WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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)), 7 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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 215 ms]
    (20) proven lower bound
    (21) LowerBoundPropagationProof [FINISHED, 0 ms]
    (22) BOUNDS(n^1, INF)


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(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:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(c) -> c6(A__C)
   MARK(b) -> c7
S tuples:
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
   MARK(c) -> c6(A__C)
   MARK(b) -> c7
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, A__C, MARK_1

Compound Symbols:   c1_2, c2, c3, c4, c5_2, c6_1, c7


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   A__C -> c4
   MARK(c) -> c6(A__C)
   MARK(b) -> c7
   A__C -> c3
   A__F(z0, z1, z2) -> c2
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(A__F(z0, mark(z1), z2), MARK(z1))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_2


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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   a__f(b, z0, c) -> a__f(z0, a__c, z0)
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, mark(z1), z2)
   mark(c) -> a__c
   mark(b) -> b

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_1


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(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
We considered the (Usable) Rules:none
And the Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(MARK(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(f(x_1, x_2, x_3)) = [1] + x_2

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(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
S tuples:none
K tuples:
   MARK(f(z0, z1, z2)) -> c5(MARK(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   MARK_1

Compound Symbols:   c5_1


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(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:

   a__f(b, X, c) -> a__f(X, a__c, X)
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
   mark(c) -> a__c
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST
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(15) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(16)
Obligation:
Innermost TRS:
Rules:
a__f(b, X, c) -> a__f(X, a__c, X)
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(c) -> a__c
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat -> b:c:f

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(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

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(18)
Obligation:
Innermost TRS:
Rules:
a__f(b, X, c) -> a__f(X, a__c, X)
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(c) -> a__c
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat -> b:c:f


Generator Equations:
gen_b:c:f2_0(0) <=> b
gen_b:c:f2_0(+(x, 1)) <=> f(b, gen_b:c:f2_0(x), b)


The following defined symbols remain to be analysed:
a__f, mark

They will be analysed ascendingly in the following order:
a__f < mark

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_b:c:f2_0(n15_0)) -> gen_b:c:f2_0(n15_0), rt in Omega(1 + n15_0)

Induction Base:
mark(gen_b:c:f2_0(0)) ->_R^Omega(1)
b

Induction Step:
mark(gen_b:c:f2_0(+(n15_0, 1))) ->_R^Omega(1)
a__f(b, mark(gen_b:c:f2_0(n15_0)), b) ->_IH
a__f(b, gen_b:c:f2_0(c16_0), b) ->_R^Omega(1)
f(b, gen_b:c:f2_0(n15_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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(20)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__f(b, X, c) -> a__f(X, a__c, X)
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3)
mark(c) -> a__c
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
b :: b:c:f
c :: b:c:f
a__c :: b:c:f
mark :: b:c:f -> b:c:f
f :: b:c:f -> b:c:f -> b:c:f -> b:c:f
hole_b:c:f1_0 :: b:c:f
gen_b:c:f2_0 :: Nat -> b:c:f


Generator Equations:
gen_b:c:f2_0(0) <=> b
gen_b:c:f2_0(+(x, 1)) <=> f(b, gen_b:c:f2_0(x), b)


The following defined symbols remain to be analysed:
mark
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(21) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(22)
BOUNDS(n^1, INF)