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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) 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) RewriteLemmaProof [LOWER BOUND(ID), 188 ms]
    (14) proven lower bound
    (15) LowerBoundPropagationProof [FINISHED, 0 ms]
    (16) BOUNDS(n^1, INF)


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

(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:

   list(Cons(x, xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(x)

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

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

(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:

   list(Cons(x, xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(x)

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

(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, 2, 3]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
True0() -> 0
isEmpty[Match]0(0) -> 0
False0() -> 0
list0(0) -> 1
notEmpty0(0) -> 2
goal0(0) -> 3
list1(0) -> 1
True1() -> 1
Nil1() -> 4
isEmpty[Match]1(4) -> 1
True1() -> 2
False1() -> 2
list1(0) -> 3
True1() -> 3
isEmpty[Match]1(4) -> 3

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

(4)
BOUNDS(1, n^1)

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

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

(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:

   list(Cons(x, xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(x)

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

(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0
isEmpty[Match]/0

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

(8)
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:

   list(Cons(xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match]
   notEmpty(Cons(xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(x)

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

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

(10)
Obligation:
Innermost TRS:
Rules:
list(Cons(xs)) -> list(xs)
list(Nil) -> True
list(Nil) -> isEmpty[Match]
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(x) -> list(x)

Types:
list :: Cons:Nil -> True:isEmpty[Match]:False
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil

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

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

(12)
Obligation:
Innermost TRS:
Rules:
list(Cons(xs)) -> list(xs)
list(Nil) -> True
list(Nil) -> isEmpty[Match]
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(x) -> list(x)

Types:
list :: Cons:Nil -> True:isEmpty[Match]:False
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
list(gen_Cons:Nil3_0(n5_0)) -> True, rt in Omega(1 + n5_0)

Induction Base:
list(gen_Cons:Nil3_0(0)) ->_R^Omega(1)
True

Induction Step:
list(gen_Cons:Nil3_0(+(n5_0, 1))) ->_R^Omega(1)
list(gen_Cons:Nil3_0(n5_0)) ->_IH
True

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

(14)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
list(Cons(xs)) -> list(xs)
list(Nil) -> True
list(Nil) -> isEmpty[Match]
notEmpty(Cons(xs)) -> True
notEmpty(Nil) -> False
goal(x) -> list(x)

Types:
list :: Cons:Nil -> True:isEmpty[Match]:False
Cons :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
True :: True:isEmpty[Match]:False
isEmpty[Match] :: True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_True:isEmpty[Match]:False1_0 :: True:isEmpty[Match]:False
hole_Cons:Nil2_0 :: Cons:Nil
gen_Cons:Nil3_0 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil3_0(0) <=> Nil
gen_Cons:Nil3_0(+(x, 1)) <=> Cons(gen_Cons:Nil3_0(x))


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

(15) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(16)
BOUNDS(n^1, INF)