/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsToCdtProof [UPPER BOUND(ID), 8 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, 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), 548 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs


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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:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
   d(g(g(0, x), y), 0) -> e(y)
   d(g(0, x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of h: d
The following defined symbols can occur below the 0th argument of d: d

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
d(g(g(0, x), y), s(z)) -> g(e(x), d(g(g(0, x), y), z))
d(g(g(0, x), y), 0) -> e(y)
d(g(0, x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))

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

(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:

   h(e(x), y) -> h(d(x, y), s(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL
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(3) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
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(4)
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:

   h(e(x), y) -> h(d(x, y), s(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   H_2, G_2

Compound Symbols:   c_1, c1_1


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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   H(e(z0), z1) -> c(H(d(z0, z1), s(z1)))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:   h_2, g_2

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   h(e(z0), z1) -> h(d(z0, z1), s(z1))
   g(e(z0), e(z1)) -> e(g(z0, z1))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(e(z0), e(z1)) -> c1(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(G(x_1, x_2)) = x_2
   POL(c1(x_1)) = x_1
   POL(e(x_1)) = [1] + x_1

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
S tuples:none
K tuples:
   G(e(z0), e(z1)) -> c1(G(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   G_2

Compound Symbols:   c1_1


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(13) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(14)
BOUNDS(1, 1)

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

(16)
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:

   h(e(x), y) -> h(d(x, y), s(y))
   d(g(g(0', x), y), s(z)) -> g(e(x), d(g(g(0', x), y), z))
   d(g(g(0', x), y), 0') -> e(y)
   d(g(0', x), y) -> e(x)
   d(g(x, y), z) -> g(d(x, z), e(y))
   g(e(x), e(y)) -> e(g(x, y))

S is empty.
Rewrite Strategy: FULL
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(17) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(18)
Obligation:
TRS:
Rules:
h(e(x), y) -> h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) -> g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') -> e(y)
d(g(0', x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: e:s:0' -> e:s:0' -> h
e :: e:s:0' -> e:s:0'
d :: e:s:0' -> e:s:0' -> e:s:0'
s :: e:s:0' -> e:s:0'
g :: e:s:0' -> e:s:0' -> e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat -> e:s:0'

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(19) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
h, d, g

They will be analysed ascendingly in the following order:
d < h
g < d

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(20)
Obligation:
TRS:
Rules:
h(e(x), y) -> h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) -> g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') -> e(y)
d(g(0', x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: e:s:0' -> e:s:0' -> h
e :: e:s:0' -> e:s:0'
d :: e:s:0' -> e:s:0' -> e:s:0'
s :: e:s:0' -> e:s:0'
g :: e:s:0' -> e:s:0' -> e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat -> e:s:0'


Generator Equations:
gen_e:s:0'3_0(0) <=> 0'
gen_e:s:0'3_0(+(x, 1)) <=> e(gen_e:s:0'3_0(x))


The following defined symbols remain to be analysed:
g, h, d

They will be analysed ascendingly in the following order:
d < h
g < d

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_e:s:0'3_0(+(1, n5_0)), gen_e:s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
g(gen_e:s:0'3_0(+(1, 0)), gen_e:s:0'3_0(+(1, 0)))

Induction Step:
g(gen_e:s:0'3_0(+(1, +(n5_0, 1))), gen_e:s:0'3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
e(g(gen_e:s:0'3_0(+(1, n5_0)), gen_e:s:0'3_0(+(1, n5_0)))) ->_IH
e(*4_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)
Complex Obligation (BEST)

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(23)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
h(e(x), y) -> h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) -> g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') -> e(y)
d(g(0', x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: e:s:0' -> e:s:0' -> h
e :: e:s:0' -> e:s:0'
d :: e:s:0' -> e:s:0' -> e:s:0'
s :: e:s:0' -> e:s:0'
g :: e:s:0' -> e:s:0' -> e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat -> e:s:0'


Generator Equations:
gen_e:s:0'3_0(0) <=> 0'
gen_e:s:0'3_0(+(x, 1)) <=> e(gen_e:s:0'3_0(x))


The following defined symbols remain to be analysed:
g, h, d

They will be analysed ascendingly in the following order:
d < h
g < d

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(24) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(25)
BOUNDS(n^1, INF)

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(26)
Obligation:
TRS:
Rules:
h(e(x), y) -> h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) -> g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') -> e(y)
d(g(0', x), y) -> e(x)
d(g(x, y), z) -> g(d(x, z), e(y))
g(e(x), e(y)) -> e(g(x, y))

Types:
h :: e:s:0' -> e:s:0' -> h
e :: e:s:0' -> e:s:0'
d :: e:s:0' -> e:s:0' -> e:s:0'
s :: e:s:0' -> e:s:0'
g :: e:s:0' -> e:s:0' -> e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat -> e:s:0'


Lemmas:
g(gen_e:s:0'3_0(+(1, n5_0)), gen_e:s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_e:s:0'3_0(0) <=> 0'
gen_e:s:0'3_0(+(x, 1)) <=> e(gen_e:s:0'3_0(x))


The following defined symbols remain to be analysed:
d, h

They will be analysed ascendingly in the following order:
d < h