
    ↳22 QDP
    ↳23 UsableRulesProof (⇔)
    ↳24 QDP
    ↳25 QReductionProof (⇔)
    ↳26 QDP
    ↳27 Instantiation (⇔)
    ↳28 QDP
    ↳29 Instantiation (⇔)
    ↳30 QDP
    ↳31 NonTerminationProof (⇔)
    ↳32 FALSE
  ↳33 PrologToPiTRSProof (⇐)
    ↳34 PiTRS
    ↳35 DependencyPairsProof (⇔)
    ↳36 PiDP
    ↳37 DependencyGraphProof (⇔)
    ↳38 PiDP
    ↳39 UsableRulesProof (⇔)
    ↳40 PiDP
    ↳41 PiDPToQDPProof (⇐)
    ↳42 QDP
    ↳43 Rewriting (⇔)
    ↳44 QDP
    ↳45 UsableRulesProof (⇔)
    ↳46 QDP
    ↳47 QReductionProof (⇔)
    ↳48 QDP
    ↳49 Narrowing (⇐)
    ↳50 QDP
    ↳51 UsableRulesProof (⇔)
    ↳52 QDP
    ↳53 QReductionProof (⇔)
    ↳54 QDP
    ↳55 Instantiation (⇔)
    ↳56 QDP
    ↳57 Instantiation (⇔)
    ↳58 QDP
    ↳59 DependencyGraphProof (⇔)
    ↳60 AND
      ↳61 QDP
        ↳62 NonTerminationProof (⇔)
        ↳63 FALSE
      ↳64 QDP
        ↳65 QDPSizeChangeProof (⇔)
        ↳66 TRUE

(0) Obligation:

Clauses:

less(0, Y) :- ','(!, =(Y, s(X2))).
less(X, Y) :- ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1))).
p(0, 0).
p(s(X), X).
=(X, X).

Queries:

less(g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

less(0, Y) :- =(Y, s(X2)).
less(X, Y) :- ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1))).
p(0, 0).
p(s(X), X).
=(X, X).

Queries:

less(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
less_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x3)

(5) 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:

LESS_IN_GA(0, Y) → U1_GA(Y, =_in_aa(Y, s(X2)))
LESS_IN_GA(0, Y) → =_IN_AA(Y, s(X2))
LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
LESS_IN_GA(X, Y) → P_IN_GA(X, X1)
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U2_GA(X, Y, p_out_ga(X, X1)) → P_IN_AA(Y, Y1)
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → U4_GA(X, Y, less_in_ga(X1, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
=_IN_AA(x1, x2)  =  =_IN_AA
U2_GA(x1, x2, x3)  =  U2_GA(x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x3, x4)
P_IN_AA(x1, x2)  =  P_IN_AA
U4_GA(x1, x2, x3)  =  U4_GA(x3)

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

(6) Obligation:

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

LESS_IN_GA(0, Y) → U1_GA(Y, =_in_aa(Y, s(X2)))
LESS_IN_GA(0, Y) → =_IN_AA(Y, s(X2))
LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
LESS_IN_GA(X, Y) → P_IN_GA(X, X1)
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U2_GA(X, Y, p_out_ga(X, X1)) → P_IN_AA(Y, Y1)
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → U4_GA(X, Y, less_in_ga(X1, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
=_IN_AA(x1, x2)  =  =_IN_AA
U2_GA(x1, x2, x3)  =  U2_GA(x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x3, x4)
P_IN_AA(x1, x2)  =  P_IN_AA
U4_GA(x1, x2, x3)  =  U4_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(8) Obligation:

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

LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga
U2_ga(x1, x2, x3)  =  U2_ga(x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x3, x4)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)

The argument filtering Pi contains the following mapping:
0  =  0
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x2)
s(x1)  =  s(x1)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x3, x4)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

LESS_IN_GA(X) → U2_GA(p_in_ga(X))
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_in_aa)
U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)
p_in_aa → p_out_aa

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(13) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(s(X)) → p_out_ga(X)
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0   
POL(LESS_IN_GA(x1)) = 2 + x1   
POL(U2_GA(x1)) = 1 + x1   
POL(U3_GA(x1, x2)) = 2 + x1 + 2·x2   
POL(p_in_aa) = 0   
POL(p_in_ga(x1)) = 1 + x1   
POL(p_out_aa) = 0   
POL(p_out_ga(x1)) = 1 + x1   
POL(s(x1)) = 1 + 2·x1   

(14) Obligation:

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

LESS_IN_GA(X) → U2_GA(p_in_ga(X))
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_in_aa)
U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)

The TRS R consists of the following rules:

p_in_aa → p_out_aa
p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(15) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U2_GA(p_out_ga(X1)) → U3_GA(X1, p_in_aa) at position [1] we obtained the following new rules [LPAR04]:

U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)

(16) Obligation:

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

LESS_IN_GA(X) → U2_GA(p_in_ga(X))
U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)

The TRS R consists of the following rules:

p_in_aa → p_out_aa
p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(17) 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.

(18) Obligation:

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

LESS_IN_GA(X) → U2_GA(p_in_ga(X))
U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(19) 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].

p_in_aa

(20) Obligation:

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

LESS_IN_GA(X) → U2_GA(p_in_ga(X))
U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(21) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESS_IN_GA(X) → U2_GA(p_in_ga(X)) at position [0] we obtained the following new rules [LPAR04]:

LESS_IN_GA(0) → U2_GA(p_out_ga(0))

(22) Obligation:

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

U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(23) 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.

(24) Obligation:

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

U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(p_out_ga(0))

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

p_in_ga(x0)

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

(25) 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].

p_in_ga(x0)

(26) Obligation:

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

U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(p_out_ga(0))

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

(27) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GA(p_out_ga(X1)) → U3_GA(X1, p_out_aa) we obtained the following new rules [LPAR04]:

U2_GA(p_out_ga(0)) → U3_GA(0, p_out_aa)

(28) Obligation:

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

U3_GA(X1, p_out_aa) → LESS_IN_GA(X1)
LESS_IN_GA(0) → U2_GA(p_out_ga(0))
U2_GA(p_out_ga(0)) → U3_GA(0, p_out_aa)

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

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GA(X1, p_out_aa) → LESS_IN_GA(X1) we obtained the following new rules [LPAR04]:

U3_GA(0, p_out_aa) → LESS_IN_GA(0)

(30) Obligation:

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

LESS_IN_GA(0) → U2_GA(p_out_ga(0))
U2_GA(p_out_ga(0)) → U3_GA(0, p_out_aa)
U3_GA(0, p_out_aa) → LESS_IN_GA(0)

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

(31) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U2_GA(p_out_ga(0)) evaluates to t =U2_GA(p_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Matcher: [ ]
  • Semiunifier: [ ]



Rewriting sequence

U2_GA(p_out_ga(0)) → U3_GA(0, p_out_aa)
with rule U2_GA(p_out_ga(0)) → U3_GA(0, p_out_aa) at position [] and matcher [ ]

U3_GA(0, p_out_aa) → LESS_IN_GA(0)
with rule U3_GA(0, p_out_aa) → LESS_IN_GA(0) at position [] and matcher [ ]

LESS_IN_GA(0) → U2_GA(p_out_ga(0))
with rule LESS_IN_GA(0) → U2_GA(p_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(32) FALSE

(33) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
less_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(34) Obligation:

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

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)

(35) 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:

LESS_IN_GA(0, Y) → U1_GA(Y, =_in_aa(Y, s(X2)))
LESS_IN_GA(0, Y) → =_IN_AA(Y, s(X2))
LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
LESS_IN_GA(X, Y) → P_IN_GA(X, X1)
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U2_GA(X, Y, p_out_ga(X, X1)) → P_IN_AA(Y, Y1)
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → U4_GA(X, Y, less_in_ga(X1, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
=_IN_AA(x1, x2)  =  =_IN_AA
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x3, x4)
P_IN_AA(x1, x2)  =  P_IN_AA
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

(36) Obligation:

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

LESS_IN_GA(0, Y) → U1_GA(Y, =_in_aa(Y, s(X2)))
LESS_IN_GA(0, Y) → =_IN_AA(Y, s(X2))
LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
LESS_IN_GA(X, Y) → P_IN_GA(X, X1)
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U2_GA(X, Y, p_out_ga(X, X1)) → P_IN_AA(Y, Y1)
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → U4_GA(X, Y, less_in_ga(X1, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x2)
=_IN_AA(x1, x2)  =  =_IN_AA
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
P_IN_GA(x1, x2)  =  P_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x3, x4)
P_IN_AA(x1, x2)  =  P_IN_AA
U4_GA(x1, x2, x3)  =  U4_GA(x1, x3)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 5 less nodes.

(38) Obligation:

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

LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

less_in_ga(0, Y) → U1_ga(Y, =_in_aa(Y, s(X2)))
=_in_aa(X, X) → =_out_aa(X, X)
U1_ga(Y, =_out_aa(Y, s(X2))) → less_out_ga(0, Y)
less_in_ga(X, Y) → U2_ga(X, Y, p_in_ga(X, X1))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_ga(X, Y, p_out_ga(X, X1)) → U3_ga(X, Y, X1, p_in_aa(Y, Y1))
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)
U3_ga(X, Y, X1, p_out_aa(Y, Y1)) → U4_ga(X, Y, less_in_ga(X1, Y1))
U4_ga(X, Y, less_out_ga(X1, Y1)) → less_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
less_in_ga(x1, x2)  =  less_in_ga(x1)
0  =  0
U1_ga(x1, x2)  =  U1_ga(x2)
=_in_aa(x1, x2)  =  =_in_aa
=_out_aa(x1, x2)  =  =_out_aa
less_out_ga(x1, x2)  =  less_out_ga(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
U3_ga(x1, x2, x3, x4)  =  U3_ga(x1, x3, x4)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
U4_ga(x1, x2, x3)  =  U4_ga(x1, x3)
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x3, x4)

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

(39) UsableRulesProof (EQUIVALENT transformation)

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

(40) Obligation:

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

LESS_IN_GA(X, Y) → U2_GA(X, Y, p_in_ga(X, X1))
U2_GA(X, Y, p_out_ga(X, X1)) → U3_GA(X, Y, X1, p_in_aa(Y, Y1))
U3_GA(X, Y, X1, p_out_aa(Y, Y1)) → LESS_IN_GA(X1, Y1)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
p_in_aa(0, 0) → p_out_aa(0, 0)
p_in_aa(s(X), X) → p_out_aa(s(X), X)

The argument filtering Pi contains the following mapping:
0  =  0
p_in_ga(x1, x2)  =  p_in_ga(x1)
p_out_ga(x1, x2)  =  p_out_ga(x1, x2)
s(x1)  =  s(x1)
p_in_aa(x1, x2)  =  p_in_aa
p_out_aa(x1, x2)  =  p_out_aa
LESS_IN_GA(x1, x2)  =  LESS_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x3, x4)

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

(41) PiDPToQDPProof (SOUND transformation)

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

(42) Obligation:

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

LESS_IN_GA(X) → U2_GA(X, p_in_ga(X))
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_in_aa)
U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)
p_in_aa → p_out_aa

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(43) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_in_aa) at position [2] we obtained the following new rules [LPAR04]:

U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)

(44) Obligation:

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

LESS_IN_GA(X) → U2_GA(X, p_in_ga(X))
U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)
p_in_aa → p_out_aa

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(45) 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.

(46) Obligation:

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

LESS_IN_GA(X) → U2_GA(X, p_in_ga(X))
U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)
p_in_aa

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

(47) 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].

p_in_aa

(48) Obligation:

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

LESS_IN_GA(X) → U2_GA(X, p_in_ga(X))
U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(49) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule LESS_IN_GA(X) → U2_GA(X, p_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))

(50) Obligation:

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

U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(51) 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.

(52) Obligation:

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

U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))

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

p_in_ga(x0)

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

(53) 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].

p_in_ga(x0)

(54) Obligation:

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

U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa)
LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))

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

(55) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GA(X, p_out_ga(X, X1)) → U3_GA(X, X1, p_out_aa) we obtained the following new rules [LPAR04]:

U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)
U2_GA(s(z0), p_out_ga(s(z0), z0)) → U3_GA(s(z0), z0, p_out_aa)

(56) Obligation:

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

U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1)
LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))
U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)
U2_GA(s(z0), p_out_ga(s(z0), z0)) → U3_GA(s(z0), z0, p_out_aa)

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

(57) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GA(X, X1, p_out_aa) → LESS_IN_GA(X1) we obtained the following new rules [LPAR04]:

U3_GA(0, 0, p_out_aa) → LESS_IN_GA(0)
U3_GA(s(z0), z0, p_out_aa) → LESS_IN_GA(z0)

(58) Obligation:

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

LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))
U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)
U2_GA(s(z0), p_out_ga(s(z0), z0)) → U3_GA(s(z0), z0, p_out_aa)
U3_GA(0, 0, p_out_aa) → LESS_IN_GA(0)
U3_GA(s(z0), z0, p_out_aa) → LESS_IN_GA(z0)

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

(59) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(60) Complex Obligation (AND)

(61) Obligation:

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

U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)
U3_GA(0, 0, p_out_aa) → LESS_IN_GA(0)
LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))

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

(62) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U3_GA(0, 0, p_out_aa) evaluates to t =U3_GA(0, 0, p_out_aa)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

U3_GA(0, 0, p_out_aa) → LESS_IN_GA(0)
with rule U3_GA(0, 0, p_out_aa) → LESS_IN_GA(0) at position [] and matcher [ ]

LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0))
with rule LESS_IN_GA(0) → U2_GA(0, p_out_ga(0, 0)) at position [] and matcher [ ]

U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)
with rule U2_GA(0, p_out_ga(0, 0)) → U3_GA(0, 0, p_out_aa)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(63) FALSE

(64) Obligation:

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

LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))
U2_GA(s(z0), p_out_ga(s(z0), z0)) → U3_GA(s(z0), z0, p_out_aa)
U3_GA(s(z0), z0, p_out_aa) → LESS_IN_GA(z0)

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

(65) 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:

  • U2_GA(s(z0), p_out_ga(s(z0), z0)) → U3_GA(s(z0), z0, p_out_aa)
    The graph contains the following edges 1 >= 1, 2 > 1, 1 > 2, 2 > 2

  • U3_GA(s(z0), z0, p_out_aa) → LESS_IN_GA(z0)
    The graph contains the following edges 1 > 1, 2 >= 1

  • LESS_IN_GA(s(x0)) → U2_GA(s(x0), p_out_ga(s(x0), x0))
    The graph contains the following edges 1 >= 1

(66) TRUE