/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 1 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)), 40 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) SlicingProof [LOWER BOUND(ID), 0 ms]
    (18) CpxTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 411 ms]
    (24) proven lower bound
    (25) LowerBoundPropagationProof [FINISHED, 0 ms]
    (26) BOUNDS(n^1, INF)


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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:

   :(:(x, y), z) -> :(x, :(y, z))
   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
:(:(x, y), z) -> :(x, :(y, z))

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(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:

   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

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:

   :(+(x, y), z) -> +(:(x, z), :(y, z))
   :(z, +(x, f(y))) -> :(g(z, y), +(x, a))

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:
   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))
Tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))
S tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))
K tuples:none
Defined Rule Symbols:   :_2

Defined Pair Symbols:   :'_2

Compound Symbols:   c_2, c1_1


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(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   :'(z0, +(z1, f(z2))) -> c1(:'(g(z0, z2), +(z1, a)))

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(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))
Tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
S tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
K tuples:none
Defined Rule Symbols:   :_2

Defined Pair Symbols:   :'_2

Compound Symbols:   c_2


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(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   :(+(z0, z1), z2) -> +(:(z0, z2), :(z1, z2))
   :(z0, +(z1, f(z2))) -> :(g(z0, z2), +(z1, a))

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
S tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   :'_2

Compound Symbols:   c_2


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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.
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(:'(x_1, x_2)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
S tuples:none
K tuples:
   :'(+(z0, z1), z2) -> c(:'(z0, z2), :'(z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:   :'_2

Compound Symbols:   c_2


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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.
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(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:

   :(:(x, y), z) -> :(x, :(y, z))
   :(+'(x, y), z) -> +'(:(x, z), :(y, z))
   :(z, +'(x, f(y))) -> :(g(z, y), +'(x, a))

S is empty.
Rewrite Strategy: FULL
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(17) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
f/0
g/0
g/1

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(18)
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:

   :(:(x, y), z) -> :(x, :(y, z))
   :(+'(x, y), z) -> +'(:(x, z), :(y, z))
   :(z, +'(x, f)) -> :(g, +'(x, a))

S is empty.
Rewrite Strategy: FULL
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(19) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(20)
Obligation:
TRS:
Rules:
:(:(x, y), z) -> :(x, :(y, z))
:(+'(x, y), z) -> +'(:(x, z), :(y, z))
:(z, +'(x, f)) -> :(g, +'(x, a))

Types:
: :: +':f:g:a -> +':f:g:a -> +':f:g:a
+' :: +':f:g:a -> +':f:g:a -> +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat -> +':f:g:a

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(21) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
:
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(22)
Obligation:
TRS:
Rules:
:(:(x, y), z) -> :(x, :(y, z))
:(+'(x, y), z) -> +'(:(x, z), :(y, z))
:(z, +'(x, f)) -> :(g, +'(x, a))

Types:
: :: +':f:g:a -> +':f:g:a -> +':f:g:a
+' :: +':f:g:a -> +':f:g:a -> +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat -> +':f:g:a


Generator Equations:
gen_+':f:g:a2_0(0) <=> f
gen_+':f:g:a2_0(+(x, 1)) <=> +'(gen_+':f:g:a2_0(x), f)


The following defined symbols remain to be analysed:
:
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(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)) -> *3_0, rt in Omega(n4_0)

Induction Base:
:(gen_+':f:g:a2_0(+(1, 0)), gen_+':f:g:a2_0(b))

Induction Step:
:(gen_+':f:g:a2_0(+(1, +(n4_0, 1))), gen_+':f:g:a2_0(b)) ->_R^Omega(1)
+'(:(gen_+':f:g:a2_0(+(1, n4_0)), gen_+':f:g:a2_0(b)), :(f, gen_+':f:g:a2_0(b))) ->_IH
+'(*3_0, :(f, gen_+':f:g:a2_0(b)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(24)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
:(:(x, y), z) -> :(x, :(y, z))
:(+'(x, y), z) -> +'(:(x, z), :(y, z))
:(z, +'(x, f)) -> :(g, +'(x, a))

Types:
: :: +':f:g:a -> +':f:g:a -> +':f:g:a
+' :: +':f:g:a -> +':f:g:a -> +':f:g:a
f :: +':f:g:a
g :: +':f:g:a
a :: +':f:g:a
hole_+':f:g:a1_0 :: +':f:g:a
gen_+':f:g:a2_0 :: Nat -> +':f:g:a


Generator Equations:
gen_+':f:g:a2_0(0) <=> f
gen_+':f:g:a2_0(+(x, 1)) <=> +'(gen_+':f:g:a2_0(x), f)


The following defined symbols remain to be analysed:
:
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(25) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(26)
BOUNDS(n^1, INF)