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Outline

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Assumptions

↑ Back to Outline As part of a modular ABCI application, CCV interacts with both the consensus engine (via ABCI) and other application modules (e.g, the Staking module). As an IBC application, CCV interacts with external relayers (defined in ICS 18). In this section we specify what we assume about these other components. A more thorough discussion of the environment in which CCV operates is given in the section Placing CCV within an ABCI Application.
Intuition: CCV safety relies on the Safe Blockchain assumption, i.e., neither Live Blockchain and Correct Relayer are required for safety. Note though that CCV liveness relies on both Live Blockchain and Correct Relayer assumptions; furthermore, the Correct Relayer assumption relies on both Safe Blockchain and Live Blockchain assumptions. The Validator Update Provision, Unbonding Safety, Slashing Warranty, and Distribution Warranty assumptions define what is needed from the ABCI application of the provider chain. The Evidence Provision assumptions defines what is needed from the ABCI application of the consumer chains.
  • Safe Blockchain: Both the provider and the consumer chains are safe. This means that, for every chain, the underlying consensus engine satisfies safety (e.g., the chain does not fork) and the execution of the state machine follows the described protocol.
  • Live Blockchain: Both the provider and the consumer chains are live. This means that, for every chain, the underlying consensus engine satisfies liveness (i.e., new blocks are eventually added to the chain).
    Note: Both Safe Blockchain and Live Blockchain assumptions require the consensus engine’s assumptions to hold, e.g., less than a third of the voting power is Byzantine. For an example, take a look at the Tendermint Paper.
  • Correct Relayer: There is at least one correct, live relayer between the provider and consumer chains. This assumption has the following implications.
    • The opening handshake messages on the CCV channel are relayed before the Channel Initialization subprotocol times out (see initTimeout).
    • Every packet sent on the CCV channel is relayed to the receiving end before the packet timeout elapses (see both vscTimeout and ccvTimeoutTimestamp).
    • A correct relayer will eventually relay packets on the token transfer channel. Clearly, the CCV protocol is responsible of setting the timeouts (see ccvTimeoutTimestamp, vscTimeout, initTimeout in the CCV State), such that the Correct Relayer assumption is feasible.
    Discussion: IBC relies on timeouts to signal that a sent packet is not going to be received on the other end. Once an ordered IBC channel timeouts, the channel is closed (see ICS 4). The Correct Relayer assumption is necessary to ensure that the CCV channel cannot ever timeout and, as a result, cannot transit to the closed state. In practice, the Correct Relayer assumption is realistic since any validator could play the role of the relayer and it is in the best interest of correct validators to successfully relay packets. The following strategy is a practical example of how to ensure the Correct Relayer assumption holds. Let S denote the sending chain and D the destination chain; and let drift(S,D) be the time drift between S and D, i.e., drift(S,D) = S.currentTimestamp() - D.currentTimestamp() (drift(S,D) > 0 means that S is “ahead” of D). For every packet, S only sets timeoutTimestamp = S.currentTimestamp() + to, with to an application-level parameter. The timeoutTimestamp indicates a timestamp on the destination chain after which the packet will no longer be processed (cf. ICS 4). Therefore, the packet MUST be relayed within a time period of to - drift(S,D), i.e., to - drift(S,D) > RTmax, where RTmax is the maximum relaying time across all packet. Theoretically, choosing the value of to requires knowing the value of drift(S,D) (i.e., to > drift(S,D)); yet, drift(S,D) is not known at a chain level. In practice, choosing to such that to >> drift(S,D) and to >> RTmax, e.g., to = 4 weeks, makes the Correct Relayer assumption feasible.
  • Validator Update Provision: Let {U1, U2, ..., Ui} be a batch of validator updates applied (by the provider Staking module) to the validator set of the provider chain at block height h. Then, the batch of validator updates obtained (by the provider CCV module) from the provider Staking module at height h MUST be exactly the batch {U1, U2, ..., Ui}.
  • Unbonding Safety: Let uo be any unbonding operation that starts with an unbonding transaction being executed and completes with the event that returns the corresponding stake; let U(uo) be the validator update caused by initiating uo; let vsc(uo) be the VSC that contains U(uo). Then,
    • (unbonding initiation) the provider CCV module MUST be notified of uo’s initiation before receiving U(uo);
    • (unbonding completion) uo MUST NOT complete on the provider chain before the provider chain registers notifications of vsc(uo)’s maturity from all consumer chains.
    Note: Depending on the implementation, the (unbonding initiation) part of the Unbonding Safety MAY NOT be necessary for validator unbonding operations.
  • Slashing Warranty: If the provider ABCI application (e.g., the Slashing module) receives a request to slash a validator val that misbehaved at block height h, then it slashes the amount of tokens val had bonded at height h except the amount that has already completely unbonded.
  • Evidence Provision: If the consumer ABCI application receives a valid evidence of misbehavior at block height h, then it MUST submit it to the consumer CCV module exactly once and at the same height h. Furthermore, the consumer ABCI application MUST NOT submit invalid evidence to the consumer CCV module.
    Note: What constitutes a valid evidence of misbehavior depends on the type of misbehavior and it is outside the scope of this specification.
  • Distribution Warranty: The provider ABCI application (e.g., the Distribution module) distributes the tokens from the distribution module account among the validators that are part of the validator set.

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Desired Properties

The following properties are concerned with one provider chain providing security to multiple consumer chains. Between the provider chain and each consumer chain, a separate (unique) CCV channel is established.
Note: Except for liveness properties — Channel Liveness, Apply VSC Liveness, Register Maturity Liveness, and Distribution Liveness — none of the properties of CCV require the Correct Relayer assumption to hold. Nonetheless, the Correct Relayer assumption is necessary to guarantee the systems properties (except for Validator Set Replication) — Bond-Based Consumer Voting Power, Slashable Consumer Misbehavior, and Consumer Rewards Distribution.

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System Properties

↑ Back to Outline We use the following notations:
  • ts(h) is the timestamp of a block with height h, i.e., ts(h) = B.currentTimestamp(), where B is the block at height h;
  • pBonded(h,val) is the number of tokens bonded by validator val on the provider chain at block height h;
  • pUnbonding(h,val) is the number of tokens a validator val starts unbonding on the provider at block height h;
  • VP(T) is the voting power associated to a number T of tokens;
  • Power(c,h,val) is the voting power granted to a validator val on a chain c at block height h;
  • Token(power) is the amount of tokens necessary to be bonded (on the provider chain) by a validator to be granted power voting power, i.e., Token(VP(T)) = T;
  • slash(val, h, hi, sf) is the amount of token slashed from a validator val on the provider chain (i.e., pc) at height h for an infraction (with a slashing fraction of sf) committed at (provider) height hi, i.e., slash(val, h, hi, sf) = sf * Token(Power(pc,hi,val)); note that the infraction can be committed also on a consumer chain, in which case hi is the corresponding height on the provider chain.
Also, we use ha << hb to denote an order relation between heights, i.e., the block at height ha happens before the block at height hb. For heights on the same chain, << is equivalent to <, i.e., ha << hb entails hb is larger than ha. For heights on two different chains, << is establish by the packets sent over an order channel between two chains, i.e., if a chain A sends at height ha a packet to a chain B and B receives it at height hb, then ha << hb.
Note: << is transitive, i.e., ha << hb and hb << hc entail ha << hc. Note: The block on the proposer chain that handles a governance proposal to spawn a new consumer chain cc happens before all the blocks of cc.
CCV provides the following system properties.
  • Validator Set Replication: Every validator set on any consumer chain MUST either be or have been a validator set on the provider chain.
  • Bond-Based Consumer Voting Power: Let val be a validator, cc be a consumer chain, both hc and hc' be heights on cc, and both hp and hp' be heights on the provider chain, such that
    • val has Power(cc,hc,val) voting power on cc at height hc;
    • hc' is the smallest height on cc that satisfies ts(hc') >= ts(hc) + UnbondingPeriod, i.e., val cannot completely unbond on cc before hc';
    • hp is the largest height on the provider chain that satisfies hp << hc, i.e., Power(pc,hp,val) = Power(cc,hc,val), where pc is the provider chain;
    • hp' is the smallest height on the provider chain that satisfies hc' << hp', i.e., val cannot completely unbond on the provider chain before hp';
    • sumUnbonding(hp, h, val) is the sum of all tokens of val that start unbonding on the provider at all heights hu and are still unbonding at height h, such that hp < hu <= h
    • sumSlash(hp, h, val) is the sum of the slashes of val at all heights hs for infractions committed at hp, such that hp < hs <= h.
    Then for all heights h on the provider chain, 
    hp <= h < hp': 
Power(cc,hc,val) <= VP( pBonded(h,val) + sumUnbonding(hp, h, val) + sumSlash(hp, h, val) )
    Note: The reason for + sumUnbonding(hp, h, val) in the above inequality is that tokens that val start unbonding after hp have contributed to the power granted to val at height hc on cc (i.e., Power(cc,hc,val)). As a result, these tokens should be available for slashing until hp'. Note: The reason for + sumSlash(hp, h, val) in the above inequality is that slashing val reduces its locked tokens (i.e., pBonded(h,val) and sumUnbonding(hp, h, val)), however it does not reduce the power already granted to it at height hc on cc (i.e., Power(cc,hc,val)). Intuition: The Bond-Based Consumer Voting Power property ensures that validators that validate on the consumer chains have enough tokens bonded on the provider chain for a sufficient amount of time such that the security model holds. This means that if the validators misbehave on the consumer chains, their tokens bonded on the provider chain can be slashed during the unbonding period. For example, if one unit of voting power requires 1.000.000 bonded tokens (i.e., VP(1.000.000)=1), then a validator that gets one unit of voting power on a consumer chain must have at least 1.000.000 tokens bonded on the provider chain until the unbonding period elapses on the consumer chain. Note: When an existing chain becomes a consumer chain (see Channel Initialization: Existing Chains), the existing validator set is replaced by the provider validator set. For safety, the stake bonded by the existing validator set must remain bonded until the unbonding period elapses. Thus, the existing Staking module must be kept for at least the unbonding period.
  • Slashable Consumer Misbehavior: If a validator val commits an infraction, with a slashing fraction of sf, on a consumer chain cc at a block height hi, then any evidence of misbehavior that is received by cc at height he, such that ts(he) < ts(hi) + UnbondingPeriod, MUST results in exactly the amount of tokens sf*Token(Power(cc,hi,val)) to be slashed on the provider chain. Furthermore, val MUST NOT be slashed more than once for the same misbehavior.
    Note: Unlike in single-chain validation, in CCV the tokens sf*Token(Power(cc,hi,val)) MAY be slashed even if the evidence of misbehavior is received at height he such that ts(he) >= ts(hi) + UnbondingPeriod, since unbonding operations need to reach maturity on both the provider and all the consumer chains. Note: The Slashable Consumer Misbehavior property also ensures that if a delegator starts unbonding an amount x of tokens from val before height hi, then x will not be slashed, since x is not part of Token(Power(c,hi,val)).
  • Consumer Rewards Distribution: If a consumer chain sends to the provider chain an amount T of tokens as reward for providing security, then
    • T (equivalent) tokens MUST be eventually minted on the provider chain and then distributed among the validators that are part of the validator set;
    • the total supply of tokens MUST be preserved, i.e., the T (original) tokens are escrowed on the consumer chain.

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CCV Channel

↑ Back to Outline
  • Channel Uniqueness: The channel between the provider chain and a consumer chain MUST be unique.
  • Channel Validity: If a packet P is received by one end of a CCV channel, then P MUST have been sent by the other end of the channel.
  • Channel Order: If a packet P1 is sent over a CCV channel before a packet P2, then P2 MUST NOT be received by the other end of the channel before P1.
  • Channel Liveness: Every packet sent over a CCV channel MUST eventually be received by the other end of the channel.

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Validator Sets, Validator Updates and VSCs

↑ Back to Outline In this section, we provide a short discussion on how the validator set, the validator updates, and the VSCs relates in the context of multiple chains. Every chain consists of a sequence of blocks. At the end of each block, validator updates (i.e., changes in the validators voting power) results in changes in the validator set of the next block. Thus, the sequence of blocks produces a sequence of validator updates and a sequence of validator sets. Furthermore, the sequence of validator updates on the provider chain results in a sequence of VSCs to all consumer chains. Ideally, this sequence of VSCs is applied by every consumer chain, resulting in a sequence of validator sets identical to the one on the provider chain. However, in general this need not be the case. The reason is twofold:
  • first, given any two chains A and B, we cannot assume that A’s rate of adding new block is the same as B’s rate (i.e., we consider the sequences of blocks of any two chains to be completely asynchronous);
  • and second, due to relaying delays, we cannot assume that the rate of sending VSCs matches the rate of receiving VSCs.
As a result, it is possible for multiple VSCs to be received by a consumer chain within the same block and be applied together at the end of the block, i.e., the validator updates within the VSCs are being aggregated by keeping only the latest update per validator. As a consequence, some validator sets on the provider chain are not existing on all consumer chains. In other words, the validator sets on each consumer chain form a subsequence of the validator sets on the provider chain. Nonetheless, as a requirement of CCV, all the validator updates on the provider chain MUST be included in the sequence of validator sets on all consumer chains. This is possible since every validator update contains the absolute voting power of that validator. Given a validator val, the sequence of validator updates targeting val (i.e., updates of the voting power of val) is the prefix sum of the sequence of relative changes of the voting power of val. Thus, given a validator update U targeting val that occurs at a block height h, U sums up all the relative changes of the voting power of val that occur until height h, i.e., U = c_1+c_2+...+c_i, such that c_i is the last relative change that occurs by h. Note that relative changes are integer values. As a consequence, CCV can rely on the following property:
  • Validator Update Inclusion: Let U1 and U2 be two validator updates targeting the same validator val. If U1 occurs before U2, then U2 sums up all the changes of the voting power of val that are summed up by U1, i.e.,
    • U1 = c_1+c_2+...+c_i and
    • U2 = c_1+c_2+...+c_i+c_(i+1)+...+c_j.
The Validator Update Inclusion property enables CCV to aggregate multiple VSCs. It is sufficient for the consumer chains to apply only the last update per validator. Since the last update of a validator includes all the previous updates of that validator, once it is applied, all the previous updates are also applied.

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Staking Module Interface

↑ Back to Outline The following properties define the guarantees of CCV on providing VSCs to the consumer chains as a consequence of validator updates on the provider chain.
  • Validator Update To VSC Validity: Every VSC provided to a consumer chain MUST contain only validator updates that were applied to the validator set of the provider chain (i.e., resulted from a change in the amount of bonded tokens on the provider chain).
  • Validator Update To VSC Order: Let U1 and U2 be two validator updates on the provider chain. If U1 occurs before U2, then U2 MUST NOT be included in a provided VSC before U1. Note that the order within a single VSC is not relevant.
  • Validator Update To VSC Liveness: Every update of a validator in the validator set of the provider chain MUST eventually be included in a VSC provided to all consumer chains.
Note that as a consequence of the Validator Update To VSC Liveness property, CCV guarantees the following property:
  • Provide VSC uniformity: If the provider chain provides a VSC to a consumer chain, then it MUST eventually provide that VSC to all consumer chains.

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Validator Set Update

↑ Back to Outline The provider chain providing VSCs to the consumer chains has two desired outcomes: the consumer chains apply the VSCs; and the provider chain registers VSC maturity notifications from every consumer chain. Thus, for clarity, we split the properties of VSCs in two: properties of applying provided VSCs on the consumer chains; and properties of registering VSC maturity notifications on the provider chain. For simplicity, we focus on a single consumer chain. The following properties define the guarantees of CCV on applying on the consumer chain VSCs provided by the provider chain.
  • Apply VSC Validity: Every VSC applied by the consumer chain MUST be provided by the provider chain.
  • Apply VSC Order: If a VSC vsc1 is provided by the provider chain before a VSC vsc2, then the consumer chain MUST NOT apply the validator updates included in vsc2 before the validator updates included in vsc1.
  • Apply VSC Liveness: If the provider chain provides a VSC vsc, then the consumer chain MUST eventually apply all validator updates included in vsc.
The following properties define the guarantees of CCV on registering on the provider chain maturity notifications (from the consumer chain) of VSCs provided by the provider chain to the consumer chain.
  • Register Maturity Validity: If the provider chain registers a maturity notification of a VSC from the consumer chain, then the provider chain MUST have provided that VSC to the consumer chain.
  • Register Maturity Timeliness: The provider chain MUST NOT register a maturity notification of a VSC vsc before UnbondingPeriod has elapsed on the consumer chain since the consumer chain applied vsc.
  • Register Maturity Order: If a VSC vsc1 was provided by the provider chain before another VSC vsc2, then the provider chain MUST NOT register the maturity notification of vsc2 before the maturity notification of vsc1.
  • Register Maturity Liveness: If the provider chain provides a VSC vsc to the consumer chain, then the provider chain MUST eventually register a maturity notification of vsc from the consumer chain.

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Consumer Initiated Slashing

↑ Back to Outline
  • Consumer Slashing Warranty: Let cc be a consumer chain, such that its CCV module receives at height he evidence that a validator val misbehaved on cc at height hi. Let hv be the height when the CCV module of cc receives the first VSC from the provider CCV module, i.e., the height when the CCV channel is established. Then, the CCV module of cc MUST send (to the provider CCV module) exactly one SlashPacket P, such that
    • P is sent at height h = max(he, hv);
    • P.val = val and P.id = HtoVSC[hi], i.e., the ID of the latest VSC that updated the validator set on cc at height hi or 0 if such a VSC does not exist (if hi < hv).
    Note: A consequence of the Consumer Slashing Warranty property is that the initial validator set on a consumer chain cannot be slashed during the initialization of the CCV channel. Therefore, consumer chains SHOULD NOT allow user transactions before the CCV channel is established. Note that once the CCV channel is established (i.e., a VSC is received from the provider CCV module), CCV enables the slashing of the initial validator set for infractions committed during channel initialization.
  • Provider Slashing Warranty: If the provider CCV module receives from a consumer chain cc a SlashPacket containing a validator val and a VSC ID vscId, then it MUST make exactly one request to the provider Slashing module to slash val for misbehaving at height h, such that
    • if vscId = 0, h is the height of the block when the provider chain established a CCV channel to cc;
    • otherwise, h is the height of the block immediately subsequent to the block when the provider chain provided to cc the VSC with ID vscId. Furthermore, the provider CCV module MUST make this slash request before registering any maturity notifications received from cc after the SlashPacket.
  • VSC Maturity and Slashing Order: If a consumer chain sends to the provider chain a SlashPacket before a maturity notification of a VSC, then the provider chain MUST NOT receive the maturity notification before the SlashPacket.
    Note: VSC Maturity and Slashing Order requires the VSC maturity notifications to be sent through their own IBC packets (i.e., VSCMaturedPackets) instead of e.g., through acknowledgements of VSCPackets.

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Reward Distribution

↑ Back to Outline
  • Distribution Liveness: If the CCV module on a consumer chain sends to the distribution module account on the provider chain an amount T of tokens as reward for providing security, then T (equivalent) tokens are eventually minted in the distribution module account on the provider chain.

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Correctness Reasoning

↑ Back to Outline In this section we argue the correctness of the CCV protocol described in the Technical Specification, i.e., we informally prove the properties described in the previous section.
  • Channel Uniqueness: The consumer chain side of the CCV channel is established when the consumer CCV module receives the first ChanOpenAck message that is successfully executed; all subsequent ChanOpenAck messages will fail (cf. Safe Blockchain). Let ccvChannel denote this channel. Then, ccvChannel is the only OPEN channel that can be connected to a port owned by the consumer CCV module. The provider chain side of the CCV channel is established when the provider CCV module receives the first ChanOpenConfirm message that is successfully executed; all subsequent ChanOpenConfirm messages will fail (cf. Safe Blockchain). The ccvChannel is the only channel for which ChanOpenConfirm can be successfully executed (cf. Safe Blockchain, i.e., IBC channel opening handshake guarantee). As a result, ccvChannel is unique. Moreover, ts existence is guaranteed by the Correct Relayer assumption.
  • Channel Validity: Follows directly from the Safe Blockchain assumption.
  • Channel Order: The provider chain accepts only ordered channels when receiving a ChanOpenTry message (cf. Safe Blockchain). Similarly, the consumer chain accepts only ordered channels when receiving ChanOpenInit messages (cf. Safe Blockchain). Thus, the property follows directly from the fact that the CCV channel is ordered.
  • Channel Liveness: The property follows from the Correct Relayer assumption.
  • Validator Update To VSC Validity: The provider CCV module provides only VSCs that contain validator updates obtained from the Staking module, i.e., by calling the GetValidatorUpdates() method (cf. Safe Blockchain). Furthermore, these validator updates were applied to the validator set of the provider chain (cf. Validator Update Provision).
  • Validator Update To VSC Order: We prove the property through contradiction. Given two validator updates U1 and U2, with U1 occurring on the provider chain before U2, we assume U2 is included in a provided VSC before U1. However, U2 could not have been obtained by the provider CCV module before U1 (cf. Validator Update Provision). Thus, the provider CCV module could not have provided a VSC that contains U2 before a VSC that contains U1 (cf. Safe Blockchain), which contradicts the initial assumption.
  • Validator Update To VSC Liveness: The provider CCV module eventually provides to all consumer chains VSCs containing all validator updates obtained from the provider Staking module (cf. Safe Blockchain, Life Blockchain). Thus, it is sufficient to prove that every update of a validator in the validator set of the provider chain MUST eventually be obtained from the provider Staking module. We prove this through contradiction. Given a validator update U that is applied to the validator set of the provider chain at the end of a block B with height h, we assume U is never obtained by the provider CCV module. However, at height h, the provider CCV module tries to obtain a new batch of validator updates from the provider Staking module (cf. Safe Blockchain). Thus, this batch of validator updates MUST contain all validator updates applied to the validator set of the provider chain at the end of block B, including U (cf. Validator Update Provision), which contradicts the initial assumption.
  • Apply VSC Validity: The property follows from the following two assertions.
    • The consumer chain only applies VSCs received in VSCPackets through the CCV channel (cf. Safe Blockchain).
    • The provider chain only sends VSCPackets containing provided VSCs (cf. Safe Blockchain).
  • Apply VSC Order: We prove the property through contradiction. Given two VSCs vsc1 and vsc2 such that the provider chain provides vsc1 before vsc2, we assume the consumer chain applies the validator updates included in vsc2 before the validator updates included in vsc1. The following sequence of assertions leads to a contradiction.
    • The provider chain could not have sent a VSCPacket P2 containing vsc2 before a VSCPacket P1 containing vsc1 (cf. Safe Blockchain).
    • The consumer chain could not have received P2 before P1 (cf. Channel Order).
    • Given the Safe Blockchain assumption, we distinguish two cases.
      • First, the consumer chain receives P1 during block B1 and P2 during block B2 (with B1 < B2). Then, it applies the validator updates included in vsc1 at the end of B1 and the validator updates included in vsc2 at the end of B2 (cf. Validator Update Inclusion), which contradicts the initial assumption.
      • Second, the consumer chain receives both P1 and P2 during the same block. Then, it applies the validator updates included in both vsc1 and vsc2 at the end of the block. Thus, it could not have apply the validator updates included in vsc2 before.
  • Apply VSC Liveness: The provider chain eventually sends over the CCV channel a VSCPacket containing vsc (cf. Safe Blockchain, Life Blockchain). As a result, the consumer chain eventually receives this packet (cf. Channel Liveness). Then, the consumer chain aggregates all received VSCs at the end of the block and applies all the aggregated updates (cf. Safe Blockchain, Life Blockchain). As a result, the consumer chain applies all validator updates in vsc (cf. Validator Update Inclusion).
  • Register Maturity Validity: The property follows from the following sequence of assertions.
    • The provider chain only registers VSC maturity notifications when receiving on the CCV channel a VSCMaturedPackets notifying the maturity of those VSCs (cf. Safe Blockchain).
    • The provider chain receives on the CCV channel only packets sent by the consumer chain (cf. Channel Validity).
    • The consumer chain only sends VSCMaturedPackets matching the VSCPackets it receives on the CCV channel (cf. Safe Blockchain).
    • The consumer chain receives on the CCV channel only packets sent by the provider chain (cf. Channel Validity).
    • The provider chain only sends VSCPackets containing provided VSCs (cf. Safe Blockchain).
  • Register Maturity Timeliness: We prove the property through contradiction. Given a VSC vsc provided by the provider chain to the consumer chain, we assume that the provider chain registers a maturity notification of vsc before UnbondingPeriod has elapsed on the consumer chain since the consumer chain applied vsc. The following sequence of assertions leads to a contradiction.
    • The provider chain could not have register a maturity notification of vsc before receiving on the CCV channel a VSCMaturedPacket P with P.id = vsc.id (cf. Safe Blockchain).
    • The provider chain could not have received P on the CCV channel before the consumer chain sent it (cf. Channel Validity).
    • The consumer chain could not have sent P before at least UnbondingPeriod has elapsed since receiving a VSCPacket P' with P'.id = P.id on the CCV channel (cf. Safe Blockchain). Note that since time is measured in terms of the block time, the time of receiving P' is the same as the time of applying vsc.
    • The consumer chain could not have received P' on the CCV channel before the provider chain sent it (cf. Channel Validity).
    • The provider chain could not have sent P' before providing vsc.
    • Since the duration of sending packets through the CCV channel cannot be negative, the provider chain could not have registered a maturity notification of vsc before UnbondingPeriod has elapsed on the consumer chain since the consumer chain applied vsc.
  • Register Maturity Order: We prove the property through contradiction. Given two VSCs vsc1 and vsc2 such that the provider chain provides vsc1 before vsc2, we assume the provider chain registers the maturity notification of vsc2 before the maturity notification of vsc1. The following sequence of assertions leads to a contradiction.
    • The provider chain could not have sent a VSCPacket P2, with P2.updates = C2, before a VSCPacket P1, with P1.updates = C1 (cf. Safe Blockchain).
    • The consumer chain could not have received P2 before P1 (cf. Channel Order).
    • The consumer chain could not have sent a VSCMaturedPacket P2', with P2'.id = P2.id, before a VSCMaturedPacket P1', with P1'.id = P1.id (cf. Safe Blockchain).
    • The provider chain could not have received P2' before P1' (cf. Channel Order).
    • The provider chain could not have registered the maturity notification of vsc2 before the maturity notification of vsc1 (cf. Safe Blockchain).
  • Register Maturity Liveness: The property follows from the following sequence of assertions.
    • The provider chain eventually sends on the CCV channel a VSCPacket P, with P.updates = C (cf. Safe Blockchain, Life Blockchain).
    • The consumer chain eventually receives P on the CCV channel (cf. Channel Liveness).
    • The consumer chain eventually sends on the CCV channel a VSCMaturedPacket P', with P'.id = P.id (cf. Safe Blockchain, Life Blockchain).
    • The provider chain eventually receives P' on the CCV channel (cf. Channel Liveness).
    • The provider chain eventually registers the maturity notification of vsc (cf. Safe Blockchain, Life Blockchain).
  • Consumer Slashing Warranty: Follows directly from Safe Blockchain.
  • Provider Slashing Warranty: Follows directly from Safe Blockchain.
  • VSC Maturity and Slashing Order: Follows directly from Channel Order.
  • Distribution Liveness: The CCV module on the consumer chain sends to the provider chain an amount T of tokens through an IBC token transfer packet (as defined in ICS 20). Thus, if the packet is relayed within the timeout period, then T (equivalent) tokens are minted on the provider chain. Otherwise, the T tokens are refunded to the consumer CCV module account. In this case, the T tokens will be part of the next token transfer packet. Eventually, a correct relayer will relay a token transfer packet containing the T tokens (cf. Correct Relayer, Life Blockchain). As a result, T (equivalent) tokens are eventually minted on the provider chain.
  • Validator Set Replication: The property follows from the Safe Blockchain assumption and both the Apply VSC Validity and Validator Update To VSC Validity properties.
  • Bond-Based Consumer Voting Power: The existence of hp is given by construction, i.e., the block on the proposer chain that handles a governance proposal to spawn a new consumer chain cc happens before all the blocks of cc. The existence of hc' and hp' is given by Life Blockchain and Channel Liveness. To prove the Bond-Based Consumer Voting Power property, we use the following property that follows directly from the design of the protocol (cf. Safe Blockchain, Life Blockchain).
    • Property1: Let val be a validator; let Ua and Ub be two updates of val that are applied subsequently by a consumer chain cc, at block heights ha and hb, respectively (i.e., no other updates of val are applied in between). Then, Power(cc,ha,val) = Power(cc,h,val), for all block heights h, such that ha <= h < hb (i.e., the voting power granted to val on cc in the period between ts(ha) and ts(hb) is constant).
    We prove the Bond-Based Consumer Voting Power property through contradiction. We assume there exist a height h on the provider chain between hp and hp' such that Power(cc,hc,val) > VP( pBonded(h,val) + sumUnbonding(hp, h, val) + sumSlash(hp, h, val) ). The following sequence of assertions leads to a contradiction.
    • Let U1 be the latest update of val that is applied by cc before or not later than block height hc (i.e., U1 is the update that sets Power(cc,hc,val) for val). Let hp1 be the height at which U1 occurs on the provider chain; let hc1 be the height at which U1 is applied on cc. Then, hp1 << hc1 <= hc, hp1 <= hp, and Power(cc,hc,val) = Power(cc,hc1,val) = VP(pBonded(hp1,val)). This means that some of the tokens bonded by val at height hp1 were completely unbonded before or not later than height hp' (cf. Power(cc,hc,val) > VP( pBonded(h,val) + sumUnbonding(hp, h, val) + sumSlash(hp, h, val) )).
    • Let uo be the first such unbonding operation that is initiated on the provider chain at height hp2, such that hp1 < hp2 <= hp'. Note that at height hp2, the tokens unbonded by uo are part of pUnbonding(hp2,val). Let U2 be the validator update caused by initiating uo. Let hc2 be the height at which U2 is applied on cc; clearly, Power(cc,hc2,val) < Power(cc,hc,val). Note that the existence of hc2 is ensured by Validator Update To VSC Liveness and Apply VSC Liveness. Then, hc2 > hc1 (cf. hp2 > hp1, Validator Update To VSC Order, Apply VSC Order).
    • Power(cc,hc,val) = Power(cc,hc1,val) = Power(cc,h,val), for all heights h, such that hc1 <= h < hc2 (cf. Property1). Thus, hc2 > hc (cf. Power(cc,hc2,val) < Power(cc,hc,val)).
    • uo cannot complete before ts(hc2) + UnbondingPeriod, which means it cannot complete before hc' and thus it cannot complete before hp' (cf. hc' << hp').
  • Slashable Consumer Misbehavior: The second part of the Slashable Consumer Misbehavior property (i.e., val is not slashed more than once for the same misbehavior) follows directly from Evidence Provision, Channel Validity, Consumer Slashing Warranty, Provider Slashing Warranty. To prove the first part of the Slashable Consumer Misbehavior property (i.e., exactly the amount of tokens sf*Token(Power(cc,hi,val)) are slashed on the provider chain), we consider the following sequence of statements.
    • The CCV module of cc receives at height he the evidence that val misbehaved on cc at height hi (cf. Evidence Provision, Safe Blockchain, Life Blockchain).
    • Let hv be the height when the CCV module of cc receives the first VSC from the provider CCV module. Then, the CCV module of cc sends at height h = max(he, hv) to the provider chain a SlashPacket P, such that P.val = val and P.id = HtoVSC[hi] (cf. Consumer Slashing Warranty).
    • The provider CCV module eventually receives P (cf. Channel Liveness).
    • The provider CCV module requests the provider Slashing module to slash val for misbehaving at height hp = VSCtoH[P.id] before handling any further maturity notifications received from the CCV module of cc (cf. Provider Slashing Warranty).
    • The provider Slashing module slashes the amount of tokens val had bonded at height hp except the amount that has already completely unbonded (cf. Slashing Warranty). Thus, it remains to be proven that Token(Power(cc,hi,val)) = pBonded(hp,val), with hp = VSCtoH[HtoVSC[hi]]. We distinguish two cases:
    • HtoVSC[hi] != 0, which means that by definition HtoVSC[hi] is the ID of the last VSC that update Power(cc,hi,val). Also by definition, this VSC contains the last updates to the voting power at height VSCtoH[HtoVSC[hi]] on the provider. Thus, Token(Power(cc,hi,val)) = pBonded(hp,val).
    • HtoVSC[hi] == 0, which means that by definition Power(cc,hi,val) was setup at genesis during Channel Initialization. Also by definition, this is the same voting power of the provider chain block when the first VSC was provided to that consumer chain, i.e., VSCtoH[HtoVSC[hi]]. Thus, Token(Power(cc,hi,val)) = pBonded(hp,val).
  • Consumer Rewards Distribution: The first part of the Consumer Rewards Distribution property (i.e., the tokens are eventually minted on the provider chain and then distributed among the validators) follows directly from Distribution Liveness and Distribution Warranty. The second part of the Consumer Rewards Distribution property (i.e., the total supply of tokens is preserved) follows directly from the Supply property of the Fungible Token Transfer protocol (see ICS 20).