Termination w.r.t. Q proof of AProVE_06

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

0 QTRS

↳1 DependencyPairsProof (⇔, 0 ms)

↳2 QDP

↳3 DependencyGraphProof (⇔, 0 ms)

↳4 AND

    ↳5 QDP

        ↳6 MNOCProof (⇔, 0 ms)

        ↳7 QDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 QDP

        ↳10 QReductionProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 QDP

        ↳15 MNOCProof (⇔, 0 ms)

        ↳16 QDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QReductionProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 QDP

        ↳24 MNOCProof (⇔, 0 ms)

        ↳25 QDP

        ↳26 UsableRulesProof (⇔, 0 ms)

        ↳27 QDP

        ↳28 QReductionProof (⇔, 0 ms)

        ↳29 QDP

        ↳30 QDPSizeChangeProof (⇔, 0 ms)

        ↳31 YES

    ↳32 QDP

        ↳33 MNOCProof (⇔, 0 ms)

        ↳34 QDP

        ↳35 UsableRulesProof (⇔, 0 ms)

        ↳36 QDP

        ↳37 QReductionProof (⇔, 0 ms)

        ↳38 QDP

        ↳39 TransformationProof (⇔, 0 ms)

        ↳40 QDP

        ↳41 DependencyGraphProof (⇔, 0 ms)

        ↳42 QDP

        ↳43 UsableRulesProof (⇔, 0 ms)

        ↳44 QDP

        ↳45 QReductionProof (⇔, 0 ms)

        ↳46 QDP

        ↳47 TransformationProof (⇔, 0 ms)

        ↳48 QDP

        ↳49 UsableRulesProof (⇔, 0 ms)

        ↳50 QDP

        ↳51 TransformationProof (⇔, 0 ms)

        ↳52 QDP

        ↳53 UsableRulesProof (⇔, 0 ms)

        ↳54 QDP

        ↳55 QReductionProof (⇔, 0 ms)

        ↳56 QDP

        ↳57 TransformationProof (⇔, 0 ms)

        ↳58 QDP

        ↳59 UsableRulesProof (⇔, 0 ms)

        ↳60 QDP

        ↳61 QReductionProof (⇔, 0 ms)

        ↳62 QDP

        ↳63 TransformationProof (⇔, 0 ms)

        ↳64 QDP

        ↳65 QDPSizeChangeProof (⇔, 0 ms)

        ↳66 YES

(0) Obligation:

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

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

PLUS(s(x), y) → PLUS(x, y)
TIMES(s(x), y) → PLUS(y, times(x, y))
TIMES(s(x), y) → TIMES(x, y)
MINUS(x, s(y)) → P(minus(x, y))
MINUS(x, s(y)) → MINUS(x, y)
FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x))
FACITER(x, y) → ISZERO(x)
FACITER(x, y) → MINUS(x, s(0))
FACITER(x, y) → TIMES(y, x)
IF(false, x, y, z) → FACITER(x, z)
FACTORIAL(x) → FACITER(x, s(0))

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 6 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

MINUS(x, s(y)) → MINUS(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

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

(6) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(7) Obligation:

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

MINUS(x, s(y)) → MINUS(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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:

MINUS(x, s(y)) → MINUS(x, y)

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

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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].

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

(11) Obligation:

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

MINUS(x, s(y)) → MINUS(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:

MINUS(x, s(y)) → MINUS(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:

PLUS(s(x), y) → PLUS(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

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

(15) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(16) Obligation:

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

PLUS(s(x), y) → PLUS(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

We have to consider all minimal (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:

PLUS(s(x), y) → PLUS(x, y)

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

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

We have to consider all minimal (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].

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

(20) Obligation:

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

PLUS(s(x), y) → PLUS(x, y)

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

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

(22) YES

(23) Obligation:

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

TIMES(s(x), y) → TIMES(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

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

(24) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(25) Obligation:

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

TIMES(s(x), y) → TIMES(x, y)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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

(26) 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.

(27) Obligation:

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

TIMES(s(x), y) → TIMES(x, y)

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

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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

(28) 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].

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

(29) Obligation:

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

TIMES(s(x), y) → TIMES(x, y)

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

(30) 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:

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

(31) YES

(32) Obligation:

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

FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

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

(33) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(34) Obligation:

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

FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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

(35) 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.

(36) Obligation:

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

FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))
facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

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

(37) 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].

facIter(x0, x1)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
factorial(x0)

(38) Obligation:

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

FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))

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

(39) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule FACITER(x, y) → IF(isZero(x), minus(x, s(0)), y, times(y, x)) at position [0] we obtained the following new rules [LPAR04]:

FACITER(0, y1) → IF(true, minus(0, s(0)), y1, times(y1, 0)) → FACITER(0, y1) → IF(true, minus(0, s(0)), y1, times(y1, 0))
FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0))) → FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0)))

(40) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(0, y1) → IF(true, minus(0, s(0)), y1, times(y1, 0))
FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0)))

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))

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

(41) DependencyGraphProof (EQUIVALENT transformation)

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

(42) Obligation:

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

FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0)))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))

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

(43) 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.

(44) Obligation:

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

FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0)))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(0, x) → 0
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))
isZero(0)
isZero(s(x0))

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

(45) 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].

isZero(0)
isZero(s(x0))

(46) Obligation:

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

FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0)))
IF(false, x, y, z) → FACITER(x, z)

The TRS R consists of the following rules:

minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(0, x) → 0
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

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

(47) TransformationProof (EQUIVALENT transformation)

By narrowing [LPAR04] the rule FACITER(s(x0), y1) → IF(false, minus(s(x0), s(0)), y1, times(y1, s(x0))) at position [1] we obtained the following new rules [LPAR04]:

FACITER(s(y0), y1) → IF(false, p(minus(s(y0), 0)), y1, times(y1, s(y0))) → FACITER(s(y0), y1) → IF(false, p(minus(s(y0), 0)), y1, times(y1, s(y0)))

(48) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, p(minus(s(y0), 0)), y1, times(y1, s(y0)))

The TRS R consists of the following rules:

minus(x, s(y)) → p(minus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))
minus(0, x) → 0
minus(x, 0) → x
p(s(x)) → x
p(0) → 0

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

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

(49) 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.

(50) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, p(minus(s(y0), 0)), y1, times(y1, s(y0)))

The TRS R consists of the following rules:

minus(x, 0) → x
p(s(x)) → x
p(0) → 0
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

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

(51) TransformationProof (EQUIVALENT transformation)

By rewriting [LPAR04] the rule FACITER(s(y0), y1) → IF(false, p(minus(s(y0), 0)), y1, times(y1, s(y0))) at position [1,0] we obtained the following new rules [LPAR04]:

FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0))) → FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0)))

(52) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0)))

The TRS R consists of the following rules:

minus(x, 0) → x
p(s(x)) → x
p(0) → 0
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

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

(53) 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.

(54) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0)))

The TRS R consists of the following rules:

p(s(x)) → x
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)
minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

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

(55) 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].

minus(x0, 0)
minus(0, x0)
minus(x0, s(x1))

(56) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0)))

The TRS R consists of the following rules:

p(s(x)) → x
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)

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

(57) TransformationProof (EQUIVALENT transformation)

By rewriting [LPAR04] the rule FACITER(s(y0), y1) → IF(false, p(s(y0)), y1, times(y1, s(y0))) at position [1] we obtained the following new rules [LPAR04]:

FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0))) → FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))

(58) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))

The TRS R consists of the following rules:

p(s(x)) → x
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)

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

(59) 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.

(60) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))

The TRS R consists of the following rules:

times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
p(s(x0))
p(0)

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

(61) 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(s(x0))
p(0)

(62) Obligation:

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

IF(false, x, y, z) → FACITER(x, z)
FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))

The TRS R consists of the following rules:

times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)

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

(63) TransformationProof (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule IF(false, x, y, z) → FACITER(x, z) we obtained the following new rules [LPAR04]:

IF(false, s(y_0), x1, x2) → FACITER(s(y_0), x2) → IF(false, s(y_0), x1, x2) → FACITER(s(y_0), x2)

(64) Obligation:

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

FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))
IF(false, s(y_0), x1, x2) → FACITER(s(y_0), x2)

The TRS R consists of the following rules:

times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, x) → x
plus(s(x), y) → s(plus(x, y))

The set Q consists of the following terms:

plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)

We have to consider all minimal (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:

IF(false, s(y_0), x1, x2) → FACITER(s(y_0), x2)
The graph contains the following edges 2 >= 1, 4 >= 2
FACITER(s(y0), y1) → IF(false, y0, y1, times(y1, s(y0)))
The graph contains the following edges 1 > 2, 2 >= 3

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) MNOCProof (EQUIVALENT transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) MNOCProof (EQUIVALENT transformation)

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) QReductionProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) MNOCProof (EQUIVALENT transformation)

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) QReductionProof (EQUIVALENT transformation)

(29) Obligation:

(30) QDPSizeChangeProof (EQUIVALENT transformation)

(31) YES

(32) Obligation:

(33) MNOCProof (EQUIVALENT transformation)

(34) Obligation:

(35) UsableRulesProof (EQUIVALENT transformation)

(36) Obligation:

(37) QReductionProof (EQUIVALENT transformation)

(38) Obligation:

(39) TransformationProof (EQUIVALENT transformation)

(40) Obligation:

(41) DependencyGraphProof (EQUIVALENT transformation)

(42) Obligation:

(43) UsableRulesProof (EQUIVALENT transformation)

(44) Obligation:

(45) QReductionProof (EQUIVALENT transformation)

(46) Obligation:

(47) TransformationProof (EQUIVALENT transformation)

(48) Obligation:

(49) UsableRulesProof (EQUIVALENT transformation)

(50) Obligation:

(51) TransformationProof (EQUIVALENT transformation)

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) QReductionProof (EQUIVALENT transformation)

(56) Obligation:

(57) TransformationProof (EQUIVALENT transformation)

(58) Obligation:

(59) UsableRulesProof (EQUIVALENT transformation)

(60) Obligation:

(61) QReductionProof (EQUIVALENT transformation)

(62) Obligation:

(63) TransformationProof (EQUIVALENT transformation)

(64) Obligation:

(65) QDPSizeChangeProof (EQUIVALENT transformation)

(66) YES