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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 41 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 1 ms]
(8) YES


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

(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))
   c(X) -> d(X)
   h(X) -> c(d(X))

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set
d: empty set
h: {1}

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

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))
   c(X) -> d(X)
   h(X) -> c(d(X))

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set
d: empty set
h: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(c(x_1)) = x_1
   POL(d(x_1)) = x_1
   POL(f(x_1)) = 1 + x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   h(X) -> c(d(X))




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

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))
   c(X) -> d(X)

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set
d: empty set

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

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))
   c(X) -> d(X)

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set
d: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

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

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

   c(X) -> d(X)




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set

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

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   f(f(X)) -> c(f(g(f(X))))

The replacement map contains the following entries:

f: {1}
c: empty set
g: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

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

   f(f(X)) -> c(f(g(f(X))))




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

(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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

(7) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(8)
YES