YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 38 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 13 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 3 ms]
(10) RelTRS
(11) RIsEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

The relative TRS consists of the following S rules:

   rand(x) -> x
   rand(x) -> rand(s(x))


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

(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(evenodd(x_1, x_2)) = 1 + x_1 + x_2
   POL(false) = 1
   POL(not(x_1)) = x_1
   POL(rand(x_1)) = 1 + x_1
   POL(s(x_1)) = x_1
   POL(true) = 1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

   rand(x) -> x




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

(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(0, s(0)) -> false
   evenodd(s(x), s(0)) -> evenodd(x, 0)

The relative TRS consists of the following S rules:

   rand(x) -> rand(s(x))


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

(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(0) = 1
   POL(evenodd(x_1, x_2)) = 1 + x_1 + x_2
   POL(false) = 1
   POL(not(x_1)) = x_1
   POL(rand(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(true) = 1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   evenodd(0, s(0)) -> false
Rules from S:
none




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

(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   not(true) -> false
   not(false) -> true
   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(s(x), s(0)) -> evenodd(x, 0)

The relative TRS consists of the following S rules:

   rand(x) -> rand(s(x))


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

(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(not(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(true) =  	[[0], [1]]
>>>

   <<<
 POL(false) =  	[[0], [1]]
>>>

   <<<
 POL(evenodd(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   not(true) -> false
   not(false) -> true
Rules from S:
none




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

(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   evenodd(x, 0) -> not(evenodd(x, s(0)))
   evenodd(s(x), s(0)) -> evenodd(x, 0)

The relative TRS consists of the following S rules:

   rand(x) -> rand(s(x))


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

(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(evenodd(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(not(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   evenodd(s(x), s(0)) -> evenodd(x, 0)
Rules from S:
none




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

(8)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   evenodd(x, 0) -> not(evenodd(x, s(0)))

The relative TRS consists of the following S rules:

   rand(x) -> rand(s(x))


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

(9) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(evenodd(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [1]]
>>>

   <<<
 POL(not(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(rand(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   evenodd(x, 0) -> not(evenodd(x, s(0)))
Rules from S:
none




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

(10)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   rand(x) -> rand(s(x))


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

(11) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(12)
YES