WORST_CASE(?,O(n^1))
proof of input_743DUHiUNN.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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, 1 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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z2))
S tuples:
   OR(true, z0) -> c
   OR(z0, true) -> c1
   OR(false, false) -> c2
   MEM(z0, nil) -> c3
   MEM(z0, set(z1)) -> c4
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   OR_2, MEM_2

Compound Symbols:   c, c1, c2, c3, c4, c5_2, c6_2


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   MEM(z0, nil) -> c3
   OR(false, false) -> c2
   MEM(z0, set(z1)) -> c4
   OR(true, z0) -> c
   OR(z0, true) -> c1

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(OR(mem(z0, z1), mem(z0, z2)), MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_2, c6_2


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(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))
K tuples:none
Defined Rule Symbols:   or_2, mem_2

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_1, c6_1


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(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   or(true, z0) -> true
   or(z0, true) -> true
   or(false, false) -> false
   mem(z0, nil) -> false
   mem(z0, set(z1)) -> =(z0, z1)
   mem(z0, union(z1, z2)) -> or(mem(z0, z1), mem(z0, z2))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))
S tuples:
   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   MEM_2

Compound Symbols:   c5_1, c6_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:

   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))

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:

   MEM(z0, union(z1, z2)) -> c5(MEM(z0, z1))
   MEM(z0, union(z1, z2)) -> c6(MEM(z0, z2))

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: 
union0(0, 0) -> 0
c50(0) -> 0
c60(0) -> 0
MEM0(0, 0) -> 1
MEM1(0, 0) -> 2
c51(2) -> 1
MEM1(0, 0) -> 3
c61(3) -> 1
c51(2) -> 2
c51(2) -> 3
c61(3) -> 2
c61(3) -> 3

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(14)
BOUNDS(1, n^1)