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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 14 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, 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) RewriteLemmaProof [LOWER BOUND(ID), 277 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 71 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (22) BOUNDS(1, INF)


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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
half(double(x)) -> x

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   if(0, y, z) -> y
   if(s(x), y, z) -> z

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
00() -> 0
s0(0) -> 0
double0(0) -> 1
half0(0) -> 2
-0(0, 0) -> 3
if0(0, 0, 0) -> 4
01() -> 1
double1(0) -> 6
s1(6) -> 5
s1(5) -> 1
01() -> 2
half1(0) -> 7
s1(7) -> 2
-1(0, 0) -> 3
01() -> 6
s1(5) -> 6
01() -> 7
s1(7) -> 7
0 -> 3
0 -> 4

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(6)
BOUNDS(1, n^1)

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(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   if(0', y, z) -> y
   if(s(x), y, z) -> z
   half(double(x)) -> x

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

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

(10)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
half :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
if :: 0':s -> if -> if -> if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat -> 0':s

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

(12)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
half :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
if :: 0':s -> if -> if -> if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
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:
double, half, -
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)

Induction Base:
double(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s3_0(n5_0)))) ->_IH
s(s(gen_0':s3_0(*(2, c6_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
half :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
if :: 0':s -> if -> if -> if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
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:
double, half, -
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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
half :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
if :: 0':s -> if -> if -> if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)


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:
half, -
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s3_0(*(2, n249_0))) -> gen_0':s3_0(n249_0), rt in Omega(1 + n249_0)

Induction Base:
half(gen_0':s3_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s3_0(*(2, +(n249_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s3_0(*(2, n249_0)))) ->_IH
s(gen_0':s3_0(c250_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
if(0', y, z) -> y
if(s(x), y, z) -> z
half(double(x)) -> x

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
half :: 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
if :: 0':s -> if -> if -> if
hole_0':s1_0 :: 0':s
hole_if2_0 :: if
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
double(gen_0':s3_0(n5_0)) -> gen_0':s3_0(*(2, n5_0)), rt in Omega(1 + n5_0)
half(gen_0':s3_0(*(2, n249_0))) -> gen_0':s3_0(n249_0), rt in Omega(1 + n249_0)


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:
-
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) -> gen_0':s3_0(0), rt in Omega(1 + n676_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n676_0, 1)), gen_0':s3_0(+(n676_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n676_0), gen_0':s3_0(n676_0)) ->_IH
gen_0':s3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(22)
BOUNDS(1, INF)