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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 333 ms]
    (12) proven lower bound
    (13) LowerBoundPropagationProof [FINISHED, 0 ms]
    (14) BOUNDS(n^1, INF)


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


The TRS R consists of the following rules:

   eq0(S(x'), S(x)) -> eq0(x', x)
   eq0(S(x), 0) -> 0
   eq0(0, S(x)) -> 0
   eq0(0, 0) -> S(0)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   eq0(S(x'), S(x)) -> eq0(x', x)
   eq0(S(x), 0) -> 0
   eq0(0, S(x)) -> 0
   eq0(0, 0) -> S(0)

S is empty.
Rewrite Strategy: INNERMOST
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(3) 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]
transitions: 
S0(0) -> 0
00() -> 0
eq00(0, 0) -> 1
eq01(0, 0) -> 1
01() -> 1
01() -> 2
S1(2) -> 1

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(4)
BOUNDS(1, n^1)

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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   eq0(S(x'), S(x)) -> eq0(x', x)
   eq0(S(x), 0') -> 0'
   eq0(0', S(x)) -> 0'
   eq0(0', 0') -> S(0')

S is empty.
Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
eq0(S(x'), S(x)) -> eq0(x', x)
eq0(S(x), 0') -> 0'
eq0(0', S(x)) -> 0'
eq0(0', 0') -> S(0')

Types:
eq0 :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
eq0
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(10)
Obligation:
Innermost TRS:
Rules:
eq0(S(x'), S(x)) -> eq0(x', x)
eq0(S(x), 0') -> 0'
eq0(0', S(x)) -> 0'
eq0(0', 0') -> S(0')

Types:
eq0 :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
eq0
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) -> gen_S:0'2_1(0), rt in Omega(1 + n4_1)

Induction Base:
eq0(gen_S:0'2_1(+(1, 0)), gen_S:0'2_1(0)) ->_R^Omega(1)
0'

Induction Step:
eq0(gen_S:0'2_1(+(1, +(n4_1, 1))), gen_S:0'2_1(+(n4_1, 1))) ->_R^Omega(1)
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) ->_IH
gen_S:0'2_1(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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(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
eq0(S(x'), S(x)) -> eq0(x', x)
eq0(S(x), 0') -> 0'
eq0(0', S(x)) -> 0'
eq0(0', 0') -> S(0')

Types:
eq0 :: S:0' -> S:0' -> S:0'
S :: S:0' -> S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat -> S:0'


Generator Equations:
gen_S:0'2_1(0) <=> 0'
gen_S:0'2_1(+(x, 1)) <=> S(gen_S:0'2_1(x))


The following defined symbols remain to be analysed:
eq0
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)