YES Termination w.r.t. Q proof of AProVE_07_otto03.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 Overlay + Local Confluence (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 4 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 QReductionProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 QDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 QDP
        ↳17 QReductionProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 NonInfProof (⇔, 29 ms)
        ↳20 AND
            ↳21 QDP
                ↳22 DependencyGraphProof (⇔, 0 ms)
                ↳23 TRUE
            ↳24 QDP
                ↳25 DependencyGraphProof (⇔, 0 ms)
                ↳26 TRUE

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

LT(s(x), s(y)) → LT(x, y)
FAC(x) → HELP(x, 0)
HELP(x, c) → IF(lt(c, x), x, c)
HELP(x, c) → LT(c, x)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

LT(s(x), s(y)) → LT(x, y)

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(8) 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.

(9) Obligation:

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

LT(s(x), s(y)) → LT(x, y)

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

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(10) 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].

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

(11) Obligation:

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

LT(s(x), s(y)) → LT(x, y)

R is empty.
Q is empty.
We have to consider all minimal (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:

  • LT(s(x), s(y)) → LT(x, y)
    The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

HELP(x, c) → IF(lt(c, x), x, c)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(15) 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.

(16) Obligation:

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

HELP(x, c) → IF(lt(c, x), x, c)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))
fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

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

(17) 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].

fac(x0)
help(x0, x1)
if(true, x0, x1)
if(false, x0, x1)

(18) Obligation:

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

HELP(x, c) → IF(lt(c, x), x, c)
IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))

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

(19) NonInfProof (EQUIVALENT transformation)

The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair HELP(x, c) → IF(lt(c, x), x, c) the following chains were created:
  • We consider the chain IF(true, x2, x3) → HELP(x2, s(x3)), HELP(x4, x5) → IF(lt(x5, x4), x4, x5) which results in the following constraint:
    (1)    (HELP(x2, s(x3))=HELP(x4, x5) ⇒ HELP(x4, x5)≥IF(lt(x5, x4), x4, x5))


    We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
    (2)    (HELP(x2, s(x3))≥IF(lt(s(x3), x2), x2, s(x3)))






For Pair IF(true, x, c) → HELP(x, s(c)) the following chains were created:
  • We consider the chain HELP(x6, x7) → IF(lt(x7, x6), x6, x7), IF(true, x8, x9) → HELP(x8, s(x9)) which results in the following constraint:
    (1)    (IF(lt(x7, x6), x6, x7)=IF(true, x8, x9) ⇒ IF(true, x8, x9)≥HELP(x8, s(x9)))


    We simplified constraint (1) using rules (I), (II), (III) which results in the following new constraint:
    (2)    (lt(x7, x6)=true ⇒ IF(true, x6, x7)≥HELP(x6, s(x7)))


    We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on lt(x7, x6)=true which results in the following new constraints:
    (3)    (true=true ⇒ IF(true, s(x12), 0)≥HELP(s(x12), s(0)))

    (4)    (lt(x15, x14)=true∧(lt(x15, x14)=true ⇒ IF(true, x14, x15)≥HELP(x14, s(x15))) ⇒ IF(true, s(x14), s(x15))≥HELP(s(x14), s(s(x15))))


    We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
    (5)    (IF(true, s(x12), 0)≥HELP(s(x12), s(0)))


    We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (lt(x15, x14)=true ⇒ IF(true, x14, x15)≥HELP(x14, s(x15))) with σ = [ ] which results in the following new constraint:
    (6)    (IF(true, x14, x15)≥HELP(x14, s(x15)) ⇒ IF(true, s(x14), s(x15))≥HELP(s(x14), s(s(x15))))






To summarize, we get the following constraints P≥ for the following pairs.
  • HELP(x, c) → IF(lt(c, x), x, c)
    • (HELP(x2, s(x3))≥IF(lt(s(x3), x2), x2, s(x3)))

  • IF(true, x, c) → HELP(x, s(c))
    • (IF(true, s(x12), 0)≥HELP(s(x12), s(0)))
    • (IF(true, x14, x15)≥HELP(x14, s(x15)) ⇒ IF(true, s(x14), s(x15))≥HELP(s(x14), s(s(x15))))




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [NONINF]:

POL(0) = 0   
POL(HELP(x1, x2)) = 1 + x1 - x2   
POL(IF(x1, x2, x3)) = x1 + x2 - x3   
POL(c) = -1   
POL(false) = 0   
POL(lt(x1, x2)) = 0   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following pairs are in P>:

HELP(x, c) → IF(lt(c, x), x, c)
The following pairs are in Pbound:

IF(true, x, c) → HELP(x, s(c))
The following rules are usable:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)

(20) Complex Obligation (AND)

(21) Obligation:

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

IF(true, x, c) → HELP(x, s(c))

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))

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

(22) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(23) TRUE

(24) Obligation:

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

HELP(x, c) → IF(lt(c, x), x, c)

The TRS R consists of the following rules:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)

The set Q consists of the following terms:

lt(0, s(x0))
lt(x0, 0)
lt(s(x0), s(x1))

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(26) TRUE