/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 240 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 73 ms]
        (14) typed CpxTrs


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(0)
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:

   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   min(X, 0) -> X
   min(s(X), s(Y)) -> min(X, Y)
   min(min(X, Y), Z) -> min(X, plus(Y, Z))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

S is empty.
Rewrite Strategy: FULL
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
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:

   plus(0', Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   min(X, 0') -> X
   min(s(X), s(Y)) -> min(X, Y)
   min(min(X, Y), Z) -> min(X, plus(Y, Z))
   quot(0', s(Y)) -> 0'
   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

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

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

(4)
Obligation:
TRS:
Rules:
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
min(X, 0') -> X
min(s(X), s(Y)) -> min(X, Y)
min(min(X, Y), Z) -> min(X, plus(Y, Z))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z -> 0':s:Z -> 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z -> 0':s:Z
min :: 0':s:Z -> 0':s:Z -> 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z -> 0':s:Z -> 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat -> 0':s:Z

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, min, quot

They will be analysed ascendingly in the following order:
plus < min
min < quot

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(6)
Obligation:
TRS:
Rules:
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
min(X, 0') -> X
min(s(X), s(Y)) -> min(X, Y)
min(min(X, Y), Z) -> min(X, plus(Y, Z))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z -> 0':s:Z -> 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z -> 0':s:Z
min :: 0':s:Z -> 0':s:Z -> 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z -> 0':s:Z -> 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat -> 0':s:Z


Generator Equations:
gen_0':s:Z2_0(0) <=> 0'
gen_0':s:Z2_0(+(x, 1)) <=> s(gen_0':s:Z2_0(x))


The following defined symbols remain to be analysed:
plus, min, quot

They will be analysed ascendingly in the following order:
plus < min
min < quot

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) -> gen_0':s:Z2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
plus(gen_0':s:Z2_0(0), gen_0':s:Z2_0(b)) ->_R^Omega(1)
gen_0':s:Z2_0(b)

Induction Step:
plus(gen_0':s:Z2_0(+(n4_0, 1)), gen_0':s:Z2_0(b)) ->_R^Omega(1)
s(plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b))) ->_IH
s(gen_0':s:Z2_0(+(b, c5_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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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
min(X, 0') -> X
min(s(X), s(Y)) -> min(X, Y)
min(min(X, Y), Z) -> min(X, plus(Y, Z))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z -> 0':s:Z -> 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z -> 0':s:Z
min :: 0':s:Z -> 0':s:Z -> 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z -> 0':s:Z -> 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat -> 0':s:Z


Generator Equations:
gen_0':s:Z2_0(0) <=> 0'
gen_0':s:Z2_0(+(x, 1)) <=> s(gen_0':s:Z2_0(x))


The following defined symbols remain to be analysed:
plus, min, quot

They will be analysed ascendingly in the following order:
plus < min
min < quot

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
min(X, 0') -> X
min(s(X), s(Y)) -> min(X, Y)
min(min(X, Y), Z) -> min(X, plus(Y, Z))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z -> 0':s:Z -> 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z -> 0':s:Z
min :: 0':s:Z -> 0':s:Z -> 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z -> 0':s:Z -> 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat -> 0':s:Z


Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) -> gen_0':s:Z2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:Z2_0(0) <=> 0'
gen_0':s:Z2_0(+(x, 1)) <=> s(gen_0':s:Z2_0(x))


The following defined symbols remain to be analysed:
min, quot

They will be analysed ascendingly in the following order:
min < quot

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) -> gen_0':s:Z2_0(0), rt in Omega(1 + n457_0)

Induction Base:
min(gen_0':s:Z2_0(0), gen_0':s:Z2_0(0)) ->_R^Omega(1)
gen_0':s:Z2_0(0)

Induction Step:
min(gen_0':s:Z2_0(+(n457_0, 1)), gen_0':s:Z2_0(+(n457_0, 1))) ->_R^Omega(1)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) ->_IH
gen_0':s:Z2_0(0)

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

(14)
Obligation:
TRS:
Rules:
plus(0', Y) -> Y
plus(s(X), Y) -> s(plus(X, Y))
min(X, 0') -> X
min(s(X), s(Y)) -> min(X, Y)
min(min(X, Y), Z) -> min(X, plus(Y, Z))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z -> 0':s:Z -> 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z -> 0':s:Z
min :: 0':s:Z -> 0':s:Z -> 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z -> 0':s:Z -> 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat -> 0':s:Z


Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) -> gen_0':s:Z2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) -> gen_0':s:Z2_0(0), rt in Omega(1 + n457_0)


Generator Equations:
gen_0':s:Z2_0(0) <=> 0'
gen_0':s:Z2_0(+(x, 1)) <=> s(gen_0':s:Z2_0(x))


The following defined symbols remain to be analysed:
quot