/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(4) QTRS
(5) DependencyPairsProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) QDP
(9) TransformationProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) TransformationProof [EQUIVALENT, 0 ms]
(14) QDP
(15) UsableRulesProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QReductionProof [EQUIVALENT, 0 ms]
(18) QDP
(19) TransformationProof [EQUIVALENT, 0 ms]
(20) QDP
(21) TransformationProof [EQUIVALENT, 0 ms]
(22) QDP
(23) DependencyGraphProof [EQUIVALENT, 0 ms]
(24) QDP
(25) UsableRulesProof [EQUIVALENT, 0 ms]
(26) QDP
(27) QReductionProof [EQUIVALENT, 0 ms]
(28) QDP
(29) TransformationProof [EQUIVALENT, 0 ms]
(30) QDP
(31) DependencyGraphProof [EQUIVALENT, 0 ms]
(32) QDP
(33) TransformationProof [EQUIVALENT, 0 ms]
(34) QDP
(35) DependencyGraphProof [EQUIVALENT, 0 ms]
(36) TRUE


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

(0)
Obligation:
Conditional term rewrite system:
The TRS R is empty.
The conditional TRS C consists of the following conditional rules:

   App(App(App(S, a), b), c) -> App(App(a, c), App(b, c)) <= a -> I, b -> I, c -> I
   App(App(K, a), b) -> a <= a -> I, b -> I
   App(I, a) -> a <= a -> I


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

(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
----------------------------------------

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

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

Q is empty.

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

(3) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)


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

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U1^1(I, a, b, c) -> U2^1(b, a, b, c)
   U2^1(I, a, b, c) -> U3^1(c, a, b, c)
   U3^1(I, a, b, c) -> APP(App(a, c), App(b, c))
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   APP(App(K, a), b) -> U4^1(a, a, b)
   U4^1(I, a, b) -> U5^1(b, a)
   APP(I, a) -> U6^1(a, a)

The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1^1(I, a, b, c) -> U2^1(b, a, b, c)
   U2^1(I, a, b, c) -> U3^1(c, a, b, c)
   U3^1(I, a, b, c) -> APP(App(a, c), App(b, c))
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)

The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(9) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U1^1(I, a, b, c) -> U2^1(b, a, b, c) we obtained the following new rules [LPAR04]:

   (U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2),U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2))


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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U2^1(I, a, b, c) -> U3^1(c, a, b, c)
   U3^1(I, a, b, c) -> APP(App(a, c), App(b, c))
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)

The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(11) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U2^1(I, a, b, c) -> U3^1(c, a, b, c) we obtained the following new rules [LPAR04]:

   (U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1),U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1))


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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U3^1(I, a, b, c) -> APP(App(a, c), App(b, c))
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)

The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(13) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U3^1(I, a, b, c) -> APP(App(a, c), App(b, c)) we obtained the following new rules [LPAR04]:

   (U3^1(I, I, I, I) -> APP(App(I, I), App(I, I)),U3^1(I, I, I, I) -> APP(App(I, I), App(I, I)))


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

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

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, I, I, I) -> APP(App(I, I), App(I, I))

The TRS R consists of the following rules:

   App(App(App(S, a), b), c) -> U1(a, a, b, c)
   U1(I, a, b, c) -> U2(b, a, b, c)
   U2(I, a, b, c) -> U3(c, a, b, c)
   U3(I, a, b, c) -> App(App(a, c), App(b, c))
   App(App(K, a), b) -> U4(a, a, b)
   U4(I, a, b) -> U5(b, a)
   U5(I, a) -> a
   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(15) 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.
----------------------------------------

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

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, I, I, I) -> APP(App(I, I), App(I, I))

The TRS R consists of the following rules:

   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   App(App(K, x0), x1)
   U4(I, x0, x1)
   U5(I, x0)
   App(I, x0)
   U6(I, x0)

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

(17) 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].

   U1(I, x0, x1, x2)
   U2(I, x0, x1, x2)
   U3(I, x0, x1, x2)
   U4(I, x0, x1)
   U5(I, x0)


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

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

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, I, I, I) -> APP(App(I, I), App(I, I))

The TRS R consists of the following rules:

   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)
   U6(I, x0)

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

(19) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule U3^1(I, I, I, I) -> APP(App(I, I), App(I, I)) at position [0] we obtained the following new rules [LPAR04]:

   (U3^1(I, I, I, I) -> APP(U6(I, I), App(I, I)),U3^1(I, I, I, I) -> APP(U6(I, I), App(I, I)))


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

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

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, I, I, I) -> APP(U6(I, I), App(I, I))

The TRS R consists of the following rules:

   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)
   U6(I, x0)

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

(21) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule U3^1(I, I, I, I) -> APP(U6(I, I), App(I, I)) at position [0] we obtained the following new rules [LPAR04]:

   (U3^1(I, I, I, I) -> APP(I, App(I, I)),U3^1(I, I, I, I) -> APP(I, App(I, I)))


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

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

   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(a, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, I, I, I) -> APP(I, App(I, I))

The TRS R consists of the following rules:

   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)
   U6(I, x0)

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

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

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

   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, a, b, c) -> APP(a, c)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(b, c)

The TRS R consists of the following rules:

   App(I, a) -> U6(a, a)
   U6(I, a) -> a

The set Q consists of the following terms:

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)
   U6(I, x0)

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

(25) 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.
----------------------------------------

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

   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, a, b, c) -> APP(a, c)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(b, c)

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

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)
   U6(I, x0)

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

(27) 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].

   U6(I, x0)


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

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, a, b, c) -> APP(a, c)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(b, c)

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

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)

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

(29) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U3^1(I, a, b, c) -> APP(a, c) we obtained the following new rules [LPAR04]:

   (U3^1(I, I, I, I) -> APP(I, I),U3^1(I, I, I, I) -> APP(I, I))


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

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U3^1(I, a, b, c) -> APP(b, c)
   U3^1(I, I, I, I) -> APP(I, I)

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

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)

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

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

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

   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   U3^1(I, a, b, c) -> APP(b, c)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)

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

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)

We have to consider all minimal (P,Q,R)-chains.
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(33) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U3^1(I, a, b, c) -> APP(b, c) we obtained the following new rules [LPAR04]:

   (U3^1(I, I, I, I) -> APP(I, I),U3^1(I, I, I, I) -> APP(I, I))


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

   U2^1(I, I, I, z1) -> U3^1(z1, I, I, z1)
   APP(App(App(S, a), b), c) -> U1^1(a, a, b, c)
   U1^1(I, I, z1, z2) -> U2^1(z1, I, z1, z2)
   U3^1(I, I, I, I) -> APP(I, I)

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

   App(App(App(S, x0), x1), x2)
   App(App(K, x0), x1)
   App(I, x0)

We have to consider all minimal (P,Q,R)-chains.
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(35) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
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(36)
TRUE