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

(0) Obligation:

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

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, 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:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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(s(x), y) → IF(gt(s(x), y), x, y)
MINUS(s(x), y) → GT(s(x), y)
IF(true, x, y) → MINUS(x, y)
GCD(x, y) → IF1(ge(x, y), x, y)
GCD(x, y) → GE(x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(true, x, y) → GT(y, 0)
IF1(false, x, y) → GCD(y, x)
IF2(true, x, y) → GCD(minus(x, y), y)
IF2(true, x, y) → MINUS(x, y)
GT(s(x), s(y)) → GT(x, y)
GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(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 4 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

GE(s(x), s(y)) → GE(x, y)

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

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

(11) Obligation:

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

GE(s(x), s(y)) → GE(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:

• GE(s(x), s(y)) → GE(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:

GT(s(x), s(y)) → GT(x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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:

GT(s(x), s(y)) → GT(x, y)

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

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(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].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

(18) Obligation:

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

GT(s(x), s(y)) → GT(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:

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

(20) YES

(21) Obligation:

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

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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:

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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].

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

(25) Obligation:

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

IF(true, x, y) → MINUS(x, y)
MINUS(s(x), y) → IF(gt(s(x), y), x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))

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

(26) 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), y) → IF(gt(s(x), y), x, y)
  The graph contains the following edges 1 > 2, 2 >= 3

• IF(true, x, y) → MINUS(x, y)
  The graph contains the following edges 2 >= 1, 3 >= 2

(27) YES

(28) Obligation:

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

IF2(true, x, y) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → GCD(y, x)

The TRS R consists of the following rules:

minus(s(x), y) → if(gt(s(x), y), x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0
gcd(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → gcd(y, x)
if2(true, x, y) → gcd(minus(x, y), y)
if2(false, x, y) → x
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(29) 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.

(30) Obligation:

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

IF2(true, x, y) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → GCD(y, x)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(31) 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].

gcd(x0, x1)
if1(true, x0, x1)
if1(false, x0, x1)
if2(true, x0, x1)
if2(false, x0, x1)

(32) Obligation:

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

IF2(true, x, y) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → GCD(y, x)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(33) 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 IF2(true, x, y) → GCD(minus(x, y), y) the following chains were created:

• We consider the chain IF1(true, x, y) → IF2(gt(y, 0), x, y), IF2(true, x, y) → GCD(minus(x, y), y) which results in the following constraint:
  

  (1)    (IF2(gt(x5, 0), x4, x5)=IF2(true, x6, x7) ⇒ IF2(true, x6, x7)≥GCD(minus(x6, x7), x7))

  
  
  We simplified constraint (1) using rules (I), (II), (III), (VII) which results in the following new constraint:
  

  (2)    (0=x42∧gt(x5, x42)=true ⇒ IF2(true, x4, x5)≥GCD(minus(x4, x5), x5))

  
  
  We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on gt(x5, x42)=true which results in the following new constraints:
  

  (3)    (true=true∧0=0 ⇒ IF2(true, x4, s(x44))≥GCD(minus(x4, s(x44)), s(x44)))

  
  

  (4)    (gt(x46, x45)=true∧0=s(x45)∧(∀x47:gt(x46, x45)=true∧0=x45 ⇒ IF2(true, x47, x46)≥GCD(minus(x47, x46), x46)) ⇒ IF2(true, x4, s(x46))≥GCD(minus(x4, s(x46)), s(x46)))

  
  
  We simplified constraint (3) using rules (I), (II) which results in the following new constraint:
  

  (5)    (IF2(true, x4, s(x44))≥GCD(minus(x4, s(x44)), s(x44)))

  
  
  We solved constraint (4) using rules (I), (II).

For Pair GCD(x, y) → IF1(ge(x, y), x, y) the following chains were created:

• We consider the chain IF2(true, x, y) → GCD(minus(x, y), y), GCD(x, y) → IF1(ge(x, y), x, y) which results in the following constraint:
  

  (6)    (GCD(minus(x10, x11), x11)=GCD(x12, x13) ⇒ GCD(x12, x13)≥IF1(ge(x12, x13), x12, x13))

  
  
  We simplified constraint (6) using rules (I), (II), (III), (IV) which results in the following new constraint:
  

  (7)    (GCD(x12, x11)≥IF1(ge(x12, x11), x12, x11))

  
  
  
• We consider the chain IF1(false, x, y) → GCD(y, x), GCD(x, y) → IF1(ge(x, y), x, y) which results in the following constraint:
  

  (8)    (GCD(x19, x18)=GCD(x20, x21) ⇒ GCD(x20, x21)≥IF1(ge(x20, x21), x20, x21))

  
  
  We simplified constraint (8) using rules (I), (II), (III) which results in the following new constraint:
  

  (9)    (GCD(x19, x18)≥IF1(ge(x19, x18), x19, x18))

  
  
  

For Pair IF1(true, x, y) → IF2(gt(y, 0), x, y) the following chains were created:

• We consider the chain GCD(x, y) → IF1(ge(x, y), x, y), IF1(true, x, y) → IF2(gt(y, 0), x, y) which results in the following constraint:
  

  (10)    (IF1(ge(x24, x25), x24, x25)=IF1(true, x26, x27) ⇒ IF1(true, x26, x27)≥IF2(gt(x27, 0), x26, x27))

  
  
  We simplified constraint (10) using rules (I), (II), (III) which results in the following new constraint:
  

  (11)    (ge(x24, x25)=true ⇒ IF1(true, x24, x25)≥IF2(gt(x25, 0), x24, x25))

  
  
  We simplified constraint (11) using rule (V) (with possible (I) afterwards) using induction on ge(x24, x25)=true which results in the following new constraints:
  

  (12)    (true=true ⇒ IF1(true, x48, 0)≥IF2(gt(0, 0), x48, 0))

  
  

  (13)    (ge(x51, x50)=true∧(ge(x51, x50)=true ⇒ IF1(true, x51, x50)≥IF2(gt(x50, 0), x51, x50)) ⇒ IF1(true, s(x51), s(x50))≥IF2(gt(s(x50), 0), s(x51), s(x50)))

  
  
  We simplified constraint (12) using rules (I), (II) which results in the following new constraint:
  

  (14)    (IF1(true, x48, 0)≥IF2(gt(0, 0), x48, 0))

  
  
  We simplified constraint (13) using rule (VI) where we applied the induction hypothesis (ge(x51, x50)=true ⇒ IF1(true, x51, x50)≥IF2(gt(x50, 0), x51, x50)) with σ = [ ] which results in the following new constraint:
  

  (15)    (IF1(true, x51, x50)≥IF2(gt(x50, 0), x51, x50) ⇒ IF1(true, s(x51), s(x50))≥IF2(gt(s(x50), 0), s(x51), s(x50)))

  
  
  

For Pair IF1(false, x, y) → GCD(y, x) the following chains were created:

• We consider the chain GCD(x, y) → IF1(ge(x, y), x, y), IF1(false, x, y) → GCD(y, x) which results in the following constraint:
  

  (16)    (IF1(ge(x34, x35), x34, x35)=IF1(false, x36, x37) ⇒ IF1(false, x36, x37)≥GCD(x37, x36))

  
  
  We simplified constraint (16) using rules (I), (II), (III) which results in the following new constraint:
  

  (17)    (ge(x34, x35)=false ⇒ IF1(false, x34, x35)≥GCD(x35, x34))

  
  
  We simplified constraint (17) using rule (V) (with possible (I) afterwards) using induction on ge(x34, x35)=false which results in the following new constraints:
  

  (18)    (false=false ⇒ IF1(false, 0, s(x53))≥GCD(s(x53), 0))

  
  

  (19)    (ge(x55, x54)=false∧(ge(x55, x54)=false ⇒ IF1(false, x55, x54)≥GCD(x54, x55)) ⇒ IF1(false, s(x55), s(x54))≥GCD(s(x54), s(x55)))

  
  
  We simplified constraint (18) using rules (I), (II) which results in the following new constraint:
  

  (20)    (IF1(false, 0, s(x53))≥GCD(s(x53), 0))

  
  
  We simplified constraint (19) using rule (VI) where we applied the induction hypothesis (ge(x55, x54)=false ⇒ IF1(false, x55, x54)≥GCD(x54, x55)) with σ = [ ] which results in the following new constraint:
  

  (21)    (IF1(false, x55, x54)≥GCD(x54, x55) ⇒ IF1(false, s(x55), s(x54))≥GCD(s(x54), s(x55)))

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• IF2(true, x, y) → GCD(minus(x, y), y)
  
  • (IF2(true, x4, s(x44))≥GCD(minus(x4, s(x44)), s(x44)))
    
  
  
• GCD(x, y) → IF1(ge(x, y), x, y)
  
  • (GCD(x12, x11)≥IF1(ge(x12, x11), x12, x11))
    
  • (GCD(x19, x18)≥IF1(ge(x19, x18), x19, x18))
    
  
  
• IF1(true, x, y) → IF2(gt(y, 0), x, y)
  
  • (IF1(true, x48, 0)≥IF2(gt(0, 0), x48, 0))
    
  • (IF1(true, x51, x50)≥IF2(gt(x50, 0), x51, x50) ⇒ IF1(true, s(x51), s(x50))≥IF2(gt(s(x50), 0), s(x51), s(x50)))
    
  
  
• IF1(false, x, y) → GCD(y, x)
  
  • (IF1(false, 0, s(x53))≥GCD(s(x53), 0))
    
  • (IF1(false, x55, x54)≥GCD(x54, x55) ⇒ IF1(false, s(x55), s(x54))≥GCD(s(x54), s(x55)))
    
  
  

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation [NONINF]:

POL(0) = 0   
POL(GCD(x₁, x₂)) = -1 + x₂   
POL(IF1(x₁, x₂, x₃)) = -1 - x₁ + x₃   
POL(IF2(x₁, x₂, x₃)) = -1 - x₁ + x₃   
POL(c) = -1   
POL(false) = 0   
POL(ge(x₁, x₂)) = 0   
POL(gt(x₁, x₂)) = 0   
POL(if(x₁, x₂, x₃)) = 0   
POL(minus(x₁, x₂)) = 0   
POL(s(x₁)) = 1 + x₁   
POL(true) = 0   

The following pairs are in P_>:

IF1(false, x, y) → GCD(y, x)

The following pairs are in P_bound:

IF2(true, x, y) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF1(false, x, y) → GCD(y, x)

The following rules are usable:

true → ge(x, 0)
false → ge(0, s(x))
ge(x, y) → ge(s(x), s(y))
false → gt(0, y)
true → gt(s(x), 0)

(34) Obligation:

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

IF2(true, x, y) → GCD(minus(x, y), y)
GCD(x, y) → IF1(ge(x, y), x, y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(35) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule GCD(x, y) → IF1(ge(x, y), x, y) at position [0] we obtained the following new rules [LPAR04]:

GCD(x0, 0) → IF1(true, x0, 0)
GCD(0, s(x0)) → IF1(false, 0, s(x0))
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))

(36) Obligation:

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

IF2(true, x, y) → GCD(minus(x, y), y)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
GCD(x0, 0) → IF1(true, x0, 0)
GCD(0, s(x0)) → IF1(false, 0, s(x0))
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(37) DependencyGraphProof (EQUIVALENT transformation)

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

(38) Obligation:

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

GCD(x0, 0) → IF1(true, x0, 0)
IF1(true, x, y) → IF2(gt(y, 0), x, y)
IF2(true, x, y) → GCD(minus(x, y), y)
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(39) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule IF1(true, x, y) → IF2(gt(y, 0), x, y) at position [0] we obtained the following new rules [LPAR04]:

IF1(true, y0, 0) → IF2(false, y0, 0)
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))

(40) Obligation:

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

GCD(x0, 0) → IF1(true, x0, 0)
IF2(true, x, y) → GCD(minus(x, y), y)
GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, 0) → IF2(false, y0, 0)
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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 2 less nodes.

(42) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, x, y) → GCD(minus(x, y), y)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(43) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule IF2(true, x, y) → GCD(minus(x, y), y) at position [0] we obtained the following new rules [LPAR04]:

IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)

(44) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, y0, s(x0)) → IF2(true, y0, s(x0))
IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(45) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IF1(true, y0, s(x0)) → IF2(true, y0, s(x0)) we obtained the following new rules [LPAR04]:

IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

(46) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1)
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(47) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IF2(true, s(x0), x1) → GCD(if(gt(s(x0), x1), x0, x1), x1) we obtained the following new rules [LPAR04]:

IF2(true, s(z0), s(z1)) → GCD(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1))

(48) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(49) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule IF2(true, s(z0), s(z1)) → GCD(if(gt(s(z0), s(z1)), z0, s(z1)), s(z1)) at position [0,0] we obtained the following new rules [LPAR04]:

IF2(true, s(z0), s(z1)) → GCD(if(gt(z0, z1), z0, s(z1)), s(z1))

(50) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))
IF2(true, s(z0), s(z1)) → GCD(if(gt(z0, z1), z0, s(z1)), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(51) Induction-Processor (SOUND transformation)

This DP could be deleted by the Induction-Processor:
IF2(true, s(z0'), s(z1')) → GCD(if(gt(z0', z1'), z0', s(z1')), s(z1'))


This order was computed:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(GCD(x₁, x₂)) = x₁   
POL(IF1(x₁, x₂, x₃)) = x₂   
POL(IF2(x₁, x₂, x₃)) = x₂   
POL(false) = 1   
POL(ge(x₁, x₂)) = 1 + x₁ + x₂   
POL(gt(x₁, x₂)) = 1   
POL(if(x₁, x₂, x₃)) = x₁ + x₂   
POL(minus(x₁, x₂)) = x₁   
POL(s(x₁)) = 1 + x₁   
POL(true) = 1   

At least one of these decreasing rules is always used after the deleted DP:
if(false, x683, y503) → 0


The following formula is valid:
z0':sort[a23],z1':sort[a23].if'(gt(z0' , z1' ), z0' , s(z1' ))=true


The transformed set:
minus'(s(x41), y29) → if'(gt(s(x41), y29), x41, y29)
if'(true, x59, y43) → minus'(x59, y43)
if'(false, x68, y50) → true
minus'(0, x1) → false
gt(0, y) → false
gt(s(x5), 0) → true
ge(x14, 0) → true
ge(0, s(x23)) → false
ge(s(x32), s(y22)) → ge(x32, y22)
minus(s(x41), y29) → if(gt(s(x41), y29), x41, y29)
gt(s(x50), s(y36)) → gt(x50, y36)
if(true, x59, y43) → s(minus(x59, y43))
if(false, x68, y50) → 0
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true


The proof given by the theorem prover:
The following program was given to the internal theorem prover:

   [x, x0, x1, x68, y50, x41, y43, y29, y, x5, x50, y36, x14, x23, x32, y22]
   equal_bool(true, false) -> false
   equal_bool(false, true) -> false
   equal_bool(true, true) -> true
   equal_bool(false, false) -> true
   true and x -> x
   false and x -> false
   true or x -> true
   false or x -> x
   not(false) -> true
   not(true) -> false
   isa_true(true) -> true
   isa_true(false) -> false
   isa_false(true) -> false
   isa_false(false) -> true
   equal_sort[a23](0, 0) -> true
   equal_sort[a23](0, s(x0)) -> false
   equal_sort[a23](s(x0), 0) -> false
   equal_sort[a23](s(x0), s(x1)) -> equal_sort[a23](x0, x1)
   equal_sort[a40](witness_sort[a40], witness_sort[a40]) -> true
   if'(false, x68, y50) -> true
   if'(true, s(x41), y43) -> if'(gt(s(x41), y43), x41, y43)
   if'(true, 0, y43) -> false
   minus'(0, x1) -> false
   equal_bool(gt(s(x41), y29), true) -> true | minus'(s(x41), y29) -> minus'(x41, y29)
   equal_bool(gt(s(x41), y29), true) -> false | minus'(s(x41), y29) -> true
   gt(0, y) -> false
   gt(s(x5), 0) -> true
   gt(s(x50), s(y36)) -> gt(x50, y36)
   ge(x14, 0) -> true
   ge(0, s(x23)) -> false
   ge(s(x32), s(y22)) -> ge(x32, y22)
   if(false, x68, y50) -> 0
   if(true, s(x41), y43) -> s(if(gt(s(x41), y43), x41, y43))
   if(true, 0, y43) -> s(0)
   minus(0, x1) -> 0
   equal_bool(gt(s(x41), y29), true) -> true | minus(s(x41), y29) -> s(minus(x41, y29))
   equal_bool(gt(s(x41), y29), true) -> false | minus(s(x41), y29) -> 0

The following output was given by the internal theorem prover:

proof of internal
# AProVE Commit ID: 9a00b172b26c9abb2d4c4d5eaf341e919eb0fbf1 nowonder 20100222 unpublished dirty


Partial correctness of the following Program

   [x, x0, x1, x68, y50, x41, y43, y29, y, x5, x50, y36, x14, x23, x32, y22]
   equal_bool(true, false) -> false
   equal_bool(false, true) -> false
   equal_bool(true, true) -> true
   equal_bool(false, false) -> true
   true and x -> x
   false and x -> false
   true or x -> true
   false or x -> x
   not(false) -> true
   not(true) -> false
   isa_true(true) -> true
   isa_true(false) -> false
   isa_false(true) -> false
   isa_false(false) -> true
   equal_sort[a23](0, 0) -> true
   equal_sort[a23](0, s(x0)) -> false
   equal_sort[a23](s(x0), 0) -> false
   equal_sort[a23](s(x0), s(x1)) -> equal_sort[a23](x0, x1)
   equal_sort[a40](witness_sort[a40], witness_sort[a40]) -> true
   if'(false, x68, y50) -> true
   if'(true, s(x41), y43) -> if'(gt(s(x41), y43), x41, y43)
   if'(true, 0, y43) -> false
   minus'(0, x1) -> false
   equal_bool(gt(s(x41), y29), true) -> true | minus'(s(x41), y29) -> minus'(x41, y29)
   equal_bool(gt(s(x41), y29), true) -> false | minus'(s(x41), y29) -> true
   gt(0, y) -> false
   gt(s(x5), 0) -> true
   gt(s(x50), s(y36)) -> gt(x50, y36)
   ge(x14, 0) -> true
   ge(0, s(x23)) -> false
   ge(s(x32), s(y22)) -> ge(x32, y22)
   if(false, x68, y50) -> 0
   if(true, s(x41), y43) -> s(if(gt(s(x41), y43), x41, y43))
   if(true, 0, y43) -> s(0)
   minus(0, x1) -> 0
   equal_bool(gt(s(x41), y29), true) -> true | minus(s(x41), y29) -> s(minus(x41, y29))
   equal_bool(gt(s(x41), y29), true) -> false | minus(s(x41), y29) -> 0

using the following formula:
z0':sort[a23],z1':sort[a23].if'(gt(z0', z1'), z0', s(z1'))=true

could be successfully shown:
(0) Formula
(1) Induction by algorithm [EQUIVALENT]
(2) AND
    (3) Formula
        (4) Symbolic evaluation [EQUIVALENT]
        (5) YES
    (6) Formula
        (7) Symbolic evaluation [EQUIVALENT]
        (8) Formula
        (9) Induction by data structure [EQUIVALENT]
        (10) AND
            (11) Formula
                (12) Symbolic evaluation [EQUIVALENT]
                (13) YES
            (14) Formula
                (15) Symbolic evaluation under hypothesis [EQUIVALENT]
                (16) YES
    (17) Formula
        (18) Symbolic evaluation [EQUIVALENT]
        (19) Formula
        (20) Hypothesis Lifting [EQUIVALENT]
        (21) Formula
        (22) Inverse Substitution [SOUND]
        (23) Formula
        (24) Inverse Substitution [SOUND]
        (25) Formula
        (26) Induction by algorithm [EQUIVALENT]
        (27) AND
            (28) Formula
                (29) Symbolic evaluation [EQUIVALENT]
                (30) YES
            (31) Formula
                (32) Symbolic evaluation [EQUIVALENT]
                (33) YES
            (34) Formula
                (35) Symbolic evaluation under hypothesis [EQUIVALENT]
                (36) YES


----------------------------------------

(0)
Obligation:
Formula:
z0':sort[a23],z1':sort[a23].if'(gt(z0', z1'), z0', s(z1'))=true

There are no hypotheses.




----------------------------------------

(1) Induction by algorithm (EQUIVALENT)
Induction by algorithm gt(z0', z1') generates the following cases:

1. Base Case:
Formula:
y:sort[a23].if'(gt(0, y), 0, s(y))=true

There are no hypotheses.





2. Base Case:
Formula:
x5:sort[a23].if'(gt(s(x5), 0), s(x5), s(0))=true

There are no hypotheses.





1. Step Case:
Formula:
x50:sort[a23],y36:sort[a23].if'(gt(s(x50), s(y36)), s(x50), s(s(y36)))=true

Hypotheses:
x50:sort[a23],y36:sort[a23].if'(gt(x50, y36), x50, s(y36))=true






----------------------------------------

(2)
Complex Obligation (AND)

----------------------------------------

(3)
Obligation:
Formula:
y:sort[a23].if'(gt(0, y), 0, s(y))=true

There are no hypotheses.




----------------------------------------

(4) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(5)
YES

----------------------------------------

(6)
Obligation:
Formula:
x5:sort[a23].if'(gt(s(x5), 0), s(x5), s(0))=true

There are no hypotheses.




----------------------------------------

(7) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
x5:sort[a23].if'(gt(x5, 0), x5, s(0))=true
----------------------------------------

(8)
Obligation:
Formula:
x5:sort[a23].if'(gt(x5, 0), x5, s(0))=true

There are no hypotheses.




----------------------------------------

(9) Induction by data structure (EQUIVALENT)
Induction by data structure sort[a23] generates the following cases:



1. Base Case:
Formula:
if'(gt(0, 0), 0, s(0))=true

There are no hypotheses.





1. Step Case:
Formula:
n:sort[a23].if'(gt(s(n), 0), s(n), s(0))=true

Hypotheses:
n:sort[a23].if'(gt(n, 0), n, s(0))=true






----------------------------------------

(10)
Complex Obligation (AND)

----------------------------------------

(11)
Obligation:
Formula:
if'(gt(0, 0), 0, s(0))=true

There are no hypotheses.




----------------------------------------

(12) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Formula:
n:sort[a23].if'(gt(s(n), 0), s(n), s(0))=true

Hypotheses:
n:sort[a23].if'(gt(n, 0), n, s(0))=true




----------------------------------------

(15) Symbolic evaluation under hypothesis (EQUIVALENT)
Could be shown using symbolic evaluation under hypothesis, by using the following hypotheses:

n:sort[a23].if'(gt(n, 0), n, s(0))=true

----------------------------------------

(16)
YES

----------------------------------------

(17)
Obligation:
Formula:
x50:sort[a23],y36:sort[a23].if'(gt(s(x50), s(y36)), s(x50), s(s(y36)))=true

Hypotheses:
x50:sort[a23],y36:sort[a23].if'(gt(x50, y36), x50, s(y36))=true




----------------------------------------

(18) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
x50:sort[a23],y36:sort[a23].if'(gt(x50, y36), s(x50), s(s(y36)))=true
----------------------------------------

(19)
Obligation:
Formula:
x50:sort[a23],y36:sort[a23].if'(gt(x50, y36), s(x50), s(s(y36)))=true

Hypotheses:
x50:sort[a23],y36:sort[a23].if'(gt(x50, y36), x50, s(y36))=true




----------------------------------------

(20) Hypothesis Lifting (EQUIVALENT)
Formula could be generalised by hypothesis lifting to the following new obligation:
Formula:
x50:sort[a23],y36:sort[a23].(if'(gt(x50, y36), x50, s(y36))=true->if'(gt(x50, y36), s(x50), s(s(y36)))=true)

There are no hypotheses.




----------------------------------------

(21)
Obligation:
Formula:
x50:sort[a23],y36:sort[a23].(if'(gt(x50, y36), x50, s(y36))=true->if'(gt(x50, y36), s(x50), s(s(y36)))=true)

There are no hypotheses.




----------------------------------------

(22) Inverse Substitution (SOUND)
The formula could be generalised by inverse substitution to:
n:bool,x50:sort[a23],y36:sort[a23].(if'(n, x50, s(y36))=true->if'(n, s(x50), s(s(y36)))=true)

Inverse substitution used:
[gt(x50, y36)/n]


----------------------------------------

(23)
Obligation:
Formula:
n:bool,x50:sort[a23],y36:sort[a23].(if'(n, x50, s(y36))=true->if'(n, s(x50), s(s(y36)))=true)

There are no hypotheses.




----------------------------------------

(24) Inverse Substitution (SOUND)
The formula could be generalised by inverse substitution to:
n:bool,x50:sort[a23],n':sort[a23].(if'(n, x50, n')=true->if'(n, s(x50), s(n'))=true)

Inverse substitution used:
[s(y36)/n']


----------------------------------------

(25)
Obligation:
Formula:
n:bool,x50:sort[a23],n':sort[a23].(if'(n, x50, n')=true->if'(n, s(x50), s(n'))=true)

There are no hypotheses.




----------------------------------------

(26) Induction by algorithm (EQUIVALENT)
Induction by algorithm if'(n, x50, n') generates the following cases:

1. Base Case:
Formula:
x68:sort[a23],y50:sort[a23].(if'(false, x68, y50)=true->if'(false, s(x68), s(y50))=true)

There are no hypotheses.





2. Base Case:
Formula:
y43:sort[a23].(if'(true, 0, y43)=true->if'(true, s(0), s(y43))=true)

There are no hypotheses.





1. Step Case:
Formula:
x41:sort[a23],y43:sort[a23].(if'(true, s(x41), y43)=true->if'(true, s(s(x41)), s(y43))=true)

Hypotheses:
x41:sort[a23],y43:sort[a23].(if'(gt(s(x41), y43), x41, y43)=true->if'(gt(s(x41), y43), s(x41), s(y43))=true)






----------------------------------------

(27)
Complex Obligation (AND)

----------------------------------------

(28)
Obligation:
Formula:
x68:sort[a23],y50:sort[a23].(if'(false, x68, y50)=true->if'(false, s(x68), s(y50))=true)

There are no hypotheses.




----------------------------------------

(29) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(30)
YES

----------------------------------------

(31)
Obligation:
Formula:
y43:sort[a23].(if'(true, 0, y43)=true->if'(true, s(0), s(y43))=true)

There are no hypotheses.




----------------------------------------

(32) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(33)
YES

----------------------------------------

(34)
Obligation:
Formula:
x41:sort[a23],y43:sort[a23].(if'(true, s(x41), y43)=true->if'(true, s(s(x41)), s(y43))=true)

Hypotheses:
x41:sort[a23],y43:sort[a23].(if'(gt(s(x41), y43), x41, y43)=true->if'(gt(s(x41), y43), s(x41), s(y43))=true)




----------------------------------------

(35) Symbolic evaluation under hypothesis (EQUIVALENT)
Could be shown using symbolic evaluation under hypothesis, by using the following hypotheses:

x41:sort[a23],y43:sort[a23].(if'(gt(s(x41), y43), x41, y43)=true->if'(gt(s(x41), y43), s(x41), s(y43))=true)

----------------------------------------

(36)
YES

(52) Complex Obligation (AND)

(53) Obligation:

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

GCD(s(x0), s(x1)) → IF1(ge(x0, x1), s(x0), s(x1))
IF1(true, s(z0), s(z1)) → IF2(true, s(z0), s(z1))

The TRS R consists of the following rules:

gt(0, y) → false
gt(s(x), 0) → true
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(s(x), y) → if(gt(s(x), y), x, y)
gt(s(x), s(y)) → gt(x, y)
if(true, x, y) → s(minus(x, y))
if(false, x, y) → 0

The set Q consists of the following terms:

minus(s(x0), x1)
if(true, x0, x1)
if(false, x0, x1)
gt(0, x0)
gt(s(x0), 0)
gt(s(x0), s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))

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

(54) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(55) TRUE

(56) Obligation:

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

minus'(s(x41), y29) → if'(gt(s(x41), y29), x41, y29)
if'(true, x59, y43) → minus'(x59, y43)
if'(false, x68, y50) → true
minus'(0, x1) → false
gt(0, y) → false
gt(s(x5), 0) → true
ge(x14, 0) → true
ge(0, s(x23)) → false
ge(s(x32), s(y22)) → ge(x32, y22)
minus(s(x41), y29) → if(gt(s(x41), y29), x41, y29)
gt(s(x50), s(y36)) → gt(x50, y36)
if(true, x59, y43) → s(minus(x59, y43))
if(false, x68, y50) → 0
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

Q is empty.

(57) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
minus'(x1, x2)  =  minus'(x1, x2)
s(x1)  =  s(x1)
if'(x1, x2, x3)  =  if'(x1, x2, x3)
gt(x1, x2)  =  gt(x1, x2)
true  =  true
false  =  false
0  =  0
ge(x1, x2)  =  ge(x1, x2)
minus(x1, x2)  =  minus(x1, x2)
if(x1, x2, x3)  =  if(x1, x2, x3)
equal_bool(x1, x2)  =  equal_bool(x1, x2)
and(x1, x2)  =  and(x1, x2)
or(x1, x2)  =  or(x1, x2)
not(x1)  =  not(x1)
isa_true(x1)  =  x1
isa_false(x1)  =  isa_false(x1)
equal_sort[a23](x1, x2)  =  equal_sort[a23](x1, x2)
equal_sort[a40](x1, x2)  =  equal_sort[a40](x1, x2)
witness_sort[a40]  =  witness_sort[a40]

Recursive path order with status [RPO].
Quasi-Precedence:

and₂ > [minus'₂, if'₃, false, 0] > [true, ge₂, equal_bool2] > [minus₂, if₃] > [s₁, gt₂, equal_sort[a23]₂]
or₂ > [true, ge₂, equal_bool2] > [minus₂, if₃] > [s₁, gt₂, equal_sort[a23]₂]
not₁ > [minus'₂, if'₃, false, 0] > [true, ge₂, equal_bool2] > [minus₂, if₃] > [s₁, gt₂, equal_sort[a23]₂]
isa_false1 > [minus'₂, if'₃, false, 0] > [true, ge₂, equal_bool2] > [minus₂, if₃] > [s₁, gt₂, equal_sort[a23]₂]
equal_sort[a40]₂ > [true, ge₂, equal_bool2] > [minus₂, if₃] > [s₁, gt₂, equal_sort[a23]₂]

Status:

minus'₂: [2,1]
s₁: multiset
if'₃: [3,2,1]
gt₂: [1,2]
true: multiset
false: multiset
0: multiset
ge₂: [2,1]
minus₂: [1,2]
if₃: [2,3,1]
equal_bool2: multiset
and₂: [1,2]
or₂: [2,1]
not₁: [1]
isa_false1: [1]
equal_sort[a23]₂: multiset
equal_sort[a40]₂: multiset
witness_sort[a40]: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

minus'(s(x41), y29) → if'(gt(s(x41), y29), x41, y29)
if'(true, x59, y43) → minus'(x59, y43)
if'(false, x68, y50) → true
minus'(0, x1) → false
gt(0, y) → false
gt(s(x5), 0) → true
ge(x14, 0) → true
ge(0, s(x23)) → false
ge(s(x32), s(y22)) → ge(x32, y22)
minus(s(x41), y29) → if(gt(s(x41), y29), x41, y29)
gt(s(x50), s(y36)) → gt(x50, y36)
if(true, x59, y43) → s(minus(x59, y43))
if(false, x68, y50) → 0
minus(0, x1) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a23](0, 0) → true
equal_sort[a23](0, s(x0)) → false
equal_sort[a23](s(x0), 0) → false
equal_sort[a23](s(x0), s(x1)) → equal_sort[a23](x0, x1)
equal_sort[a40](witness_sort[a40], witness_sort[a40]) → true

(58) Obligation:

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

isa_true(true) → true
isa_true(false) → false

Q is empty.

(59) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(false) = 1   
POL(isa_true(x₁)) = 2 + 2·x₁   
POL(true) = 1   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isa_true(true) → true
isa_true(false) → false

(60) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(61) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(62) YES