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Erasure.agda
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open import Data.List using (List; []; _∷_)
open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
open import Data.Product using (_×_; _,_; ∃-syntax; Σ-syntax)
open import Relation.Binary.PropositionalEquality using (refl; _≡_)
open import Relation.Nullary using (yes; no)
open import Grove.Ident
open import Grove.Marking.STyp
open import Grove.Marking.Typ
open import Grove.Marking.UExp
open import Grove.Marking.MExp
module Grove.Marking.Erasure where
mutual
_⇒□ : ∀ {Γ τ} → (ě : Γ ⊢⇒ τ) → UExp
(⊢_^_ {x = x} ∋x u) ⇒□ = - x ^ u
(⊢λ x ∶ τ ∙ ě ^ u) ⇒□ = -λ x ∶ τ ∙ (ě ⇒□s) ^ u
(⊢ ě₁ ∙ ě₂ [ τ▸ ]^ u) ⇒□ = - (ě₁ ⇒□s) ∙ (ě₂ ⇐□s) ^ u
(⊢⸨ ě₁ ⸩∙ ě₂ [ τ!▸ ]^ u) ⇒□ = - (ě₁ ⇒□s) ∙ (ě₂ ⇐□s) ^ u
(⊢ℕ n ^ u) ⇒□ = -ℕ n ^ u
(⊢ ě₁ + ě₂ ^ u) ⇒□ = - (ě₁ ⇐□s) + (ě₂ ⇐□s) ^ u
(⊢⟦_⟧^_ {y = y} ∌y u) ⇒□ = - y ^ u
(⊢⋎^ w ^ v) ⇒□ = -⋎^ w ^ v
(⊢↻^ w ^ v) ⇒□ = -↻^ w ^ v
_⇒□s : ∀ {Γ τ} → (ě : Γ ⊢⇒s τ) → UChildExp
(⊢□ s) ⇒□s = -□ s
(⊢∶ (w , ě)) ⇒□s = -∶ (w , ě ⇒□)
(⊢⋏ s ė*) ⇒□s = -⋏ s (ė* ⇒□s*)
_⇒□s* : ∀ {Γ} → (ė* : List (EdgeId × ∃[ τ ] Γ ⊢⇒ τ)) → List UChildExp'
[] ⇒□s* = []
((w , _ , ě) ∷ ė*) ⇒□s* = (w , ě ⇒□) ∷ (ė* ⇒□s*)
_⇐□ : ∀ {Γ τ} → (ě : Γ ⊢⇐ τ) → UExp
(⊢λ x ∶ τ ∙ ě [ τ₃▸ ∙ τ~τ₁ ]^ u) ⇐□ = -λ x ∶ τ ∙ (ě ⇐□s) ^ u
(⊢⸨λ x ∶ τ ∙ ě ⸩[ τ'!▸ ]^ u) ⇐□ = -λ x ∶ τ ∙ (ě ⇐□s) ^ u
(⊢λ x ∶⸨ τ ⸩∙ ě [ τ₃▸ ∙ τ~̸τ₁ ]^ u) ⇐□ = -λ x ∶ τ ∙ (ě ⇐□s) ^ u
(⊢⋎^ w ^ v) ⇐□ = -⋎^ w ^ v
(⊢↻^ w ^ v) ⇐□ = -↻^ w ^ v
⊢⸨ ě ⸩[ τ~̸τ' ∙ su ] ⇐□ = ě ⇒□
⊢∙ ě [ τ~τ' ∙ su ] ⇐□ = ě ⇒□
_⇐□s : ∀ {Γ τ} → (ě : Γ ⊢⇐s τ) → UChildExp
(⊢□ s) ⇐□s = -□ s
(⊢∶ (w , ě)) ⇐□s = -∶ (w , ě ⇐□)
(⊢⋏ s ė*) ⇐□s = -⋏ s (ė* ⇐□s*)
_⇐□s* : ∀ {Γ τ} → (ė* : List (EdgeId × Γ ⊢⇐ τ)) → List UChildExp'
[] ⇐□s* = []
((w , ě) ∷ ė*) ⇐□s* = (w , ě ⇐□) ∷ (ė* ⇐□s*)
mutual
⊢⇐-⊢⇒ : ∀ {Γ τ} → (ě : Γ ⊢⇐ τ) → ∃[ τ' ] Σ[ ě' ∈ Γ ⊢⇒ τ' ] ě ⇐□ ≡ ě' ⇒□
⊢⇐-⊢⇒ (⊢λ x ∶ τ ∙ ě [ τ₃▸ ∙ τ~τ₁ ]^ u)
with τ' , ě' , eq ← ⊢⇐s-⊢⇒s ě rewrite eq
= (τ △s) -→ τ' , ⊢λ x ∶ τ ∙ ě' ^ u , refl
⊢⇐-⊢⇒ (⊢⸨λ x ∶ τ ∙ ě ⸩[ τ'!▸ ]^ u)
with τ' , ě' , eq ← ⊢⇐s-⊢⇒s ě rewrite eq
= (τ △s) -→ τ' , ⊢λ x ∶ τ ∙ ě' ^ u , refl
⊢⇐-⊢⇒ (⊢λ x ∶⸨ τ ⸩∙ ě [ τ₃▸ ∙ τ~̸τ₁ ]^ u)
with τ' , ě' , eq ← ⊢⇐s-⊢⇒s ě rewrite eq
= (τ △s) -→ τ' , ⊢λ x ∶ τ ∙ ě' ^ u , refl
⊢⇐-⊢⇒ (⊢⋎^ w ^ v) = unknown , (⊢⋎^ w ^ v) , refl
⊢⇐-⊢⇒ (⊢↻^ w ^ v) = unknown , (⊢↻^ w ^ v) , refl
⊢⇐-⊢⇒ (⊢⸨_⸩[_∙_] {τ' = τ'} ě τ~̸τ' su) = τ' , ě , refl
⊢⇐-⊢⇒ (⊢∙_[_∙_] {τ' = τ'} ě τ~τ' su) = τ' , ě , refl
⊢⇐s-⊢⇒s : ∀ {Γ τ} → (ě : Γ ⊢⇐s τ) → ∃[ τ' ] Σ[ ě' ∈ Γ ⊢⇒s τ' ] ě ⇐□s ≡ ě' ⇒□s
⊢⇐s-⊢⇒s (⊢□ s) = unknown , (⊢□ s) , refl
⊢⇐s-⊢⇒s (⊢∶ (w , ě))
with τ' , ě' , eq ← ⊢⇐-⊢⇒ ě rewrite eq
= τ' , (⊢∶ (w , ě')) , refl
⊢⇐s-⊢⇒s (⊢⋏ s ė*)
with ė*' , eq* ← ⊢⇐s-⊢⇒s* ė*
rewrite ⊢⇐s-⊢⇒s*-eq eq* = unknown , (⊢⋏ s ė*') , refl
⊢⇐s-⊢⇒s* : ∀ {Γ τ}
→ (ė* : List (EdgeId × Γ ⊢⇐ τ))
→ Σ[ ė*' ∈ List (EdgeId × ∃[ τ' ] Γ ⊢⇒ τ') ] Pointwise (λ { (w , ě) (w' , _ , ě') → w ≡ w' × ě ⇐□ ≡ ě' ⇒□ }) ė* ė*'
⊢⇐s-⊢⇒s* [] = [] , []
⊢⇐s-⊢⇒s* ((w , ě) ∷ ė*)
with τ' , ě' , eq ← ⊢⇐-⊢⇒ ě
with ė*' , eq* ← ⊢⇐s-⊢⇒s* ė*
= (w , τ' , ě') ∷ ė*' , (refl , eq) ∷ eq*
⊢⇐s-⊢⇒s*-eq : ∀ {Γ τ} {ė* : List (EdgeId × Γ ⊢⇐ τ)}
→ {ė*' : List (EdgeId × ∃[ τ' ] Γ ⊢⇒ τ')}
→ (eq* : Pointwise (λ { (w , ě) (w' , _ , ě') → w ≡ w' × ě ⇐□ ≡ ě' ⇒□ }) ė* ė*')
→ ė* ⇐□s* ≡ ė*' ⇒□s*
⊢⇐s-⊢⇒s*-eq [] = refl
⊢⇐s-⊢⇒s*-eq ((weq , eq) ∷ eq*) rewrite weq | eq | ⊢⇐s-⊢⇒s*-eq eq* = refl
private
⊢⇒-⊢⇐-subsume : ∀ {Γ τ τ'} → (ě : Γ ⊢⇒ τ) → (su : MSubsumable ě) → Σ[ ě' ∈ Γ ⊢⇐ τ' ] ě ⇒□ ≡ ě' ⇐□
⊢⇒-⊢⇐-subsume {τ = τ} {τ' = τ'} ě su
with τ' ~? τ
... | yes τ'~τ = ⊢∙ ě [ τ'~τ ∙ su ] , refl
... | no τ'~̸τ = ⊢⸨ ě ⸩[ τ'~̸τ ∙ su ] , refl
mutual
⊢⇒-⊢⇐ : ∀ {Γ τ τ'} → (ě : Γ ⊢⇒ τ) → Σ[ ě' ∈ Γ ⊢⇐ τ' ] ě ⇒□ ≡ ě' ⇐□
⊢⇒-⊢⇐ ě@(⊢ ∋x ^ u) = ⊢⇒-⊢⇐-subsume ě MSuVar
⊢⇒-⊢⇐ {τ' = τ'} (⊢λ x ∶ τ ∙ ě ^ u)
with τ' ▸-→?
... | no τ'!▸ with ě' , eq ← ⊢⇒s-⊢⇐s ě rewrite eq
= ⊢⸨λ x ∶ τ ∙ ě' ⸩[ τ'!▸ ]^ u , refl
... | yes (τ₁ , τ₂ , τ'▸) with (τ △s) ~? τ₁
... | yes τ~τ₁ with ě' , eq ← ⊢⇒s-⊢⇐s ě rewrite eq
= ⊢λ x ∶ τ ∙ ě' [ τ'▸ ∙ τ~τ₁ ]^ u , refl
... | no τ~̸τ₁ with ě' , eq ← ⊢⇒s-⊢⇐s ě rewrite eq
= ⊢λ x ∶⸨ τ ⸩∙ ě' [ τ'▸ ∙ τ~̸τ₁ ]^ u , refl
⊢⇒-⊢⇐ ě@(⊢ ě₁ ∙ ě₂ [ τ▸ ]^ u) = ⊢⇒-⊢⇐-subsume ě MSuAp1
⊢⇒-⊢⇐ ě@(⊢⸨ ě₁ ⸩∙ ě₂ [ τ!▸ ]^ u) = ⊢⇒-⊢⇐-subsume ě MSuAp2
⊢⇒-⊢⇐ ě@(⊢ℕ n ^ u) = ⊢⇒-⊢⇐-subsume ě MSuNum
⊢⇒-⊢⇐ ě@(⊢ ě₁ + ě₂ ^ u) = ⊢⇒-⊢⇐-subsume ě MSuPlus
⊢⇒-⊢⇐ ě@(⊢⟦ ∌y ⟧^ u) = ⊢⇒-⊢⇐-subsume ě MSuFree
⊢⇒-⊢⇐ ě@(⊢⋎^ w ^ v) = (⊢⋎^ w ^ v) , refl
⊢⇒-⊢⇐ ě@(⊢↻^ w ^ v) = (⊢↻^ w ^ v) , refl
⊢⇒s-⊢⇐s : ∀ {Γ τ τ'} → (ě : Γ ⊢⇒s τ) → Σ[ ě' ∈ Γ ⊢⇐s τ' ] ě ⇒□s ≡ ě' ⇐□s
⊢⇒s-⊢⇐s (⊢□ s) = ⊢□ s , refl
⊢⇒s-⊢⇐s (⊢∶ (w , ě))
with ě' , eq ← ⊢⇒-⊢⇐ ě
rewrite eq = (⊢∶ (w , ě')) , refl
⊢⇒s-⊢⇐s (⊢⋏ s ė*)
with ė*' , eq* ← ⊢⇒s-⊢⇐s* ė*
rewrite ⊢⇒s-⊢⇐s*-eq eq* = (⊢⋏ s ė*') , refl
⊢⇒s-⊢⇐s* : ∀ {Γ τ}
→ (ė* : List (EdgeId × ∃[ τ' ] Γ ⊢⇒ τ'))
→ Σ[ ė*' ∈ List (EdgeId × Γ ⊢⇐ τ) ] Pointwise (λ { (w , _ , ě) (w' , ě') → w ≡ w' × ě ⇒□ ≡ ě' ⇐□ }) ė* ė*'
⊢⇒s-⊢⇐s* [] = [] , []
⊢⇒s-⊢⇐s* ((w , τ , ě) ∷ ė*)
with ě' , eq ← ⊢⇒-⊢⇐ ě
with ė*' , eq* ← ⊢⇒s-⊢⇐s* ė*
= (w , ě') ∷ ė*' , (refl , eq) ∷ eq*
⊢⇒s-⊢⇐s*-eq : ∀ {Γ τ} {ė* : List (EdgeId × ∃[ τ' ] Γ ⊢⇒ τ')}
→ {ė*' : List (EdgeId × Γ ⊢⇐ τ)}
→ (eq* : Pointwise (λ { (w , _ , ě) (w' , ě') → w ≡ w' × ě ⇒□ ≡ ě' ⇐□ }) ė* ė*')
→ ė* ⇒□s* ≡ ė*' ⇐□s*
⊢⇒s-⊢⇐s*-eq [] = refl
⊢⇒s-⊢⇐s*-eq ((weq , eq) ∷ eq*) rewrite weq | eq | ⊢⇒s-⊢⇐s*-eq eq* = refl