WORST_CASE(?,O(n^1))
proof of input_q4pkXIsEzS.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(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) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTRS
(11) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxTRS
(13) CpxTrsMatchBoundsTAProof [FINISHED, 5 ms]
(14) BOUNDS(1, n^1)


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(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(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: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST 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:
   MARK(b) -> c7
   MARK(c) -> c6(A__C)
   A__C -> c4
   A__F(z0, z1, z2) -> c2
   A__F(b, z0, c) -> c1(A__F(z0, a__c, z0), A__C)
   A__C -> c3

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(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
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(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

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(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) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(10)
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:

   MARK(f(z0, z1, z2)) -> c5(MARK(z1))

S is empty.
Rewrite Strategy: INNERMOST
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(11) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(12)
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:

   MARK(f(z0, z1, z2)) -> c5(MARK(z1))

S is empty.
Rewrite Strategy: INNERMOST
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(13) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
f0(0, 0, 0) -> 0
c50(0) -> 0
MARK0(0) -> 1
MARK1(0) -> 2
c51(2) -> 1
c51(2) -> 2

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(14)
BOUNDS(1, n^1)