Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 PisEmptyProof (⇔)
    ↳21 TRUE
  ↳22 QDP

(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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(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:

LE(s(x), s(y)) → LE(x, y)
ID(s(x)) → ID(x)
MINUS(s(x), s(y)) → MINUS(x, y)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))
MOD(x, y) → ZERO(x)
MOD(x, y) → ZERO(y)
MOD(x, y) → LE(y, x)
MOD(x, y) → ID(x)
MOD(x, y) → ID(y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
IF2(false, b2, x, y) → IF3(b2, x, y)
IF3(true, x, y) → MOD(minus(x, y), s(y))
IF3(true, x, 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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(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 4 SCCs with 6 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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


MINUS(s(x), s(y)) → MINUS(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MINUS(x1, x2)  =  x2
s(x1)  =  s(x1)
le(x1, x2)  =  le
0  =  0
true  =  true
false  =  false
zero(x1)  =  zero
id(x1)  =  x1
minus(x1, x2)  =  x1
mod(x1, x2)  =  mod(x1)
if_mod(x1, x2, x3, x4, x5)  =  if_mod(x4)
if2(x1, x2, x3, x4)  =  if2(x3)
if3(x1, x2, x3)  =  if3(x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:
le > [true, zero] > s1 > false > [0, mod1, ifmod1, if21, if31]

Status:
s1: [1]
le: []
0: []
true: []
false: []
zero: []
mod1: [1]
ifmod1: [1]
if21: [1]
if31: [1]


The following usable rules [FROCOS05] were oriented:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

(9) Obligation:

Q DP problem:
P is empty.
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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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

ID(s(x)) → ID(x)

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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ID(s(x)) → ID(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ID(x1)  =  ID(x1)
s(x1)  =  s(x1)
le(x1, x2)  =  le
0  =  0
true  =  true
false  =  false
zero(x1)  =  zero
id(x1)  =  x1
minus(x1, x2)  =  x1
mod(x1, x2)  =  x1
if_mod(x1, x2, x3, x4, x5)  =  x4
if2(x1, x2, x3, x4)  =  x3
if3(x1, x2, x3)  =  x2

Lexicographic path order with status [LPO].
Quasi-Precedence:
[ID1, s1] > false > 0
le > true > 0
le > false > 0
zero > true > 0
zero > false > 0

Status:
ID1: [1]
s1: [1]
le: []
0: []
true: []
false: []
zero: []


The following usable rules [FROCOS05] were oriented:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

(14) Obligation:

Q DP problem:
P is empty.
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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) 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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LE(s(x), s(y)) → LE(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LE(x1, x2)  =  x2
s(x1)  =  s(x1)
le(x1, x2)  =  le
0  =  0
true  =  true
false  =  false
zero(x1)  =  zero
id(x1)  =  x1
minus(x1, x2)  =  x1
mod(x1, x2)  =  x1
if_mod(x1, x2, x3, x4, x5)  =  x4
if2(x1, x2, x3, x4)  =  x3
if3(x1, x2, x3)  =  x2

Lexicographic path order with status [LPO].
Quasi-Precedence:
le > true > s1 > false > 0
zero > true > s1 > false > 0

Status:
s1: [1]
le: []
0: []
true: []
false: []
zero: []


The following usable rules [FROCOS05] were oriented:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

(19) Obligation:

Q DP problem:
P is empty.
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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

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

MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
IF2(false, b2, x, y) → IF3(b2, x, y)
IF3(true, x, y) → MOD(minus(x, y), s(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)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

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