Ledger.Conway.Conformance.Certs

{-# OPTIONS --safe #-}

open import Ledger.Prelude
open import Ledger.Conway.Specification.Gov.Base
import Ledger.Conway.Specification.Certs

module Ledger.Conway.Conformance.Certs
  (gs : _) (open GovStructure gs)
  where


open import Ledger.Conway.Specification.Gov.Actions gs
private module Certs = Ledger.Conway.Specification.Certs gs
open Certs public
  hiding (DState; GState; CertState; HasCast-DState; HasCast-GState; HasCast-CertState;
          _⊢_⇀⦇_,DELEG⦈_; _⊢_⇀⦇_,GOVCERT⦈_;
          _⊢_⇀⦇_,CERT⦈_; _⊢_⇀⦇_,PRE-CERT⦈_; _⊢_⇀⦇_,POST-CERT⦈_; _⊢_⇀⦇_,CERTS⦈_; ⟦_,_,_⟧ᵈ)
open RewardAddress

record DState : Type where
  constructor ⟦_,_,_,_⟧ᵈ
  field
    voteDelegs   : VoteDelegs
    stakeDelegs  : Credential ⇀ KeyHash
    rewards      : Rewards
    deposits     : Deposits

record GState : Type where
  constructor ⟦_,_,_⟧ᵛ
  field
    dreps      : Credential ⇀ Epoch
    ccHotKeys  : Credential ⇀ Maybe Credential
    deposits   : Deposits

record CertState : Type where
  constructor ⟦_,_,_⟧ᶜˢ
  field
    dState : DState
    pState : PState
    gState : GState

instance
  unquoteDecl HasCast-DState HasCast-GState HasCast-CertState = derive-HasCast
    (   (quote DState , HasCast-DState)
    ∷   (quote GState , HasCast-GState)
    ∷ [ (quote CertState , HasCast-CertState) ])

certDeposit : DCert → PParams → Deposits
certDeposit (delegate c _ _ v) _   = ❴ CredentialDeposit c , v ❵
certDeposit (regdrep c v _)    _   = ❴ DRepDeposit c , v ❵
certDeposit (reg c v)          pp  = ❴ CredentialDeposit c , pp .PParams.keyDeposit ❵
certDeposit _                  _   = ∅
-- handled in the Utxo module:
-- certDeposit (regpool kh _)     pp  = ❴ PoolDeposit kh , pp .poolDeposit ❵

certRefund : DCert → ℙ DepositPurpose
certRefund (dereg c _)      = ❴ CredentialDeposit c ❵
certRefund (deregdrep c _)  = ❴ DRepDeposit c ❵
certRefund _                = ∅

updateCertDeposit  : PParams → DCert → Deposits → Deposits
updateCertDeposit pp (delegate c _ _ v) deposits = deposits ∪⁺ ❴ CredentialDeposit c , v ❵
updateCertDeposit pp (reg c _)          deposits = deposits ∪⁺ ❴ CredentialDeposit c , pp .PParams.keyDeposit ❵
updateCertDeposit pp (regdrep c v _)    deposits = deposits ∪⁺ ❴ DRepDeposit c , v ❵
updateCertDeposit pp (dereg c _)        deposits = deposits ∣ ❴ CredentialDeposit c ❵ ᶜ
updateCertDeposit pp (deregdrep c _)    deposits = deposits ∣ ❴ DRepDeposit c ❵ ᶜ
updateCertDeposit pp (regpool kh _)     deposits = deposits ∪⁺ ❴ PoolDeposit kh , pp .PParams.poolDeposit ❵
updateCertDeposit _ (retirepool _ _)    deposits = deposits
updateCertDeposit _ (ccreghot _ _)      deposits = deposits
-- updateCertDeposit pp cert deposits
--   = (deposits ∪⁺ certDeposit cert pp) ∣ certRefund cert ᶜ

private variable
  rwds rewards           : Rewards
  dReps                  : Credential ⇀ Epoch
  sDelegs stakeDelegs    : Credential ⇀ KeyHash
  ccKeys ccHotKeys       : Credential ⇀ Maybe Credential
  vDelegs voteDelegs     : VoteDelegs
  pools fPools           : Pools
  retiring               : KeyHash ⇀ Epoch
  wdrls                  : Withdrawals

  an             : Anchor
  Γ              : CertEnv
  d              : Coin
  md             : Maybe Coin
  c              : Credential
  mc             : Maybe Credential
  delegatees cc  : ℙ Credential
  dCert          : DCert
  dep ddep gdep  : Deposits
  e              : Epoch
  vs             : List GovVote
  kh             : KeyHash
  mkh            : Maybe KeyHash
  poolParams     : StakePoolParams
  pp             : PParams
  mv             : Maybe VDeleg

  stᵈ stᵈ' : DState
  stᵍ stᵍ' : GState
  stᵖ stᵖ' : PState

open GovVote

data _⊢_⇀⦇_,DELEG⦈_ : DelegEnv → DState → DCert → DState → Type

data _⊢_⇀⦇_,DELEG⦈_ where
  DELEG-delegate : let open PParams pp in
    ∙ (c ∉ dom rwds → d ≡ keyDeposit)
    ∙ (c ∈ dom rwds → d ≡ 0)
    ∙ mv ∈ mapˢ (just ∘ vDelegCredential) delegatees ∪
        fromList
          ( nothing
          ∷ just vDelegAbstain
          ∷ just vDelegNoConfidence
          ∷ []
          )
    ∙ mkh ∈ mapˢ just (dom pools) ∪ ❴ nothing ❵
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{4096}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2956}{\htmlId{4101}{\htmlClass{Generalizable}{\text{pools}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3245}{\htmlId{4109}{\htmlClass{Generalizable}{\text{delegatees}}}}\, \end{pmatrix}$ ⊢
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4132}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4142}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4152}{\htmlClass{Generalizable}{\text{rwds}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{4159}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$
      ⇀⦇ delegate c mv mkh d ,DELEG⦈
      $\begin{pmatrix} \,\href{Axiom.Set.Map.html#9262}{\htmlId{4210}{\htmlClass{Function}{\text{insertIfJust}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4223}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3509}{\htmlId{4225}{\htmlClass{Generalizable}{\text{mv}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4228}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \\ \,\href{Axiom.Set.Map.html#9262}{\htmlId{4238}{\htmlClass{Function}{\text{insertIfJust}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4251}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3414}{\htmlId{4253}{\htmlClass{Generalizable}{\text{mkh}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4257}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4267}{\htmlClass{Generalizable}{\text{rwds}}}}\, \,\href{Axiom.Set.Map.html#7640}{\htmlId{4272}{\htmlClass{Function Operator}{\text{∪ˡ}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4275}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4277}{\htmlClass{Generalizable}{\text{c}}}}\, , \,\htmlId{4281}{\htmlClass{Number}{\text{0}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4283}{\htmlClass{Field Operator}{\text{❵}}}}\,
      \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{4293}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{4311}{\htmlClass{Generalizable}{\text{pp}}}}\, \,\htmlId{4314}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2137}{\htmlId{4315}{\htmlClass{InductiveConstructor}{\text{delegate}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4324}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3509}{\htmlId{4326}{\htmlClass{Generalizable}{\text{mv}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3414}{\htmlId{4329}{\htmlClass{Generalizable}{\text{mkh}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3125}{\htmlId{4333}{\htmlClass{Generalizable}{\text{d}}}}\,\,\htmlId{4334}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{4336}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

  DELEG-dereg :
    ∙ (c , 0) ∈ rwds
    ∙ (CredentialDeposit c , d) ∈ dep
    ∙ md ≡ nothing ⊎ md ≡ just d
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{4498}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2956}{\htmlId{4503}{\htmlClass{Generalizable}{\text{pools}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3245}{\htmlId{4511}{\htmlClass{Generalizable}{\text{delegatees}}}}\, \end{pmatrix}$ ⊢
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4534}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4544}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4554}{\htmlClass{Generalizable}{\text{rwds}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{4561}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$
      ⇀⦇ dereg c md ,DELEG⦈
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4603}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4611}{\htmlClass{Function Operator}{\text{∣}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4613}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4615}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4617}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4619}{\htmlClass{Function Operator}{\text{ᶜ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4623}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4631}{\htmlClass{Function Operator}{\text{∣}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4633}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4635}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4637}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4639}{\htmlClass{Function Operator}{\text{ᶜ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4643}{\htmlClass{Generalizable}{\text{rwds}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4648}{\htmlClass{Function Operator}{\text{∣}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4650}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4652}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4654}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{4656}{\htmlClass{Function Operator}{\text{ᶜ}}}}\,
      \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{4666}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{4684}{\htmlClass{Generalizable}{\text{pp}}}}\, \,\htmlId{4687}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2210}{\htmlId{4688}{\htmlClass{InductiveConstructor}{\text{dereg}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4694}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3149}{\htmlId{4696}{\htmlClass{Generalizable}{\text{md}}}}\,\,\htmlId{4698}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{4700}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

  DELEG-reg : let open PParams pp in
    ∙ c ∉ dom rwds
    ∙ d ≡ keyDeposit ⊎ d ≡ 0
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{4839}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2956}{\htmlId{4844}{\htmlClass{Generalizable}{\text{pools}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3245}{\htmlId{4852}{\htmlClass{Generalizable}{\text{delegatees}}}}\, \end{pmatrix}$ ⊢
        $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4877}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4887}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4897}{\htmlClass{Generalizable}{\text{rwds}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{4904}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$ ⇀⦇ reg c d ,DELEG⦈
        $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2918}{\htmlId{4939}{\htmlClass{Generalizable}{\text{vDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2813}{\htmlId{4949}{\htmlClass{Generalizable}{\text{sDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2732}{\htmlId{4959}{\htmlClass{Generalizable}{\text{rwds}}}}\, \,\href{Axiom.Set.Map.html#7640}{\htmlId{4964}{\htmlClass{Function Operator}{\text{∪ˡ}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4967}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{4969}{\htmlClass{Generalizable}{\text{c}}}}\, , \,\htmlId{4973}{\htmlClass{Number}{\text{0}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{4975}{\htmlClass{Field Operator}{\text{❵}}}}\,
        \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{4987}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{5005}{\htmlClass{Generalizable}{\text{pp}}}}\, \,\htmlId{5008}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2829}{\htmlId{5009}{\htmlClass{InductiveConstructor}{\text{reg}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5013}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3125}{\htmlId{5015}{\htmlClass{Generalizable}{\text{d}}}}\,\,\htmlId{5016}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5018}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

data _⊢_⇀⦇_,GOVCERT⦈_ : CertEnv → GState → DCert → GState → Type where
  GOVCERT-regdrep : ∀ {pp} → let open PParams pp in
    ∙ (d ≡ drepDeposit × c ∉ dom dReps) ⊎ (d ≡ 0 × c ∈ dom dReps)
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{5261}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#5119}{\htmlId{5265}{\htmlClass{Bound}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{5270}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{5275}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{5283}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5298}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5306}{\htmlClass{Generalizable}{\text{ccKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5315}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$
        ⇀⦇ regdrep c d an ,GOVCERT⦈
      $\begin{pmatrix} \,\href{Class.HasSingleton.html#288}{\htmlId{5365}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5367}{\htmlClass{Generalizable}{\text{c}}}}\, , \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{5371}{\htmlClass{Generalizable}{\text{e}}}}\, \,\href{Class.HasAdd.Core.html#162}{\htmlId{5373}{\htmlClass{Field Operator}{\text{+}}}}\, \,\href{Ledger.Conway.Specification.PParams.html#3446}{\htmlId{5375}{\htmlClass{Field}{\text{drepActivity}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{5388}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#7640}{\htmlId{5390}{\htmlClass{Function Operator}{\text{∪ˡ}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5393}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5401}{\htmlClass{Generalizable}{\text{ccKeys}}}}\,
      \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{5416}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#5119}{\htmlId{5434}{\htmlClass{Bound}{\text{pp}}}}\, \,\htmlId{5437}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2348}{\htmlId{5438}{\htmlClass{InductiveConstructor}{\text{regdrep}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5446}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3125}{\htmlId{5448}{\htmlClass{Generalizable}{\text{d}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3072}{\htmlId{5450}{\htmlClass{Generalizable}{\text{an}}}}\, \,\htmlId{5453}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5455}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

  GOVCERT-deregdrep :
    ∙ c ∈ dom dReps
    ∙ (DRepDeposit c , d) ∈ dep
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{5583}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{5587}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{5592}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{5597}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{5605}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5614}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5622}{\htmlClass{Generalizable}{\text{ccKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5631}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$
          ⇀⦇ deregdrep c d ,GOVCERT⦈
          $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5686}{\htmlClass{Generalizable}{\text{dReps}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{5692}{\htmlClass{Function Operator}{\text{∣}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{5694}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5696}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{5698}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#13606}{\htmlId{5700}{\htmlClass{Function Operator}{\text{ᶜ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5704}{\htmlClass{Generalizable}{\text{ccKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{5713}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{5731}{\htmlClass{Generalizable}{\text{pp}}}}\, \,\htmlId{5734}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2399}{\htmlId{5735}{\htmlClass{InductiveConstructor}{\text{deregdrep}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5745}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3125}{\htmlId{5747}{\htmlClass{Generalizable}{\text{d}}}}\,\,\htmlId{5748}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5750}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

  GOVCERT-ccreghot :
    ∙ (c , nothing) ∉ ccKeys
    ∙ c ∈ cc
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{5867}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{5871}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{5876}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{5881}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{5889}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5898}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5906}{\htmlClass{Generalizable}{\text{ccKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{5915}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$
          ⇀⦇ ccreghot c mc ,GOVCERT⦈
          $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{5970}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Class.HasSingleton.html#288}{\htmlId{5978}{\htmlClass{Field Operator}{\text{❴}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{5980}{\htmlClass{Generalizable}{\text{c}}}}\, , \,\href{Ledger.Conway.Conformance.Certs.html#3209}{\htmlId{5984}{\htmlClass{Generalizable}{\text{mc}}}}\, \,\href{Class.HasSingleton.html#288}{\htmlId{5987}{\htmlClass{Field Operator}{\text{❵}}}}\, \,\href{Axiom.Set.Map.html#7640}{\htmlId{5989}{\htmlClass{Function Operator}{\text{∪ˡ}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#2861}{\htmlId{5992}{\htmlClass{Generalizable}{\text{ccKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#1868}{\htmlId{6001}{\htmlClass{Function}{\text{updateCertDeposit}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{6019}{\htmlClass{Generalizable}{\text{pp}}}}\, \,\htmlId{6022}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Certs.html#2441}{\htmlId{6023}{\htmlClass{InductiveConstructor}{\text{ccreghot}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3179}{\htmlId{6032}{\htmlClass{Generalizable}{\text{c}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3209}{\htmlId{6034}{\htmlClass{Generalizable}{\text{mc}}}}\,\,\htmlId{6036}{\htmlClass{Symbol}{\text{)}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3302}{\htmlId{6038}{\htmlClass{Generalizable}{\text{dep}}}}\, \end{pmatrix}$

data _⊢_⇀⦇_,CERT⦈_ : CertEnv → CertState → DCert → CertState → Type where
  CERT-deleg :
    ∙ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{6142}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Specification.Certs.html#3941}{\htmlId{6147}{\htmlClass{Field}{\text{PState.pools}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6160}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Class.IsSet.html#916}{\htmlId{6166}{\htmlClass{Function}{\text{dom}}}}\, \,\htmlId{6170}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Conformance.Certs.html#849}{\htmlId{6171}{\htmlClass{Field}{\text{GState.dreps}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6184}{\htmlClass{Generalizable}{\text{stᵍ}}}}\,\,\htmlId{6187}{\htmlClass{Symbol}{\text{)}}}\, \end{pmatrix}$ ⊢ stᵈ ⇀⦇ dCert ,DELEG⦈ stᵈ'
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{6266}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{6270}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{6275}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{6280}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{6288}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3542}{\htmlId{6297}{\htmlClass{Generalizable}{\text{stᵈ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6303}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6309}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$ ⇀⦇ dCert ,CERT⦈ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3546}{\htmlId{6333}{\htmlClass{Generalizable}{\text{stᵈ'}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6340}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6346}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$

  CERT-pool :
    ∙ pp ⊢ stᵖ ⇀⦇ dCert ,POOL⦈ stᵖ'
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{6450}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{6454}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{6459}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{6464}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{6472}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3542}{\htmlId{6481}{\htmlClass{Generalizable}{\text{stᵈ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6487}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6493}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$ ⇀⦇ dCert ,CERT⦈ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3542}{\htmlId{6517}{\htmlClass{Generalizable}{\text{stᵈ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3586}{\htmlId{6523}{\htmlClass{Generalizable}{\text{stᵖ'}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6530}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$

  CERT-vdel :
    ∙ Γ ⊢ stᵍ ⇀⦇ dCert ,GOVCERT⦈ stᵍ'
      ────────────────────────────────
      Γ ⊢ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3542}{\htmlId{6640}{\htmlClass{Generalizable}{\text{stᵈ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6646}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{6652}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$ ⇀⦇ dCert ,CERT⦈ $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3542}{\htmlId{6676}{\htmlClass{Generalizable}{\text{stᵈ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{6682}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3566}{\htmlId{6688}{\htmlClass{Generalizable}{\text{stᵍ'}}}}\, \end{pmatrix}$

data _⊢_⇀⦇_,PRE-CERT⦈_ : CertEnv → CertState → ⊤ → CertState → Type where

  CERT-pre :
    let open PParams pp
        refresh         = mapPartial (isGovVoterDRep ∘ voter) (fromList vs)
        refreshedDReps  = mapValueRestricted (const (e + drepActivity)) dReps refresh
        wdrlCreds       = mapˢ stake (dom wdrls)
    in
    ∙ filterˢ isKeyHash wdrlCreds ⊆ dom voteDelegs
    ∙ mapˢ (map₁ stake) (wdrls ˢ) ⊆ rewards ˢ
      ────────────────────────────────
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{7170}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{7174}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{7179}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{7184}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{7192}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$ ⊢ $\begin{pmatrix} \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2926}{\htmlId{7203}{\htmlClass{Generalizable}{\text{voteDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2821}{\htmlId{7216}{\htmlClass{Generalizable}{\text{stakeDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2737}{\htmlId{7230}{\htmlClass{Generalizable}{\text{rewards}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3306}{\htmlId{7240}{\htmlClass{Generalizable}{\text{ddep}}}}\, \end{pmatrix} \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{7249}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2767}{\htmlId{7257}{\htmlClass{Generalizable}{\text{dReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2868}{\htmlId{7265}{\htmlClass{Generalizable}{\text{ccHotKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3311}{\htmlId{7277}{\htmlClass{Generalizable}{\text{gdep}}}}\, \end{pmatrix} \end{pmatrix}$
      ⇀⦇ _ ,PRE-CERT⦈
      $\begin{pmatrix} \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2926}{\htmlId{7318}{\htmlClass{Generalizable}{\text{voteDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2821}{\htmlId{7331}{\htmlClass{Generalizable}{\text{stakeDelegs}}}}\, \\ \,\href{Axiom.Set.Map.html#11988}{\htmlId{7345}{\htmlClass{Function}{\text{constMap}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#6978}{\htmlId{7354}{\htmlClass{Bound}{\text{wdrlCreds}}}}\, \,\htmlId{7364}{\htmlClass{Number}{\text{0}}}\, \,\href{Axiom.Set.Map.html#7640}{\htmlId{7366}{\htmlClass{Function Operator}{\text{∪ˡ}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#2737}{\htmlId{7369}{\htmlClass{Generalizable}{\text{rewards}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3306}{\htmlId{7379}{\htmlClass{Generalizable}{\text{ddep}}}}\, \end{pmatrix} \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{7388}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#6892}{\htmlId{7396}{\htmlClass{Bound}{\text{refreshedDReps}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2868}{\htmlId{7413}{\htmlClass{Generalizable}{\text{ccHotKeys}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3311}{\htmlId{7425}{\htmlClass{Generalizable}{\text{gdep}}}}\, \end{pmatrix} \end{pmatrix}$

data _⊢_⇀⦇_,POST-CERT⦈_ : CertEnv → CertState → ⊤ → CertState → Type where

  CERT-post :
      $\begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#3330}{\htmlId{7533}{\htmlClass{Generalizable}{\text{e}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3482}{\htmlId{7537}{\htmlClass{Generalizable}{\text{pp}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3355}{\htmlId{7542}{\htmlClass{Generalizable}{\text{vs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3032}{\htmlId{7547}{\htmlClass{Generalizable}{\text{wdrls}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3256}{\htmlId{7555}{\htmlClass{Generalizable}{\text{cc}}}}\, \end{pmatrix}$
      ⊢ $\begin{pmatrix} \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2926}{\htmlId{7572}{\htmlClass{Generalizable}{\text{voteDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2821}{\htmlId{7585}{\htmlClass{Generalizable}{\text{stakeDelegs}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#2737}{\htmlId{7599}{\htmlClass{Generalizable}{\text{rewards}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3306}{\htmlId{7609}{\htmlClass{Generalizable}{\text{ddep}}}}\, \end{pmatrix} \\ \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{7618}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, \\ \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{7624}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$
        ⇀⦇ _ ,POST-CERT⦈
        $\begin{pmatrix} \begin{pmatrix} \,\href{Ledger.Conway.Conformance.Certs.html#2926}{\htmlId{7667}{\htmlClass{Generalizable}{\text{voteDelegs}}}}\, \,\href{Axiom.Set.Map.html#17775}{\htmlId{7678}{\htmlClass{Function Operator}{\text{∣\^{}}}}}\, \,\htmlId{7681}{\htmlClass{Symbol}{\text{(}}}\,\,\href{abstract-set-theory.FiniteSetTheory.html#519}{\htmlId{7682}{\htmlClass{Function}{\text{mapˢ}}}}\, \,\href{Ledger.Conway.Specification.Gov.Actions.html#2592}{\htmlId{7687}{\htmlClass{InductiveConstructor}{\text{vDelegCredential}}}}\, \,\htmlId{7704}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Class.IsSet.html#916}{\htmlId{7705}{\htmlClass{Function}{\text{dom}}}}\, \,\htmlId{7709}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Conformance.Certs.html#849}{\htmlId{7710}{\htmlClass{Field}{\text{GState.dreps}}}}\, \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{7723}{\htmlClass{Generalizable}{\text{stᵍ}}}}\,\,\htmlId{7726}{\htmlClass{Symbol}{\text{))}}}\, \,\href{Axiom.Set.html#9137}{\htmlId{7729}{\htmlClass{Function Operator}{\text{∪}}}}\, \,\href{Axiom.Set.html#5903}{\htmlId{7731}{\htmlClass{Function}{\text{fromList}}}}\, \,\htmlId{7740}{\htmlClass{Symbol}{\text{(}}}\,\,\href{Ledger.Conway.Specification.Gov.Actions.html#2665}{\htmlId{7741}{\htmlClass{InductiveConstructor}{\text{vDelegNoConfidence}}}}\, \,\href{Agda.Builtin.List.html#199}{\htmlId{7760}{\htmlClass{InductiveConstructor Operator}{\text{∷}}}}\, \,\href{Ledger.Conway.Specification.Gov.Actions.html#2635}{\htmlId{7762}{\htmlClass{InductiveConstructor}{\text{vDelegAbstain}}}}\, \,\href{Agda.Builtin.List.html#199}{\htmlId{7776}{\htmlClass{InductiveConstructor Operator}{\text{∷}}}}\, \,\href{Agda.Builtin.List.html#184}{\htmlId{7778}{\htmlClass{InductiveConstructor}{\text{[]}}}}\,\,\htmlId{7780}{\htmlClass{Symbol}{\text{))}}}\,
          , \,\href{Ledger.Conway.Conformance.Certs.html#2821}{\htmlId{7795}{\htmlClass{Generalizable}{\text{stakeDelegs}}}}\, , \,\href{Ledger.Conway.Conformance.Certs.html#2737}{\htmlId{7809}{\htmlClass{Generalizable}{\text{rewards}}}}\, , \,\href{Ledger.Conway.Conformance.Certs.html#3306}{\htmlId{7819}{\htmlClass{Generalizable}{\text{ddep}}}}\, \end{pmatrix} , \,\href{Ledger.Conway.Conformance.Certs.html#3582}{\htmlId{7828}{\htmlClass{Generalizable}{\text{stᵖ}}}}\, , \,\href{Ledger.Conway.Conformance.Certs.html#3562}{\htmlId{7834}{\htmlClass{Generalizable}{\text{stᵍ}}}}\, \end{pmatrix}$

_⊢_⇀⦇_,CERTS⦈_  : CertEnv → CertState  → List DCert  → CertState  → Type
_⊢_⇀⦇_,CERTS⦈_ = RunTraceAfterAndThen _⊢_⇀⦇_,PRE-CERT⦈_ _⊢_⇀⦇_,CERT⦈_ _⊢_⇀⦇_,POST-CERT⦈_