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 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 DependencyGraphProof (⇔)
    ↳21 TRUE

(0) Obligation:

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

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(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:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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:

MINUS(X, s(Y)) → PRED(minus(X, Y))
MINUS(X, s(Y)) → MINUS(X, Y)
LE(s(X), s(Y)) → LE(X, Y)
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
GCD(s(X), s(Y)) → LE(Y, X)
IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
IF(true, s(X), s(Y)) → MINUS(X, Y)
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
IF(false, s(X), s(Y)) → MINUS(Y, X)

The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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 3 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) 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:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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.


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)  =  LE(x1)
s(x1)  =  s(x1)
minus(x1, x2)  =  minus(x1)
pred(x1)  =  x1
0  =  0
le(x1, x2)  =  le
false  =  false
true  =  true
gcd(x1, x2)  =  gcd(x1, x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
0 > [LE1, s1, minus1] > [gcd2, if2]
[le, true] > false > [LE1, s1, minus1] > [gcd2, if2]


The following usable rules [FROCOS05] were oriented:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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:

MINUS(X, s(Y)) → MINUS(X, Y)

The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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.


MINUS(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)  =  MINUS(x2)
s(x1)  =  s(x1)
minus(x1, x2)  =  x1
pred(x1)  =  x1
0  =  0
le(x1, x2)  =  le(x2)
false  =  false
true  =  true
gcd(x1, x2)  =  gcd(x1, x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
[MINUS1, s1, le1, false, true, gcd2, if2] > 0


The following usable rules [FROCOS05] were oriented:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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:

IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))

The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(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.


IF(true, s(X), s(Y)) → GCD(minus(X, Y), s(Y))
IF(false, s(X), s(Y)) → GCD(minus(Y, X), s(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF(x1, x2, x3)  =  IF(x2, x3)
true  =  true
s(x1)  =  s(x1)
GCD(x1, x2)  =  GCD(x1, x2)
minus(x1, x2)  =  minus(x1)
le(x1, x2)  =  le
false  =  false
pred(x1)  =  x1
0  =  0
gcd(x1, x2)  =  gcd(x1, x2)
if(x1, x2, x3)  =  if(x2, x3)

Recursive Path Order [RPO].
Precedence:
[true, le, false] > s1 > [IF2, GCD2] > [gcd2, if2]
[true, le, false] > s1 > minus1 > [gcd2, if2]
0 > s1 > [IF2, GCD2] > [gcd2, if2]
0 > s1 > minus1 > [gcd2, if2]


The following usable rules [FROCOS05] were oriented:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

(19) Obligation:

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

GCD(s(X), s(Y)) → IF(le(Y, X), s(X), s(Y))

The TRS R consists of the following rules:

minus(X, s(Y)) → pred(minus(X, Y))
minus(X, 0) → X
pred(s(X)) → X
le(s(X), s(Y)) → le(X, Y)
le(s(X), 0) → false
le(0, Y) → true
gcd(0, Y) → 0
gcd(s(X), 0) → s(X)
gcd(s(X), s(Y)) → if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) → gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) → gcd(minus(Y, X), s(X))

The set Q consists of the following terms:

minus(x0, s(x1))
minus(x0, 0)
pred(s(x0))
le(s(x0), s(x1))
le(s(x0), 0)
le(0, x0)
gcd(0, x0)
gcd(s(x0), 0)
gcd(s(x0), s(x1))
if(true, s(x0), s(x1))
if(false, s(x0), s(x1))

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

(20) DependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE