/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 16 ms]
(4) QDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) QDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) QDP
        (10) MNOCProof [EQUIVALENT, 0 ms]
        (11) QDP
        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
        (13) QDP
        (14) QDPOrderProof [EQUIVALENT, 159 ms]
        (15) QDP
        (16) QDPOrderProof [EQUIVALENT, 164 ms]
        (17) QDP
        (18) QDPOrderProof [EQUIVALENT, 38 ms]
        (19) QDP
        (20) PisEmptyProof [EQUIVALENT, 0 ms]
        (21) YES
    (22) QDP
        (23) QDPOrderProof [EQUIVALENT, 34 ms]
        (24) QDP
        (25) UsableRulesProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (28) YES


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

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

   p(0(x1)) -> 0(s(s(p(x1))))
   p(s(x1)) -> x1
   p(p(s(x1))) -> p(x1)
   f(s(x1)) -> g(s(x1))
   g(x1) -> i(s(half(x1)))
   i(x1) -> f(p(x1))
   half(0(x1)) -> 0(s(s(half(x1))))
   half(s(s(x1))) -> s(half(p(p(s(s(x1))))))
   0(x1) -> x1
   rd(0(x1)) -> 0(0(0(0(0(0(rd(x1)))))))

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   0(p(x1)) -> p(s(s(0(x1))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))
   0(half(x1)) -> half(s(s(0(x1))))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))

Q is empty.

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

(3) DependencyPairsProof (EQUIVALENT)
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:

   0^1(p(x1)) -> S(s(0(x1)))
   0^1(p(x1)) -> S(0(x1))
   0^1(p(x1)) -> 0^1(x1)
   S(f(x1)) -> S(g(x1))
   S(f(x1)) -> G(x1)
   G(x1) -> S(i(x1))
   G(x1) -> I(x1)
   0^1(half(x1)) -> S(s(0(x1)))
   0^1(half(x1)) -> S(0(x1))
   0^1(half(x1)) -> 0^1(x1)
   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
   S(s(half(x1))) -> S(p(p(half(s(x1)))))
   S(s(half(x1))) -> S(x1)
   0^1(rd(x1)) -> 0^1(0(0(0(0(0(x1))))))
   0^1(rd(x1)) -> 0^1(0(0(0(0(x1)))))
   0^1(rd(x1)) -> 0^1(0(0(0(x1))))
   0^1(rd(x1)) -> 0^1(0(0(x1)))
   0^1(rd(x1)) -> 0^1(0(x1))
   0^1(rd(x1)) -> 0^1(x1)

The TRS R consists of the following rules:

   0(p(x1)) -> p(s(s(0(x1))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))
   0(half(x1)) -> half(s(s(0(x1))))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))

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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   S(f(x1)) -> G(x1)
   G(x1) -> S(i(x1))
   S(f(x1)) -> S(g(x1))
   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
   S(s(half(x1))) -> S(x1)

The TRS R consists of the following rules:

   0(p(x1)) -> p(s(s(0(x1))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))
   0(half(x1)) -> half(s(s(0(x1))))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))

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

(8) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

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

   S(f(x1)) -> G(x1)
   G(x1) -> S(i(x1))
   S(f(x1)) -> S(g(x1))
   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
   S(s(half(x1))) -> S(x1)

The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

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

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

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

   S(f(x1)) -> G(x1)
   G(x1) -> S(i(x1))
   S(f(x1)) -> S(g(x1))
   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
   S(s(half(x1))) -> S(x1)

The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

The set Q consists of the following terms:

   s(p(x0))
   s(s(half(x0)))
   s(f(x0))
   g(x0)
   i(x0)

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

(12) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
   S(f(x1)) -> S(g(x1))
   S(s(half(x1))) -> S(x1)

The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

The set Q consists of the following terms:

   s(p(x0))
   s(s(half(x0)))
   s(f(x0))
   g(x0)
   i(x0)

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

(14) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   S(s(half(x1))) -> S(s(p(p(half(s(x1))))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, 0A, 0A], [0A, 0A, 0A], [-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(half(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(p(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [0A, -I, 0A], [0A, -I, -I]] 	* 	x_1
>>>

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

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

   <<<
 POL(i(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))


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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(f(x1)) -> S(g(x1))
   S(s(half(x1))) -> S(x1)

The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

The set Q consists of the following terms:

   s(p(x0))
   s(s(half(x0)))
   s(f(x0))
   g(x0)
   i(x0)

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

(16) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   S(s(half(x1))) -> S(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(g(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

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

   <<<
 POL(half(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(i(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(p(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   g(x1) -> half(s(i(x1)))
   s(f(x1)) -> s(g(x1))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   i(x1) -> p(f(x1))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)


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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(f(x1)) -> S(g(x1))

The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

The set Q consists of the following terms:

   s(p(x0))
   s(s(half(x0)))
   s(f(x0))
   g(x0)
   i(x0)

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

(18) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   S(f(x1)) -> S(g(x1))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( S_1(x_1) ) = max{0, 2x_1 - 2}
POL( g_1(x_1) ) = 0
POL( half_1(x_1) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( i_1(x_1) ) = 0
POL( f_1(x_1) ) = x_1 + 2
POL( p_1(x_1) ) = max{0, 2x_1 - 2}

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   g(x1) -> half(s(i(x1)))


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

(19)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))

The set Q consists of the following terms:

   s(p(x0))
   s(s(half(x0)))
   s(f(x0))
   g(x0)
   i(x0)

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

(20) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(21)
YES

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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(half(x1)) -> 0^1(x1)
   0^1(p(x1)) -> 0^1(x1)
   0^1(rd(x1)) -> 0^1(0(0(0(0(0(x1))))))
   0^1(rd(x1)) -> 0^1(0(0(0(0(x1)))))
   0^1(rd(x1)) -> 0^1(0(0(0(x1))))
   0^1(rd(x1)) -> 0^1(0(0(x1)))
   0^1(rd(x1)) -> 0^1(0(x1))
   0^1(rd(x1)) -> 0^1(x1)

The TRS R consists of the following rules:

   0(p(x1)) -> p(s(s(0(x1))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))
   0(half(x1)) -> half(s(s(0(x1))))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))

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

(23) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   0^1(rd(x1)) -> 0^1(0(0(0(0(0(x1))))))
   0^1(rd(x1)) -> 0^1(0(0(0(0(x1)))))
   0^1(rd(x1)) -> 0^1(0(0(0(x1))))
   0^1(rd(x1)) -> 0^1(0(0(x1)))
   0^1(rd(x1)) -> 0^1(0(x1))
   0^1(rd(x1)) -> 0^1(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(0(x_1)) = x_1
   POL(0^1(x_1)) = x_1
   POL(f(x_1)) = 1 + x_1
   POL(g(x_1)) = 1 + x_1
   POL(half(x_1)) = x_1
   POL(i(x_1)) = 1 + x_1
   POL(p(x_1)) = x_1
   POL(rd(x_1)) = 1 + x_1
   POL(s(x_1)) = x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   0(p(x1)) -> p(s(s(0(x1))))
   0(half(x1)) -> half(s(s(0(x1))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))


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

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(half(x1)) -> 0^1(x1)
   0^1(p(x1)) -> 0^1(x1)

The TRS R consists of the following rules:

   0(p(x1)) -> p(s(s(0(x1))))
   s(p(x1)) -> x1
   s(p(p(x1))) -> p(x1)
   s(f(x1)) -> s(g(x1))
   g(x1) -> half(s(i(x1)))
   i(x1) -> p(f(x1))
   0(half(x1)) -> half(s(s(0(x1))))
   s(s(half(x1))) -> s(s(p(p(half(s(x1))))))
   0(x1) -> x1
   0(rd(x1)) -> rd(0(0(0(0(0(0(x1)))))))

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

(25) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   0^1(half(x1)) -> 0^1(x1)
   0^1(p(x1)) -> 0^1(x1)

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

(27) 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:
*0^1(half(x1)) -> 0^1(x1)
The graph contains the following edges 1 > 1


*0^1(p(x1)) -> 0^1(x1)
The graph contains the following edges 1 > 1


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

(28)
YES