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docs/zkEVM/protocol/flow-of-assets.md

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For example, the interaction between the Consensus Contract and the Sequencer is omitted in the figure.
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For a more wholistic view of the interaction between the Consensus Contract and the Sequencer, the reader is referred to earlier subsections of this documentation, specifically on the [<ins>Consensus Contract</ins>](../architecture/architecture.md).
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For a more wholistic view of the interaction between the Consensus Contract and the Sequencer, the reader is referred to earlier subsections of this documentation, specifically on the [<ins>Consensus Contract</ins>](../architecture/index.md).
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![A end-to-end flow of assets between L1 and L2](../../img/zkEVM/06pzb-complete-asset-flow-l1-l2.png)

docs/zkEVM/protocol/submit-transaction.md

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!!!info
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This document series describes in detail the various forms and stages that L2 users' transactions go through, from the time they are created in users' wallets to the time they are finally verified with indisputable evidence on L1.
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docs/zkEVM/specifications/pil/configuration-files.md

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## Dependency Inclusion Feature
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Let's consider a scenario. If the PIL code of Secondary SMs reflects unique properties such as the maximum length (for example, the length $\mathtt{2^{10}}$ of the Multiplier SM as seen in the first line of [the `multiplier.pil` code](/zkevm/PIL/pil-compile.md)), such properties can easily become magic numbers which attackers could use as distinguishers of which computation is running at a given point in time.
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Let's consider a scenario. If the PIL code of Secondary SMs reflects unique properties such as the maximum length (for example, the length $\mathtt{2^{10}}$ of the Multiplier SM as seen in the first line of [the `multiplier.pil` code](compiling-using-pilcom.md)), such properties can easily become magic numbers which attackers could use as distinguishers of which computation is running at a given point in time.
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This is where the **dependency inclusion feature** comes in.
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docs/zkEVM/specifications/pil/cyclicity-in-pil.md

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In order to synchronize the execution trace of a given program with the subgroup $G$ of the multiplicative group $\mathbb{F}^*$, over which interpolation is performed, an extra constant polynomial (or precompiled column) is added to the trace.
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An explanation of what this group $G = \langle g \rangle$ is, and why it is naturally a cyclic group, was discussed in the [Basic Concepts](/zkevm/zkProver/mfibonacci-example.md) section of the zkProver.
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An explanation of what this group $G = \langle g \rangle$ is, and why it is naturally a cyclic group, was discussed in the [Basic Concepts](../../concepts/mfibonacci-example.md) section of the zkProver.
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## Non-cyclic SM Example
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docs/zkEVM/specifications/pil/generating-proofs.md

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If the output of the `starkVerify` function is `true`, the proof is valid. Otherwise, the verifier should invalidate the proof sent by the prover.
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A `pil-stark` DIY guide is given [here](/zkevm/zkProver/pil-stark-demo.md).
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A `pil-stark` DIY guide is given [here](../../concepts/pil-stark-demo.md).

docs/zkEVM/specifications/pil/inclusion-arguments.md

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\texttt{a}\ \text{ is } \textbf{contained in }\ \texttt{b}\ \text{ if for all }\ i ∈ [n],\ \text{ there exists a }\ j ∈ [m]\ \text{ such that }\ a_i = b_j.
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$$
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In other words, if one thinks of $\texttt{a}$ and $\texttt{b}$ as multisets and reduce them to sets (by removing the multiplicity), then $\texttt{a}$ is contained in $\texttt{b}$ if $\texttt{a}$ is a subset of $\texttt{b}$. See [this document](/zkevm/zkProver/intro-generic-sm.md) for more details on multisets and Plookup.
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In other words, if one thinks of $\texttt{a}$ and $\texttt{b}$ as multisets and reduce them to sets (by removing the multiplicity), then $\texttt{a}$ is contained in $\texttt{b}$ if $\texttt{a}$ is a subset of $\texttt{b}$. See [this document](../../concepts/intro-generic-sm.md) for more details on multisets and Plookup.
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A protocol $(\mathcal{P},\mathcal{V})$ is an **inclusion argument** if the protocol can be used by $\mathcal{P}$ to prove to $\mathcal{V}$ that one vector is contained in another vector.
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In the PIL context, the implemented inclusion argument is the same as the $\text{Plookup}$ method provided in [[GW20](https://eprint.iacr.org/2020/315.pdf)], also discussed [here](/zkevm/zkProver/intro-generic-sm.md). Other "alternative" method exists such as the $\text{PlonkUp}$ described in [[PFM+22](https://eprint.iacr.org/2022/086.pdf)].
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In the PIL context, the implemented inclusion argument is the same as the $\text{Plookup}$ method provided in [[GW20](https://eprint.iacr.org/2020/315.pdf)], also discussed [here](../../concepts/intro-generic-sm.md). Other "alternative" method exists such as the $\text{PlonkUp}$ described in [[PFM+22](https://eprint.iacr.org/2022/086.pdf)].
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An inclusion argument is invoked in PIL with the "$\texttt{in}$" keyword.
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