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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(4) TRS for Loop Detection
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(ID), 0 ms]
    (8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
(13) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxWeightedTrs
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedTrs
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CompletionProof [UPPER BOUND(ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
        (21) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxTypedWeightedCompleteTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(4)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0) -> 0
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0') -> 0'
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
<=/0
<=/1

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

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=, x, y, z)
   g(true, x, y, z) -> z
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
   p(0') -> 0'
   p(s(x)) -> x

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
TRS:
Rules:
f(x, y, z) -> g(<=, x, y, z)
g(true, x, y, z) -> z
g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
f :: 0':s -> 0':s -> 0':s -> 0':s
g :: <=:true:false -> 0':s -> 0':s -> 0':s -> 0':s
<= :: <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_<=:true:false2_0 :: <=:true:false
gen_0':s3_0 :: Nat -> 0':s

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f
----------------------------------------

(12)
Obligation:
TRS:
Rules:
f(x, y, z) -> g(<=, x, y, z)
g(true, x, y, z) -> z
g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
f :: 0':s -> 0':s -> 0':s -> 0':s
g :: <=:true:false -> 0':s -> 0':s -> 0':s -> 0':s
<= :: <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_<=:true:false2_0 :: <=:true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
f
----------------------------------------

(13) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(14)
Obligation:
The Runtime Complexity (full) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z) [1]
   g(true, x, y, z) -> z [1]
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

Rewrite Strategy: FULL
----------------------------------------

(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z) [1]
   g(true, x, y, z) -> z [1]
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
f :: 0:s -> 0:s -> 0:s -> 0:s
g :: <=:true:false -> 0:s -> 0:s -> 0:s -> 0:s
<= :: 0:s -> 0:s -> <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s

Rewrite Strategy: FULL
----------------------------------------

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   g(v0, v1, v2, v3) -> null_g [0]
   p(v0) -> null_p [0]

And the following fresh constants:    null_g, null_p

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

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z) [1]
   g(true, x, y, z) -> z [1]
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   g(v0, v1, v2, v3) -> null_g [0]
   p(v0) -> null_p [0]

The TRS has the following type information:
f :: 0:s:null_g:null_p -> 0:s:null_g:null_p -> 0:s:null_g:null_p -> 0:s:null_g:null_p
g :: <=:true:false -> 0:s:null_g:null_p -> 0:s:null_g:null_p -> 0:s:null_g:null_p -> 0:s:null_g:null_p
<= :: 0:s:null_g:null_p -> 0:s:null_g:null_p -> <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0:s:null_g:null_p -> 0:s:null_g:null_p
0 :: 0:s:null_g:null_p
s :: 0:s:null_g:null_p -> 0:s:null_g:null_p
null_g :: 0:s:null_g:null_p
null_p :: 0:s:null_g:null_p

Rewrite Strategy: FULL
----------------------------------------

(19) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   f_3
   p_1
   g_4

Due to the following rules being added:

   g(v0, v1, v2, v3) -> 0 [0]

And the following fresh constants: none

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

(20)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z) [1]
   g(true, x, y, z) -> z [1]
   g(false, x, y, z) -> f(f(p(x), y, z), f(p(y), z, x), f(p(z), x, y)) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   g(v0, v1, v2, v3) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> 0:s -> 0:s -> 0:s
g :: <=:true:false -> 0:s -> 0:s -> 0:s -> 0:s
<= :: 0:s -> 0:s -> <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s

Rewrite Strategy: FULL
----------------------------------------

(21) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(22)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x, y, z) -> g(<=(x, y), x, y, z) [1]
   g(true, x, y, z) -> z [1]
   g(false, 0, 0, 0) -> f(f(0, 0, 0), f(0, 0, 0), f(0, 0, 0)) [4]
   g(false, 0, 0, s(x2)) -> f(f(0, 0, s(x2)), f(0, s(x2), 0), f(x2, 0, 0)) [4]
   g(false, 0, s(x''), 0) -> f(f(0, s(x''), 0), f(x'', 0, 0), f(0, 0, s(x''))) [4]
   g(false, 0, s(x''), s(x3)) -> f(f(0, s(x''), s(x3)), f(x'', s(x3), 0), f(x3, 0, s(x''))) [4]
   g(false, s(x'), 0, 0) -> f(f(x', 0, 0), f(0, 0, s(x')), f(0, s(x'), 0)) [4]
   g(false, s(x'), 0, s(x4)) -> f(f(x', 0, s(x4)), f(0, s(x4), s(x')), f(x4, s(x'), 0)) [4]
   g(false, s(x'), s(x1), 0) -> f(f(x', s(x1), 0), f(x1, 0, s(x')), f(0, s(x'), s(x1))) [4]
   g(false, s(x'), s(x1), s(x5)) -> f(f(x', s(x1), s(x5)), f(x1, s(x5), s(x')), f(x5, s(x'), s(x1))) [4]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   g(v0, v1, v2, v3) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> 0:s -> 0:s -> 0:s
g :: <=:true:false -> 0:s -> 0:s -> 0:s -> 0:s
<= :: 0:s -> 0:s -> <=:true:false
true :: <=:true:false
false :: <=:true:false
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s

Rewrite Strategy: FULL