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[Dijkstra] Certs PoV: discharge the five CERTS-side LEDGER-PoV parameters
Certs.Properties.PoVLemmas proves the per-CERT-step facts: rewards preservation, closed-form deposit accounting under PoolDepositsRegistered, preservation of that invariant, and the pool-set tracking of the newCertDeposits fold (via seed-certificate shift lemmas, a ∷-split, and ≡ᵉ-congruence in the pool-set argument), plus refundCertDeposits-++. Certs.Properties.PoV lifts these along the reflexive-transitive closure, with statements matching the LEDGER-PoV module parameters verbatim: CERTS-rewards-pov, CERTS-deposits-pov, CERTS-deposits-registered, CERTS-new-thread, refundCertDeposits-++. AI-assisted: Claude Fable 5 (Anthropic)
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‎src/Ledger/Dijkstra/Specification/Certs/Properties.lagda.md‎

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@@ -14,4 +14,6 @@ module Ledger.Dijkstra.Specification.Certs.Properties where
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```agda
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open import Ledger.Dijkstra.Specification.Certs.Properties.Computational
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open import Ledger.Dijkstra.Specification.Certs.Properties.PoVLemmas
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open import Ledger.Dijkstra.Specification.Certs.Properties.PoV
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```

‎src/Ledger/Dijkstra/Specification/Certs/Properties/PoV.lagda.md‎

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# `CERTS`{.AgdaDatatype} Properties: Preservation of Value {#sec:certs-pov}
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This module provides the `CERTS`{.AgdaDatatype}-level value-accounting facts
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consumed by `Ledger.Properties.PoV`{.AgdaModule}: it discharges the
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`CERTS-rewards-pov`, `CERTS-deposits-pov`, `CERTS-deposits-registered`,
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`CERTS-new-thread` and `refundCertDeposits-++` module parameters of the
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`LEDGER-PoV`{.AgdaModule} consumer, with statements matching those parameters
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verbatim.
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`CERTS`{.AgdaDatatype} is the reflexive–transitive closure of the
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single-certificate rule `CERT`{.AgdaDatatype}, so each fact is an induction
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over the closure whose step case is the corresponding per-step lemma of
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`Certs.Properties.PoVLemmas`{.AgdaModule}. Composing the deposit facts across
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steps rests on the pool-set threading explained there: peeling the first
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certificate off `newCertDeposits`{.AgdaFunction}
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(`newCertDeposits-∷`{.AgdaFunction}) leaves the tail charged against the
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evolved accumulator, which `CERT-poolSet`{.AgdaFunction} identifies — up to
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extensional equality, absorbed by `newCertDeposits-cong`{.AgdaFunction} — with
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the pool set of the state after the step.
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<!--
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```agda
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{-# OPTIONS --safe #-}
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open import Ledger.Dijkstra.Specification.Gov.Base using (GovStructure)
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module Ledger.Dijkstra.Specification.Certs.Properties.PoV
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(gs : GovStructure) (open GovStructure gs) where
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open import Data.Nat.Properties using (+-assoc; +-comm)
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open import Ledger.Prelude
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open import Ledger.Dijkstra.Specification.Gov.Actions gs hiding (yes; no)
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open import Ledger.Dijkstra.Specification.Certs gs
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open import Ledger.Dijkstra.Specification.Certs.Properties.PoVLemmas gs
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hiding (refundCertDeposits-++)
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open import Ledger.Dijkstra.Specification.Certs.Properties.PoVLemmas gs
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using (refundCertDeposits-++) public
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open import Interface.STS
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open ≡-Reasoning
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```
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-->
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## Rewards preservation
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A `CERTS`{.AgdaDatatype} run leaves the rewards balance unchanged, by chaining
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`CERT-rewards-pov`{.AgdaFunction} along the closure.
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```agda
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CERTS-rewards-pov : {Γ : CertEnv} {s s' : CertState} {dCerts : List DCert}
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→ Γ ⊢ s ⇀⦇ dCerts ,CERTS⦈ s' → coinFromRewards s ≡ coinFromRewards s'
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CERTS-rewards-pov (BS-base Id-nop) = refl
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CERTS-rewards-pov (BS-ind step rest) =
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trans (CERT-rewards-pov step) (CERTS-rewards-pov rest)
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```
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## Preservation of the pool-deposit registration invariant
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```agda
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CERTS-deposits-registered : {Γ : CertEnv} {s s' : CertState} {dCerts : List DCert}
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→ PoolDepositsRegistered s
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→ Γ ⊢ s ⇀⦇ dCerts ,CERTS⦈ s'
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→ PoolDepositsRegistered s'
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CERTS-deposits-registered reg (BS-base Id-nop) = reg
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CERTS-deposits-registered reg (BS-ind step rest) =
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CERTS-deposits-registered (CERT-deposits-registered reg step) rest
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```
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## Splitting `newCertDeposits`{.AgdaFunction} at a run boundary
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For a `CERTS`{.AgdaDatatype} run followed by further certificates
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`ys`{.AgdaBound}, the new-deposit total splits at the boundary, with the
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trailing certificates charged against the run's *final* pool set. This is
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what lets per-step accounting compose across a batch.
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```agda
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CERTS-new-thread : {Γ : CertEnv} {s s' : CertState} {dCerts : List DCert}
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→ Γ ⊢ s ⇀⦇ dCerts ,CERTS⦈ s'
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→ (ys : List DCert)
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→ newCertDeposits (PParamsOf Γ) (dom (PoolsOf s)) (dCerts ++ ys)
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≡ newCertDeposits (PParamsOf Γ) (dom (PoolsOf s)) dCerts
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+ newCertDeposits (PParamsOf Γ) (dom (PoolsOf s')) ys
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CERTS-new-thread (BS-base Id-nop) ys = refl
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CERTS-new-thread {Γ = Γ} (BS-ind {s = s₀} {sig = c} {s' = s₁} {sigs = cs} {s'' = s₂} step rest) ys =
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begin
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nc P₀ (c ∷ (cs ++ ys))
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≡⟨ newCertDeposits-∷ pp P₀ c (cs ++ ys) ⟩
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nc P₀ (c ∷ []) + nc (nextPools P₀ c) (cs ++ ys)
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≡⟨ cong (nc P₀ (c ∷ []) +_) (newCertDeposits-cong pp (cs ++ ys) (CERT-poolSet step)) ⟩
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nc P₀ (c ∷ []) + nc P₁ (cs ++ ys)
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≡⟨ cong (nc P₀ (c ∷ []) +_) (CERTS-new-thread rest ys) ⟩
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nc P₀ (c ∷ []) + (nc P₁ cs + nc P₂ ys)
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≡˘⟨ +-assoc (nc P₀ (c ∷ [])) (nc P₁ cs) (nc P₂ ys) ⟩
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nc P₀ (c ∷ []) + nc P₁ cs + nc P₂ ys
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≡˘⟨ cong (_+ nc P₂ ys) (head-split step) ⟩
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nc P₀ (c ∷ cs) + nc P₂ ys
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∎
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where
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pp = PParamsOf Γ
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nc = newCertDeposits pp
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P₀ = dom (PoolsOf s₀)
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P₁ = dom (PoolsOf s₁)
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P₂ = dom (PoolsOf s₂)
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head-split : Γ ⊢ s₀ ⇀⦇ c ,CERT⦈ s₁ → nc P₀ (c ∷ cs) ≡ nc P₀ (c ∷ []) + nc P₁ cs
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head-split st = trans (newCertDeposits-∷ pp P₀ c cs)
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(cong (nc P₀ (c ∷ []) +_) (newCertDeposits-cong pp cs (CERT-poolSet st)))
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```
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## Closed-form deposit accounting
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Over a `CERTS`{.AgdaDatatype} run from a state satisfying
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`PoolDepositsRegistered`{.AgdaFunction}, the deposit pots satisfy
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*pre + new ≡ post + refunds* in the closed form of the `UTXO`{.AgdaDatatype}
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batch balance. The step case splits the new-deposit total as above, applies
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the per-step accounting, and recombines the refunds with
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`refundCertDeposits-++`{.AgdaFunction}; the invariant is threaded through the
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run by `CERT-deposits-registered`{.AgdaFunction}.
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```agda
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CERTS-deposits-pov : {Γ : CertEnv} {s s' : CertState} {dCerts : List DCert}
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→ PoolDepositsRegistered s
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→ Γ ⊢ s ⇀⦇ dCerts ,CERTS⦈ s'
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→ coinFromDeposits s + newCertDeposits (PParamsOf Γ) (dom (PoolsOf s)) dCerts
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≡ coinFromDeposits s' + refundCertDeposits (PParamsOf Γ) dCerts
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CERTS-deposits-pov _ (BS-base Id-nop) = refl
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CERTS-deposits-pov {Γ = Γ} reg (BS-ind {s = s₀} {sig = c} {s' = s₁} {sigs = cs} {s'' = s₂} step rest) =
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begin
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D₀ + nc P₀ (c ∷ cs)
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≡⟨ cong (D₀ +_) (trans (newCertDeposits-∷ pp P₀ c cs)
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(cong (nc P₀ (c ∷ []) +_)
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(newCertDeposits-cong pp cs (CERT-poolSet step)))) ⟩
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D₀ + (nc P₀ (c ∷ []) + nc P₁ cs)
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≡˘⟨ +-assoc D₀ (nc P₀ (c ∷ [])) (nc P₁ cs) ⟩
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D₀ + nc P₀ (c ∷ []) + nc P₁ cs
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≡⟨ cong (_+ nc P₁ cs) (CERT-deposits-pov reg step) ⟩
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D₁ + refund (c ∷ []) + nc P₁ cs
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≡⟨ +-assoc D₁ (refund (c ∷ [])) (nc P₁ cs) ⟩
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D₁ + (refund (c ∷ []) + nc P₁ cs)
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≡⟨ cong (D₁ +_) (+-comm (refund (c ∷ [])) (nc P₁ cs)) ⟩
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D₁ + (nc P₁ cs + refund (c ∷ []))
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≡˘⟨ +-assoc D₁ (nc P₁ cs) (refund (c ∷ [])) ⟩
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D₁ + nc P₁ cs + refund (c ∷ [])
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≡⟨ cong (_+ refund (c ∷ []))
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(CERTS-deposits-pov (CERT-deposits-registered reg step) rest) ⟩
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D₂ + refund cs + refund (c ∷ [])
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≡⟨ +-assoc D₂ (refund cs) (refund (c ∷ [])) ⟩
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D₂ + (refund cs + refund (c ∷ []))
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≡⟨ cong (D₂ +_) (+-comm (refund cs) (refund (c ∷ []))) ⟩
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D₂ + (refund (c ∷ []) + refund cs)
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≡˘⟨ cong (D₂ +_) (refundCertDeposits-++ pp (c ∷ []) cs) ⟩
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D₂ + refund (c ∷ cs)
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∎
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where
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pp = PParamsOf Γ
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nc = newCertDeposits pp
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refund = refundCertDeposits pp
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D₀ = coinFromDeposits s₀
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D₁ = coinFromDeposits s₁
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D₂ = coinFromDeposits s₂
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P₀ = dom (PoolsOf s₀)
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P₁ = dom (PoolsOf s₁)
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```

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