/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 10 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) MNOCProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) QReductionProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) MNOCProof [EQUIVALENT, 0 ms]
        (16) QDP
        (17) UsableRulesProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) QReductionProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES
    (23) QDP
        (24) MNOCProof [EQUIVALENT, 0 ms]
        (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QReductionProof [EQUIVALENT, 0 ms]
        (29) QDP
        (30) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (31) YES
    (32) QDP
        (33) MNOCProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) UsableRulesProof [EQUIVALENT, 0 ms]
        (36) QDP
        (37) QReductionProof [EQUIVALENT, 0 ms]
        (38) QDP
        (39) TransformationProof [EQUIVALENT, 0 ms]
        (40) QDP
        (41) DependencyGraphProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) UsableRulesProof [EQUIVALENT, 0 ms]
        (44) QDP
        (45) QReductionProof [EQUIVALENT, 0 ms]
        (46) QDP
        (47) TransformationProof [EQUIVALENT, 0 ms]
        (48) QDP
        (49) UsableRulesProof [EQUIVALENT, 0 ms]
        (50) QDP
        (51) TransformationProof [EQUIVALENT, 0 ms]
        (52) QDP
        (53) UsableRulesProof [EQUIVALENT, 0 ms]
        (54) QDP
        (55) QReductionProof [EQUIVALENT, 0 ms]
        (56) QDP
        (57) TransformationProof [EQUIVALENT, 0 ms]
        (58) QDP
        (59) UsableRulesProof [EQUIVALENT, 0 ms]
        (60) QDP
        (61) QReductionProof [EQUIVALENT, 0 ms]
        (62) QDP
        (63) TransformationProof [EQUIVALENT, 0 ms]
        (64) QDP
        (65) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (66) YES


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

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 6 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

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

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

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

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

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(8) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

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

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

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

(10) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUS(x, s(y)) -> MINUS(x, y)

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

(12) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*MINUS(x, s(y)) -> MINUS(x, y)
The graph contains the following edges 1 >= 1, 2 > 2


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

(13)
YES

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(s(x), y) -> PLUS(x, y)

The TRS R consists of the following rules:

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

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

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(s(x), y) -> PLUS(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(17) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(s(x), y) -> PLUS(x, y)

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

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

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

(19) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(s(x), y) -> PLUS(x, y)

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

(21) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PLUS(s(x), y) -> PLUS(x, y)
The graph contains the following edges 1 > 1, 2 >= 2


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

(22)
YES

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

(23)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(s(x), y) -> TIMES(x, y)

The TRS R consists of the following rules:

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

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

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

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(s(x), y) -> TIMES(x, y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(26) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(s(x), y) -> TIMES(x, y)

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

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

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

(28) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(29)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(s(x), y) -> TIMES(x, y)

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

(30) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*TIMES(s(x), y) -> TIMES(x, y)
The graph contains the following edges 1 > 1, 2 >= 2


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

(31)
YES

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

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

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

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(35) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(37) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

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

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


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

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

(43) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   minus(x, s(y)) -> p(minus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))
   minus(0, x) -> 0
   minus(x, 0) -> x
   p(s(x)) -> x
   p(0) -> 0

The set Q consists of the following terms:

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

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

(45) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   isZero(0)
   isZero(s(x0))


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

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   minus(x, s(y)) -> p(minus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))
   minus(0, x) -> 0
   minus(x, 0) -> x
   p(s(x)) -> x
   p(0) -> 0

The set Q consists of the following terms:

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

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

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

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


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

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   minus(x, s(y)) -> p(minus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))
   minus(0, x) -> 0
   minus(x, 0) -> x
   p(s(x)) -> x
   p(0) -> 0

The set Q consists of the following terms:

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

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

(49) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   minus(x, 0) -> x
   p(s(x)) -> x
   p(0) -> 0
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

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

   (FACITER(s(y0), y1) -> IF(false, p(s(y0)), y1, times(y1, s(y0))),FACITER(s(y0), y1) -> IF(false, p(s(y0)), y1, times(y1, s(y0))))


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

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   minus(x, 0) -> x
   p(s(x)) -> x
   p(0) -> 0
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

(53) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   p(s(x)) -> x
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

(55) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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


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

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   p(s(x)) -> x
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

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

   (FACITER(s(y0), y1) -> IF(false, y0, y1, times(y1, s(y0))),FACITER(s(y0), y1) -> IF(false, y0, y1, times(y1, s(y0))))


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

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   p(s(x)) -> x
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

(59) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

(61) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   p(s(x0))
   p(0)


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

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

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

   (IF(false, s(y_0), x1, x2) -> FACITER(s(y_0), x2),IF(false, s(y_0), x1, x2) -> FACITER(s(y_0), x2))


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

(64)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, x) -> x
   plus(s(x), y) -> s(plus(x, y))

The set Q consists of the following terms:

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

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

(65) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*IF(false, s(y_0), x1, x2) -> FACITER(s(y_0), x2)
The graph contains the following edges 2 >= 1, 4 >= 2


*FACITER(s(y0), y1) -> IF(false, y0, y1, times(y1, s(y0)))
The graph contains the following edges 1 > 2, 2 >= 3


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

(66)
YES