/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 71 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 258 ms]
    (22) proven lower bound
    (23) LowerBoundPropagationProof [FINISHED, 0 ms]
    (24) 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:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
S tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   FROM_1, AFTER_2, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4_1, c5


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   ACTIVATE(z0) -> c5
   FROM(z0) -> c1
   FROM(z0) -> c
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   AFTER(0, z0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_2


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   activate_1, from_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
We considered the (Usable) Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
And the Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(AFTER(x_1, x_2)) = x_1 + x_2
   POL(activate(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(from(x_1)) = [1]
   POL(n__from(x_1)) = [1]
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:none
K tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
Defined Rule Symbols:   activate_1, from_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)

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

(14)
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:

   from(X) -> cons(X, n__from(s(X)))
   after(0', XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

(15) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
from/0
cons/0
n__from/0

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

(16)
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:

   from -> cons(n__from)
   after(0', XS) -> XS
   after(s(N), cons(XS)) -> after(N, activate(XS))
   from -> n__from
   activate(n__from) -> from
   activate(X) -> X

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

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

(18)
Obligation:
Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s

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

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

(20)
Obligation:
Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons3_0(0) <=> n__from
gen_n__from:cons3_0(+(x, 1)) <=> cons(gen_n__from:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
after(gen_0':s4_0(n6_0), gen_n__from:cons3_0(1)) -> gen_n__from:cons3_0(1), rt in Omega(1 + n6_0)

Induction Base:
after(gen_0':s4_0(0), gen_n__from:cons3_0(1)) ->_R^Omega(1)
gen_n__from:cons3_0(1)

Induction Step:
after(gen_0':s4_0(+(n6_0, 1)), gen_n__from:cons3_0(1)) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), activate(gen_n__from:cons3_0(0))) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), from) ->_R^Omega(1)
after(gen_0':s4_0(n6_0), cons(n__from)) ->_IH
gen_n__from:cons3_0(1)

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

(22)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
from -> cons(n__from)
after(0', XS) -> XS
after(s(N), cons(XS)) -> after(N, activate(XS))
from -> n__from
activate(n__from) -> from
activate(X) -> X

Types:
from :: n__from:cons
cons :: n__from:cons -> n__from:cons
n__from :: n__from:cons
after :: 0':s -> n__from:cons -> n__from:cons
0' :: 0':s
s :: 0':s -> 0':s
activate :: n__from:cons -> n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_0':s2_0 :: 0':s
gen_n__from:cons3_0 :: Nat -> n__from:cons
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_n__from:cons3_0(0) <=> n__from
gen_n__from:cons3_0(+(x, 1)) <=> cons(gen_n__from:cons3_0(x))
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

(24)
BOUNDS(n^1, INF)