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

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 UsableRulesProof (⇔)
    ↳9 QDP
    ↳10 QReductionProof (⇔)
    ↳11 QDP
    ↳12 QDPSizeChangeProof (⇔)
    ↳13 TRUE
  ↳14 QDP
    ↳15 UsableRulesProof (⇔)
    ↳16 QDP
    ↳17 QReductionProof (⇔)
    ↳18 QDP
    ↳19 QDPSizeChangeProof (⇔)
    ↳20 TRUE
  ↳21 QDP
    ↳22 UsableRulesProof (⇔)
    ↳23 QDP
    ↳24 QReductionProof (⇔)
    ↳25 QDP
    ↳26 Narrowing (⇔)
    ↳27 QDP
    ↳28 DependencyGraphProof (⇔)
    ↳29 QDP
    ↳30 UsableRulesProof (⇔)
    ↳31 QDP
    ↳32 QReductionProof (⇔)
    ↳33 QDP
    ↳34 Narrowing (⇔)
    ↳35 QDP
    ↳36 DependencyGraphProof (⇔)
    ↳37 QDP
    ↳38 Narrowing (⇔)
    ↳39 QDP
    ↳40 Instantiation (⇔)
    ↳41 QDP
    ↳42 DependencyGraphProof (⇔)
    ↳43 QDP
    ↳44 ForwardInstantiation (⇔)
    ↳45 QDP
    ↳46 DependencyGraphProof (⇔)
    ↳47 QDP
    ↳48 QDPOrderProof (⇔)
    ↳49 QDP
    ↳50 DependencyGraphProof (⇔)
    ↳51 TRUE

(0) Obligation:

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

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:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

(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:

LE(s(x), s(y)) → LE(x, y)
MINUS(s(x), s(y)) → MINUS(x, y)
MOD(x, y) → IF_MOD(isZero(y), le(y, x), x, y, minus(x, y))
MOD(x, y) → ISZERO(y)
MOD(x, y) → LE(y, x)
MOD(x, y) → MINUS(x, y)
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MINUS(s(x), s(y)) → MINUS(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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(s(x), s(y)) → MINUS(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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].

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

(11) Obligation:

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

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

(13) TRUE

(14) Obligation:

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

LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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:

LE(s(x), s(y)) → LE(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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].

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

(18) Obligation:

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

LE(s(x), s(y)) → LE(x, y)

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

(19) 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:

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

(20) TRUE

(21) Obligation:

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

MOD(x, y) → IF_MOD(isZero(y), le(y, x), x, y, minus(x, y))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
isZero(0) → true
isZero(s(x)) → false
mod(x, y) → if_mod(isZero(y), le(y, x), x, y, minus(x, y))
if_mod(true, b, x, y, z) → divByZeroError
if_mod(false, false, x, y, z) → x
if_mod(false, true, x, y, z) → mod(z, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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

(22) 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.

(23) Obligation:

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

MOD(x, y) → IF_MOD(isZero(y), le(y, x), x, y, minus(x, y))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minus(x0, x0)
minus(x0, 0)
minus(0, x0)
minus(s(x0), s(x1))
isZero(0)
isZero(s(x0))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

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

(24) 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].

mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, false, x0, x1, x2)
if_mod(false, true, x0, x1, x2)

(25) Obligation:

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

MOD(x, y) → IF_MOD(isZero(y), le(y, x), x, y, minus(x, y))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)

The set Q consists of the following terms:

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

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

(26) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule MOD(x, y) → IF_MOD(isZero(y), le(y, x), x, y, minus(x, y)) at position [0] we obtained the following new rules [LPAR04]:

MOD(y0, 0) → IF_MOD(true, le(0, y0), y0, 0, minus(y0, 0))
MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0)))

(27) Obligation:

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

IF_MOD(false, true, x, y, z) → MOD(z, y)
MOD(y0, 0) → IF_MOD(true, le(0, y0), y0, 0, minus(y0, 0))
MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0)))

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)

The set Q consists of the following terms:

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

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0)))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

isZero(0) → true
isZero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)

The set Q consists of the following terms:

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

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

(30) 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.

(31) Obligation:

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

MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0)))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(32) 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))

(33) Obligation:

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

MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0)))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(34) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule MOD(y0, s(x0)) → IF_MOD(false, le(s(x0), y0), y0, s(x0), minus(y0, s(x0))) at position [1] we obtained the following new rules [LPAR04]:

MOD(0, s(x0)) → IF_MOD(false, false, 0, s(x0), minus(0, s(x0)))
MOD(s(x1), s(x0)) → IF_MOD(false, le(x0, x1), s(x1), s(x0), minus(s(x1), s(x0)))

(35) Obligation:

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

IF_MOD(false, true, x, y, z) → MOD(z, y)
MOD(0, s(x0)) → IF_MOD(false, false, 0, s(x0), minus(0, s(x0)))
MOD(s(x1), s(x0)) → IF_MOD(false, le(x0, x1), s(x1), s(x0), minus(s(x1), s(x0)))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MOD(s(x1), s(x0)) → IF_MOD(false, le(x0, x1), s(x1), s(x0), minus(s(x1), s(x0)))
IF_MOD(false, true, x, y, z) → MOD(z, y)

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(38) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule MOD(s(x1), s(x0)) → IF_MOD(false, le(x0, x1), s(x1), s(x0), minus(s(x1), s(x0))) at position [4] we obtained the following new rules [LPAR04]:

MOD(s(y0), s(y0)) → IF_MOD(false, le(y0, y0), s(y0), s(y0), 0)
MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))

(39) Obligation:

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

IF_MOD(false, true, x, y, z) → MOD(z, y)
MOD(s(y0), s(y0)) → IF_MOD(false, le(y0, y0), s(y0), s(y0), 0)
MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(40) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IF_MOD(false, true, x, y, z) → MOD(z, y) we obtained the following new rules [LPAR04]:

IF_MOD(false, true, s(z0), s(z0), 0) → MOD(0, s(z0))
IF_MOD(false, true, s(z0), s(z1), y_1) → MOD(y_1, s(z1))

(41) Obligation:

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

MOD(s(y0), s(y0)) → IF_MOD(false, le(y0, y0), s(y0), s(y0), 0)
MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))
IF_MOD(false, true, s(z0), s(z0), 0) → MOD(0, s(z0))
IF_MOD(false, true, s(z0), s(z1), y_1) → MOD(y_1, s(z1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(42) DependencyGraphProof (EQUIVALENT transformation)

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

(43) Obligation:

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

IF_MOD(false, true, s(z0), s(z1), y_1) → MOD(y_1, s(z1))
MOD(s(y0), s(y0)) → IF_MOD(false, le(y0, y0), s(y0), s(y0), 0)
MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(44) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule IF_MOD(false, true, s(z0), s(z1), y_1) → MOD(y_1, s(z1)) we obtained the following new rules [LPAR04]:

IF_MOD(false, true, s(x0), s(x1), s(y_0)) → MOD(s(y_0), s(x1))

(45) Obligation:

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

MOD(s(y0), s(y0)) → IF_MOD(false, le(y0, y0), s(y0), s(y0), 0)
MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))
IF_MOD(false, true, s(x0), s(x1), s(y_0)) → MOD(s(y_0), s(x1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(46) DependencyGraphProof (EQUIVALENT transformation)

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

(47) Obligation:

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

MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))
IF_MOD(false, true, s(x0), s(x1), s(y_0)) → MOD(s(y_0), s(x1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MOD(s(x0), s(x1)) → IF_MOD(false, le(x1, x0), s(x0), s(x1), minus(x0, x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(IF_MOD(x1, x2, x3, x4, x5)) = x5   
POL(MOD(x1, x2)) = x1   
POL(false) = 0   
POL(le(x1, x2)) = 0   
POL(minus(x1, x2)) = x1   
POL(s(x1)) = 1 + x1   
POL(true) = 0   

The following usable rules [FROCOS05] were oriented:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
minus(0, x) → 0
minus(x, x) → 0

(49) Obligation:

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

IF_MOD(false, true, s(x0), s(x1), s(y_0)) → MOD(s(y_0), s(x1))

The TRS R consists of the following rules:

le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, x) → 0
minus(0, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
le(0, y) → true

The set Q consists of the following terms:

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

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

(50) DependencyGraphProof (EQUIVALENT transformation)

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

(51) TRUE